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Discrete target recognition in polarimetric SAR imagery Heal, John Russell 1989

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D I S C R E T E T A R G E T R E C O G N I T I O N IN P O L A R I M E T R I C S A R I M A G E R Y JOHN RUSSELL HEAL B. Sc. (Electrical Engineering) Queen's University, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R O F A P P L I E D S C I E N C E in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF E L E C T R I C A L E N G I N E E R I N G We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1989 © JOHN RUSSELL H E A L , 1989 In presenting this thesis in partial f u l f i l l m e n t , of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ELECTRICAL ENGINEERING The University of British Columbia 1956 Main Mall Vancouver, Canada Date: Abstract The recognition of discrete man-made targets in remotely sensed imagery is an important prob-lem for strategic and tactical applications. The objective of this thesis is to examine whether the extra information content in polarimetric radar imagery will overcome the difficult problems in remotely sensed Synthetic Aperture Radar (SAR) data and improve target recognition capa-bilities with respect to single channel SAR. In conventional SAR these problems are mainly the result of speckle and receiver noise adversely affecting the limited information available in the single channel of data. To meet this objective, target samples from different target classes have been identified in real, polarimetric SAR data. These targets have a strong backscatter relative to the background clutter and are about three to six pixels in size. Target classes are denned by their polarization signature and available ground truth data. By exploiting the polarimetric properties of these targets it is possible to demonstrate an improvement in target detectability. A large number of measurements extracted from the polarimetric properties of scatterers are examined and reduced sets of these features have been selected. To discriminate identified targets in a SAR image, a supervised classification algorithm has been implemented. Optimal weighting of the feature sets to improve classification was not implemented due to the low confidence placed on the target feature distribution estimates as a result of the sparse training set. However, a comparison of classification results using the polarimetric data with trials performed on single channel SAR data synthesized from the same data set, clearly demonstrates a significant performance benefit of polarimetric radar. A polarimetric target model has been developed to estimate the sensitivity of the polarimetric classifier to several of the adverse properties of SAR polarimetry. Throughout this thesis, the observations are compared with other current research in this area and several related conclusions can be reached. Keywords: SAR, polarimetric radar imagery, target recognition, detection, classification. ii Table o f Contents Abs t rac t i i L is t o f Tables v i i i L is t o f Figures x L is t o f Abbrev ia t ions x i i Acknowledgements x i i i 1 I N T R O D U C T I O N 1 1.1 Motivation for Study 1 1.2 Literature Review of Related Research 3 1.3 Importance of this work 6 1.4 Introduction to Synthetic Aperture Radar Imagery 7 1.4.1 Noise in SAR detection and classification 7 1.4.2 Improving the Detection and Classification Process 10 1.5 Outline of Thesis 12 2 T H E O R Y OF R A D A R P O L A R I M E T R Y 13 2.1 Introduction to Radar Polarimetry 13 2.2 Concepts in Polarimetry 14 2.2.1 Coordinate Representation 14 2.2.2 Electromagnetic Wave Polarization 14 2.2.3 Polarization Ellipse 16 2.2.4 The Poincare sphere 17 iii 2.2.5 Stokes' vector 19 2.2.6 Partially polarized waves 20 2.3 Properties of the Received Signal 21 2.3.1 Formulation of received power 21 2.3.2 The scattering matrix 21 2.3.3 The Stokes' matrix 23 2.3.4 The cross-products of the received channels 27 2.3.5 Polarization phase difference 27 2.3.6 The scattered Stokes' vector 28 2.4 Analysis tools 31 2.4.1 Polarization synthesis 31 2.4.2 Polarization signatures 31 2.4.3 Coefficient of variation 33 2.4.4 Theoretical scattering matrix models 35 2.5 Summary 37 3 P O L A R I M E T R I C D A T A A N D T A R G E T F E A T U R E S 38 3.1 Introduction to Polarimetric Data and Target Features 38 3.2 Polarimetric Data Set 39 3.2.1 JPL Radar Polarimeter 39 3.2.2 JPL Compressed Data Format 39 3.3 Primary Polarimetric Feature Definitions 42 3.3.1 Stokes' Matrix Feature Vector and Span 43 3.3.2 Scattering Matrix Cross Products 43 3.3.3 Polarimetric Feature Normalization 43 3.4 Secondary Polarimetric Feature Definitions 46 3.4.1 Polarization phase difference 46 3.4.2 Synthesized intensities 47 iv 3.5 Summary of Polarimetric Features 48 3.6 Spatial Functions of Polarimetric Features 49 3.6.1 Description of Spatial Functions 49 3.6.2 Definition of Spatial Functions 50 3.6.3 Polarimetric versus conventional radar features 52 4 D E T E C T I O N OF DISCRETE TARGETS 53 4.1 Definition and Segmentation of Discrete Targets 53 4.1.1 Target Definition 53 4.1.2 Detection Methods 54 4.1.3 Polarization Signatures and Description of Target Classes 56 4.1.4 Polarization Signatures and Description of Clutter Backgrounds 61 4.2 Improving Target Detectability 65 4.2.1 Optimum Polarization to Maximize T C R 65 4.2.2 Effect of Polarization on Target Contrast 66 4.2.3 Examples of Optimum T C R 68 4.2.4 Predicted P D improvement in SAR 75 4.2.5 Comparison of Results with other Research 78 4.3 Summary 80 5 T A R G E T CLASSIFICATION 81 5.1 Classification Methodology 82 5.1.1 Minimum Distance Classification Algorithm 82 5.1.2 Classification Procedure 83 5.1.3 Classifier performance criteria 84 5.1.4 Target Class Definitions 84 5.2 Correlation Analysis of Polarimetric Features 86 5.2.1 Definition of Correlation Coefficient . . . 86 v 5.2.2 Correlation Effects of SPAN Normalization 87 5.2.3 Correlation of Single Channel Radar Data 89 5.3 Analysis and Reduction of Polarimetric Features 91 5.3.1 Definition of Data Analysis Tools 91 5.3.2 Analysis of Primary Polarimetric Features 93 5.3.3 Analysis of Secondary Polarimetric Features 100 5.3.4 Feature Selection Summary 104 5.4 Classification of Conventional SAR Data 105 5.4.1 Pixel averaging of conventional SAR data 105 5.4.2 Spatial Functions of Conventional SAR data 106 5.4.3 Summary of conventional SAR classification 108 5.5 Classification of Polarimetric SAR Data 109 5.5.1 Pixel Averaging of Polarimetric Features 109 5.5.2 Spatial Functions of Polarimetric Features 112 5.5.3 Best Polarimetric Feature Sets 115 5.6 Experimental Improvement in S A R Classification 117 5.7 Error Sensitivity of Classification Algorithm 119 5.7.1 Polarimetric Target Model Description 119 5.7.2 Model Functions 122 5.7.3 Model Results 123 5.7.4 Conclusions of polarimetric target model observations 130 6 C O N C L U S I O N S 131 6.1 Advantages of Polarimetry to SAR Target Recognition 131 6.1.1 Detection advantages with polarimetry 131 6.1.2 Improvement in Target Classification in SAR Imagery 132 6.1.3 Classification sensitivity to T C R and phase distortion 133 6.2 Evaluation of Polarimetric Features and Spatial Functions 133 vi 6.3 Further Study 134 References 136 A F E A T U R E DATA 142 A.l Polarimetric Feature Data /i3X3 142 A.2 Summary of Polarimetric Feature Data 148 B SPAN PROFILES A N D POLARIZATION SIGNATURES OF TARGETS 157 C INTERCLASS DISTANCE M E A S U R E M E N T S 169 D DESCRIPTION OF SOFTWARE 197 vii L i s t of Tables 2.1 Handedness of Travelling Elliptical Wave 17 2.2 Example of Wave Polarizations 17 2.3 Multi-polarization Intensity Feature Set 33 2.4 Scattering Mechanisms 34 2.5 Scattering matrix model parameters 35 3.1 List of Polarimetric Features 48 3.2 List of Spatial Functions 50 4.1 Scattering Mechanisms in Background Clutter 63 4.2 T C R for single channel and optimum SAR 72 4.3 T C R for single channel and optimum SAR 74 4.4 Summary of A T C R from single channel to optimum SAR 77 4.5 Probability of detection for single channel and optimum SAR 78 5.1 Target Class Definitions 85 5.2 Correlation of unnormalized cross products 87 5.3 Correlation of normalized cross products 88 5.4 Correlation of Synthesized Intensities (power scaling) 89 5.5 Correlation of Synthesized Intensities (logarithmic scaling) 90 5.6 Summary of Primary Polarimetric Data - ^ 3 X 3 93 5.7 Inter-Class Distance of Primary Polarimetric Features 95 5.8 Summary of Secondary Polarimetric Data 100 5.9 Inter-Class Distance for / ^3 X 3 of Secondary Polarimetric Features 101 5.10 Correlation of ShhS^ vs. Co-PPD 104 viii 5.11 Classification of Conventional SAR Data 105 5.12 List of Spatial Functions 106 5.13 Classification using Spatial Functions in Conventional SAR 107 5.14 List of Polarimetric Features 109 5.15 Classification by Primary Polarimetric Features 110 5.16 Classification by Secondary Polarimetric Features Ill 5.17 Classification by Primary and Secondary Polarimetric Features 112 5.18 Classification using Spatial Functions of Primary Polarimetric Features 113 5.19 Classification using Spatial Functions of Secondary Polarimetric Features . . . . 114 5.20 Classification using best polarimetric features 116 ix L i s t of F igures 1.1 Target Recognition using Radar Imagery 2 2.1 Coordinate System for Radar Polarimetry 15 2.2 Elliptical Polarization Representation 16 2.3 The Poincar^  sphere 18 2.4 The "unwrapped" Poincare sphere 19 2.5 San Francisco Image with HH antenna polarization 32 2.6 Polarization signature of a ship 34 2.7 Polarization Signatures of Scattering Models 36 3.1 Span Image of San Francisco scene 42 4.1 Target Radar Cross Section profile of a SHIP 54 4.2 Target Polarization Signatures: SHIP and ROCK 57 4.3 Target Polarization Signatures: CAR 58 4.4 Target Polarization Signatures: BUILDING and BACKSTOP 59 4.5 Polarization Signatures of Clutter 64 4.6 San Francisco Image with VV antenna polarization 67 4.7 San Francisco Image with HV antenna polarization 68 4.8 San Francisco Image with cross-pol RHC/LHC antenna polarization 69 4.9 San Francisco Image with co-pol RHC/RHC antenna polarization 70 4.10 RCS profiles of targets in clutter 71 4.11 Probability of Detection for Wi.3 75 4.12 Probability of Detection for W2.o 76 5.1 Summary of Classification Results 117 x 5.2 Polarimetric Target Model Power (HH) Profiles 125 5.3 Classifier Distance (1-10:1) versus PTM Target-Clutter Ratio 126 5.4 Co-polarization Signatures of PTM versus Phase De-calibration 128 5.5 Cross-polarization Signatures of PTM versus Phase De-calibration 129 5.6 Classifier Distance (1-10:1) versus PTM Phase De-calibration . . . 130 xi L i s t o f A b b r e v i a t i o n s ADC Analogue to Digital Converter CCW Counter Clockwise CFAR Constant False Alarm Rate CW Clockwise DPCA Displaced Phase Centre Antenna EM Electromagnetic FAR False Alarm Rate JPL Jet Propulsion Lab LHC Left Hand Circular MTI Moving Target Indication OPD Optimum Polarimetric Detector PD Probability of Detection Pf Polarimetric Feature PFA Probability of False Alarm (FAR) PMF Polarimetric Matched Filter PTM Polarimetric Target Model RCS Radar Cross Section RHC Right Hand Circular SAR Synthetic Aperture Radar sf Spatial Function SNR Signal to Noise Ratio TCR Target to Clutter Ratio xii Acknowledgements I wish, to express my sincere gratitude to the superb guidance generously provided to me by my thesis supervisor Dr. Ian Cumming. I also wish to thank Dr. Mark Scivier for his patience and assistance in understanding the polarimetric theory. Both Dr. Cumming's and Dr. Scivier's time and other computing and administrative support have been provided by MacDonald Det-twiler & Associates, Richmond, BC. Many thanks to the Jet Propulsion Laboratory, Pasedena, CA for making available to me the polarimetric data used in this thesis. Financial assistance has been provided to me through scholarships and research operating grants from the following organizations: NSERC, Science Council of British Columbia, Computing Devices Company of Ottawa, Ontario, and the University of British Columbia. I also wish to thank my co-supervisor, Dr. Mabo R. Ito for his assistance, constructive criticism and encouragement during the course of my work. Mr Robert Ross deserves much thanks for his patient and helpful manner with answering so many of my questions. Finally and most important, I dedicate this thesis to my fiancee, Sheilah, who has been with me from the start of this program and through her incredible thoughtfulness and encouragement has helped me with every step. John Russell Heal August 1989 xiii Chapter 1 I N T R O D U C T I O N 1.1 Motivation for Study Widespread use of Synthetic Aperture Radar (SAR) to remotely sense the earth's surface is now almost common practice. From estimating ice flow to assist marine navigation, to mapping the surface of Jupiter, SAR imagery is an increasingly important scientific and economic tool. In the next generation of SAR [1], it will be possible to apply this technology to the strategically important problem of automatic recognition of man-made targets. A complete target recognition system consists of three stages. First detection of potential targets is performed. Secondly, discrimination further separates target-like returns from interesting targets and perform some target class segmentation. Finally, target classification is performed to segregate the detected samples into known or unknown classes. For all three of these stages, polarimetry is useful in improving the capabilities of the system. Improvement in recognition is achieved either by finding better or optimum use of the classification information presently available or by the acquisition of more information about the target. Radar polarimetry is the science of using polarization diversity in transmission and reception to form a more complete model of the scattering properties of targets and clutter. A conventional single channel SAR is capable of producing a high resolution, remotely sensed, digital image. Recently, the Jet Propulsion Laboratory(JPL), Pasadena, CA, has de-veloped an Imaging Radar Polarimeter [2] that not only measures the amplitude of the radar backscatter, but also the relative phase for every electromagnetic wave polarization state. These complex measurements lead to the nine independent real elements of the Stokes' matrix de-scribing how the scattering mechanisms in each pixel transform the illuminating E M wave back 1 Chapter 1. INTRODUCTION 2 to the receiving antenna. The polarization signature [2], derived from the Stokes' matrix, is the radar cross section as a function of the antenna polarization state and is useful in interpreting the scattering mechanisms within a resolution element. In addition to the polarization signa-ture, many features describing the scattering mechanisms within a pixel can be derived from the Stokes' matrix. Figure 1.1: Target Recognition using Radar Imagery The principle objective of this thesis is to use theory and data analysis to determine whether the extra information content in polarimetric radar imagery will improve the target recogni-t o r capabilities of discrete objects compared with conventional, single channel SAR. A further goal is to gain more insight into the polarimetric scattering properties of targets and deter-mine the relative usefulness of polarimetric features through their performance in a supervised classification algorithm. Chapter 1. INTRODUCTION 3 1.2 Literature Review of Related Research Previously, SAR data analysis has dealt primarily with image enhancement for visual interpre-tation. Machine classification of conventional SAR images presents many challenges because of receiver noise and coherent speckle adversely affecting the single channel of data; subsequently, little evidence of successful applications exists in this area. An interesting area of image classi-fication is the problem of automatically detecting and recognizing discrete man-made targets. Synthetic aperture radar has several inherent factors associated with its image formation that degrades the image information content. However the significant improvement in image resolution over conventional radar makes target recognition in SAR possible. An example of target recognition of single channel SAR was shown by Singhal [3] to be effective in a well defined training region. The use of real aperture radar polarimetry for the task of target detection has been in-vestigated in Huynen [4], Ioannidis [5], and Giuli [6]. In recently published work by Novak et al [7], several target detection algorithms which make use of polarimetric radar information were evaluated. In [7] it is briefly reported on the ability to discriminate target types by exploiting differences in the polarimetric scattering properties. Favourable results were achieved in the discrimination between two interesting targets in varying degrees of ground clutter. Normal-ized polarimetric properties containing amplitude and phase information formed the feature vector. This work concluded that target discrimination appears promising and improved with pixel averaging and increased target to clutter ratio (TCR). It does not, however, compare the classification results with single channel radar which will be done in this classification analysis. In addition, classes consist of only land based man-made targets while this work has sea and land based man-made and natural target classes. The recent introduction of SAR polarimetry has led to several investigations of pixel by pixel image classification to discriminate such terrain classes as: ice [8], forest [9], vegetation [10], and geology [11, 12, 13]. An unsupervised classifier developed at JPL [13] requires no operator training and segregates Chapter 1. INTRODUCTION 4 samples into classes based upon polarimetric properties associated with the extracted data. This algorithm classifies terrain elements based on the relationship between the polarization phase difference and the handedness of the transmitting and receiving polarization states. The class definitions are the number of bounces in the received signal: odd, even, or diffuse. This simple but powerful algorithm has been shown to effectively identify several terrain classes. Since this algorithm uses no ground truth, it is highly applicable to remotely sensed data where little or no ground detail is available. It is very useful then is the first step of a complex classification system. A successful supervised classification algorithm for polarimetric SAR imagery has already been examined. This maximum likelihood classifier, implemented by Lim et al [12] for pixel by pixel classification, utilized normalized and unnormalized multi-polarization intensity feature sets. This scheme required training the classifier in three regions: ocean, urban and park land, and then running it on the whole image. Various feature sets derived from the polarimetric data were evaluated using the incorrectly classified pixels in the training region as an error measure. However, this classifier used a Bayes technique and assumes that the polarimetric data has a multivariate Gaussian distribution. While this method may be able to work well in target recognition, it is not certain whether the probability distribution assumption is valid with the sparse training sets. SAR polarimetry research conducted at MacDonald Dettwiler & Associates evaluated var-ious supervised and unsupervised classification methods already proven to be effective with Landsat multispectral SAR images [14, 15]. This study determined what linear combinations of polarimetric features are the most useful in the images examined [16,17] in an effort to reduce the computational requirements of classification. While demonstrating how classification may be conducted and the useful interpretations revealed by the polarimetric data, no quantitative error estimate of the classifier results was presented. This was the result of inadequate ground truth information of the uncontrolled experimental data set. Further work by dimming and van Zyl [16] studied the utility of polarimetric features Chapter 1. INTRODUCTION 5 using a minimum distance classifier. Various polarimetric radar schemes were studied with quantitative estimates of feature utility presented. The features studied are identical to the primary polarimetric features implemented in this work. However, conclusions drawn about the utility of features are not always directly applicable here since the sparse nature of the training sets of target classes results in a relatively poor estimate of feature statistics. The minimum distance classification method used in this prior study can be applied to target recognition here since it implies no distribution relationships of the feature data. It was found in the proceeding studies that averaging of the phase matrix in a rectangular neighbourhood was necessary to reduce noise effects and improve classification results. However, none of the proceeding studies involve multiple uses of the spatial functions of the polarimetric data defined in Section 3.6. This review has discussed a few of the more related papers that have guided this research. Many more papers have been published recently on the use on radar polarimetry in classification. It is apparent from these that the search for effective automatic target recognition algorithms is an active research interest presently being pursued by several institutions. Chapter 1. INTRODUCTION 6 1.3 Importance of this work The performance of target detection and classification in real polarimetric SAR imagery is compared with the performance achieved by single channel SAR data synthesized from the same data set. The approach taken in this thesis is to use a classical classification method in which to evaluate the new polarimetric features. In doing this, it is assumed that the relative classification performances will be roughly independent of the method. The classification algo-rithm [18, 16] that has been implemented will be used on real polarimetric data and targets rather than theoretical target and clutter models. This thesis will show how the methods of fea-ture extraction and selection presented in [3], when coupled with polarimetric information, can reduce the effects from speckle and receiver noise to significantly improve target recognition in SAR imagery. The relative comparison results are significant since the experimental conditions and radar characteristics are identical for both polarimetric and single channel radar. Since this work uses real remotely sensed data to estimate the target feature values, some limitations must be placed on the results as the training set is sparse and complete information of each target is not known. No assumptions are made about each target and clutter scattering properties as real remotely sensed data is used to get a set of supervised features consisting of measurements of the polarimetric information. A comparison of the target and clutter data with theoretical scattering models has lead to a different understanding of the principles of target recognition than that achieved using only models and assumptions about the data distributions as recorded in other work [7]. This new understanding provides further evidence of the utility of polarimetry in the problem of target recognition. Finally, a polarimetric target model has been developed based on the theory of polarimetric SAR imagery and the experimental observations of this thesis. This model is used to esti-mate the sensitivity of the classification algorithm to some of the adverse properties of SAR polarimetry such as phase distortion and different target to clutter ratios. Chapter 1. INTRODUCTION 7 1.4 Introduction to Synthetic Aperture Radar Imagery Synthetic Aperture Radar (SAR) is a form of radar that takes advantage of the coherent sampling of radar in order to make a small radar antenna work like one many kilometers larger than its physical size [19]. The radar antenna alternately emits linear modulated F M electromagnetic pulses and gathers the return echoes from points on the ground. The data is match filtered in range and azimuth directions substantially increasing the effective sampling resolution. The digital signal processor selectively combines these sampled data based on time intervals and Doppler frequency shifts of the signal relative to the moving radar platform. The length of the synthetic antenna aperture is the distance traveled by the radar during the processing interval. The resulting SAR output is a two dimensional monochromatic digital image. Pixel resolu-tion is equal to the range resolution in the y (or vertical) direction and the azimuth resolution in the x (or horizontal) direction. Each pixel represents the intensity of the returned signal strength and is proportional to the reflectance properties of the scattering mechanisms in that region. 1.4.1 Noise in SAR detection and classification There are a number of sources of noise and distortion which affect the ability of a radar system to detect and classify targets of interest. Clutter The dominant noise affecting SAR detection and classification is clutter from the surface of the earth. In a simple sense, clutter is defined as any radar reflection from an object or surface which is not of interest to the radar interpreter. In other words, it is that part of the radar return which clutters up the desired image. Clutter returns come from all parts of the earth's surface. The strength of the clutter return depends upon the surface material and the incidence angle of the radar beam. Most surfaces Chapter 1. INTRODUCTION 8 tend to produce significant amounts of clutter, the major exception being smooth water at incidence angles greater than 20°. The strongest clutter returns come from urban areas, forests and mountains. The effect of clutter on the radar performance is summarised in the Target to Clutter Ratio or TCR, and to a lesser extent by the distribution of the clutter. The target and clutter strengths after radar signal processing are expressed in power units, and divided to form the ratio. The TCR is often expressed in dB units. The distribution of the clutter is usually modelled by the Weibull distribution yVp(x), in which /3 expresses the skewness or variability of the clutter [20]. TCR and Wp(x) will be used in Chapter 4 to estimate the improved detectability of targets by maximizing the polarization response. Since targets and clutter can have quite different polarization signatures, radar polarimetry can help in the target separation process. Receiver noise The dominant source of random noise in a radar signal comes from the resistive noise in the front end of the radar low-noise amplifier or receiver. It is referred to as receiver noise, and is Gaussian white noise with a bandwidth given by the bandwidth of the receiver chain. Sometimes other sources of random noise such as analog to digital converter noise are lumped in with receiver noise. Receiver noise is incoherent and unpolarized. This means it has a flat polarization signature, unlike targets, and so polarimetry techniques can be used to increase the effective Target to N-'.'ie ratio (TNR). In a properly calibrated SAR, clutter dominates receiver noise except when the radar beam strikes the earth at a very large incidence angle. Speckle noise Speckle noise [21] results from the coherent addition of the returns from multiple scatterers Chapter!. INTRODUCTION 9 in a radar resolution cell. Since the range to the individual scatterers tends to be random at the wavelength scale (ie. range modulo(A/2)), the coherent addition process modulates the amplitude of the return from that cell. This effect creates localized destructive and construc-tive interference which appear in the radar image as bright and dark speckles in a region of homogeneous reflectance. Speckle noise has been modeled as a multiplicative random process nt modulating the reflected signal r(x, y) [22]. The recorded image signal I(x, y) is then: where * indicates convolution in the spatial domain and h(x,y) is the point spread function of the imaging system. n$(x,y) is commonly modeled as a stationary, white, non-Gaussian random process with x 2 probability distribution function with 2N degrees of freedom where N is the number of looks. Speckle has the effect of increasing the standard deviation/mean ratio for both targets and clutter, which leads to the requirement for a higher TCR to achieve a certain level of detection performance. Simple functions denning speckle and receiver noise will be used in a polarimetric target model presented in Chapter 5. Target m o t i o n defocusing Target motion can cause defocusing by virtue of the change in range during the exposure of the target resulting in a change in the target's Doppler history. These effects are appreciable in a SAR since the exposure times are long and the spatial cell sizes are small. Target defocusing tends to reduce the TCR, adversely affecting detectability. It also can hurt the classification process by smearing the detail of the target. (1.1) Chapter 1. INTRODUCTION 10 Target orientation In addition to the speckle effect, target orientation can cause a substantial change in the target Radar Cross Section (RCS) by changing the dominance of corner reflectors and the ground interaction. A classic example is given in Skolnik (ref. [23], page 27-9) where the RCS of an aircraft is shown to vary by 40 dB with azimuth angle. In addition to affecting detectability, this variation also affects classification by changing the expected RCS and, possibly, the polarization signature of the target. In recently published papers, Huynen [24, 25] has identified a method to separate target dependent information, such as orientation, from the polarimetric data. However the results are vague and it was beyond the scope of this thesis to investigate this method. 1.4.2 Improving the Detection and Classification Process To combat the detrimental effects of the sources of noise discussed above, there are several methods for reducing their effects and improving image clarity and information content. Multi-look or incoherent averaging Multi-look processing is the principal technique of reducing coherent fading or speckle and can involve averaging adjacent samples [19]. This averaging can take place in the range, the azimuth, or the Doppler domains, depending on the signal processing used. The RCS of a homogeneous region would ideally be constant but the multiplicative nature of speckle noise will cause the observed RCS to vary around its mean. By averaging adjacent samples, the mean value of the clutter remains fixed but its variance decreases, improving the probability of detection. In averaging samples, the target resolution may decrease, but the net effect on detectability and image interpretation may be positive. Chapter 1. INTRODUCTION 11 Frequency diversity Another method to reduce the speckle noise is transmitting at different radar frequency bands, and averaging the results. This works because the change in wavelength changes the relative phase of the various scatterers in a resolution element, thereby randomizing the speckle effect. Polarimetry An individual resolution element of a natural surface is comprised of multiple scattering mech-anisms and absorption losses. Depending on the terrain these mechanisms may or may not be of a similar nature. Thus the backscattered radar cross section will be the sum of multiple reflections which can change the polarization of the transmitted wave. The backscattered RCS is also dependent upon the polarization of the incident wave. Since a polarized plane wave can be represented by horizontal and vertical components, it is possible to fully characterize the polarization properties of a target or clutter by the generalized radar cross section a: where asv is the complex backscatter cross section in the vertical polarization when the incident wave is horizontally polarized. physical structure of the scattering elements within the pixel. This information can be used to overcome some of the adverse properties of radar signals mentioned above and significantly improve target recognition. a — (1.2) Having the full polarimetric characterization of a pixel gives further information of the Chapter 1. INTRODUCTION 12 1.5 Outline of Thesis This chapter has introduced the field of synthetic aperture radar and summarised some of the problems associated with classification in SAR. The following chapters are structured to clearly present the technology, method and results to meet the stated objectives of this thesis. Chapter 2 presents the background technology behind radar polarimetry and the analytical tools used in producing the polarimetric data set and interpreting mformation. A description of the polarimetric data and detailed definitions of polarimetric features and spatial functions is presented in Chapter 3. These features are based on the polarimetric theory presented in Chapter 2 and will be used in the target classification algorithm presented in Chapter 5. In Chapter 4, the polarization signature of several identifiable target classes and clutter regions are analyzed for the purpose of target segmentation. Using data from these targets as an illustration, it is then discussed how polarimetry can improve target detectability in radar imagery. Chapter 5 describes the classification method and the experimental methodology. This chapter includes an heuristical clustering analysis of each feature and spatial function in or-der to get an understanding of their usefulness in classification. The results of several target classification experiments are presented along with a discussion explaining the significance of each experiment. The results of this algorithm in the classification of several target classes demonstrates the performance enhancement of radar polarimetry over conventional SAR. At the end of Chapter 5, a simple polarimetric target and clutter model is presented for evaluating the error sensitivity of the polarimetric classifier to phase distortion. The significant findings of this thesis are reviewed in Chapter 6 along with a summary of the advantages of radar polarimetry to target recognition. Chapter 2 T H E O R Y O F R A D A R P O L A R I M E T R Y 2.1 Introduction to Radar Polarimetry A conventional monospectral Synthetic Aperture Radar has a single antenna configuration and is capable of generating high resolution digital images but with only one measured variable per pixel. For example, SEASAT operated with a horizontally linear polarized antenna. Motivated by recent measurements indicating that different terrain types respond differently to variable radar antenna polarizations (as summarised in Giuli [6]), JPL developed a radar polarimeter capable of producing multipolarized Synthetic Aperture Radar (SAR) images [26]. The original polarimeter measures the dependence of radar backscatter (intensity and relative phase) as a function of both transmitted and received radar electromagnetic wave polarization state. The data was acquired at L-band (wavelength 24.6 cm) simultaneously in four polarization states: horizontal transmit, horizontal receive (HH); horizontal transmit, vertical receive (HV); vertical transmit, horizontal receive (VH) and vertical transmit, vertical receive (VV). The measurements taken allow calculation of the four complex components of the scattering matrix, S, for each radar image resolution element (pixel). In this section the principles of radar polarimetry are presented along with several analysis techniques developed at JPL, MDA and UBC. Many features that can be derived from the polarimetric information are defined for use in a classification algorithm This introduction will acquaint the reader with the theoretical concepts of this thesis to will allow an understanding of the significance of each polarimetric feature used in detection and classification. 13 Chapter 2. THEORY OF RADAR POLARIMETRY 14 2.2 Concepts in Polarimetry 2.2.1 Coordinate Representation The coordinate system for the components of the electric field for scattered waves as described in reference [2] is displayed in Figure 2.1. The global Cartesian coordinate system with basis vectors x, y and z originate within the scatterer. The transverse components of the iUuminating electric field (h, i), h), originating at the transmitting antenna, is expressed in terms of this coordinate system as: h = sm(<fii)x — cos(0,)y (2.1) v = — cos(<&) cos(ft)x — sin(0i) cos(0;)y -f sin(0,)z (2.2) n = — cos(0i)sin(f?i)x —sin(0i)cos(^)y + cos(f?j)z (2.3) The components for the receiving antenna are the same with the subscript i replaced with s. For the backscatter case, the transmitting and receiving antenna coordinate systems coincide. In the subsequent presentation, the subscripts H and V represent the h (horizontal) and v (vertical) components of the electric field respectively. Transmitted waves, then, travel in the +n direction and received waves in the — n direction. 2.2.2 Electromagnetic Wave Polarization The expression for the electric field of a fully polarized electromagnetic wave propagating along the n axis is: E(r,t) = n{E0e-(wt-kz)) (2.4) The two mutually orthogonal components in equation 2.4 can be separated and rewritten as: V Ev{t,z) j = as cos(urf — kz)ti + ay cos(wt — kz + e)v (2.5) Chapter 2. THEORY OF RADAR POLARIMETRY 15 TRANSMITTING AND AJkyh RECEIVING ANTENNAS L. Figure 2.1: Coordinate System for Radar Polarimetry where w is the angular velocity, k is the wave number of the radiating wave, t is time, z is the distance along the n axis, e is the phase difference between EH and Ey components, and Off, ay are the magnitudes of the horizontal and vertical electric field components, respectively. By expanding the cosine term as follows: cos(tvf — kz + e) = cos(tvt — kz) cos e — sin (u>t — kz) sin e (2.6) and making the following substitutions, Eh cos(u;t - kz) = — (2.7) O f f HJTT cosiwt -kz + e) = — (2.8) ay equation 2.5 can be rewritten into the equation of an ellipse [26]: ( ^ ) 2 + ( ^ ) 2 - 2 ( ^ ) c o s e = sin', (2.9) OH <*H an ay As the electric field vector travels in space, it traces out an ellipse in a plane perpendicular to its direction of propagation. Chapter 2. THEORY OF RADAR POLARIMETRY 16 2.2.3 Polarization Ellipse The shape of the ellipse can be completely described by two geometrical parameters, the el-lipticity angle x and the orientation angle xp. Both of these angles are depicted in Figure 2.2. The polarization ellipse of a wave receding from an observer is denoted right handed if the electric field vector is rotating clockwise [27] and left handed if it is rotating counterclockwise. This is the situation in the case of a transmitted wave receding from the antenna. The handed-ness of the polarization is indicated by the sign of the ellipticity angle with negative (positive) values of x indicating right handed (left handed). This handedness convention is displayed in Table 2.1 where CW indicates clockwise and CCW indicates counter clockwise. i v / A /i \ / a h h Figure 2.2: Elliptical Polarization Representation When the rotating electric field vector is viewed advancing towards the viewer (as is the case when an antenna receives a signal) the field is considered right handed if it is rotating in a counter-clockwise direction and left handed if it is rotating in a clockwise direction [26]. This naming condition exists since according to IEEE convention [27], the handedness and the Chapter 2. THEORY OF RADAR POLARIMETRY 17 sign of the ellipticity angle of a circularly polarized is always determined when the wave is receding from an observer. The ellipticity and orientation angle can be denned in terms of the parameters of equation 2.9 as follows [14]: . /„ * 2a.ffavsine sm(2*) = „2 ,„2 (2-10) Off T " y tan(2tf) = 2 ° f V 7 € (2.11) Transmitted Wave Wave travelling away from viewer CW ccw - 4 5 ° < x < 0° 0° < X < 45° right handed left handed Scattered Wave Wave travelling towards the viewer ccw CW - 4 5 ° < x < 0° 0° < X < 45° right handed left handed Table 2.1: Handedness of Travelling Elliptical Wave Notice that values in the ranges —45° < X < 45° and 0° < ij) < 180° sufficiently represent all physical polarizations. Some examples of common polarizations are shown in Table 2.2. Ellipticity (x) Orientation (tp) Type of Wave 0° 0° or 180° Horizontal linear 0° 90° Vertical linear 45° 0° < tp < 180° Left hand circular - 4 5 ° 0° <ip< 180° Right hand circular Wave travelling away from viewer Table 2.2: Example of Wave Polarizations 2.2.4 The Poincar6 sphere A full display of all possible polarizations is obtained through a mapping of each polarization ellipse onto a Poincare sphere [29] (Figure 2.3). A point on this sphere having longitude 2ip Chapter 2. THEORY OF RADAR POLARIMETRY 18 Figure 2.3: The Poincar£ sphere and latitude 2\ uniquely identifies an elliptical polarization. Thus the equator represents linear polarizations, the poles represent circular polarizations, all left handed (right handed) elliptical polarizations map onto the northern (southern) hemisphere and the radius of the sphere is proportional to the power carried by the electromagnetic wave. In Figure 2.4 the surface of the Poincare sphere has been "unwrapped" to form a two dimensional representation of all possible polarization states. The set of all linear polarizations is represented by a horizontal line through the centre of the rectangle. Left and right circular polarizations lie along the top and bottom boundaries of the rectangle and all other points within represent elliptical polarizations. The unwrapped Poincare sphere is the two dimensional space that the polarization signatures of the scatterers are represented on. Chapter 2. THEORY OF RADAR POLARIMETRY 19 +45° x --45 a 0° i/> 180" Figure 2.4: The "unwrapped" Poincare sphere 2.2.5 S tokes ' vector The state of a polarized wave can be expressed in terms of the Stokes' parameters[30] as a function of the orientation and ellipticity angles ip and \ a s follows: Si = p0cos(2 )^cos(2x) (2.12) g2 = 5osin(2V)cos(2x) (2.13) g3 = 0osin(2x) (2.14) where go is the radius of the Poincar£ sphere and is proportional to the propagating wave's total power. These four components make up a four element vector G called the Stokes' vector. The Stokes' vector can also be written in terms of the time averaged components of the polarized electric field < \EH\ >, < \Ey \ > as follows: 90 = <\EH\2 + \EV\2> (2.15) 91 = < |JE7^ |2 - i^v l 2 > (2-16) g2 = 2 < \n{ErBEv}\ > (2.17) g3 = 2<\Z{E*HEV}\> (2.18) The Stokes' vector is a useful form to represent the polarization of an E M wave because it contains the polarization purity of paritally polarized waves discussed in the next section. Chapter 2. THEORY OF RADAR POLARIMETRY 20 2.2.6 Partially polarized waves The elements of the Stokes' vector satisfy the following equation: 9o > 9i + 92 + 9l (2.19) When <7Q = 9i + 92 + 9a * n e w a v e is considered fully polarized. A wave is considered completely unpolarized when the components pi, <?2></3 are equal to zero. The received Stokes' vector of a partially polarized wave can be decomposed into its com-pletely unpolarized and completely polarized components [27]. 9o ( l - d | ( d \ 9i 0 + 9o dcos(2x) cos(2ip) = 9o 92 0 rfcos(2x) sin(2^ >) \ ^  ) I 0 J ^ dsin(2x) j (2.20) where / Stokes' vector of completely polarized part of the received signal (2.21) dcos(2x)cos(2^>) dcos(2x) sin(2^) dsin(2x) The parameter d is the degree of polarization and is equal to the ratio of the magnitude of the polarized component to the unpolarized component [27]. d = y ^ i +92+93 9o (2.22) d is bounded by 0 > d > 1 and is a measure of the polarization purity of the received signal. While the degree of polarization has not been implemented as a target feature in this thesis, it a useful expression to describe the unpolarized component in the backscattered wave. It is used later in Chapter 4 and elsewhere to interprete the polarization signatures of various target and clutter classes. Chapter 2. THEORY OF RADAR POLARIMETRY 21 2.3 Properties of the Received Signal 2.3.1 Formulation of received power The JPL radar polarimeter, referenced in Section 2.1, records the amplitude and phase of the backscattered signal for each of the four linear polarization transmit and receive combinations: HH, HV, VH and VV. These terms are described mathematically as: Shh = Ahh expJ*hh (2.23) Shv = Ahv exp^1" (2.24) Svh = Avh exp-7*** (2.25) Svv = A™ exp^" (2.26) where Ahh, Ahv, Avh, Aw and <f>hhi <t>hv> 4>vhi <f>w are the amplitudes and phases of the HH, HV, V H and V V signals, respectively. Note that the first subscript designates the polarization state of the transmitting antenna and the second subscript designates the polarization state of the receiving antenna. 2.3.2 The scattering matrix The relationship between the polarization of the illuminating and backscattered electromagnetic waves depends upon the nature of the scattering elements, the incident polarization and angle, and the EM wave frequency or wavelength. Assuming the electromagnetic backscattering phenomena is linear and homogeneous, the scattered electric field polarization vector E , can be related to the transmitted or incident electric field polarization vector as follows: E , = S E t (2.27) / where Et = Em , The matrix S is the monostatic scattering matrix that describes how Evt Chapter 2. THEORY OF RADAR POLARIMETRY 22 the scattering mechanism transforms the iUuminating electromagnetic field into a scattered E M field. It is a 2 X 2 complex transformation matrix defined as: , Shh Shv ^ S = (2.28) Svh Svv i Similarly, if the electric field vector of the receiving antenna, E r , is known, then the complex amplitude of the received signal is: V = E j E , = EjSEt (2.29) Assuming ideal antennas, the incident polarization is equivalent to the polarization of the transmitting antenna, and the receiving antenna receives that component of the scattered field (directed back to the radar) given by the receiver antenna polarization. Thus, once the scatter-ing matrix is determined, a synthesised response may be computed for any desired configuration of antenna polarization states defined by E r and E*. Since the scattering properties we are seeking depend upon the relative phase but not the absolute phase, the matrix phase terms can be written relative to the phase of Shh- This leaves only seven absolute measurements. Also in the backscatter case, the symmetry of the geometry allow the number of independent terms to be reduced further. The reciprocity theorem applies here in that the electric field at a point P produced by a transmitter at point 0 is equal to the electric field at point O produced by a transmitter at point P, that is: V = EjT SEt = E f S r E P (2.30) For this to be valid the scattering matrix must be symmetric, hence: Shv = Svh (2.31) This leaves only five independent parameters: three amplitudes and two phases. In the formu-lation of the scattering matrix elements, it is assumed the received E M wave is fully polarized. Chapter 2. THEORY OF RADAR POLARIMETRY 23 Therefore in practice, the two measurements Svh and Shv are averaged together to compensate for any random or unpolarized component in the received wave. 2.3.3 The Stokes' matrix Analogous to the scattering matrix, it can be shown that a real 4x4 matrix exists that relates the Stokes' parameters of the scattered wave to the Stokes' parameters of the muminating wave. Where the scattering matrix can be used with the transmit and receive electric field vectors to synthesise the receive voltage, the Stokes' matrix can be used to determine the receive power. The power P received at the antenna is: P = VV* = (E^SE t )(EfSE t )* (2.32) When properly expanded [14, 5], Equation 2.32 can be regrouped in the following form: P = Y fWY, (2.33) where W = ShhS^h SvhS*h  ShhS*h SvvSvv Shv Svv ShhSlv ShhSZv $hvS*.h ShvS*h { Svh$hh Sw>S*h SvhS*h SWSM J Yt = EtvE*v EtHE*y V EtvE*H J (2.34) (2.35) Chapter 2. THEORY OF RADAR POLARIMETRY 24 and / etHe;h ^ (2.36) ErvE*v The matrix W is a property of the scatterer and is completely independent of the polarization configurations of the transmitting and receiving antennas. The vectors Yt and Yr describe the polarization states of the transmitting and receiving antennas in units appropriate to the power formulation. Vector Y is related to the Stokes' vector by the transformation matrix R as follows: 1. Y = ^ R _ 1 G 2 where (2.37) R = 1 1 0 o \ 1 -1 0 0 0 0 1 1 V 0 0 -j i ) Substituting Y t = 0 . 5 R _ 1 G t and Y;T = 0 .5G^(R r ) _ 1 into equation 2.33 reveals: P = ^ G f ( R r ) - 1 W R " 1 G t (2.38) (2.39) The transformation matrix R converts the complex valued matrix W to a 4 x 4 real matrix known as the Stokes' matrix by: A = ( R r ) - 1 W R - 1 (2.40) By substituting equation 2.40 into equation 2.39, the received power becomes: 1 rp P = - G ^ A G t (2.41) where Gt, G r are the transmit and receive antenna Stokes' vectors and A is the Stokes' matrix. Chapter 2. THEORY OF RADAR POLARIMETRY 25 Since the principle of reciprocity applies to both the Stokes' and scattering matrices, both are symmetrical. The elements of the Stokes' matrix are real and can then be defined in terms of the scattering matrix as follows: On = -^{ShhShh + SVVS*V + 2ShvS*.v) (2.42) 012 = (2.43) 013 = -$l{ShhShv + ShvS*v} (2.44) Ol4 - (2.45) 023 = •^{ShhShv ~ ShvS*v} (2.46) 024 = -^{ShhStv - shvs*v} (2.47) 033 = (2.48) 034 = (2.49) 044 = (2.50) O22 = ^(ShhShh + SVVS*V — 2ShvShv) = flu — 033 — 044 (2.51) Note that the terms in the above equations without the or operators are real-valued. Thus the symmetrical Stokes' matrix is: A = (2.52) On O12 Oi3 O14 Gl2 022 023 024 013 O23 O33 O34 014 O24 O34 O44 Containing basically the same information as the scattering matrix, the Stokes' matrix can be used to describe the polarimetric properties of the scatterer and to synthesise the received Chapter 2. THEORY OF RADAR POLARIMETRY 26 power for any antenna configuration. It can thus be said to define the polarization signature of the scattering mechanism. The power of the synthesised signal derived from the scattering matrix is identical to the power from the Stokes' matrix as long as the waves are fully polarized. A wave is considered to be fully polarized if it rotates in a plane perpendicular to its direction of propagation and its path in this plane can be expressed as a linear combination of the horizontal component and the vertical component. Since the electric field vector, E, is defined in terms of these two components, it is considered to be a fully polarized wave if its statistics are stationary. Associated with the electric field vector, the scattering matrix is a linear operator that assumes the scatterer does not introduce any diffuse (non-polarized) component into the backscattered wave. Thus the backscattered wave has to be fully polarized when represented by SEt. The Stokes' matrix is associated with the Stokes' vector and differs from the scattering matrix in that it allows a diffuse component in the electric field. This non-polarized component is given by [2]: 9o-(9i+922 +9l) (2-53) Consequently, the Stokes' representation of a scatterer contains the characteristics of a partially polarized wave. It can then be used to model scatterers which add random components to the scattered signal as well as receiver noise. In multilook polarimetric SAR data, the Stokes' matrix elements (aij) have been averaged. If the return is highly polarized, the degree of polarization is said to be high. An example of this is scattering from the ocean which is discussed further in Chapter 4. Likewise, a scatterer that has random structure will result in diffuse scattering with a low degree of polarization. This useful information is contained in a single pixel of a multilook Stokes' matrix. Chapter 2. THEORY OF RADAR POLARIMETRY 27 2.3.4 The cross-products of the received channels The elements of the matrix W above are the cross products of the scattering matrix. Due to the symmetry of the scattering matrix and multilook averaging of the Stokes' matrix there are only six independent complex elements: ShhShh ShhSZv ShhShv The cross products on the left hand side are real and the cross products on the right hand side are complex. As a result there are nine unique terms. These cross products can be derived from the elements of the Stokes' matrix and will be used in Chapter 5 as features in a polarimetric classification algorithm. 2.3.5 Polarization phase difference Recent research has introduced the quantity polarization phase difference [31, 32] and investi-gated how it relates to the physical properties of scattering mechanisms. This quantity has also been referred to in the literature as the co-polarization phase term [32]. Recall the co-polarization terms Shh and Svv defined in equations 2.23 through 2.26. The polarization phase difference(PPD) is defined as the phase difference between these two signals: A0 = 4>hh - 4>w (2.54) A<}> can be found from the product of the HH signal and the complex conjugate of the VV signal [31]: ShhS^ = AhhAvve>^ (2.55) Likewise, the cross polarization phase term is defined as the phase difference between the HH and HV channels: A0FC„ = <f>hh - 4>hv (2.56) Chapter 2. THEORY OF RADAR POLARIMETRY 28 and can be found from the product of the HH signal and the complex conjugate of the HV signal: The co-PPD can be used to determine the number of reflection or bounces in the return signal. Recognizing that different scatterers will cause the average E M scattered wave to bounce a different number of times before being received, it is possible to classify an image in this manner. Reflections from a flat surface will have one bounce which does not effect the phase angle between the HH and VV polarized waves. This results in a Co-PPD of 0°. Man made structures that resemble a dihedral reflector, such as the ship, will have a double bounce and add a 180° phase shift between the HH and VV waves resulting in a Co-PPD near 180°. A trihedral reflector will cause two 180° phase shifts resulting in a signature similar to single bounce scattering. Forested areas are typically made up of several scattering mechanisms and are likely to have multiple bounces in its reflected wave resulting in a random phase shift. Classification on the basis of odd bounce, even bounce or diffuse scattering requires no scene training, and is a key element in a powerful unsupervised classification method [13]. 2.3.6 The scattered Stokes' vector The scattered Stokes' vector G , defines the polarization and power in the scattered wave that is presented to the receiving antenna. It is determined from the Mueller matrix [2] M and the transmit Stokes' vector Gt as follows: The Mueller matrix is related to the Stokes' matrix but denned slightly differently. Recall that the Stokes matrix A is given by: ShhSL = AhhAhv exp(jA(t)hv) (2.57) G , = M G ( (2.58) A = ( R r ) - 1 W R " 1 (2.59) Chapter 2. THEORY OF RADAR POLARIMETRY 29 where R and W were defined previously in Section 2.3.3. The Mueller matrix is denned as: M = R W R - l (2.60) By substituting W R 1 = R r A into equation 2.60, the Mueller matrix can be expressed in terms of the Stokes' matrix: M = R R T A (2.61) The matrix R R r is a simple diagonal matrix: R R r = \ 1 0 0 0 0 0 1 0 0 0 0 0 0 - 1 (2.62) The mathematical difference between the Mueller matrix and the Stokes' matrix is only a sign change in the bottom row. The following statements emphasize the difference between the matrices used in polarimetry and clarify the distinctions between each: • the Mueller matrix M is a 4 X 4 real element matrix that maps the transmit Stokes' vector onto the scattered Stokes' vector [2] by Gt = M G t , • the Stokes matrix is a 4 x 4 real element matrix that can be used with the transmit and receive Stokes' vectors to determine the received power by P = lG;T A G t , • the scattering matrix is a 2 X 2 complex element matrix that assumes the scattering mechanism is completely homogeneous and does not add a diffuse or random component to the scattered E M wave. It can be used with the transmit and receive polarization vectors to determine the receive voltage by V = EjfSEt. • The Mueller and Stokes matrices representation includes the unpolarized components of scattered waves while the scattering matrix assumes the received signals are fully Chapter 2. THEORY OF RADAR POLARIMETRY 30 polarized. When the Stokes' matrix is originally determined from the parameters of the scattering matrix, it contains no diffuse or non-polarized component. However, once the matrices are averaged, it may contain a non-polarized part. Therefore when the Stokes' matrix elements are averaged over four adjacent azimuth pixels, the unique relationship between the scattering matrix and the Stokes' matrix is lost [2]. Chapter 2. THEORY OF RADAR POLARIMETRY 31 2.4 A n a l y s i s tools 2.4.1 P o l a r i z a t i o n synthesis The scattering properties within a pixel are often defined by the Stokes' matrix. Along with the Stokes's vectors of the transmitting and receiving antenna, Gt and G r , the Stokes' Matrix, A , can be used to synthesize the received power P for any arbitrary receive and transmit antenna configuration. For example, S E A S A T had a linear horizontally polarized antenna (HH). For this antenna the Stokes's vectors are: assuming a nominal l f i antenna impedance. Thus given the Stokes' matrix A for each pixel in an image, the image that would be received by an HH radar can be synthesized. Figure 2.5 shows the synthesized HH image constructed from the Stokes' matrix data the the San Francisco area supplied by J P L . Polarimetry allows synthesis of arbitrary antenna configurations such as cross-polarized HV and circularly polarization. Since the scattering mechanisms are sensitive to the polarization of the i l l u m i n a t i n g E M wave, it is possible to synthesize high resolution images that enhance or suppress interesting surfaces or clutter. Table 2.3 shows the polarization state of six commonly derived images. 2.4.2 P o l a r i z a t i o n signatures The polarization signature of a scatterer is the radar cross section as a function of (tpt, Xt > ipr, Xr) • In a simplified form it can be represented on the two dimensional "unwrapped" Poincere' sphere or polarization signature diagrams. A co-polarization signature diagram can be computed from 1 T P = - G r r A G t 4 (2.63) GTr = G? = (1 1 0 0) (2.64) Chapter 2. THEORY OF RADAR POLARIMETRY 32 Figure 2.5: San Francisco Image with HH antenna polarization the Stokes' matrix when the Stokes' vectors of the transmitting and receiving wave are like po-larized (ie ipr — tpt and \r = Xt)- Likewise the cross-polarization signature diagram is computed with a 90 degree rotation between the polarization ellipse orientation angles (ie tpr — %pt ± 90°) and a reversal of the handedness ( x r = —\t)- The polarization signature gives a concise visual representation of how the scatterer reacts to different incident EM wave polarizations and can be used to categorise the scattering mechanisms of the scatterer [2]. An example of the polarization signatures for a ship in the San Francisco scene (pixel=339, line=191) is shown in Figure 2.6. This signature is dominated by a double bounce scattering mechanism. In the signature plots the polarization ellipticity angle \ 1 S the short axis and the orientation angle tp is the long axis. The height of the graph is the relative power synthesized at that value of Xi ip' The co-pol signature is on the left and the cross-pol signature is on the right of Figure 2.6. Examples of the polarization signatures for simple scattering matrix models are shown later in Section 2.4.4. Chapter 2. THEORY OF RADAR POLARIMETRY 33 Antenna Polarization State Description Polarization H H 0°,0° 0°,0° co-pol: horizontal transmit, horizontal receive H V 0°,0° 0°,90° cross-pol: horizontal transmit, vertical receive V V 0°,90° 0°,90° co-pol: vertical transmit, vertical receive LL 45°, x 45°, x co-pol: left hand circular, left hand circular LR 45°, x - 4 5 ° , x cross-pol: left hand circular transmit, right hand circular receive Table 2.3: Multi-polarization Intensity Feature Set 2.4.3 Coefficient o f va r i a t ion The coefficient of variation is the ratio of the minimum value to the maximum value of the four dimensional polarization signature [2]. minimum of received power .„ . v = : J-. T—^- (2.65) maximum of received power The coefficient of variation is commonly extracted from the co- and cross-polarization signa-tures. Since the receive antenna polarization states are fixed with respect to the transmit state, this does not provide an complete measure of the RCS variation within a pixel. However, experimental practice has shown that the coefficient of variation extracted from the co- and cross-polarization signatures is adequate for describing the scattering mechanisms studied in this report. This property is proportional to the range in intensity of the backscattered signal throughout the degrees of freedom of the antenna polarization states. The smaller the value of v the more the backscatter signal can change with a change in polarization. The coefficient of variation is an indication of the number of unique scattering mechanisms within a resolution element [2, 33]. A low v indicates scattering is dominated by a single type of scatterer. In the limit, if v is zero, then the return signal is completely polarized and an antenna polarization exists that completely attenuates the return signal. A high coefficient of variation occurs when many different scattering mechanisms exist within a resolution element. Chapter 2. THEORY OF RADAR POLARIMETRY 34 Figure 2.6: Polarization signature of a snip In this case it is impossible to completely attenuate the scattered wave. A list of scattering mechanisms identified by van Zyl et ol [2, 13] is contained in Table 2.4. Scatter er Number of bounces Coeff. of Variation Specular reflection single low Slightly rough surface single low Dihedral corner reflector double low combination variable high random effects diffuse high Table 2.4: Scattering Mechanisms The coefficient of variation u is loosely inversely proportional to the degree of polarization d. If d = 1 then v = 0 and the backscattered signal is fully polarized indicating there is likely to be only one dominate scattering mechanism. Likewise, if d = 0 then v = 1 and the backscattered signal is fully unpolarized indicating that either random noise dominates the return signal or there are many different scattering mechanisms in the resolution element. Chapter 2. THEORY OF RADAR POLARIMETRY 35 2.4.4 Theoretical scattering matrix models Consider a scattering mechanism of the form: S = 0 b (2.66) where a,b are real constants and o = Shh and b = Svv. The nulls in the off diagonal indicate there are no cross polarized terms in the backscatter. In Table 2.5, the values of a and b are given for three common scattering mechanisms. Model S Matrix Notes Bragg (::) a,be^ a> 0,b< 0 b > a Real-dielectric dihedral corner reflector a,beU a > 0,6 < 0 o > 6 Three bounce or trihedral corner reflector ae& a > 0 Table 2.5: Scattering matrix model parameters These are simple models of pure scattering mechanisms. The signature for these three models are shown in Figure 2.7. The scattering matrix assumes the polarization response is fully polarized and each signature contains a null value hence the degree of polarization d of the scattered wave is 1. If d < 1, then a degree of unpolarization exist and no null value would exist in the polariation signature. In this case the variable part of the signature would appear to sit on a "pedestal". The height of the pedestal relative to the variable part of the signature is an indication of the randomness of the scattering properties within the resolution element. Chapter 2. THEORY OF RADAR POLARIMETRY 36 BRAGG DIHEDRAL Figure 2.7: Polarization Signatures of Scattering Models Chapter 2. THEORY OF RADAR POLARIMETRY 37 2.5 Summary This chapter has presented a detailed review of the concepts in polarimetry used in the anal-ysis of the target scattering mechanisms and the formulation of the polarimetric classification features. This review has described many of the current methods for analysing remotely sensed polarimetric radar data for the understanding and interpretation of the Earth. Analysis of the polarization signatures will be used to assist in the a priori classification of several discrete targets and background clutter regions. In the next chapter polarimetric features based on the scattering response of objects will be derived from a polarimetric image data set. Later, these features wil l be analyzed and used in a target classification algorithm. Chapter 3 P O L A R I M E T R I C D A T A A N D T A R G E T F E A T U R E S 3.1 Introduction to Polarimetric Data and Target Features In a SAR image a target is defined as a high reflectance object and appears in the image as a spatially concentrated bright object. The polarimetric properties of the target can be very different from its surroundings. Consequently, the polarimetric features in the target neighbourhood will vary depending on the strength, size, shape and reflectance properties of the target as well as the surrounding clutter. In order to utilize this information in target recognition, spatial functions of the polarimetric features are used. The scattering information of each pixel contained in the Stokes' matrix is the source of many classification features. This chapter describes the polarimetric data set and specific information about the sample image that is analyzed in this study. Next exact definitions for the polarimetric features that have been recognized as a means to discriminate terrain and target features and are used in this thesis are presented. Spatial functions of these features will be used in a classification algorithm described in detail in Chapter 5. 38 Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 39 3.2 Polarimetric Data Set 3.2.1 J P L Radar Polarimeter The data used to in this research has been generously supplied by the Jet Propulsion Lab-oratory, Pasedena, CA. The SAR image data was collected using a prototype L-band radar polarimeter [26] flown from the NASA CV990 aircraft. The wavelength is 24.5 cm and the incident angle varied from 25° to 55°. The location of the image is the San Francisco/Golden Gate area of California. The aircraft bearing the instrument flew along a west-east direction at an altitude of 6 to 12 km. Within the scene there exists a wide variety of targets and terrain types mcluding ocean, park land and urban. The data is nominal 10 m resolution in the across track (range) direction and 12 m four look resolution in along track (azimuth) direction. The amplitude calibration was accurately achieved in the data collection stage [34] with rms error of 0.5dB to a theoretical curve. The phase calibration between the two orthogonally polarized antennas was fairly accurately achieved by comparing scattering data to a theoretical model of ocean scattering and compen-sating the remainder of the image. 3.2.2 J P L Compressed Data Format This section presents the JPL data reduction method used in packaging their polarimetric data sets for storage memory savings and lower image synthesis time. The memory storage requirements for containing all the polarization information obtained from the imaging polarimeter is 128 megabytes per 1024 X 1024 image. This consists of four looks of the four complex elements of the scattering matrix as follows: Memory = (1024 x 1024)pixels x Alooks x ^ elements x Kbytes/element ~ 128 x IQPbytes The resulting compressed data set consists of only 10 bytes stored per pixel with a reduction ratio of 12.8:1. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 40 Using the reciprocity assumption, only three different elements of the scattering matrix are needed since Shv is s e * to ^{Shv + Svh) • For each scattering matrix, the Stokes' matrix is determined according to equations 2.42 ... 2.51. The elements of the four adjacent along track phase matrices are averaged to get one Stokes' matrix corresponding to a "four look pixel". The power derived from the average Stokes' matrix is identical to the averaged power of four consecutive Stokes' matrices. This can be shown by letting .A* be one of the four consecutive Stokes' matrices; then P% is its corresponding received power: P*' = ^GTA'Gt (3.1) Therefore the resulting power is [2]: = ^GjA'Gt t=i = - ^ ( h ^ G t (3.2) »=i Since the processes are identical methods of conducting four look averaging, the averaged Stokes' matrix is stored. This is a 4 X 4 symmetrical real matrix consisting of 10 distinct elements; nine of which are independent in the backscatter case since 0 2 2 is a combination of an, 0 3 3 and O 4 4 (equation 2.51). The first element of the Stokes' matrix, O n , is the total power or span of the electric field. All other terms in the matrix have a similar scaling and since it is also the largest element and always positive, it is coded in two bytes. The eight other independent elements are scaled by an and stored in one byte each. A further increase in the accuracy is obtained by recognizing that all the values are less than one and that storing the square root compresses the dynamic range. The terms that are stored directly and those that were "square rooted" was determined by experimental testing and error analysis. With the complete image set stored in 10 x 1024 x 1024 bytes, the resulting image synthesis Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 41 speed increased by a factor of 5 with errors of the order of 10 4 . To reproduce the averaged phase matrix elements the following operations are performed. On — (Me(2)/254 + 1.5)26j/te(1) Ol2 = byte{Z) x an/127 Ol3 = sign(byte{4)) x [byte{4)/127]2 au Ol4 = sign(byte(5)) x [6^e(5)/127]2an 023 = sign(byte(6)) x [&2/te(6)/127]2an 024 = sign(byte(7)) x [6^e(7)/127]2an 033 = byte(8) x an/127 034 = byte(9) x an/127 044 = byte(10) x a n /127 022 — on — 033 — a 4 4 In summary, the JPL compression algorithm has three key features: • data is stored as elements of the Stokes' matrix • elements of the Stokes' matrix are averaged prior to storage • data compression is used to reduce the amount of storage space required. For a complete description of the algorithm the reader is directed to Dubois et al. [35, 36]. This decompression step is a necessary part of any analysis of the polarimetric data set and has been included here for completeness. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 42 3.3 Primary Polarimetric Feature Definitions The features derived from polarimetric imagery are divided into two categories. The primary polarimetric features are defined here as the nine independent elements of the 4x4 real Stokes' matrix or simple functions of them. The first element of the matrix is the total available power or span [32]. The span and the other Stokes' matrix elements are used in linear combinations to form the nine independent cross products of the pixel scattering matrix. Figure 3.1 shows the span for each pixel in a scene of the San Francisco region (see Sec-tion 4.1.3). This image appears to have slightly less contrast than the HH image in Figure 2.5 and less variation in homogeneous regions such as the ocean. Figure 3.1: Span Image of San Francisco scene Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 43 3.3.1 Stokes' Matrix Feature Vector and Span The top left hand element of the Stokes' matrix an (equation 2.42) is equal to the total power or span of the scattered wave when the incident wave contains equal amounts of all polarizations. The span is the sum of three intensity images (HH, VV and HV) and assuming the speckle is not entirely correlated between the three images it contains less speckle than any one of the images on its own. span = an = \{ShhS*.h + SVVS*V + 2SvhS^h) (3.3) Except for an, none of the individual elements of the Stokes' matrix have any intuitive definition other than being linear combinations of the cross product terms of the scattering matrix; thus they will not be directly used in classification. 3.3.2 Scattering Matrix Cross Products The scattering matrix cross products are the six independent complex elements of the matrix W defined in equation 2.34. ShhSfrh ShhS*.v ShvS*.v ShhSfo (3.4) Svv Svv Shv §vv Each of these terms are directly related to the signals measured by each channel of the polarimetric radar. Due to the symmetry of the scattering matrix there are only six independent elements. 3.3.3 Polarimetric Feature Normalization The magnitude of each cross product term is less than the total available power or span and has roughly the same scaling. In order to separate variations in intensity from these polarization specific features, the cross products are normalized with the span. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 44 The following equations show how the cross products of the normalized scattering matrix can be determined from the phase matrix elements. I I C C * II - Q H + 022 + 2012 , . I P W ^ W J I = (3.5) On \\S~Sr„ || = Q n + G ; 2 - 2 Q l 2 (3.6) On \\SHA\\ = (3.7) an mshhs:v)\\ = ^ i p i ( 3. 8 ) Oil IM&fcOl = -^21) (3.9) Oil I I^S^JII = ^ F ^ 1 (3-10) On 11^5^)11 = (3.H) Oil I I ^ ^ ^ J H = Q l 3 + ° 2 3 (3.12) Oil 11^5^)11 = " ( Q l ^ + Q 2 4 ) (3.13) Oil In order distinguish between normalized and unnormalized cross products, the following convention will be adopted. • The unnormalized cross product of Sij will be indicated by SijS^, and • the normalized cross product of Sij will be indicated by ||5,-jS£||, where, WSijStjW = (3.14) ' span The || • || convention is not identical to Euclidean normalization although it is roughly equivalent. The main difference is that || • || as I have defined it does not mean the absolute value has been taken. Speckle noise is manifested in the digital data as variations in intensity as a result of the coherent sampling method of the SAR process. Since span normalization reduces the variation Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 45 of features with intensity, an additional benefit of this operation should be that the speckle noise content in each polarimetric feature is reduced. This speculation of the reduction of speckle noise is based upon the assumption that the speckle is correlated between Shh, Shv and Svv and span [37]. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 46 3.4 Secondary Polarimetric Feature Definitions Secondary polarimetric features are all other features that can be derived using linear and non-linear methods from the Stokes' matrix. 3.4.1 Polarization phase difference Two secondary polarimetric features are the polarization phase differences defined in Sec-tion 2.3.5. The PPD is the phase difference between the elements of the scattering matrix. The co-polarization phase term (A(j> or co-PPD) can be calculated from the cross products of the scattering matrix as follows [38]: Co-PPD = <phh-(f>vv - — The cross product 5^5*„ can be extracted from the Stokes' matrix equations 2.48, 2.49 and 2.50 as follows: ^{•SMIS1*,,} = « 3 3 - 044 ^{ShhS*vv} = -2a34 Therefore the Co-PPD can be evaluated as: Co-PPD = arctan ( ~ 2 ° 3 4 ) (3.16) \ 0 3 3 - 0 4 4 / Similar to the co-polarization phase difference, the cross polarization phase term is defined as the phase difference between the HH and HV channels. It is given by the phase angle of ShhShv. Cross-PPD = <phh - 4>hv (3.17) As in equation 3.16 Cross-PPD can be calculated from the scattering matrix cross products as follows: Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 47 = arctan f - ( Q i 4 + Q 2 4 ) \ V Ol3 + 023 / (3.19) As defined in Equations 3.15 and 3.19, the polarization phase differences are nonlinear functions of the scattering matrix cross-products. Since both representations of the polariza-tion phase properties will be evaluated in the classifier, it will be shown which representation performs better. 3.4.2 Synthesized intensities The other secondary features evaluated consist of a set of synthesized intensities of various antenna configurations. These intensities are listed later in Table 3.1. The logarithm of these unnormalized intensity features was taken in order to scale the dynamic range closer to the other features. In order to distinguish synthesized intensities from cross products derived from linear com-binations of the Stokes' matrix, the following convention is used. The synthesized power at i transmit, j receive polarization is IJIJ' where * indicates complex conjugation. Voltage is similarly indicated by IJ. For example, for h transmit and v receive, the synthesized power is HVHV* and the cross product is ShvS^. For certain linear polarizations such as HV, cross product and synthesized power are identical ShvS*.v = HVHV*. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 48 3.5 Summary of Polarimetric Features A list of polarimetric features implemented in this study is contained in Table 3.1 with an associated reference number, pf, shown on the right. The || • || indicates normalization and * indicates complex conjugation. Each feature is labeled with two numbers (pf:sf). The first number, pf, represents the label of the polarization feature. The second number, sf, is separated from the first by a colon and represents the label of the spatial function of that feature (see Section 3.6). For example, feature (4:2) is the 3 x 3 average of the normalized scattering matrix cross product ||5^„5^v ||. Polarimetric Features (pf) Primary Label Secondary Label log (span) 1: sf Co-PPD 11: sf \\ShhShh\\ 2:sf Cross-PPD 12: sf II Svv S*v || 3 : sf \og(HHHH*) 13: sf 4 : s / \o%(HVHV*) 14: sf P ( ^ 5 „ * J | | 5 : sf l o g ( W V V ) 15: sf 119(5^)11 6:sf \og(LLLL*) 16: sf l:sf log(LRLR*) 17: sf 8:sf ||^ ('5,/i«5'*1,)|| 9:s / 10:5 / Table 3.1: List of Polarimetric Features The primary polarimetric features are labeled 1 to 10 on the left of Table 3.1. The syn-thesized intensities, features 13 to 17, are analogous to values measured by a single channel SAR with that antenna configuration (R means Right hand circular and L means Left hand circular). The logarithm of the unnormalized intensity features was taken in order to scale the dynamic range closer to the other features. The observations that led to this are presented later in Section 5.3 and Chapter 5. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 49 S.6 Spatial Functions of Polarimetric Features 3.6.1 Description of Spatial Functions Since the return signal from a discrete target is generally contained in multiple pixels in the digital image, spatial functions of the polarimetric features are needed to extract all available information. Spatial functions are measures of how the primary and secondary polarimetric features vary in the sample local neighbourhood and are extracted from a 5 X 5 box about the target centre. Single channel SAR imagery provides high resolution digital images but with only one fixed antenna polarization state. In order to classify targets with only one intensity measure per pixel, we have to rely mainly on these spatial functions which measure the radiometric distribution of scatterers. Classification using spatial functions in single channel SAR was investigated by Singhal [3] and shown to be very effective in discriminating targets from a reasonably well defined training set. Spatial functions have also been used to improve terrain classification of SAR images [39]. The functions evaluated in this work are a subset of those implemented in that study. The 5x5 box size was chosen because the number of pixels containing most of the target return radar cross sections tended to be about three to five. In addition, similar work done on target recognition in single channel SAR [3] found that spatial functions from this size box are sufficient for extracting the useful information about targets. The spatial functions are determined by calculating the polarimetric feature for each pixel in the template and then applying the operator to the template of values. Table 3.2 contains a list of spatial functions that have been investigated in this study of target recognition. Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 50 Spatial Functions Type Group Function index label value sfi pf'-smoothing Pixel M3x3 P / : 2 averaging »h p / : 3 texture Standard 0 3 X 3 sU pf:4 deviation sh Pf • 5 V l _ 2 sfe p / : 6 resolution Gradient V l _ 3 sh p / : 7 V 2 - 3 sh p / : 8 Table 3.2: List of Spatial Functions 8.6.2 Definition of Spatial Functions Value The first spatial function sfi is defined as the value of the polarization feature at the centre of the target: sMx,y) = f(x,y) (3.20) where f(x,y) is the value of the polarimetric feature at pixel x in line y of the image. Smoothing Operators Smoothing operators or pixel averaging are the expected value or mean of the polarimetric features over a specified target neighbourhood and are used to further reduce the speckle noise cedent, mean of a random function is calculated as: /x=l£> (3.21) »=i where \i is the mean, N is the number of samples, and Xi is the value of the random function. The first smoothing operator sh is the mean in a 3 X 3 neighbourhood: 5/2(x,!/)4E E /(* + *.V + 3) (3-22) Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 51 and the second smoothing operator 5/3 is the mean in a 5 x 5 neighbourhood: I 2 2 5/3(x, j) = - J E f(x + i, y + j) (3.23) Texture Measures A texture or consistency measure operator is the standard deviation of the polarimetric features over a specified target neighbourhood. The number of samples that this statistical measure uses is small, hence the standard deviation is defined as: (3.24) »=i where a2 is the variance and fi is the mean defined above. For small JV, a is calculated as follows: \ J V - 1 21 (3.25) i=i - • \i=i The first texture measure 5/4 is the standard deviation in a 3 X 3 neighbourhood: E J2P(x + i,y + J)-Ut: Ef(x + i,y + j)) » = - i i = - i * v=-ij=-i / sh{x,y) = and the second texture measure s/5 is the a in a 5 X 5 neighbourhood: sfs(x,y) = (3.26) \ 24 2 2 / 2 2 ^ E E/ 2 (* + ^ + J ' ) - ^ E E /(* + *.»+;*•) i=-2j'=-2 \»=-2j=-2 y (3.27) Resolution Measures The size and shape of a target is determined using resolution measures. These are denned as the two dimensional gradient V of the polarimetric feature and is the ratio of the average change in value from the centre of the target to the outer edge over the inner value. The inner value of ring! is simply the value of the feature at the target centre /(x, y). ring2 is the average value of the feature surrounding the centre: ring2(x,y) = i ^ E E f(x + »>V + •?') ~ f(x>v)j (3.28) Chapter 3. POLARIMETRIC DATA AND TARGET FEATURES 52 Likewise, ringi is the average value of the outer ring of feature values in the 5x5 neigh-bourhood. ' 2 2 1 1 rir ingi{x,y) = - [ J f{x + i,y + j)- £ £ f(x,y) (3.29) \i=-2j=~2 t= - l j = - l The gradient spatial functions are defined in terms of f(x, y), ringl and ringZ as: sfe(x,y) = Vu(x,y) f(x, y) - ring2{x, y) f(X, y) *Mx,y) = Vi3(x,y) /(x, y) - ringZ(x, y) (3.30) (3.31) (3.32) sfs(x,y) = V23(x,y) _ ring2(x, y) - ring3(x, y) ~ ring2(x,y) 3.6.3 Polarimetric versus conventional radar features Single channel SAR imagery provides high resolution digital images but with only one fixed antenna polarization state. In order to classify targets with only one intensity measure per pixel, we have to rely mainly on these spatial functions which measure the radiometric distribution of scatterers. Classification using spatial functions in single channel SAR was investigated by Singhal [3] and shown to be effective in discriminating targets from a well defined training set. Polarimetry allows synthesis of not only the spatial functions in only one antenna configura-tion but multiple configurations. Conventional radar antenna configurations are HH and VV which are a subset of the unnormalized polarimetric features. In addition, the relative phase between these intensity measures is preserved. These phase (iifferences have been shown to be useful in describing different scattering mechanisms [31, 13]. Chapter 4 D E T E C T I O N O F D I S C R E T E T A R G E T S A significant problem in any reconnaissance system is the automated acquisition or detection of potential targets. This chapter addresses this issue using real SAR data to demonstrate how polarimetry may improve the detectability of discrete targets in clutter. Simple models for the distribution of clutter around stationary targets are assumed. Since the radar returns of targets and clutter are a function of the incident and receive radar wave polarizations, it is possible to reduce the clutter content in the received signal. The principle purpose of this research is to demonstrate the effectiveness of polarimetry in improving the classification or discrimination of different target classes. Accordingly, no au-tomatic target detection algorithm has been implemented or evaluated. Rather, the optimum polarization state has been determined by maximizing the target to clutter ratio. The subse-quent increase in TCR over that achieved by single channel radar leads to an increase in the detectability of targets in clutter. 4.1 Definition and Segmentation of Discrete Targets 4.1.1 Target Definition Interesting targets tend to be high reflectance discrete objects. In general, potential targets in a radar image are relatively small (3-6 pixels) or localized, bright areas. An example of the intensity profile of a potential target taken at HH polarization is shown in Figure 4.1. It shows the normalized three dimensional plot of the radar cross section of a ship as a function of image coordinates. The background clutter in Figure 4.1 should be homogenous due to scattering from the 53 Chapter 4. DETECTION OF DISCRETE TARGETS 54 Figure 4.1: Target Radar Cross Section profile of a SHIP relatively flat ocean. But the multiplicative noise due to speckle randomizes the backscatter. 4.1.2 Detection Methods In this section a brief description of some detection algorithms are reviewed. Since the the scope of this work is directed primarily at classification no detection method has actually been implemented. One method already investigated for the detection of targets in conventional SAR data [3] searches for a local maximum above a specified threshold operator. While sufficient for the purpose of demonstrating classification, it is too dependent on the statistical data of an image. Consequently it performs poorly in an image with varying low and high reflectance regions. Thus an adaptive threshold method must be used. For example, the threshold can be set such that any local maximum, a prescribed ratio (ie 3dB) above the surrounding mean, is selected as a potential target. In recently published work [7], several algorithms that make use of the polarimetric radar information in the detection of targets in ground clutter were evaluated using new target and Chapter 4. DETECTION OF DISCRETE TARGETS 55 clutter models. The first algorithms is the Optimal Polarimetric Detector (OPD). It uses the complete polarimetric information contained in the scattering matrix. The other algorithm is the Polarimetric Matched Filter ( P M F ) that detects targets by matching the return to simple polarimetric target types. The O P D algorithm detects targets in the data if the ratio of the likelihood of the presense of a target to the likelihood of the presence of just clutter is greater than a specified threshold. This algorithm yields the best possible probability of detection for a given false alarm by applying optimal weighting to the amplitude and phase information of the polarimetric data. While optimal in the likelihood sense, the O P D is dependent on knowing the a priori target to clutter statistics. Since the ratio of these statistics are likely to vary between different target classes, it is not a practical detection algorithm. Practical detection algorithms are reliant on the definition of targets as having a high Target to Clutter Ratio (TCR) . As such detection is based upon extracting these maxima from the background. The polarimetric matched filter is s i m i l a r to the simple algorithms described earlier in that it seeks the linear combination of the polarimetric data that provides the maximum T C R . Targets are then selected i f the T C R is greater than a specified threshold. While P M F depends upon the target statistics, it is invariant to the relative changes in the target to clutter ratios between target classes. In this chapter it will be shown how polarimetry can be used to maximize the T C R to improve target detection from different clutter backgrounds. A simple model for the distribution of clutter wil l be used to estimate the increase in the probability of detection due to the improved T C R compared with single channel radars. Finally, the results of this analysis wil l be compared with the findings by Novak et al [7] in their recent study of polarimetric detectors. Chapter 4. DETECTION OF DISCRETE TARGETS 56 4.1.3 Polarization Signatures and Description of Target Classes Many targets can be distinguished by examination of their polarization signatures. The co-pol and cross-polarization signatures show the radar cross section as a function of wave polarization. Wi th an understanding of the properties of different scatterers, these signatures can be used for the interpretation of different targets. In the San Francisco data set described in Chapter 3, samples from five identifiable target classes have been collected. Typical span profiles and polarization signatures for these target classes are shown below in Figures 4.2 to 4.4. A complete list of all the span profiles and polarizations signatures for every target is contained in Appendix B . These signatures are taken from the center pixel of each target. Target Class: SHIP Several ships can be located in the San Francisco data in the water around the Golden Gate bridge. The span profile reveals a narrow object easily identifiable from the surrounding ocean clutter. The polarization signatures reveal that its backscatter is dominated by a dihedral reflector or double bounce scatterer with a low coefficient of variation. Depending on the ship's size and rate of movement, its signature may be smeared over several pixels making classification more difficult. Target Class: R O C K Rocks are another identifiable target class found in the water. The span profile of a rock shows that it can appear similar to a ship; however its T C R may be much less than a ship. The typical polarization signature for a rock is a combination of Bragg-like scattering from the surrounding water and single scattering from the rocks. This results in a moderate coefficient of variation. The polarization signature of a rock in Figure 4.2 is strongly Bragg-like indicating a strong clutter content in its return. Rock targets tend to have higher degrees of clutter in their return than shiptargets and the polarization signatures wil l vary accordingly. Chapter 4. DETECTION OF DISCRETE TARGETS 57 Figure 4.2: Target Polarization Signatures: SHIP and ROCK Chapter 4. DETECTION OF DISCRETE TARGETS CAR CO-POL CROSS-POL Figure 4.3: Target Polarization Signatures: CAR Chapter 4. DETECTION OF DISCRETE TARGETS BUILDING BACKSTOP Figure 4.4: Target Polarization Signatures: BUILDING and BACKSTOP Chapter 4. DETECTION OF DISCRETE TARGETS 60 Target Class: C A R The car targets appear in the San Francisco image in the water having been displaced off the Golden Gate bridge. They have a low TCR and the signature is strongly Bragg-like due to the return from the water being added to the reflection from the car. However, the polarization signature from a car will have a strong HH component making it distinguishable from an ocean signature. These observations can be attributed to the shifted double bounce return signal adding to the return from the ocean. The car polarization signature in Figure 4.3 looks very similar to the trihedral corner reflector signature. Therefore it likely has single bounce scattering. Target Class: B U I L D I N G S The building target class is comprised of stationary land-based objects with large span profiles. They are typically strong double scatterers with high return energy and very low coefficient of variation. The sample signature of a building in Figure 4.4 has a stronger VV return than HH but is still clearly double bounce scatterer. Target Class: B A C K S T O P S Another identifiable target in the San Francisco image is the backstop of a baseball diamond. These are found in abundance in urban settings and relatively easy to locate from available ground truth information. The wire mesh surface of the backstop acts like a metal surface and returns a double bounce signal. The polarization signatures of backstops are distorted by single bounce scattering from the ground resulting in a moderate TCR and coefficient of variation. The polarization signature from a backstop contained in Figure 4.4 shows a mixed looking scattering mechanism. The co-pol signature looks Bragg-like on a large pedestal. Similarly, the cross-pol signature is dominated by a odd bounce return but the distortion is due to a strong even bounce return. Chapter 4. DETECTION OF DISCRETE TARGETS 61 Comment on Target Class Descriptions This section has presented a general description of each target class and displayed typical polarization signatures. Since each target was defined primarily by available ground truth, not all signatures and target profiles are identical to those in Figures 4.2 . . . 4.4. This can be due to misinformation in ground truth and target specific parameters that effect the scattering properties. Those factors will make detection and classification of these targets more difficult. 4.1.4 Polarization Signatures and Description of Clutter Backgrounds Three common natural surfaces that make up clutter backgrounds are: • ocean • rural • forest The polarization signatures of these surfaces are shown in Figure 4.5. Clutter class: O C E A N The polarization signature from the ocean is very homogeneous from pixel to pixel with very lit-tle spatial variation. The dominant scattering mechanisms are surface waves with wavelengths covering a large bandwidth in the image spectrum. These waves affect the surface slope, rough-ness and local velocity which directly impact the radar backscatter return. If the water is absolutely calm then the radar wave would be almost completely reflected away from the re-ceiver. A change in slope caused by a disturbance such as wind produces a small local incident angle resulting in greater backscatter. When the individual reflecting surfaces are interspaced at a distance equal to the radar half wavelength they add constructively to return a relatively high signal. Occurring in all sea states, this effect is known as Bragg scattering [33] and is a function of the polarization of the incident wave. The maximum occurs when the transmitted and received antennas are co-polarized and vertically linear. Chapter 4. DETECTION OF DISCRETE TARGETS 62 The signature at the top of Figure 4.5 is very similar to the simple model of Bragg-like scattering defined in Section 2.4.4. The main difference is the small pedestal that the ocean signature is set on. The pedestal is the unpolarized or diffuse component of the received signal and for the ocean return is relatively small. C l u t t e r class: R U R A L The signature from the rural or farmland surfaces, shown in the middle of Figure 4.5, is dom-inated by single-bounce Bragg-like scattering due to the rough nature of the ground surface. In addition, the ground surface has more multiple-bounce scattering than the ocean so there is a larger diffuse component. Depending on the density and height of the crops, double bounce scattering can also be observed from the incident wave bouncing off the ground and the stalks. If the vegetation is very dense, single bounce reflection from the top of the stalks can domi-nate [11]. Hence rural scenes tend to be a combination of single, double bounce and diffuse scatterers but depending on the crops state of growth one single form of scattering may domi-nate in the signature. C l u t t e r class: F O R E S T The variations of scattering mechanisms in forest scenes are of a similar nature to crops however they tend to be much more diffuse. This is indicated by the large pedestal in the forest signature shown in the bottom of Figure 4.5. Strong backscatters will come from single bounces from the ground, tree trunks or forest canopy, depending on the density of the vegetation. In marshy fc: - its strong double bounce reflections can be observed from the water to the tree trunks. Typically though, forest reflection tends to be diffuse with a high pixel to pixel variability of the polarized component of the signature. A summary of the dominant scattering mechanisms in background clutter surfaces is con-tained in Table 4.1. This table is based on observations found in this study and work done by van Zyl [13]. Chapter 4. DETECTION OF DISCRETE TARGETS 63 Surface Signature Comment ocean single Bragg-like rural - fallow single Bragg-like rural - medium double rural - thick single forest - very sparse single Bragg-like forest - marsh double tree trunks, ground forest - moderately thick diffuse no dominant scatterer forest - very thick single canopy Table 4.1: Scattering Mechanisms in Background Clutter M a n made vs. natural signatures In general, the polarization signatures from man made objects or targets are dominated by double bounce or dihedral scatterers. Since the reflectance of these objects is significantly higher than the natural surface around them, the target backscatter is dominated by a single scattering mechanism. The coefficient of variation is then very low and the return signature is highly polarized. Land clutter has several scattering mechanisms resulting in a high coefficient of variation and lower polarization purity but with a moderate return signal strength. On the other hand, the signature from the ocean background is dominated by a single scattering mechanism - a rough surface. This results in a low coefficient of variation and high polarization purity; however, the backscatter return is smaller and can be attenuated. From the observations of this study, the most common difference between targets and clutter is the dominant double bounce, high return signature of man made targets. The double bounce is distinguished by a 180° phase difference between SHB and 5yy. Therefore man made targets will have A(f> near 180° and clutter will tend to have a more random A<f> with ocean clutter A0 consistently near 0°. Chapter 4. DETECTION OF DISCRETE TARGETS OCEAN RURAL FOREST Figure 4.5: Polarization Signatures of Clutter Chapter 4. DETECTION OF DISCRETE TARGETS 65 4.2 Improving Target Detectability 4.2.1 Optimum Polarization to Maximize T C R The target to clutter ratio is defined as the ratio of the radar cross section of the target <JT to the surrounding clutter a0. TCR = ^ ' * , ^ , X 0 ( 4 1 } TCR is analogous to Signal to Noise Ratio (SNR) and expressed in dB. The clutter value must be taken from near the target but not close enough to be effected by the return from the target spilling over into the clutter zone. However, the return of the target is the superposition of the target cross section and the clutter cross section. As explained in Chapter 2, oj and a0 are functions of the incident and back scattered po-larizations. To determine the optimum TCR for detection, OTIO~0 must be maximized. Equa-tion 4.1 can be rewritten in terms of the polarimetric Stokes' matrix and vectors: < 4 - 2 ) where Gr and Gt are the normalized fully polarized received and transmitted antenna Stokes' vectors respectively. AT and A 0 are the 4x4 real Stokes' matrices for the target and clutter. Since AT and A^ are the ensemble average of several spatially connected Stokes' matrices, it is likely that the min imum power from these scatterers will be non-zero. In other work by Novak et al [7], it is shown theoretically what linear combinations of the polarimetric information will lead to the maximum target to clutter ratio. Several assumptions about the scattering behaviour were made (with some loss of generality) to reduce the com-plexity of the solution. One assumption was that expected value of ShhS^ is real, which for some targets and clutter models is accurate. In this work, no assumptions are make about the target or clutter statistics and a search method is used to search the four dimensional space of the transmit and receive antenna polarizations to find the combination that maximizes TCR. Chapter 4. DETECTION OF DISCRETE TARGETS 66 To perform this search, the downhill SIMPLX search algorithm presented by Nelder and Mead [40] is used. This algorithm is useful for multi-variate functions and does not require the evaluation of its derivatives. This algorithm was implemented by Scivier at MDA [14] for minimizing the backseatter from a pixel or region and maximizing the target to clutter ratio. This method has been adapted slightly for targets and used on several samples identified in the San Francisco image to study the effectiveness of improving TCR for a range of different target and clutter classes. The target polarimetric data is collected from the average Stokes' matrices in the 1 x 1 or 3 x 3 neighbourhood around the target centre. The clutter statistics are gathered from the surrounding two or three deep ring of pixels around the local target neighbourhood. Averaging the Stokes' matrices at the target reduces the measured TCR but increases the confidence of the measurements. 4.2.2 Effect of Polarization on Target Contrast Polarization of the radar antenna can effect the target to clutter ratio of discrete targets. Applying the SIMPLX algorithm to several of the targets in the ship class reveals that the mAYJmnm contrast of the ship to ocean occurs when the antenna is linear and cross-polarized (te HV). At this antenna state, the cross-polarization signature of the ocean has a trough. In order to demonstrate the effect of antenna polarization on the visual detection of these targets, sample scenes were synthesized at different antenna polarizations. In Figure 4.6, the San Francisco scene was synthesized from the polarimetric data at linear co-polarization, specifically vertical transmit and vertical receive (VV). In this figure the return from the ocean is relatively strong and no objects are visible in the ocean. However in Figure 4.7, the antenna is linear and cross-polarized, specifically horizontal transmit and vertical receive (HV). The ocean return, as expected by the polarization signature, is highly attenuated and several objects in the water are clearly noticeable. A comparison of the HV and VV images is a clear demonstration of the use of polarimetry in improving the target to clutter ratio. Chapter 4. DETECTION OF DISCRETE TARGETS Figure 4.6: San Francisco Image with VV antenna polarization Figures 2.5, 4.6 and 4.7 are all examples of images obtained with linearly polarized an-tennas. Another practical antenna configuration is circular antenna polarization described in Section 2.2. This antenna configuration occurs when the ellipticity angle \ - s ±45°. The return from circularly polarized antennas are invariant to rotations in EM wave orientation making them desirable in single channel radars when atmospheric effects are likely to cause severe dis-tortion of the polarization of the EM wave [41]. At \ = ±45°, the polarization signatures of the ocean (Fig. 4.5) show cross-polarization maximums and co-pol minimums. The ship signa-tures (Fig. 4.2), however, show a circular cross-pol minimum and a circular co-pol maximum. Therefore a co-pol circular antenna will have a higher TCR than a cross-pol circular antenna. Recall that for a circular antenna the direction of rotation is determined by the polarity of the ellipticity angle and is the opposite for transmitted and received waves. Figure 4.8 shows the San Francisco scene with cross-polarized RHC/LHC (\t = — 45°, Xr = 45°) antenna polarization and Figure 4.9 shows the scene with co-polarized RHC/RHC (xt = -45°, Xr = —45°). In the co-pol RHC/RHC image the ships are more discernible than in the Chapter 4. DETECTION OF DISCRETE TARGETS 68 Figure 4.7: San Francisco Image with HV antenna polarization cross-pol RHC/LHC image. This means the co-pol image has a higher target to clutter ratio and the observations from the polarization signatures predicting this have been verified. 4.2.3 Example s o f O p t i m u m T C R For the ship in the ocean the TCR is maximized when the receiving and transmitting antennas are cross polarized with the transmit antenna nearly linear polarized in a cross-pol. orientation. Xr,Xt « 0° rpr w rpt - 90° Figure 4.10 shows the cross section profile produced by conventional VV and HH radars and polarimetric SAR with TCR maximized for a ship in the water. At VV polarization, the clutter return is very strong relative to the target return. However, at HH polarization, the target is clearly noticeable above the clutter. When the optimum polarization is used, the contrast between the target and clutter is further increased. Since the probability of target Chapter 4. DETECTION OF DISCRETE TARGETS 69 Figure 4.8: San Francisco Image with cross-pol RHC/LHC antenna polarization detection is dependent upon this contrast any improvement in TCR will improve detection performance. For each cross section profile, the target to clutter in Figure 4.10, the target to clutter ratio is shown on the right. The increase in TCR from HH to optimum polarization is 4.6 dB for this target. As stated previously the cross section of the target has been averaged so this difference is less than can be measured. However, when comparing different targets, this averaging step increases the confidence in the TCR measurements. Chapter 4. DETECTION OF DISCRETE TARGETS 70 Chapter 4. DETECTION OF DISCRETE TARGETS 71 Figure 4.10: RCS profiles of targets in clutter Chapter 4. DETECTION OF DISCRETE TARGETS 72 The optimum TCR has been determined for all the identified targets in the San Francisco scene. A comparison of the TCR at several single channel antenna polarizations and at the optimum polarization is shown in Table 4.2 and Table 4.3. The location of each target as well as its optimum polarization are also shown. Location TCR(dB) Optimum Polarization TCR(dB) (pixel, line) HH VV HV LL LR Xutpt max ship (501,148) (392,183) (339,191) (393,231) (332,260) (177,392) 7.5 6.0 10.3 11.9 5.8 13.3 2.7 1.6 1.7 6.5 4.2 7.6 7.8 5.2 11.8 10.4 12.2 7.9 9.2 7.8 15.2 14.0 6.9 12.5 3.9 1.5 3.3 2.6 4.2 4.5 5,26 9,151 17,147 10,57 4,67 -4,34 0,128 8,48 26,41 0,158 -3,164 -25,139 14.2 9.8 19.2 16.5 14.9 15.7 average 9.1 4.1 9.2 10.9 3.3 8,50 10,143 15.0 car (401,206) (394,209) (397,212) 5.2 6.4 6.1 3.9 3.8 5.4 3.8 2.7 5.0 2.0 1.5 1.0 4.6 5.4 5.8 -29,11 -8,12 -8,1 -44,150 -24,136 -2,91 6.3 6.5 9.3 average 5.9 4.4 3.8 1.5 5.3 -15,8 -23,126 7.4 rock (106,218) (406,310) (408,319) (209,423) (178,445) (185,459) (122,511) (125,514) (132,531) (130,532) (137,534) 5.4 7.9 6.8 7.7 7.7 8.7 6.1 3.6 2.4 4.9 5.2 1.0 -0.9 6.9 5.1 4.1 2.4 4.4 4.8 2.7 2.0 4.3 5.9 4.7 8.2 7.4 8.7 6.9 10.2 2.0 3.0 7.2 4.8 8.2 2.6 5.8 8.4 6.4 5.4 7.0 2.2 2.4 4.8 4.2 1.8 5.1 7.0 4.9 5.4 4.7 5.1 4.8 3.5 2.7 5.0 6,140 -6,117 2,98 6,156 0,5 -30,13 1,178 -19,64 -3,75 -10,3 6,6 12,38 -3,11 -5,5 10,48 -10,7 35,70 8,97 32,152 18,13 -19,88 -15,111 12.4 9.6 9.4 10.9 9.8 11.0 11.3 5.1 3.9 8.0 6.4 average 6.0 3.3 6.3 5.2 4.5 -8,127 15,42 8.9 Table 4.2: TCR for single channel and optimum SAR In each target class there is a significant degree of variability in the change of TCR with polarization. For the ship class, A T C R with respect to HH antenna polarization varies from Chapter 4. DETECTION OF DISCRETE TARGETS 73 5.8dB to 13.3dB. The car class consists of moving targets displaced into clutter. The change in TCR is typically low for this class as the targets signatures closely resemble the ocean signature. In addition, HH is a very strong return for this class. The improvement of the optimum antenna over co-circular antenna is significant. The improvement in TCR for the rock class is less variable than the other target classes with an average of 2.9dB over HH. While rocks are stationary targets, their polarization signature can be strongly influenced by the background clutter similar to moving targets. Since the rock class has good relative improvement in TCR, other returns from targets with combined single and double bounce scattering can still be improved significantly by finding the optimum polarization. The clutter background for the building class is park land which tends to be a diffuse scatterer with a strong unpolarized component. The increase in TCR is then primarily due to an increase in the target return. Since the HH return is already very good for double bounce reflectors as shown in the co-pol signature of the building in Figure 4.4, the improvement in building TCR is not as significant as the improvement for targets in ocean. Included in Table 4.2 and Table 4.3 are the average optimum polarization. For each target class, this optimum polarization is different. This difference is due to the different physical structures of the dominant scattering mechanisms of the target and the background clutter in each class. This observation strengthens the argument for having the full polarimetric scattering matrix of radar targets. Knowledge of this information makes it possible to derive an optimum detector or matched filter for each target class. This topic is beyond the scope of this thesis and has already been investigated in [7]. In addition to the optimum polarizations, the average TCR measurements are also shown in Tables 4.2 and 4.3. It is apparent from these numbers that the optimum polarization has increased the TCR for all single channel polarizations of all target classes. This increase in the average TCR is summarised in Table 4.4. Chapter 4. DETECTION OF DISCRETE TARGETS Location TCR(dB) Optimum Polarization TCR(dB) (pixel, line) HH VV HV LL LR Xttlh Xr,i>r max building (795,297) 5.9 3.3 6.4 4.5 2.6 -17,173 11,72 8.8 (464,412) 13.7 12.9 9.8 14.6 4.3 17,152 13,48 14.8 (483,416) 7.1 10.4 1.6 6.1 10.2 21,118 -33,87 10.8 (468,425) 4.0 -0.2 0.3 1.7 3.6 -3,178 -7,112 5.6 (476,433) 8.3 7.8 5.7 8.7 6.9 6,161 40,52 9.4 (272,490) 13.4 14.8 13.1 16.7 1.8 -4,147 20,67 17.3 (214,505) 13.4 8.3 11.6 13.2 11.0 10,169 3,60 13.8 (308,505) 7.0 0.0 4.5 4.4 6.1 29,12 -18,35 7.5 (329,508) 11.9 12.7 10.6 14.6 6.0 16,131 0,40 15.0 (232,534) 13.5 2.7 11.1 11.7 10.9 -12,164 21,38 13.8 (269,611) 5.3 4.3 3.3 4.4 5.2 35,158 -23,35 5.8 average 9.4 5.4 7.0 9.1 6.2 -15,143 17,59 11.1 backstop (804,300) 2.8 7.0 4.0 4.7 1.9 -27,162 -2,79 9.7 (796,311) 10.2 2.7 8.2 9.0 9.3 25,73 0,15 11.0 (805,313) -2.9 10.1 10.1 7.2 6.5 -2,38 15,103 10.7 (776,483) 8.5 6.9 4.9 9.7 -0.5 2,66 3,160 10.7 (860,501) 2.4 0.9 2.3 2.4 -0.5 11,178 -11,76 3.9 (865,505) 8.4 6.4 6.6 8.5 5.8 19,58 11,4 9.2 (610,526) 4.7 2.3 7.1 6.2 1.6 10,80 1,174 7.9 (550,632) 5.2 7.4 5.7 7.4 6.9 25,164 -6,94 9.6 (621,634) 6.5 8.5 4.6 8.2 4.5 15,148 -19,65 10.8 (540,635) 3.5 7.4 5.4 5.9 7.0 13,26 -2,83 9.2 average 5.5 5.4 5.4 6.3 3.9 14,154 -6,128 8.4 Table 4.3: TCR for single channel and optimum SAR Chapter 4. DETECTION OF DISCRETE TARGETS 75 4.2.4 Predicted P D improvement in S A R With an estimate of the change in TCR due to maximization, it is possible to predict the improvement in target detectability. The distribution of clutter is usually modelled by the Weibull distribution [20] Wp(x), in which /? expresses the skewness or variability of the clutter. § o « I 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Probability of Detection for Weibull Distribution with Beta = 1.30 0 5 10 15 20 25 30 Target to Clutter + Noise Ratio (dB.) > ! I ! ! }r * s * • / • * : ' / / • .* / / : \ / * \ . ' • / ' ;.PFA.- l;fle-004/l '..1 / " / / . / • / ' • * : ' • : / ' . * ; \ / • * \ 7 / P E A = lie-006 ' * / ' ' ' • / ' / / , PFA= 1.0e-OO8 ; / : ' . * ' : : / : ' • ' . ; / : / . - : / ' . * : : / • ' '•/'•*; '• / t i e. / * ' . . ' : • / / • . ' ' • / * ' • * • •J * / / : / : ' X • * t : ,.Y \ \ \ 35 40 Figure 4.11: Probability of Detection for W1.3 The probability of detection for a target in Weibull clutter can be expressed as follows: P(detection) = p / 0 7 (4.3) where pfa is the probability of a false alarm. 7 is a function of /? and the TCR as follows: Chapter 4. DETECTION OF DISCRETE TARGETS 76 where O~T and o0 are the RCS of the target and clutter respectively. Probability of Detection for Weibull Distribution with Beta = 2.00 e o • •^ o H o Q <4-l o & •§ O 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 i r • • j 1 • i i .-•)••,i !' \ \ \ \ //y i \ • 1 / / / PFA=i.be'M4/ / L \ /pFA=1.0e-0 I t - : / ! / /PFA=1.0e-0 08 : / : ' ' • ' : I t : : / /• it'-' • I I ' - ' • : / / : I i - ' / t : / / . • : : : / / • ' : : . / * . ' ' . : : • / / • " ' : : i S*. * .** * ; i i i i i i i -5 0 5 10 15 20 25 Target to Clutter + Noise Ratio (dB.) 30 35 40 Figure 4.12: Probability of Detection for W2.0 In Figures 4.11 and 4.12, the probability of detection has been plotted as a function of the TCR for two different clutter variabilities. 0 = 1.3 (Figure 4.11) represents typical clutter content and 0 = 2.0 (Figure 4.12) represents detection in pure receiver noise. Probability of detection has been plotted for false alarm rates of 10 - 4 , 10~6 and 10 - 8 in each graph. The curves in Figure 4.12 are steeper than the curves in Figure 4.11. Therefore, when detecting targets in background clutter, such as the case in this thesis, the TCR must be higher to get the same level of detectability than in the case of detecting targets in just receiver noise. The probability of false alarm (PFA) is a detector design criterion. The higher the desired Chapter 4. DETECTION OF DISCRETE TARGETS 77 PFA, the higher the target to clutter ratio must be to achieve a desired level of detectability. An increase in the TCR through polarimetric methods is therefore justified by better detection performance and reliablity. Using the change in TCR from Table 4.2 it is possible to estimate the increase in detectability due to finding the optimum TCR for that target. The optimum target to clutter ratios shown in Table 4.2 and Table 4.3 are summarised in Table 4.4 displaying the average increase in target to clutter ratio from several single channel antenna configurations. In Table 4.4, the average optimum antenna polarizations for each target class as shown in Table 4.2 and Table 4.3 are not used since they do not reliably determine a TCR near the optimum for that target. This is due to the high variability of optimum polarizations within each target class. Therefore the average optimum polarizations shown here are not accurate estimates of the true average optimum polarizations. If the target class sample sizes were larger and better defined it may be possible to get a better estimate of these values. Target class A TCR(dB) TCR(dB) max HH VV HV LL LR ship 5.9 10.9 5.8 4.1 11.7 15.0 car 1.5 3.0 3.6 5.9 2.1 7.4 rock 2.9 5.6 2.6 3.7 4.4 8.9 building 1.7 5.7 4.1 2.0 4.9 11.1 backstop 2.9 3.0 3.0 2.1 4.5 8.4 Table 4.4: Surnmary of A T C R from single channel to optimum SAR Table 4.5 shows the increase in detectability of target classes by maximizing the TCR with a probability of a false alarm of 10"4 and Weibull distribution parameter of 1.3. The probability of detection for the optimal detector and several single channel radars is shown in Table 4.5. It is apparent from this table that the increase in TCR has improved target detectability. However, even at optimum polarization, PD is still low for most target class. This is partially due to the method of averaging the target Stokes' matrices in the TCR calculation therefore underestimating the true values. Because of this reason and the low confidence placed Chapter 4. DETECTION OF DISCRETE TARGETS 78 Target class PD{0 = 1.3,p/a = = IO" 4 ) H H V V H V L L L R Optimum ship 0.094 0.007 0.098 0.165 0.004 0.38 car 0.022 0.009 0.005 0.001 0.016 0.047 rock 0.023 0.004 0.028 0.015 0.009 0.088 building 0.10 0.017 0.040 0.095 0.026 0.17 backstop 0.018 0.017 0.017 0.028 0.006 0.073 Table 4.5: Probability of detection for single channel and optimum S A R in the small training set, it is felt that any estimate of the overall increase in detectability would not be reliable. 4.2.5 Comparison of Results with other Research In recently published work by Novak et al [7] the polarimetric matched filter method was used to derive the linear combination of the polarization scattering data that maximized T C R . For \Shh\ 2 = |£vv| 2> three solutions for simple situations were determined, namely: • HV - dihedral reflector oriented at ±45° relative to the horizontal, • HH + VV - trihedral corner reflector, • HH — VV - dihedral reflector oriented horizontally or vertically. In interpreting the first solution, recall that the simple dihedral scattering model has no H V component. Measurements taken on the targets in the San Francisco scene show that the H V return is small for similarly configured targets. The only reason that the H V would produce a maximum T C R is i f a null exists in the polarization response from the clutter. In Novak [7], the clutter background is ground or rural and is therefore very diffuse and not likely to contain a null. On the other hand, if the clutter is dominated by odd bounce or Bragg scattering, a null or minimum exists at all linear cross polarizations. 1In Novak [7] it is unclear what the polarization signature of the clutter background is. Chapter 4. DETECTION OF DISCRETE TARGETS 79 In work reported by Zebker et al [34], the maximum contrast for ships at sea was Xt — 0°,ipt = 175° cross polarized. This was determined by producing the cross polarized signature of the ratio from two scatterers: lighthouse and ocean. This is essentially H V polarization and reinforces the theoretical work by Novak. In Table 4.2, the average T C R at H V polarization is similar to that at H H . However, neither are as high as for the maximized T C R . Further, since the H V component of the clutter and targets is small in the recorded data and nonexistent in the theoretical model, it may be suggested that H V would not provide much useful information for target cuscrimination. However, in Chapter 5, it will be shown that H V is a very good target discriminator. A possible explanation for this is that H V is an indicator of the depolarization effect of a scatterer and that this is a function of a different target classes [37]. The reason H V appears as a good target detector in theoretical analysis is due primarily to the null in the clutter polarization signature. The second solution to the polarimetric matched filter is the sum of the H H and V V com-ponents for a trihedral corner reflector. This solution is easily interpreted from the theoretical model of the trihedral corner reflector scattering matrix. The H H and V V components are real and identical (Shh = Svv', Shh,Svv > 0; Shh,SVv e 3?) and the cross polarized component H V is completely attenuated (HV = 0). Therefore the sum HH + VV is the maximum value i f the polarization response of the clutter is flat or diffuse. If the polarization response of the clutter is Bragg-like, then | ^ ^ | > jyp^ and less weight would be applied to the V V component to get the maximum T C R . Similarly, the third solution is easily interpreted from the theoretical dihedral scattering matrix model. In this model, H H and V V are real and 180° out of phase (Shh < 0> Svv > 0; Shh, Svve$l). Since the H V component is 0, the difference HH — VV provides the maximum T C R with diffuse clutter. This is the most interesting solution as most of the interesting targets in the San Francisco scene fall into this particular category. Note the importance of the phase difference between H H and V V in distinguishing between odd and even bounce reflectors. Chapter 4. DETECTION OF DISCRETE TARGETS 80 4 . S S u m m a r y This chapter has shown how polarimetry can significantly improve the detection of targets in radar imagery. This improvement in detectability of targets depends upon the relative returns of targets and clutter as well as the polarization purity of each signature. The measurements presented in this chapter have illustrated the utility and importance of polarimetry in the detection of targets in background clutter. Once interesting targets have been detected, it is necessary to determine what class of targets each belongs to and separate interesting man-made targets such as ships and buildings from natural structures such as rocks. In the next chapter, the polarimetric features defined in Chapter 3 are analyzed and then used in a simple target classification scheme in order to demonstrate the discrimination improvements due to polarimetry over conventional imaging radar. Chapter 5 T A R G E T C L A S S I F I C A T I O N The two generalized forms of classifiers are supervised and unsupervised. Supervised classifiers require operators to train the recognition system by selecting a training set of samples and defining the class boundaries. An unsupervised classifier either requires no operator training or the computer defines classes by automatic segmentation. The latter method is computationally expensive and has the unfortunate characteristic of possible non-convergence. In this chapter the procedure and methodology of a non-parametric supervised target clas-sification algorithm is presented. The classifier will be used with data available to polarimetric radar as well as single channel radars to compare the target classification performances. The polarimetric features denned in Chapter 3 are analysed to predict their usefulness in the classi-fication algorithm and subsequently reduce the feature sets. Next, the classification results [42] are presented in a systematic manner to meet the objectives of this thesis and substantiate the conclusions. Finally, a polarimetric target model is presented to estimate the sensitivity of the polarimetric classifier to phase distortion and other known problems associated with radar imagery. 81 Chapter 5. TARGET CLASSIFICATION 82 5.1 Classification Methodology The target classes found in the San Francisco scene have extremely sparse populations. As a result no reliable estimate of the feature distributions within target classes can be made. For this reason a minimum distance classifier [18] was chosen for the target classifier. Unlike a maximum likelihood or Bayesian scheme, this algorithm does not imply any distribution relationship making it suitable for this recognition experiment. In other SAR classification work, described in Section 1.2, other forms of classification algorithms were used. In more recent work [43], a maximum likelihood classifier appeared to perform better than a niinimum distance classifier. However the difference was not significant and it is assumed that the difference in classification performance due to polarimetry will be invariant to the classification algorithm used. 5.1.1 Minimum Distance Classification Algorithm The classification algorithm implemented in this thesis seeks the minimum Euclidean distance of a sample target's multi-dimensional feature vector to the cluster centres of different target classes. The distance, dfc(x), of a sample target, x, to a particular class, u>k, is denned as: where Nf is the number of features, m^* is the mean of feature i in class k, /i(x) is the value of feature i for the target, and o~{ is the standard deviation for feature i averaged over all classes. Since the feature distance is normalized to the feature standard deviation, effects due to linear feature scaling are eliminated. Feature standard deviation is denned as: •"<= k=i where Nc is the number of classes and Oitk is the standard deviation of feature i in class k. This is done so that equal weighting is applied to each class regardless of the relative sample (5.1) (5.2) Chapter 5. TARGET CLASSIFICATION 83 populations. The sample is placed within the class having the minimum distance i f it is less than the maximum distance threshold: The decision criterion is given by: x euk if: dk(x) < di(x) for all I ^ k, and, dk{x) < (5.3) The characteristics of this minimum distance algorithm are: • non-parametric - it does not assume a probability distribution function for each class in the feature space. • supervised - it derives the class definitions and parameters of the algorithm (mtk and cr,,fc) using operator training. • normalized - it is invariant to feature scaling. • simplicity - easy to program and run. • robust - easy to adjust classification threshold, does not fail. 5.1.2 Classification Procedure In order to make the best use of the sparse number of target samples N,, the "Jack-knife" procedure [44] has been applied. Each incremental classification trial uses (JV, — 1) samples for the training set and one sample for the test set. This is repeated for al l 47 samples in the following sequence: 1. Polarimetric features and their spatial functions are extracted for each target and stored in a master feature file. 2. One target is selected as a test sample. 3. The remainder of the feature file is used as a training set and feature statistics are calcu-lated and stored in a summary file. Chapter 5. TARGET CLASSIFICATION 84 4. A feature set is then selected and the minimum distance classifier determines the most likely class for the test sample. This is repeated for a number of different feature sets with the results stored in separate classification files. 5. Steps 2-4 are repeated with a different test sample until all targets have been classified. 6. Each classification file is then compared with the a priori classifications in the master feature file and error calculations are determined. 5.1.3 Classifier performance criteria Empirical estimates of classification performance are determined by comparing the results of the min imum distance classifier with predetermined classifications. A success measure for each class, Pk[success], is defined as: where Nk is the number of samples in class k and Ck is the number of correct classifications made in class k. The overall success estimate, P[success], is the sum of all correct classifications divided by the total number of samples Nt. where Nc is the number of target classes. 5.1.4 Target Class Definitions The data used in this research is the compressed polarimetric data set of the San Francisco region supplied by JPL [35]. As such, ground truth can only be inferred from geographical maps, aerial photographs and image synthesis. In previous work [13, 12], class definitions are based on either scattering mechanisms or knowledge of geographical regions. In this study all available ground truth and the polarization signatures are used to define each class. Using this information, identifiable targets are extracted and span profile graphs are used for localization. Pk[success] = — (5.4) (5.5) Chapter 5. TARGET CLASSIFICATION 85 Next, using the polarization signature and the other information (photos, maps), samples are segmented into unique class regions. In all, 47 targets have been identified and are listed along with their locations and manual classifications in Table 4.2 and Table 4.3. Examples of span profiles and polarization signatures of representative target samples for each target class were shown in Figures 4.2 . . . 4.5. While it is clear from these figures that targets can have similar return power profiles, the polarization signatures are unique to different forms of reflectors. The 47 targets have been segmented into six classes shown in Table 5.1. This table shows the relative frequency of each class as well as a brief description of the signatures. Except for the ocean, each class consists of a strong reflector. Hence effects of noise and surrounding clutter in the immediate target neighbourhood are reduced. However, some of the classes have similar scattering properties. It will be shown how a combination of polarimetric features and spatial functions can distinguish between these different target classes in most cases (79%). # Class Population Polarization Signature 1 ship 6 low v, two distinct humps in co-pol, double bounce. 2 car 3 predominantly single bounce scatterer but high HH return. 3 rocks 11 medium v, multiple scattering mecha-nisms 4 ocean 6 low v, Bragg-like, slightly rough surface 5 building 11 very low v, some odd bounce and some even bounce scattereis 6 backstop 10 low v, distorted double bounce scat-terer Total 47 Table 5.1: Target Class Definitions Table 5.1 lists the classes, class populations, and a brief description of the signatures of the 47 targets studied. The interpretations are based upon the scattering models described by van Zyl [2, 13]. Chapter 5. TARGET CLASSIFICATION 86 5.2 Correlation Analysis of Polarimetric Features An indication of the orthogonality of features is the correlation coefficient. This parameter can be used to show the reduced dependency amongst polarimetric features due to span normaliza-tion. 5.2.1 Definition of Correlation Coefficient The correlation of feature i with feature j in class k is equal to: 1 N i ai>i* = -^J2ifiMhk(x)-f^,kfij,k] (5.6) z=l where x = target index and fi,k(x) is the value of feature i of target x in class k. Note that the correlation of a feature with itself is called the covariance and is equal to the square of the standard deviation as: <7i,i,fc = c 2^ For each pair of features, the correlation coefficient Pi,j,k within a class can be determined as: The correlation coefficient is limited to the region —1 < pijtk < 1. If Pi,j,k = 0 then feature i and j are uncorrelated in class k. If Pij,k = 1 or — 1 then the two features are completely correlated and feature j is merely a linear combination of feature i. If features i and j are independent then they are uncorrelated and pijtk = 0; however, the converse is not true in that if j f c = 0, features i and j are not necessarily independent. In order to reduce the data analysis of correlation of features, the aggregate correlation coefficient is calculated as: 1 N c pi,i = jrl2Nk\Pij,k\ (5-8) ' fc=l where Nc is the number of classes, iVt is the total number of samples and Nk is the number of samples in class h. Chapter 5. TARGET CLASSIFICATION 87 Finally, the average aggregate correlation coefficient for a feature is the sum of p ; ^ for all features i ^ j: 1 N f 1 J=I.J#» (5.9) 5.2.2 Correlation Effects of S P A N Normalization Correlation of Unnormalized Cross Products The most significant result of correlation analysis is revealing the reduced correlation amongst polarimetric features due to span normalization. Aggregate correlation coefficients of the un-normalized scattering matrix cross products are shown in Table 5.2. In this table, the highest coefficient is 03,5 = 0.84, the correlation of SVVS^, with &(Sh / iS w ) . The lowest is p2,9 = 0.40, the correlation of ShhShh with ShvS*v. This indicates all the features in this set contain some useful information. M3X3 Aggregate Correlation Coefficient log(apan) ShhS'hh SvvSvv Shv s v^ ShhS*v ShhS{v Shv S v v St ft ft log(apon) 1.00 0.84 0.82 0.79 0.77 0.51 0.58 0.52 0.50 0.53 0.84 1.00 0.52 0.67 0.66 0.54 0.68 0.53 0.40 0.49 S y v S v v 0.82 0.52 1.00 0.63 0.84 0.46 0.50 0.63 0.55 0.64 Shv S*_v 0.79 0.67 0.63 1.00 0.52 0.49 0.60 0.50 0.60 0.48 &{ShhS*v) 0.77 0.66 0.84 0.52 1.00 0.39 0.60 0.67 0.64 0.53 0.51 0.54 0.46 0.49 0.39 1.00 0.59 0.55 0.44 0.58 9i{ShhS*_v) 0.58 0.68 0.50 0.60 0.60 0.59 1.00 0.48 0.77 0.41 0.52 0.53 0.63 0.50 0.67 0.55 0.48 1.00 0.54 0.60 ^{ShvS*v) 0.50 0.40 0.55 0.60 0.64 0.44 0.77 0.54 1.00 0.54 HShvS'v) 0.53 0.49 0.64 0.48 0.53 0.58 0.41 0.60 0.54 1.00 Average 0.65 0.56 0.60 0.56 0.61 0.51 0.58 0.56 0.56 0.53 Table 5.2: Correlation of unnormalized cross products Correlation of Normalized Cross Products In Table 5.3, the aggregate correlation coefficients are shown for the polarimetric features after they have been normalized with respect to span. In this table, the range between the highest Chapter 5. TARGET CLASSIFICATION 88 coefficient (p2 ,3 = 0.90) and the lowest coefficient (p4t% = 0.24) amongst the polarization specific features has increased significantly. This indicates that after normalization one feature, pfz-HSooS^JI, is a linear combination of another feature and can be dropped from the feature vector. However, other features are much less correlated because the main correlating factor in Table 5.3 was the variation in intensity. The features with low p either are good features adding useful independent information or are merely random noise and can be safely removed. However, which features that may be removed and which should be kept cannot be determined from this correlation analysis. A feature selection algorithm introduced later will be used for this purpose. Aggregate Correlation Coefficient / * 3 x 3 log (span) lis..s;„|| ShhS*v ShhS*kv Shv Svv R 9 9 ft 9 log (span) 1.00 0.30 0.32 0.48 0.49 0.38 0.31 0.33 0.31 0.19 0.30 1.00 0.90 0.31 0.47 0.28 0.51 0.41 0.31 0.50 | |5 V „S^ V | | 0.32 0.90 1.00 0.31 0.49 0.29 0.58 0.37 0.30 0.43 115*. 5^ || 0.48 0.31 0.31 1.00 0.31 0.37 0.44 0.24 0.49 0.37 0.49 0.47 0.49 0.31 1.00 0.31 0.36 0.45 0.46 0.25 l|9(SkfcS¥\)|| 0.38 0.28 0.29 0.37 0.31 1.00 0.21 0.44 0.32 0.47 l l * (w;„ ) | | 0.31 0.51 0.58 0.44 0.36 0.21 1.00 0.37 0.57 0.31 0.33 0.41 0.37 0.24 0.45 0.44 0.37 1.00 0.44 0.37 ll*(sk„s;„)|| 0.31 0.31 0.30 0.49 0.46 0.32 0.57 0.44 1.00 0.41 l |9(s k ,s;j | | 0.19 0.50 0.43 0.37 0.25 0.47 0.31 0.37 0.41 1.00 Average 0.35 0.46 0.46 0.36 0.39 0.34 0.42 0.39 0.41 0.39 Table 5.3: Correlation of normalized cross products The most important difference between the Table 5.2 and Table 5.3 is the reduced average correlation from 0.56 to 0.40. due to span normalization. This fact is used in the justification to normalize the polarimetric features prior to classification. Lower correlation of features indicates a more diagonal correlation matrix. This means the features are likely to be more independent and better use of the available information is achieved. Chapter 5. TARGET CLASSIFICATION 89 Reduced Correlation with S P A N The next significant observation due to span normalization is the reduced correlation of fea-tures with respect to variation in target intensity. In Table 5.2 and Table 5.3, the correlation coefficients of span to normalized and unnormalized features are shown. Looking at the first column of each table, it is apparent that the correlation of span with the normalized features is less than span with the unnormalized features. This further indicates that some dependency of the polarimetric features with intensity variation has been removed. 5.2.3 Correlation of Single Channel Radar Data The correlation amongst the synthesized single channel radar data is shown in Table 5.4 and Table 5.5. In Table 5.4 the feature values have been extracted using power scaling while in Table 5.5 the logarithm of the features have been used. In both tables all the features are highly correlated because they are unnormalized and vary with power in the return signal. The lowest correlation is between the co-circular \og(LLLL*) feature and the cross-circular \og(LRLR*) feature. In addition \og(LRLR*) and \og(HVHV*) have a very low correlation coefficient. This substantiates the observations in Chapter 4 where \og(HVHV*) and \og(LLLL*) have very good TCR relative to log(VVVV*) and \og(LRLR*). M 3 x 3 Aggregate Correlation Coefficient HHHH* HVHV* WW* LLLL* LRLR* HHHH* 1.00 0.67 0.52 0.73 0.58 HVHV 0.67 1.00 0.63 0.79 0.53 WW* 0.52 0.63 1.00 0.73 0.57 LLLL* 0.73 0.79 0.73 1.00 0.47 LRLR* 0.58 0.53 0.57 0.47 1.00 Average 0.63 0.66 0.61 0.68 0.54 Overall 0.62 Table 5.4: Correlation of Synthesized Intensities (power scaling) The average aggregate correlation has increased from 0.62 to 0.67 as a result of the loga-rithmic scaling. While the logarithmic scaling reduces the orthogonality of these features, the Chapter 5. TARGET CLASSIFICATION 90 difference is not high and it will be shown later that classification results using this scaling are better than using power scaling. M3X3 Aggregate Correlation Coefficient log{HHHH*) log(HVHV') l o g ( V V W ) \og{LLLL*) log(LRLR') \og(HHHH*) 1.00 0.73 0.59 0.76 0.60 \og{HVHV) 0.73 1.00 0.68 0.81 0.56 l o g ( V V W ) 0.59 0.68 1.00 0.72 0.75 log(LLLL*) 0.76 0.81 0.72 1.00 0.53 log{LRLR*) 0.60 0.56 0.75 0.53 1.00 Average 0.67 0.69 0.68 0.70 0.61 Overall 0.67 Table 5.5: Correlation of Synthesized Intensities (logarithmic scaling) Summary of Correlation Analysis From this analysis of the correlation of polarimetric feature the following observations were made: • Correlation between normalized features is less than between unnormalized features. • Correlation of span with the normalized features is less than span with unnormalized features. • Correlation of synthesized intensities increases slightly when the features are logarithmi-cally scaled. These observations are useful and assist in the justification of feature normalization. In the next section the polarimetric features will be further analyzed and reduced feature sets determined. Chapter 5. TARGET CLASSIFICATION 91 5.3 Analysis and Reduction of Polarimetric Features 5.3.1 Definition of Data Analysis Tools The forty-seven target samples identified for classification analysis have been separated into six classes described by their polarization signatures (Section 4.1.3). In this section the polari-metric feature data is analyzed to determine the relative usefulness of each feature and select reduced sets of good features. The number of feature combinations is very large as there are 17 polarimetric features each with 8 spatial functions and six target classes. In order to analyse the data a few tools defined below are used. In presenting the analysis only a sample of the data is included in the body of this chapter. A complete summary of the analysis measurements are included in the appendices. Interclass Distance Interclass Distance is a measure of the separation between clustering centres of target classes within a feature space. In conventional pattern recognition this is a function of the class means and standard deviations [44]. In this work the number of samples per class ranges from only three to eleven. Therefore the distribution of the features cannot be reliably measured. For this reason a heuristic measure of the feature distance between classes is used to estimate the clustering of target features within a class. For a complete description of this derivation the reader is directed to Singhal [3, 45]. The interclass distance Vi of feature i is summarised from [45] as follows. Firstly, for each target class pair (k,l), a weighted rank sum measure, T>i,k,l> 1S calculated as shown in Equation 5.10. This is an interclass distance of feature i for each target class pair. Ri.h Rh.i (5.10) 1 r _ _1_ rNl+Ifk r where: Rk,i is the rank sum of samples in class k when paired with class /, and Nk is the number of samples in class A;. Chapter 5. TARGET CLASSIFICATION 92 Then the class conditional distance, V^k, is calculated as: Eg// i-ikiNk + Nt)Vitktl ZaU l*k( N>* +  Nl) (5.11) T>i,k gives an indication of how well feature i separates class k from all the other classes. Finally, Vi, the overall interclass distance for feature i is the average class conditional distance: where Nc is the total number of classes. A complete list of all the inter-class distance measurements is included in Appendix C. Feature Selection Algorithm Since this experiment involved the study of a very large feature set with a small training set of targets, it was helpful to use a systematic method to analyze and select features. For this purpose, a feature selection algorithm recently developed at UBC [45] for assessing the usefulness of features in sparse data was adapted to handle polarimetric data. This algorithm is shown to be useful in eliminating highly correlated features and features that are highly affected by noise. The features eliminated add little to the classification result and if included would only increase complexity of the classifier and add noise. The features selected are less correlated with one another and are better at separating the targets into their predefined classes. This algorithm assigns utility based upon how well a feature improves the partitioning of classes. It then assigns a performance estimate of the resulting feature set based upon not only the interclass distance but also the feature variance within each class. Applying this heuristic algorithm to a large number of features should produce a smaller set of features with good classification accuracy, thereby reducing the complexity of the data set and the classification algorithm. Where this algorithm has been applied in the following section, the results are supported with analysis of the data and interclass distance. Several classification trials on the reduced c k=i (5.12) Chapter 5. TARGET CLASSIFICATION 93 feature set were performed to determine more precisely the final number of features to be included in the reduced sets. 5.3.2 Analysis of Primary Polarimetric Features Feature data Table 5.6 contains a statistical summary of the feature data for each class. It shows the mean and standard deviation for each primary polarimetric feature of 3/2 - the mean of target feature data in a 3 X 3 window about the target centre. feature /*3x3 span log() 1:2 H5**5Jt f c|| 2:2 H^vv^vvH 3:2 l l & , s z j | 4:2 ShhSvv ShhSfa 1 ft 5:2 6:2 ft 7:2  8:2 ft 9:2 10:2 ship <Ti:2,l 1.141 0.210 0.503 0.111 0.424 0.084 0.073 0.056 0.117 0.404 -0.106 0.144 0.112 0.044 -0.035 0.049 -0.010 0.033 -0.045 0.071 car Pi-2,2 Ci.2,2 0.902 0.176 0.423 0.073 0.558 0.080 0.019 0.007 0.740 0.126 -0.033 0.330 0.066 0.034 -0.041 0.007 0.050 0.029 0.042 0.027 rock M»:2,3 <7t:2,3 0.980 0.166 0.374 0.103 0.520 0.121 0.106 0.066 0.418 0.188 0.168 0.191 0.053 0.087 -0.052 0.048 0.031 0.076 0.080 0.063 ocean A*t:2,4 0«:2,4 0.571 0.303 0.286 0.048 0.695 0.046 0.019 0.010 0.833 0.064 0.223 0.088 0.037 0.011 -0.036 0.029 0.051 0.019 0.068 0.044 building M>:2,5 <»i:2,5 1.899 0.467 0.487 0.142 0.402 0.137 0.111 0.053 -0.285 0.358 -0.004 0.418 0.196 0.069 0.020 0.050 -0.103 0.092 0.024 0.073 backstop Pi.2,6 1.165 0.296 0.422 0.117 0.415 0.119 0.163 0.064 -0.191 0.210 0.208 0.304 0.132 0.075 0.001 0.085 -0.030 0.064 0.066 0.078 Table 5.6: Summary of Primary Polarimetric Data - p,3X3 In analyzing the primary polarimetric data, it appears that there is poor separation of the target classes by log(span). While the building class has the strongest average return, it has a very large variance. The ship and backstop classes have about the same strength of return. In column 4:2 the cross-pol terms are considerably weaker than the co-pol features 2:2 and Chapter 5. TARGET CLASSIFICATION 94 3:2. Of the complex cross-products, ||5,/l/l5**t, || has the strongest return. The stronger return is important since the signal to receiver noise ratio in the feature will be higher. ||5J(5fcH'S'^ t))|| and H t^'SMS t^,)!! can be positive or negative resulting in a lower absolute value of the average. For example, the ship class is a double bounce scatterer and will likely have a stronger real component and a relatively weak imaginary component. While the data shows they are about the same average strength the standard deviation of ||3?(5'fcfc5^ t,)|| is very high indicating the average would be quite large. This apparent discrepancy will be compensated in the m i n i n - m r r i distance classification algorithm described later. The greater distance between the center of the feature cluster will be reduced by the greater standard deviation. Also the relative change in magnitudes or scale between different features will be compensated in the algorithm. This is not always desirable since the lower scaled features will have a greater noise content. For this reason it will be useful to eliminate those features that do not add useful information to the classifier and are highly affected by noise. Interclass distance of primary polarimetric features The overall interclass distance for each spatial function of every polarimetric feature is shown in Table 5.7. The interclass distance is larger for the pixel averaging and texture measure spatial functions while the resolution or gradient measures have lower interclass distance. While the distances tend to be larger for the first five polarimetric features there are a few spatial functions of the later features that have good interclass distance. The class conditional interclass distances for each polarimetric feature are contained in Appendix C. Studying these distance measures le;. s to a reduced set of features that have the best interdistance for all the classes. These feature sets tend to be the same as selected by the feature selection algorithm briefly described in Section 5.3.1. Reducing the number of features in a classifier reduces the complexity of the algorithm thereby increasing the computational efficiency. Chapter 5. TARGET CLASSIFICATION 95 Overall Interclass Distance Vi Feature Spatial Functions Avg. value ^3x3 £*5x5 0"3X3 0 5 X 5 Vi_2 V l _ 3 V2-3 log(span) 0.79 0.70 0.65 0.50 0.70 0.26 0.39 0.47 0.56 \\shhszh\\ 0.45 0.46 0.55 0.44 0.58 0.41 0.50 0.49 0.49 \\SVVS*V\\ 0.50 0.57 0.75 0.46 0.53 0.47 0.54 0.58 0.55 WSHAW 0.56 0.67 0.74 0.65 0.62 0.17 0.29 0.25 0.49 W(ShhS;v)\\ 0.69 0.79 0.84 0.71 0.72 0.65 0.72 0.66 0.72 P(W„*J| | 0.43 0.38 0.36 0.45 0.60 0.28 0.45 0.36 0.41 0.48 0.61 0.64 0.61 0.75 0.40 0.35 0.21 0.50 119(^5^)11 0.32 0.39 0.35 0.69 0.78 0.22 0.13 0.46 0.42 P ( ^ v 5 v * „ ) | | 0.40 0.61 0.58 0.66 0.66 0.39 0.25 0.42 0.50 | |3(5 f c,5^)|| 0.53 0.41 0.36 0.62 0.62 0.26 0.39 0.32 0.44 average 0.52 0.56 0.58 0.58 0.66 0.35 0.40 0.42 0.51 Table 5.7: Inter-Class Distance of Primary Polarimetric Features Reduced feature sets The feature selection algorithm was used to select a reduced set of polarimetric features for the first three spatial functions. Shown below are lists of reduced feature sets with the features class conditional distance and overall interclass distance measures on the right. The order of appearance in the list is that in which the features where selected by the feature selection algorithm. Reduced sets of polarimetric features The following three lists of reduced features are for spatial functions 1-value, 2-/Z3X3, and 3-/^5x5- These functions represent pixel averaging of the feature data in different sized windows. Chapter 5. TARGET CLASSIFICATION 96 Feature Interclass Distance value ship car rock ocean bldg bkstp overall 1:1 log(span) 0.64 0.82 0.76 0.91 0.96 0.68 0.79 4:1 \\shvsu\ 0.43 0.89 0.38 0.74 0.39 0.54 0.56 5:1 W(shhs*vv)\\ 0.52 0.82 0.82 0.81 0.59 0.57 0.69 10:1 \mshvs*vv)\\ 0.80 0.45 0.42 0.46 0.35 0.70 0.53 Feature Interclass Distance 1^3x3 ship car rock ocean bldg bkstp overall 1:2 log (span) 0.59 0.67 0.60 0.89 0.94 0.62 0.70 5:2 W(shhs*vv)\\ 0.64 0.89 0.84 0.92 0.72 0.73 0.79 4:2 wsnvsm 0.57 0.83 0.52 0.87 0.54 0.70 0.56 3:2 \\SVVS*,V\\ 0.48 0.62 0.50 0.96 0.45 0.43 0.57 7:2 W(shhs*hv)\\ 0.58 0.61 0.50 0.69 0.77 0.52 0.48 10:2 \mshvs*v)\\ 0.81 0.36 0.37 0.35 0.33 0.27 0.53 Feature Interclass Distance #5x5 ship car rock ocean bldg bkstp overall 3:3 \\SwS*.v\\ 0.71 0.67 0.71 1.00 0.68 0.72 0.75 5:3 W(shhs*v)\\ 0.72 0.91 0.81 0.93 0.82 0.83 0.84 1:3 log(span) 0.54 0.51 0.53 0.77 0.96 0.62 0.65 4:3 WShvSU 0.67 0.85 0.69 0.88 0.65 0.72 0.74 The reduced sets consist of features that provide good interclass distance for all the classes. While the features are listed according to classification utility, the overall interclass distances will not necessarily be in descending order of size since the features are selected in order of their ability to partition targets better than the existing set. For example, s/3-psxs contains a good Chapter 5. TARGET CLASSIFICATION 97 set of only four polarimetric features. This set has good interclass distance and low complexity. The number of features in these reduced sets varies according to each features ability to add useful information. The feature selection algorithm uses a marginal utility algorithm to heuristically select the next best feature. When this utility drops significantly, no more features should be added. It was found during later classification trials that the algorithm was poor at determining the best number of features to choose. Therefore the number of features in the reduced feature sets were determined through classification trials of three to seven features. Thus the number of features in a reduced set are not based solely on the heuristic feature selection but also on empirical experimental observations. Reduced sets of spatial functions In order to evaluate the usefulness of the different spatial functions of the primary polarimetric features, reduced sets of spatial functions were determined for each feature. These sets of functions are summarised below without the interclass distance measures shown. Reduced Sets of Spatial Functions log(spon) \\shh .5** II \\s„ .5^ 11 \\shv 1:1 value 2:3 A 5 X 5 3:3 1*5X5 4:3 1*5x5 5:3 ^5X5 1:5 o*5x5 2:5 0 5 x 5 3:5 0*5X5 4:1 value 5:4 0 3 x 3 1:3 / / 5 x 5 2:8 V 2 _ 3 3:1 value 4:5 0 5 X 5 5:1 value 2:4 0 " 3 X 3 3:2 1*3x3 4:4 0 3 X 3 5:8 V 2 - 3 2:2 M3x3 4:2 4:8 1*3x3 V 2 _ 3 Chapter 5. TARGET CLASSIFICATION 98 Reduced Sets of Spatial Functions 119(^ 5 )^11 6:5 05x5 6:2 ^3x3 6:4 03x3 6:8 V 2 _ 3 I I ^ S D I I 7:5 c75x5 7:3 p5xs 7:4 03x3 7:2 p3x3 119(^ 5 )^11 8:5 05x5 8:4 0-3x3 8:1 value 8:8 V 2 _ 3 P(^5„*J|| 9:5 05x5 9:1 value 9:8 V 2 _ 3 9:3 p5x5 \mshvs*vv)\\ 10:5 0-5x5 10:2 /U3x3 10:4 03x3 10:1 value These feature sets list the best spatial functions in descending order of utility. The best function over the 10 runs is 5/5-05x5 Having been selected 9 out of 10 trials and usually placing first or second. The best averaging appears to be feature 5/3-/X5X5. Of the gradient functions 5/8-V2-3 is selected the most with V i _ 2 not selected at all. Bes t set of primary polarimetric features Analyzing the interclass distance tables and applying the feature selection algorithm to the set of all spatial functions of all ten primary polarimetric features a near best feature set can be determined. This best feature set is shown below with the interclass distances shown on the right. Best Feature Interclass Distance List ship car rock ocean bldg bkstp overall 1:1 log(span) value 0.64 0.82 0.76 0.91 0.96 0.68 0.79 5:3 W(SHHS*VV)\\ # 5 X 5 0.72 0.91 0.81 0.93 0.82 0.83 0.84 8:5 mshhs*hv)\\ C"5x5 0.66 0.81 0.68 0.86 0.75 0.91 0.78 3:3 \\SVVS*V\\ #5X5 0.71 0.67 0.71 1.00 0.68 0.72 0.75 5:4 11^ (^ 5^ )11 03x3 0.81 0.75 0.59 0.97 0.63 0.52 0.71 This reduced set has higher interclass distance than the reduced set of polarimetric features of s/3. The overall best feature is log(span)-value (1:1). This is reasonable since span was used in the initial a priori classifications and contains the total power in the return signals. The next best feature as determined using all the criterion of the feature selection algorithm is Chapter 5. TARGET CLASSIFICATION 99 ||3?(5^^5*v)|| with two spatial functions psx5 and 03x3. This feature contains the polarization phase difference information and provides excellent separation of not only targets and clutter classes but between different target classes as well. ||S(5'hh5' t^))|| is a relatively weak feature and in other polarimetric data analysis it does not appear to add much useful information [16, 17]. However, the higher interclass distance measures in this thesis were determined on targets rather than regions and indicates ||3(Sw,S£„)|| has good separation between different target classes. ||5w»5^||-/isx5 has a high interclass distance due to the very small variance of feature values within each class ( see Appendix C). In the 5 x 5 window, the land based target classes have a lower ratio of V V in the return signal while the ocean based targets have a higher level. This can be attributed to the large VV component in the ocean clutter total power. Therefore this feature is particularly good at separating ocean based targets from land based targets. Chapter 5. TARGET CLASSIFICATION 100 5.3.3 Analysis of Secondary Polarimetric Features Secondary feature data The ^3x3 of the secondary features for each class is summarised in Table 5.8. feature M 3 X 3 Co-PPD (radians) 11:2 Cr-PPD (radians) 12:2 HHHH* (log) 13:2 HVHV (log) 14:2 WW (log) 15:2 LLLL* (log) 16:2 LRLR* (log) 17:2 ship fr.2,1 <"i :2 , l 3.168 0.741 2.476 1.188 0.810 0.256 -0.482 0.488 0.708 0.177 0.340 0.343 0.461 0.213 car tH.2,2 <"»:2,2 3.767 2.070 1.255 0.143 0.509 0.137 -1.214 0.087 0.629 0.248 0.337 0.131 0.532 0.201 rock Pi-2,3 <"t:2,3 4.062 0.599 2.470 0.799 0.489 0.209 -0.454 0.450 0.645 0.181 0.149 0.289 0.469 0.152 ocean A*.:2,4 <~t:2,4 5.337 0.482 1.454 0.744 0.012 0.368 -1.534 0.265 0.410 0.286 -0.691 0.229 0.226 0.317 building Pi.2,5 <""i:2,5 3.314 0.989 3.679 1.510 1.543 0.518 0.532 0.442 1.409 0.519 1.429 0.587 0.920 0.357 backstop Pi.2,6 <Ti:2,6 3.473 0.807 3.204 1.061 0.725 0.270 -0.018 0.356 0.715 0.335 0.697 0.336 0.293 0.371 Table 5.8: Summary of Secondary Polarimetric Data Interclass distance of Secondary Features The interclass distance measurement of the secondary polarimetric features are shown in Ta-ble 5.9. The logarithmic scaling of these features had little effect on the interclass distance measurements used in this analysis since the distance measure is heuristically determined and is invariant to both linear and non-linear feature scaling. In Table 5.9 the features with the highest interclass distances were: pf\Q-\og(LLLL*) and pfi4-\og(HVHV). This can be partially attributed to the good target to clutter ratios of feature 16 and feature 14 as shown in Chapter 4. Chapter 5. TARGET CLASSIFICATION 101 Overall Interclass Distance T>i Feature Spatial Functions Avg. value #3x3 # 5 X 5 03x3 0 5 X 5 V i _ 2 V l _ 3 V 2 - 3 Co-PPD 0.52 0.48 0.57 0.51 0.63 0.40 0.55 0.33 0.50 Cross-PPD 0.39 0.57 0.70 0.46 0.61 0.25 0.41 0.27 0.37 \og(HHHH*) 0.73 0.73 0.76 0.55 0.72 0.63 0.67 0.69 0.69 \og{HVHV) 0.81 0.79 0.81 0.67 0.69 0.63 0.70 0.58 0.71 log(VVVV) 0.52 0.46 0.48 0.47 0.68 0.35 0.45 0.39 0.48 \og{LLLL*) 0.83 0.85 0.85 0.67 0.73 0.72 0.76 0.66 0.76 \og(LRLR*) 0.51 0.50 0.52 0.46 0.59 0.39 0.48 0.43 0.43 Average 0.62 0.63 0.67 0.54 0.66 0.48 0.57 0.48 0.57 Table 5.9: Inter-Class Distance for / i 3 X 3 of Secondary Polarimetric Features The spatial functions of the polarization phase difference features show little interclass distance except for 12:Z-Cross-PPD:p-x-. This is understandable since several of the target classes are dominated by double bounce scatters; hence the polarization phase differences will be similar. This suggests that Co-PPD may be good for distinguishing man made targets from natural targets and clutter but Cross-PPD may be good for distingiiishing between man made targets. However, this observation is not proven here. Reduced feature sets Reduced set of conventional S A R spatial functions The feature selection algorithm was used to reduce the set of spatial functions on conventional SAR data - HH and VV antenna polarizations. The reduced sets consist of the following features: Feature Interclass Distance \og(HHHH*) ship car rock ocean bldg bkstp overall 13:3 /i5x5 0.66 0.64 0.70 0.84 0.96 0.76 0.76 13:5 05x5 0.56 0.74 0.67 0.94 0.75 0.66 0.72 13:2 #3x3 0.68 0.60 0.64 0.91 0.94 0.62 0.73 13:4 03x3 0.39 0.62 0.50 0.81 0.59 0.41 0.55 Chapter 5. TARGET CLASSIFICATION 102 Feature Interclass Distance l o g ( V V W ) ship car rock ocean bldg bkstp overall 15:5 ( 7 5 x 5 0.46 0.69 0.62 0.93 0.71 0.70 0.68 15:2 ^3x3 0.39 0.29 0.36 0.63 0.81 0.32 0.46 15:4 0 3 x 3 0.37 0.52 0.42 0.48 0.51 0.53 0.47 15:3 psx5 0.39 0.48 0.38 0.46 0.89 0.30 0.48 The interclass distance for the HH antenna data is much better than the V V antenna data. This is due to the higher return from clutter in the VV data reducing the signal to noise ratio thereby increasing the randomness of the target data. It is interesting that for HH and VV the same spatial functions are selected. Since pixel averaging has a better interclass distance than just the centre value s/i, classification perfor-mance should improve by increasing the number of looks at the data. This is consistent with the result found by Novak et al [7] where classification performance improved with an increase in the number of looks. Reduced sets of spatial functions Reduced feature sets were selected from the remaining secondary features. Features 14, 16 and 17 represented data from different antenna arrange-ments not normally configured in a SAR. However there are some recent studies [7] that show target detection is better with these antenna arrangements. HV and LL antennas give very good target to clutter ratio as shown in the analysis of the polarization signatures and TCR in Chapter 4. The LL antenna would intuitively he a better antenna configuration than the HV since the average return from the LL is still very strong while the HV return is mostly attenuated as shown in Table 5.8. In this table it can be seen that the LL return from targets such as ship and building is stronger than the HV return. Conversely, the LL return from ocean is weaker than the HV return from ocean. A stronger LL return results in higher signal to receiver noise and therefore less randomness in the feature space. Chapter 5. TARGET CLASSIFICATION 103 Reduced sets of spatial functions Co-•PPD Cross-PPD log{HVHV*) log(LLLL*) log{LRLR*) 11:5 05x5 12:3 # 5 X 5 14:3 #5X5 16:3 #5x5 17:5 cr 5 X 5 11:2 #3X3 12:5 0 5 X 5 14:4 03x3 16:4 03x3 17:3 <r 3 x 3 11:7 V l _ 3 12:2 #3x3 14:1 value 16:1 value 17:7 V i _ 3 11:3 #5X5 12:4 03x3 14:2 #3X3 16:2 #3X3 17:4 c r 3 x 3 11:4 03X3 14:5 14:7 05X5 V l _ 3 Bes t set of secondary polarimetric features Either from examination of the interclass distance tables in Appendix C or using the feature selection algorithm, it is possible to determine a reduced set of secondary features that provides good separation for all target classes. This best set of secondary polarimetric features are shown below: Best Feature Interclass Distance List ship car rock ocean bldg bkstp overall 16:3 \og{LLLL*) # 5 X 5 0.72 0.84 0.74 0.95 0.95 0.90 0.85 14:4 log(HVHV*) 0 3 X 3 0.78 0.75 0.52 0.76 0.59 0.63 0.67 16:1 \og(LLLL*) value 0.70 0.86 0.89 0.88 0.90 0.77 0.83 14:2 \og{HVHV*) # 3 x 3 0.59 0.93 0.68 0.98 0.89 0.70 0.79 12:3 Cross-PPD # 5 x 5 0.51 0.82 0.65 0.87 0.79 0.59 0.70 This reduced set consists of a smoothing and a texture spatial function of the two best synthesized intensities HV and LL. It is interesting that Cross-PPD is selected over Co-PPD for the reduced set. In comparing this set to the reduced primary polarimetric feature set it can be seen that Cross-PPD corresponds to ||9(SMS£„)||. In Table 5.10 the correlation coefficients for the polarimetric features containing the phase difference information are shown. It shows that Co-PPD and Cross-PPD are strongly correlated with their imaginary components while not as correlated with their real components. Since the reduced set of primary polarimetric Chapter 5. TARGET CLASSIFICATION 104 features contained ||9(5wi5£v)|| it is not surprising then that Cross-PPDh&s been selected for this reduced feature set. Aggregate Correlation Coefficient ^ 3 x 3 log (span) 115M s*vJ Co-PPD ll5ju.5J.ll Cross-PPD & » log(span) 1.00 0.49 0.38 0.41 0.31 0.33 0.36 0.49 1.00 0.31 0.41 0.36 0.45 0.33 119(5^ 5^ )11 0.38 0.31 1.00 0.84 0.21 0.44 0.42 Co-PPD 0.41 0.41 0.84 1.00 0.30 0.33 0.42 W(shhs*hv)\\ 0.31 0.36 0.21 0.30 1.00 0.37 0.38 0.33 0.45 0.44 0.33 0.37 1.00 0.75 Cross-PPD 0.36 0.33 0.42 0.42 0.38 0.75 1.00 Average 0.38 0.39 0.43 0.45 0.32 0.45 0.44 Table 5.10: Correlation of ShhS*v vs. Co-PPD 5.3.4 Fea ture Select ion S u m m a r y Several combinations of spatial functions of polarimetric features have been proposed in the lists above to form reduced feature sets. These feature sets have been determined from the interclass distance, a heuristic feature selection algorithm and classification algorithms. In the next section these feature sets will be used in a classification algorithm to evaluate the performance advantages of polarimetry in target discrimination. These feature sets consist of suboptimal linear combinations of the polarimetric data since they were selected by a heuristic algorithm. Using the feature statistics it is possible to get a covariance matrix for the feature data. From this, linear weights for each feature can be determined that maximizes the classification performance of these reduced feature sets. The principle reasons for not using statistical methods to optimize the feature sets is due to the lack of confidence in the data as a result of the small training sets and the large number of possible feature combinations that were analyzed. In addition, it will be shown that this further analysis was not necessary for demonstrating the objectives of this thesis. Chapter 5. TARGET CLASSIFICATION 105 5.4 Classification of Conventional S A R Data A conventional SAR has one of two co-polarization antenna configurations: horizontally linear, HH, or vertically linear, VV. For example, SEASAT used a single channel HH antenna. The secondary polarimetric features log(HHHH*) and log(VVVV*) are equivalent to data received from these single channel radars. In this section, the results of several classification trials using only data available to conventional radars are presented. 5.4.1 Pixel averaging of conventional S A R data Pixel averaging is similar to look averaging and should reduce the coherent noise content in the digital data; however, pixel averaging also reduces the dynamic range of features. Table 5.11 shows the results of using a single band of SAR data with no spatial functions of the data other than pixel averaging. On the left hand side of the table is the feature set. For example, 13:1 means sf\-value of the polarimetric feature log(HHHH*). To the right of the feature sets, the ratio of unclassified targets and the classification success measure for each target class is shown. The total probability of a correct decision is shown on the far right. Class unch is the probability of getting an unclassified target, that is, one with a distance statistic larger than the confidence threshold. Run Area Feature Probability of Success by Class Total Set uncls ship car rock ocean bldg bkstp P W HH antenna 1 1 X 1 13:1 0.04 0.50 0.00 0.09 0.50 0.54 0.30 0.34 2 3 x 3 13:2 0.06 0.50 0.00 0.36 0.67 0.54 0.20 0.40 3 5 x 5 13:3 0.04 0.00 0.00 0.27 0.23 0.63 0.60 0.38 VV antenna 4 1 x 1 13:1 0.06 0.17 0.00 0.00 0.17 0.45 0.40 0.23 5 3 x 3 13:2 0.06 0.00 0.00 0.27 0.83 0.54 0.00 0.30 6 5 x 5 13:3 0.04 0.17 0.00 0.09 0.23 0.54 0.00 0.31 Table 5.11: Classification of Conventional SAR Data Chapter 5. TARGET CLASSIFICATION 106 The results in Table 5.11 show that HH antenna configuration is better for target classifica-tion than V V radar. While pixel averaging improves the results slightly, the classifier is unable to recognize any class at a satisfactory level. Using the single band of HH data averaged in a 3 X 3 neighbourhood around the target centre, the classification result was 40% successful. This 40% success rate is the best result for conventional radar and is achieved with only one channel of data and no measures of the targets radiometric distribution. 5.4.2 Spatial Functions of Conventional S A R data In this section spatial functions of features are used to improve classification. The list of spatial functions is repeated in Table 5.12. Spatial Functions Group Function index label value sh pf:l Pixel P3x3 sh P/ :2 averaging M5X5 sh p / : 3 Standard 0*3x3 SJA p / : 4 deviation 05x5 sh p / : 5 Vi_2 sh p / : 6 Gradient V i _ 3 sh P / : 7 V 2 _ 3 sh p / : 8 Table 5.12: List of Spatial Functions The results of classification using spatial functions is shown in Table 5.13. The first trials used the complete set of eight spatial functions for HH and VV data. For HH data, the success rates for classes: car, rock and ocean improved considerably with the total success increasing 32% to 0.53. In order to eliminate spatial functions that provide little benefit to the classifier, the reduced set of functions selected in Section 5.3.2 is used to form the feature vector (trials 3&4). The reduced set, consisting of all the smoothing and texture measures, improved classification in Chapter 5. TARGET CLASSIFICATION 107 Run Antenna Config. Feature Set uncls Probability of Success by Class Total P[s) ship car rock ocean bldg bkstp Complete set of functions 1 2 \og{HHHH*) \og{WW*) 13:1-8 15:1-8 0.06 0.06 0.50 0.00 0.67 0.33 0.64 0.00 0.83 0.50 0.45 0.54 0.30 0.20 0.53 0.26 1 .educed set of functions 3 4 log(HHHH') \og{VVVV*) 13:3,5,2,4 15:5,2,4,3 0.06 0.04 0.50 0.00 0.67 0.33 0.54 0.18 0.67 0.67 0.64 0.45 0.50 0.40 0.57 0.34 J limited set of functions 5 6 lo^HHHH*) l o g ( V V V V ) 13:1,3,5,8 15:1,3,5,8 0.09 0.04 0.50 0.00 0.67 0.33 0.64 0.09 0.67 0.50 0.45 0.34 0.50 0.50 0.55 0.30 Power scaling of Feature 7 8 HHHH* WW' 13:1,3,5,8 15:1,3,5,8 0.06 0.06 0.67 0.00 0.00 0.00 0.54 0.09 0.83 0.83 0.27 0.27 0.20 0.10 0.43 0.22 Table 5.13: Classification using Spatial Functions in Conventional SAR HH data of the classes building and backstop at the expense of ocean and rock classes. The overall classification success increased slightly to 0.57. An alternate set of spatial functions is selected from the best spatial function from each type of statistical measure. This limited feature set consists of: value, # 5 X 5 > 0 5 x 5 ) and V2_3-The results from this limited set (shown in trials 5 and 6) are marginally less than the reduced set. It appears that the gradient measure and centre value do not provide useful information. In order to justify the use of logarithmic scaling of features, classification results from power scaled data for the limited set of functions is shown in trials 7 & 8. The success measure is much less than trial 5 & 6 and is only marginally better than using only pixel averaging. Further evidence of the better classification success achieved with logarithmic scaling will be presented later. Target classification using V V data in all the trials of Table 5.13 is very poor especially for the targets in the ocean: ship, car, rock. This is consistent with the target detection observations in Chapter 4 where V V has a very low target to clutter ratio. Chapter 5. TARGET CLASSIFICATION 108 5.4.3 Summary of conventional S A R classification Of the two conventional SAR configurations, HH and VV, the reduced feature set of spatial functions of HH gave the better result of 0.57. In comparison with the previous section when only pixel averaging was used to reduce noise effects in the single band of HH data, the classification result was only 0.40. This shows that use of spatial functions in conventional SAR can reduce the effects of noise and make better use of the available information. The improvement in classification accuracy is about 40% with the use of spatial functions. The following list summarises the significant observations: • Classification success for conventional SAR without spatial functions is 40%. • Classification success using spatial functions of conventional SAR data is 57%. • Logarithmic scaling of features appears to provide better classification between target classes. • Classification success is better for HH data rather than V V data. Chapter 5. TARGET CLASSIFICATION 109 5.5 Classification of Polarimetric S A R Data Polarimetry provides much more information about the scattering mechanisms in a target. In this section the polarimetric features identified in Chapter 3 are evaluated in the minimum distance classifier. These polarimetric features are summarised in Table 5.14. Polarimetric Features (pf) Primary Label Secondary Label log(span) 1 : 5 / Co-PPD 11: sf H'SfcfcSwJI 2 : 5 / Cross-PPD 12: sf 3 : 5 / log(HHHH*) 13: sf 4: s / \og(HVHV*) 14: sf liao&fcOl 5 : 5 / \og(VVVV*) 15: sf 6: s/ log(LLLL*) 16: sf 7 : 3 / \og(LRLR*) 17: sf P ( ^ v ) | | 8:s / 9:s/ P(5fcu5u*v)|| 10: sf Table 5.14: List of Polarimetric Features 5.5.1 Pixel Averaging of Polarimetric Features In this section classification success of polarimetric features using only pixel averaging in the target window is evaluated. Primary polarimetric features The classification results for pixel averaging of primary polarimetric features is shown in Ta-ble 5.15. Trials 1, 2, and 3 of Table 5.15 show the results for the complete set of primary polarimetric features. In these trials, pixel averaging does not have a significant effect on the results. In trials 4, 5 and 6 of Table 5.15, the reduced sets of primary polarimetric features selected in Section 5.3.2 are used in the classifier. For the reduced feature set, the features are written Chapter 5. TARGET CLASSIFICATION 110 in descending order of utility as determined by the feature selection algorithm. By using fewer features selected in a systematic manner, the results improve slightly and pixel averaging does have a slight benefit on the results. Run Area Feature Probability of Success by Class Total Set uncle ship car rock ocean bldg bkstp P[s] Complete Feature Set 1 l x l 1-10:1 0.02 0.67 1.00 0.45 1.00 0.54 0.60 0.64 2 3x3 1-10:2 0.04 0.67 0.67 0.45 1.00 0.45 0.70 0.62 3 5x5 1-10:3 0.04 0.33 1.00 0.54 1.00 0.64 0.60 0.64 Reduced] feature Set 4 l x l 1,4,5,10:1 0.02 0.67 1.00 0.54 0.50 0.67 0.60 0.62 5 3x3 1,5,4,3,7,10:2 0.02 0.50 1.00 0.65 0.83 0.73 0.70 0.66 6 5x5 3,5,1,4:3 0.02 0.23 1.00 0.54 0.83 0.64 0.90 0.68 Table 5.15: Classification by Primary Polarimetric Features The polarimetric features have improved the classification of the ocean clutter class and car target class. In addition the other class success measures are all at reasonable levels indicating good separation of target class feature centers in the feature space. The number of unclassified targets is very low and constant indicating the maximum distance threshold is at an effective level. Comparing primary polarimetric classification with the results in Table 5.11 shows that polarimetric information improves the best result from single channel radar (3x3 HH data) from 0.40 to 0.66. This is a significant 65% improvement in classification accuracy. Secondary polarimetric features The secondary polarimetric features are related to the primary features in the following man-ner. The synthesized intensities, features 13-15, are the unnormalized logarithm of features 2-4. The polarization phase terms, features 11-12, are non-linear functions (arctan) of the complex polarimetric features 5-10. Features 16-17 are co- and cross- circular polarized antenna inten-sities synthesized from the Stokes' matrix. These secondary features (11-17) contain the same Chapter 5. TARGET CLASSIFICATION 111 information as polarimetric features 1-10 but in a different form. The results of these trials are contained in Table 5.16. The classification results of the secondary polarimetric features and the span varied con-siderably due to changes in scale representation. Using the features in units of power (P) as synthesized from the Stokes' matrix, the classification was very poor relative to the results shown here. When the features where represented in voltage units (y/P) the classification re-sults improved. However, the best results occurred using logarithmic scaling (log 1 0(P)). In Table 5.16 two versions of feature scaling are shown to compare the classification results. Run Area Feature Probability of Success by Class Total Set uncls ship car rock ocean bldg bkstp P[s] log. scaled features 1 l x l 11-17:1 0.04 0.50 0.67 0.54 0.67 0.63 0.50 0.57 2 3x3 11-17:2 0.04 0.23 0.67 0.63 0.83 0.54 0.50 0.57 3 5x5 11-17:3 0.02 0.00 0.67 0.36 0.67 0.82 0.70 0.55 power scaled features 4 l x l 11-16:1 0.09 0.50 0.67 0.27 1.00 0.00 0.50 0.40 5 3x3 11-16:2 0.09 0.33 0.67 0.64 1.00 0.18 0.40 0.49 6 5x5 11-16:3 0.06 0.17 0.33 0.54 1.00 0.36 0.40 0.47 Table 5.16: Classification by Secondary Polarimetric Features In trials 1-3 the logarithmically scaled secondary polarimetric features form the feature vector using different sized windows of pixel averaging in each trial. Classification using pixel averaging of secondary features is about 10% less accurate than using the primary polarimetric features. However, secondary features are still considerably better than conventional radar with a 42% improvement in accuracy. Using power scaling of the secondary features, overall classification accuracy drops relative to log scaling especially for the building class. However, the ocean clutter class has perfect success. Because of the larger contrast between target features, the clutter class (ocean) is better separated from the target classes in the feature space. However, the increase in dynamic range of the feature values has led to an increase in the number of unclassified targets. In Table 5.16, it Chapter 5. TARGET CLASSIFICATION 112 appears that the power scaled features have more difficulty distmguishing between the different target classes thereby degrading performance. While the maximum distance threshold can be adjusted to compensate for the increase in the number of unclassified targets, it will not necessarily lead to an increase in classifier performance. Primary and secondary polarimetric features Run Area Feature Set uncls Probability of Success by Class Total P[.] ship car rock ocean bldg bkstp 1 1 x 1 1,16,14,13:1 0.06 0.50 0.00 0.63 0.50 0.54 0.40 0.55 2 3 x 3 16,4,13,14,5:2 0.04 0.23 1.00 0.54 0.83 0.54 0.70 0.62 3 5 x 5 16,3,5,13,4:3 0.02 0.50 1.00 0.54 0.83 0.63 0.80 0.68 Table 5.17: Classification by Primary and Secondary Polarimetric Features In Table 5.17, the classification results using the reduced feature set of all the polarimetric features are shown. The results of this mixed reduced feature set are similar to the results of the reduced set of primary polarimetric features in Table 5.15. The feature selection algorithm is using a super set of features from the previous experiment, and should have achieved the same or better results. Since the results did not improve for these sets, the secondary features must not be as reliable in classification as the primary polarimetric features. Therefor to achieve better classification results the polarimetric features should be normalized. Comparing trials 1-3 shows that pixel averaging can improve classification accuracy although the effect is not large. The reduced features sets in trials 2 and 3 have the polarimetric features ||K(5/lh5'*v)|| and ||<S/M;SJM)||. F ° r m o s t discrete targets | |5/i, ,5£j| has good target to clutter ratio and ||9£(S«iS*„)|| contains important phase information about the target's scattering mechanisms. Since ||3?(5'/,/lS'*u)|| has been selected over Co-PPD it appears that ||3?(5^5*v)|| is the more useful representation of the co-polarization phase difference in a classifier. 5.5.2 Spatial Functions of Polarimetric Features Chapter 5. TARGET CLASSIFICATION 113 Primary Complete Reduced Limited Run Polarimetric Set of P[s) Set of P[s] Set of PW Feature Functions Functions Functions 1 log(span) 1 1-8 0.47 1:1,5,3 0.47 1:1,3,5,8 0.45 2 \\ShhSlh\\ 2 1-8 0.51 2:3,5,8,4,2 0.43 2:1,3,5,8 0.53 3 \\SVVS*V\\ 3 1-8 0.45 3:3,5,1,2 0.55 3:1,3,5,8 0.55 4 \\SKA\\ 4 1-8 0.28 4:3,1,5,4,2,8 0.30 4:1,3,5,8 0.34 5 \mshhs;v)\\ 5 1-8 0.62 5:3,4,1,8 0.64 5:1,3,5,8 0.57 6 \\*(SHhS;v)\\ 6 1-8 0.38 6:5,2,4,8 0.36 6:1,3,5,8 0.43 7 7 1-8 0.45 7:5,3,4,2 7:1,3,5,8 0.51 8 mshkS<hv)\\ 8 1-8 0.28 8:5,4,1,8 0.32 8:1,3,5,8 0.36 9 \Mshvs;v)\\ 9 1-8 0.57 9:5,1,8,3 0.53 9:1,3,5,8 0.49 10 \MshvS;v)\\ 10:1-8 0.26 10:5,2,4,1 0.38 10:1,3,5,8 0.32 Average 0.43 Average 0.44 Average 0.46 Table 5.18: Classification using Spatial Functions of Primary Polarimetric Features The usefulness of spatial functions in classification is determined through a series of trials using spatial functions of only one polarimetric feature. The classification results for spatial functions of the primary polarimetric features are shown in Table 5.18 and the results for spatial functions of secondary polarimetric features are shown in Table 5.19. The reduced features sets are shown in the centre of Table 5.18 and Table 5.19. The feature set lists the best spatial functions in descending order of utility as determined by the feature selection algorithm. The best function over the 17 runs is 5/5 - 05x5 having been selected 15 out of 17 trials and usually placing first or second. The best averaging feature is s/3 - ^5x5. Of the gradient functions sfe- V2-3 is selected the most with s/6-Vi_2 not selected at all. The best spatial functions from each type of measurement are chosen to form a limited set of functions and classification trials performed. The results of these limited sets of functions are shown in the right hand columns. Using the complete set of spatial functions, the polarimetric features giving the best results are the co-polarized term \o%(LLLL*) (16 : sf) and the cross-polarized term \og(HVHV) (14 : sf), the complex component H ^ S M . ? * , , ) ! ! (5 : sf) and the co-polarized termlog(.ff#iz'.rI*) (13 : sf). The span (1 : sf) feature provides only average performance. Using the limited set of spatial functions, the overall results improve slightly. Selecting Chapter 5. TARGET CLASSIFICATION 114 Secondary Complete Reduced Limited Run Polarimetric Set of P[a) Set of P[°) Set of P[M] Feature Functions Functions Functions 1 Co-PPD 11:1-8 0.51 11:5,2,7,3,4 0.57 11:1,3,5,8 0.55 2 Cross-PPD 12:1-8 0.38 12:3,5,2,4 0.45 12:1,3,5,8 0.38 3 \og{HHHH') 13:1-8 0.53 13:3,5,2,4 0.57 13:1,3,5,8 0.55 4 \og{HVHV) 14:1-8 0.57 14:3,4,1,2,5,7 0.62 14:1,3,5,8 0.57 5 \og(VVVV*) 15:1-8 0.26 15:5,2,4,3 0.34 15:1,3,5,8 0.30 6 \og(LLLL») 16:1-8 0.62 16:3,4,1,2 0.68 16:1,3,5,8 0.66 7 \og(LRLR*) 17:1-8 0.32 17:5,3,7,4 0.32 17:1,3,5,8 0.34 Average 0.46 Average 0.51 Average 0.48 Table 5.19: Classification using Spatial Functions of Secondary Polarimetric Features the spatial functions with the feature selection algorithm increased the overall accuracy a little more. In analyzing the effectiveness of the selection algorithm, it is apparent that it tends to perform best when dealing with a set of "solid" features. For instance, in trial 6 of Table 5.19, the four features selected by the algorithm produced an improvement of the accuracy with the good success rate of 0.68. Since the classification results from the limited feature set is close to the results using the reduced feature set, they may be considered a good set of functions for describing the spatial distributions of target features. Classification using Single Channel Radars Trials 3-7 in Table 5.19 represent accuracies attainable using data from a single channel SAR. Of these, the probability of success in trials 13, 14 and 16 are well above average with the co-polarized polarimetric feature log(LLLL*) giving the best result of 0.68. The co-polarized feature log(LLLL*) and cross-polarized feature log(HVHV*) gave superior results than the cross-polarized feature log(LilI,R*)and co-polarized features \og(HHHH*) and log(VVVV*). This corresponds with earlier observations in Section 4.2.2 where co-pol circular and cross-pol linear polarizations have much higher target to clutter ratios than cross-pol circular and co-pol linear polarizations. Therefore they provide more useful information about each target and Chapter 5. TARGET CLASSIFICATION 115 should be less susceptible to variations from noise. The feature selection algorithm was used to determine the reduced feature set for the spatial functions of a dual polarized LL,LR radar. However, none of the spatial functions of LR data was selected. As the TCR and feature data show, the LR data does not appear to provide any useful information in classifying targets. This is understandable since the return from double bounce reflectors is mostly attenuated by an LR antenna. Classification using Polarization Phase Difference In comparing the polarization phase differences (features 11, 12), co-PPD is more useful in target classification than the Cross-PPD. This may be attributed to the higher signal to noise ratio in the Co-PPD feature. The Co-PPD information is contained primarily in the primary polarimetric feature ||K(5fcfc5^ I>)||. Using spatial functions of feature ||3?(5h/>S'*t,)|| provides better classification than not only the Co-PPD feature but also from single channel HH data. This demonstrates the importance of phase information in target classification. 5.5.3 Best Polarimetric Feature Sets To assess the classification performance of unrestricted combinations of spatial functions with polarimetric features, the feature selection algorithm was applied to all spatial functions of every polarimetric feature. Firstly, only the 80 spatial functions of the primary polarization features were used as the basis set for the algorithm. The algorithm selected the most useful spatial functions of four different polarimetric features. The classification result produced by this set of primary polarimetric features is shown in the first trial of Table 5.20. This result is far better than that yet achieved with P[success] = 0.79. Compared with classification using only pixel averaging to reduce noise in polarimetric features (P[success] = 0.68), a 16% increase in classifier performance is achieved by adding spatial functions. The second trial of unrestricted feature sets consisted of the 56 spatial functions of the secondary polarimetric features. The reduced set as selected by the feature selection algorithm Chapter 5. TARGET CLASSIFICATION 116 Feature Set Probability of Success by Class Total P[s] ttncls ship car rock ocean bldg bkstp Primary polarimetric features (lrlMSrSMfrSMStS),^*) 0.00 0.67 0.67 0.54 1.00 0.82 1.00 0.79 Secondary polarimetric features (16:3),(14:4),(16:1),(14:2),(12:3) 0.02 0.50 1.00 0.54 0.83 0.73 0.60 0.66 Polarimetric features (16:3),(1:4),(5:3),(1:1),(8:5),(3:3) || 0.00 0.50 1.00 0.54 0.83 0.82 0.90 0.74 Table 5.20: Classification using best polarimetric features consisted of functions of only three different secondary features. The classification result of this feature set, shown in trial 2 of Table 5.20, is 0.66. While this success rate is similar to the best result in Table 5.15, it is not as good as the first trial. The last trial consisted of a feature set selected from the spatial functions of the primary polarimetric features and the best secondary features: log(LLLL*) and \og(HV HV*). Adding these secondary features to the primary reduced feature set gave a result of 0.74. Since the addition of secondary features to the primary features did not improve classification accuracy, it is felt that the primary polarization features are all that are necessary for supervised target recognition. This is a reasonable conclusion since the primary polarimetric features are less correlated and represent a complete definition of the polarization signature. Chapter 5. TARGET CLASSIFICATION 5.6 Experimental Improvement in S A R Classification 117 SINGLE C H A N N E L P O L A R I M E T R I C SAR SAR Figure 5.1: Summary of Classification Results Shown in Figure 5.1 is a summary of the significant results comparing polarimetric SAR with single channel SAR, and spatial functions with pixel averaging only. The arrows illustrate the progression from conventional SAR on the left to polarimetric SAR on the right and from only pixel averaging on the top to using spatial functions on the bottom. When only pixel averaging is used to reduce noise, single channel radar gives a success rate of 0.40 while polarimetric SAR increases the rate by about 65% to 0.66. Selecting a combination of the most useful spatial functions further reduces the effects of noise such that classification success is improved by another 40% for single channel and 16% for polarimetric radar. The overall classification Chapter 5. TARGET CLASSIFICATION 118 result for polarimetric radar is 39% higher than that attainable using spatial functions of single channel radar. The LL antenna configuration has been shown to be very good for target detection in Chapter 4 as well as in [7]. Classification on using a single band of LL target was very successful with P[s] = 0.68. This is a significant improvement over conventional HH SAR but is not quite as good as using the full polarimetric data set. In addition, LL is not useful as a single channel SAR because it is cross-pol and will give poor image quality. Further, it may be possible to further improve classification performance by determining the optimum weighting vector for the polarimetric feature vector. This would increase the performance difference between the co-circular LL radar and the polarimetric radar. Chapter 5. TARGET CLASSIFICATION 119 5.7 Error Sensitivity of Classification Algorithm The results of the classification algorithm have been determined from real remotely sensed polarimetric SAR data. In addition to several target related conditions that can affect the polarimetric classification features (namely target orientation, size and angle of incidence) radar calibration is a system related condition that can adversely affect the measurements. Amplitude calibration is an important problem in any SAR in order to compensate for noise, terrain reflectance etc. In a polarimeter there is an additional problem of mamtaining proper phase calibration between the two orthogonally polarized antennas and their associated data paths. Improper phase calibration causes errors in the phase relationships leading to possible false conclusions about the scattering behavior of pixels. In order to evaluate the sensitivity of the classification algorithm to phase calibration error and other parameters, a Polarimetric Target Model (PTM) of a double bounce target in Bragg scattering clutter has been developed. The a priori target classes of the San Francisco Scene have been defined, for the most part, by their ground truth detail. However, as has been explained in Chapter 4, it is possible to relate the target and clutter classes in terms of simple scattering models defined in Section 2.4.4. The model described below incorporates many of the SAR target and clutter parameters discussed in the proceeding work. The relative strength of the scattering matrix elements, speckle and receiver noise, target-clutter ratio, and the distribution of the target cross section in the model are based on the theory and observations already presented in this thesis. 5.7.1 Polarimetric Target Model Description Scattering Matrix The true scattering matrix for a resolution element is given by S = (5.13) Chapter 5. TARGET CLASSIFICATION 120 where 5,-j is the uncorrupted scattering amplitude and phase for i receive, j transmit polar-ization. This model assumes each term in the scattering matrix is real, not complex. This assumption is consistent with the simple scattering models described in Chapter 2. The model accepts real numbers for the polarimetric scattering data (S = [Shh, Svv, Shv]) for the target and clutter. For the results shown later a dihedral reflector is used as the target S(target) = [—6,3,0.5] and Bragg scattering is used for the clutter S(clutter) = [3,6,0.2]. These scattering models are similar to the ideal scattering models of a ship in the ocean. The relative sizes of Sij in each scattering matrix are based on the average primary polarimetric features contained in Appendix A . The relative signal strength between the target and the clutter can be set by a factor K to adjust the T C R . K is applied to all terms in S(target). Phase Distortion of Polarimeters This section briefly defines phase distortion in terms of its effect on the scattering model as fully presented in Zebker et al. [34]. When the radar data is corrupted by path differences in the system hardware, the received matrix R is given by ( Shh expXc^h + <Pr,h) Shv exp j(<pt,v + 4>r,h) R = ^ Svh exp j(<ptih + (pr,v) S„v exp j(<pt<v + <pr,v) where the phase factors correspond to path lengths for the transmit and receive polarized waves within the polarimeter. The overall phase of the scattering matrix is not important so it may be factored out of the matrix. For simplicity the phase of Svv is removed leaving: ( Shhexpj(<pt + <pr) Shvexpj(<pr) R — ^ Svh*xpj(<Pt) Svv where <pt = <fitth — <j>t,v arid <j)r = <pTth — <j>r,v The two calibration terms fa and <pr correspond to the differences of the respective path lengths in the system hardware. Good design practice is to minimize these distance differences; however, at higher frequencies, such as C-band operated Chapter 5. TARGET CLASSIFICATION 121 by the JPL's multifrequency polarimeter [46], even very slight changes in the path lengths may cause serious phase errors. Model Profile The target model covers a 15 x 15 pixel region. The target scattering data is convolved over this region using a two dimensional Gaussian function. This distribution is used since it resembles the typical profile of ship targets in ocean clutter. The distribution is given by Dij{x,y) = Kexv~(x2+y2) Retarget) + R-jiclutter) (5.16) where x,y are spatial coordinates and —l<x<l,—l<y<l. R'+j is the de-calibrated version of the original scattering matrix element. This Gaussian distribution leads to a target size of 5 to 9 pixels depending on the target gain term n. K is a target gain term used to adjust the TCR within the model. Receiver and Speckle Noise Receiver noise is modelled as a uniformly distributed, zero mean, random function. It is assumed that the receiver noise between the vertical and horizontal antennas are independent so two noise terms are used n^u and nTth where r means receiver. The receiver noise content can be increased by a factor a as follows: nr,h(zi) = a(zi - 0.50), 0 < zx < 1 (5.17) TirA 2*) = a(z2 - 0-50), 0 < z2 < 1 (5.18) where z\ and z2 are independent random functions. For the results shown in this section a has been set to 1.0. The speckle noise between the horizontal and vertical antennas are assumed to be perfectly correlated but between the co-pol and cross-pol channel only 50% correlated. Therefore two speckle noise parameters are used nt<i and nt>2 where s means speckle. Speckle noise is modelled Chapter 5. TARGET CLASSIFICATION 122 as a random function given by the following equations: n . , i ( * 3 ) = 0.75 + 0.5*3, 0 < z3 < 1 (5.19) n,,2{z3,z4) = 0.75 + 0.5(^3 + z4), 0 < z3 < 1, 0 < z4 < 1 (5.20) where z3 and Z\ are independent random functions. Speckle noise is multiplicative so nty\ and n,(2 are multiplied with the phase corrupted scattering models. The receiver noise is added and the speckle noise is multiplied to the model channels as follows: Mhh(x,y) = Dhh(x,y)nt,i + nr,h (5.21) Mhv{x,y) = Dhv(x,y)n,,2 + ^(nr,h + nT,v) (5.22) Mvh(x,y) = Dvhix^n^ + ^irirfi + jir^) (5.23) Mvv(x,y) = Dvv(x,y)n,,i + n,,„ (5.24) where x, y are the spatial coordinates of the model. Mij(x,y) is the model scattering matrix element. The final step in the model is to convert the model scattering matrix elements Mitj(x,y) into Stokes' matrix elements for each pixel using Equations 2.42,... ,2.51. 5.7.2 Model Functions Several functions have been implemented to interpret the polarimetric target model. They are: 1. calculate target to clutter ratio, 2. compute target profile for any polarization, 3. compute polarization signatures, 4. extract polarimetric features and spatial functions, 5. determine the Euclidean distance statistics used in the minimum distance classifier. Chapter 5. TARGET CLASSIFICATION 123 The polarimetric target model incorporates several parameters of SAR polarimetry: • polarization diversity, • phase calibration, • speckle noise, • signal to receiver noise, and • target-clutter ratio. Except for speckle each of these parameters can be adjusted to allow analysis of their effect on the scattering behavior of targets in clutter. A major factor not modelled is multi-look averaging. The Stokes' matrix for each pixel in the model is single look while the compressed polarimetric data has four-look averaging for each pixel. As a result the Stokes' matrix for each pixel in the SAR data contains a diffuse component whilst each pixel in this model does not have a diffuse component. However, the results presented using this model consist of a man made target (ship) in ocean clutter. Both of these scattering models have a very low coefficient of variation, therefore the single-look model should be very consistent with the four-look real data. 5 . 7 . 3 Model Results This section presents visual and graphical results of the polarimetric classifier to demonstrate the effect of changing TCR and phase calibration on the polarization signature and the classifier performance. Target profile versus T C R Figure 5.2 shows the target profile of the model for different target-clutter ratios. The profiles were synthesized at HH polarization and the power levels have been normalized with respect to the maximum amplitude in the window. This figure visualizes the difference in target signal Chapter 5. TARGET CLASSIFICATION 124 strength relative to the surrounding clutter. The upper profiles clearly show the speckle effect on the uniformly distributed clutter region. The lower profiles show the distribution of the target in the clutter background. As TCR increases the target is easier to detect and the pixel lay over into other pixels increases. In Figure 5.2 the indicated TCR values appear smaller than they really are since the Stokes' matrix has been measures in the 3 x 3 neighbourhood about the target centre. Class i f ica t ion Dis tance versus T C R The graph in Figure 5.3 shows the Euclidean distance (dciat») divided by the maximum distance threshold for three classes (ship,car and rock) versus TCR. Nf is the number of features in the feature vector which consists of spatial function 1 of the ten primary polarimetric features (1-10:1). The distance statistic shown is the Euclidean distance of the target model features to the same training data used in the previous classification experiments. At TCR=-10dB the target is lost in the background clutter and the distance statistics are all about the same high level. As TCR increases d,hip decreases while dear and drock increase slowly. This decrease is due to the ship scattering properties dominating over the clutter. TCR — 20dB, dthip is clearly the smallest statistic and successful classification is likely. According to this graph it appears that ship classification is possible in the range 3dB < TCR < 40dJ9. At TCR = 20dl? the maximum difference between the class distances is observed and beyond TCR = 40dB the distance statistics increase exponentially as the intensity feature span increases. If log(span) was not included in the feature vector then distances statistics would be relatively constant b-.j ond TCR=20dB and there would be no upper bound on the TCR for effective classification. This graph shows that effective classification appears best in the 3dB to 40dB range. The TCR measures here are similar to the the measures in Chapter 4 because the target has been averaged in a 3 X 3 neighbourhood. This reduces the maximum intensity and the TCR accord-ingly. As such, the drop off of d,hip may actually occur at a higher level of TCR than reported in this graph. Chapter 5. TARGET CLASSIFICATION 125 TCR = -IQdB TCR = ldB Figure 5.2: Polarimetric Target Model Power (HH) Profiles Chapter 5. TARGET CLASSIFICATION 126 0 20 40 Target to Clutter Ratio (dB) Figure 5.3: Classifier Distance (1-10:1) versus PTM Target-Clutter Ratio Chapter 5. TARGET CLASSIFICATION 127 Polarization Signatures versus Phase Distortion Several co-pol and cross-pol signatures are shown in Figure 5.4 and Figure 5.5 respectively with 0 = <pt + 4>r and <pt = <pr • These figures reveal the transformation from a pure dihedral scattering signature (0 = 0°) in the upper left to a pure odd bounce scattering signature (0 = 180°) in the lower right. The signatures are fully polarized and have a degree of polarization d = 1 since they are only single look and therefore have no unpolarized component. The TCR at HH polarization is l ldB for all the signatures. As 0 increases the location of the nulls and maximums change. These figures give a visual understanding of the effect of phase de-calibration in the received scattering relationship of targets. With 0 = 20°, only a marginal change is noticed. However with <fi > 45° considerable distortion is apparent. This would suggest that classification using the polarimetric feature would be reasonably intolerant of phase distortion of less than 20°. Classification distance versus phase de-calibration The graph in Figure 5.6 shows the Euclidean distance used by the minimum distance algorithm to classify targets versus phase de-calibration. Since the scattering matrix of the target model resembles a ship, when 0 ~ 0°, the ship distance statistic d,hip is near the minimum and the smallest of the statistics. Likewise, at 0 ~ 0° the car statistic is the highest. The rock class is a combination of single and double bounce scattering and has an intermediate value. At 0 = 90°, dthip has changed dramatically as suggested by the considerable distortion of its signatures in Figures 5.4 and 5.5. At this point the ship could be misclassified as a rock or vice versa. At 0 = 180° the car statistic is lowest as expected by the change in sign of S^h due to the phase shift. Phase distortion of 180° is comparable to adding another bounce in the scattered wave. For the set of primary polarimetric features (1-10:1), the classifier is very insensitive to phase de-calibration in the region —35° < 0 < 35. The minimum of d,hip occurs at 0 = —20° rather than the intuitive location of 0°. This can either be due to a calibration error of the polarimetric data set or a factor in the ship target class not allowed for in this target model. Chapter 5. TARGET CLASSIFICATION 129 Figure 5.5: Cross-polarization Signatures of PTM versus Phase De-calibration Chapter 5. TARGET CLASSIFICATION 130 Figure 5.6: Classifier Distance (1-10:1) versus P T M Phase De-calibration The ship training class consists of many random factor not allowed for in the model. In either case this offset in the minimum of the ship distance statistic does not affect the conclusions. 5.7.4 Conclusions of polarimetric target model observations A polarimetric target model has been developed that demonstrates how various factors related to targets in SAR polarimetry affect classification. It was shown that classification using po-larimetric data can be insensitive to poor phase calibration within the range —35° < 4> < 35°. In addition target classification will only be successful on targets with a TCR greater than 4dB at HH polarization. While the TCR value here is lower than its actual value, this result suggest that utilization of polarimetric information may lead to successful target classification at a lower TCR than would be possible with other information. Chapter 6 C O N C L U S I O N S 6.1 Advantages of Polarimetry to S A R Target Recognition Polarimetry has been shown by many other researchers to be a useful and practical tool in several remote sensing applications. This thesis has studied the topic of target recognition of discrete man made targets and demonstrated some significant advantages of SAR polarimetry over conventional imaging radar. These advantages are three fold: • improvement in target detectability and clutter segmentation, • improvement in target classification, and • good classification sensitivity to target/clutter ratio and phase distortion. 6.1.1 Detection advantages with polarimetry In Chapter 4 it was shown how polarimetry improves the target to clutter ratio, thereby in-creasing the detectability of targets for a given false alarm rate. Target detection in coherent radar data can be improved by polarimetry because the en-ergy received from targets and background clutter are, among other factors, a function of the polarization of the radar antenna. Target detection involves the extraction of localized vari-able reflectance regions from the background clutter. Target to background contrast can be improved by finding the optimum ratio of the target cross section to background as a function of the antenna polarization. For some target classes, the TCR improvement is on average 6dB. It was shown that with a false alarm rate of 10 - 4 , a significant increase in the detectability of targets in background clutter can be achieved by polarimetric radar. 131 Chapter 6. CONCLUSIONS 132 In addition man-made targets tend to be strong double bounce reflectors which can be distinguished from typical clutter by the phase difference between the scattered elements SJJH and Svv- Polarimetry preserves this relative phase information which is unavailable with single channel radar. The unsupervised classification algorithm developed by van Zyl [13] is an example of a robust method to separate background clutter and targets into three broad classes. These classes are defined by the number of bounces in the dominant scattering mechanisms (odd, even or diffuse). While most interesting targets tend to be high reflectance objects with a single dominant scattering mechanism, it is possible that the same class of targets may reflect the E M wave with one, two or three bounces depending of the target orientation with the imaging radar and the background clutter. Since the unsupervised classifier has been shown (in other work [13]) to be effective at classifying terrain, another possible use would be to segment detected targets into classes broadly denned by the surrounding clutter background. Then an algorithm can be optimized to classify targets in known clutter background, thereby possibly improving classification performance. 6.1.2 Improvement in Target Classification in S A R Imagery After separating targets into these broad classifications, further analysis will be necessary to distinguish between different targets. In Chapters 3 and 5 it was demonstrated how polarimetry improves the classification performance over conventional single channel radar. The empirical quantitative results of this report clearly demonstrate the advantages of both polarimetric properties and spatial functions in target classification. When only pixel averaging is used to reduce noise, single channel radar gives a success rate of only 40% while polarimetric SAR increases the rate by about 65% to 66%. Selecting a combination of the most useful spatial functions further reduces the effects of noise and improves classification in both single channel and polarimetric radar. Overall polarimetric radar improved target classification 40% over that achieved by conventional radar with a 79% success rate. Chapter 6. CONCLUSIONS 133 The addition of polarimetric features together with spatial averaging and texture measures has been shown to significantly reduce the effects of limited information, speckle and receiver noise, as measured by single channel radar. This combination significantly enhances the recog-nition of discrete targets in synthetic aperture radar imagery. 6.1.3 Classification sensitivity to T C R and phase distortion A polarimetric target model that incorporates many factors related to targets in SAR po-larimetry has been developed to predict the sensitivity of the classification results to different target/clutter ratios and phase distortion. It was observed that target classification will be successful on targets with TCR greater than 4dB (taken at HH polarization). Classification based on the polarimetric features is relatively invariant to phase distortion within the range —35° < (<pt + 4>r) < 35°. This is a significant result as it allows considerable phase error which is very likely in the radar polarimeters presently operational or intended for the future. 6.2 Evaluation of Polarimetric Features and Spatial Functions The analysis of the polarimetric features and classification results has led to several observations on the use of the polarimetric features in addition to the important conclusions described above. These observations are summarised below: • The span of a target is a significant feature for classification as it contains the important relative signal strength for each target class. • The set of normalized cross products of the scattering matrix provide more useful infor-mation to the classifier than the set of secondary features. • The polarization phase differences are extremely useful in both target detection and clas-sification. Chapter 6. CONCLUSIONS 134 • The relative phase information is better represented in cross-product form than in radi-ans. This is useful since converting to radians is a computationally expensive non-linear operation. • The logarithmic scaling of features appears to provide better discrimination between tar-get classes although this has not been proven. • The texture measure standard deviation of polarimetric features is a very useful spatial function. • Pixel averaging of the primary polarimetric features is useful in reducing variations in intensity due to noise and improves target recognition. • The best single channel or dual channel SAR would be LL, or LL,LR. This antenna configuration leads to very good detection and classification although its performance is not as good as that achieved using the full polarimetric data set. In addition, LL would have limited imaging capabilities. 6.3 Further Study Although the conclusions presented in this report are significant, the sample data is not large or diverse enough to evaluate target recognition precisely. Even though all available information was used to verify the a priori target classifications, a reasonable degree of uncertainty exists. This uncertainty places a limitation on the quantitative conclusions in the thesis and the empir-ical performance estimates. While the target classes identified were sufficient for demonstrating the objectives of this research, they are not necessarily the type pertinent to evaluate target recognition algorithms with a very high degree of confidence. In order to confidently quantify the performance of target classification algorithms using real data, experiments must be performed in a controlled environment. In these experiments, the incident angle, orientation, size and class of each target are known. In addition, the targets must be sampled against different clutter backgrounds in which the physical structure is known. Chapter 6. CONCLUSIONS 135 This analysis will form polarimetric signatures for various classes of targets and clutter pertinent to the target recognition system requirements [47], and determine the degree of target and clutter separation afforded by the technology of polarimetry. This knowledge will be used to determine the performance bounds of an SAR system with polarimetry, and to provide information for the system design stage. This type of experiment is the focus of a research effort presently under way at the Canadian Defence Research Establishment in Ottawa, Ontario [48]. References [1] Bryan L. Huneycutt. Spaceborne Imaging Radar - C Instrument. IEEE Transactions on Geoscience and Remote Sensing, Vol 27(2):164-169, March 1989. [2] Jakob J. van Zyl, Howard A. Zebker, and Charles Elachi. Imaging Radar Polarization Signatures: Theory and Observation. Radio Science, Vol 22(4):529-543, July-August 1987. [3] Sanjay H. Singhal. 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Wintjes, C.E. Livingstone, M. R. Vant, and R. Austin. The development of a polarimetric SAR for the detection of man made objects. In Proceedings of IGARSS'89 Symposium, pages 2221-2224, University of British Columbia, Vancouver, BC, Canada, July 1989. Appendix A F E A T U R E D A T A A . l Polarimetric Feature Data /isxs Primary Polarimetric Data for class: ship pixel line span \\SVVS*V\\ I  5 h « ^ h K I  HWfcJI l!5h»5«t»ll (dB) a 9f 8? 1:2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 10:2 501 148 1.272 0.467 0.464 0.070 0.498 -0.026 0.088 -0.032 0.001 0.011 392 183 0.801 0.518 0.462 0.020 0.331 -0.198 0.114 -0.026 0.007 -0.015 339 191 1.155 0.568 0.365 0.067 0.075 0.010 0.160 -0.051 -0.034 0.005 393 231 1.078 0.526 0.448 0.026 -0.188 0.033 0.096 -0.011 -0.064 -0.001 332 260 1.111 0.308 0.517 0.174 0.493 -0.343 0.166 0.030 0.027 -0.153 177 392 1.430 0.633 0.284 0.083 -0.508 -0.112 0.050 -0.117 0.002 -0.118 M»:2,l 1.141 0.503 0.424 0.073 0.117 -0.106 0.112 -0.035 -0.010 -0.045 °i:2,l 0.210 0.111 0.084 0.056 0.404 0.144 0.044 0.049 0.033 0.071 Secondary Polarimetric Data for class: ship pixel line Co-PPD Cross-PPD HH HV V V LL LR (radians) (radians) (dB) (dB) (dB) (dB) (dB) 11:2 12:2 13:2 14:2 15:2 16:2 17:2 501 148 3.705 1.138 -1.207 -2.369 -1.222 -1.815 -1.297 392 183 3.099 2.421 -1.621 -3.379 -1.712 -2.216 -1.818 339 191 3.859 2.191 -1.218 -2.498 -1.468 -1.701 -1.629 393 231 3.611 1.841 -1.315 -2.955 -1.386 -1.584 -1.844 332 260 1.870 4.658 -1.507 -2.160 -1.285 -2.028 -1.500 177 392 2.864 2.607 -0.892 -2.153 -1.303 -1.240 -1.768 #«:2,1 3.168 2.476 -1.293 -2.586 -1.396 -1.764 -1.643 <7»:2,1 0.741 1.188 0.256 0.488 0.177 0.343 0.213 142 Appendix A. FEATURE DATA 143 Primary Polarimetric Data for class: car pixel line span I|5fcfc5fc\ll l |5fc«5fcj l|5fc. (dB) ft ft ft 9 1:2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 10:2 401 206 1.059 0.407 0.577 0.016 0.884 0.322 0.049 -0.048 0.052 0.073 394 209 0.712 0.503 0.470 0.027 0.689 -0.089 0.105 -0.041 0.077 0.031 397 212 0.934 0.359 0.627 0.014 0.647 -0.331 0.043 -0.035 0.020 0.023 W:2,2 0.902 0.423 0.558 0.019 0.740 -0.033 0.066 -0.041 0.050 0.042 0"«:2,2 0.176 0.073 0.080 0.007 0.126 0.330 0.034 0.007 0.029 0.027 Secondary Polarimetric Data for class: car pixel line Co-PPD Cross-PPD HH HV W LL LR (radians) (radians) (dB) (dB) (dB) (dB) (dB) 11:2 12:2 13:2 14:2 15:2 16:2 17:2 401 206 6.105 1.358 -1.441 -3.225 -1.287 -2.466 -1.376 394 209 2.169 1.091 -1.706 -3.330 -1.756 -2.558 -1.778 397 212 3.025 1.316 -1.637 -3.398 -1.382 -2.300 -1.561 Mt:2,2 3.767 1.255 -1.595 -3.317 -1.475 -2.441 -1.572 Ci.2,2 2.070 0.143 0.137 0.087 0.248 0.131 0.201 Appendix A. FEATURE DATA 144 Primary Polarimetric Data for class: rock pixel line span ll i^xS^JI l|5ht,SfcJ| ll 5/»h5«JI \\SHH \\SH»S;J (dB) ft ft 9 ft 1:2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 10:2 106 218 0.907 0.518 0.425 0.057 0.404 0.136 0.160 -0.020 -0.015 0.057 406 310 0.786 0.451 0.524 0.025 0.646 0.080 0.031 -0.085 0.043 0.057 408 319 0.833 0.298 0.655 0.048 0.723 0.401 0.004 -0.013 0.002 0.035 209 423 0.818 0.411 0.522 0.068 0.193 0.395 0.024 -0.079 -0.004 0.093 178 445 0.853 0.416 0.497 0.087 0.320 0.091 0.076 -0.021 0.053 0.023 185 459 1.105 0.526 0.306 0.168 0.111 0.178 0.072 -0.154 0.044 0.006 122 511 1.229 0.319 0.496 0.185 0.370 -0.185 -0.118 0.025 -0.101 0.108 125 514 1.115 0.269 0.683 0.048 0.573 0.266 -0.005 -0.048 -0.037 0.050 132 531 1.076 0.262 0.607 0.131 0.512 0.411 0.089 -0.036 0.100 0.126 130 532 1.199 0.409 0.364 0.227 0.295 0.117 0.211 -0.084 0.184 0.091 137 534 0.859 0.231 0.642 0.127 0.454 -0.047 0.038 -0.060 0.073 0.234 Pi:2,3 0.980 0.374 0.520 0.106 0.418 0.168 0.053 -0.052 0.031 0.080 0»:2,3 0.166 0.103 0.121 0.066 0.188 0.191 0.087 0.048 0.076 0.063 Secondary Polarimetric Data for class: rock pixel line Co-PPD Cross-PPD H H H V W L L L R (radians) (radians) (dB) (dB) (dB) (dB) (dB) 11:2 12:2 13:2 14:2 15:2 16:2 17:2 106 218 4.475 3.028 -1.519 -2.905 -1.659 -2.033 -1.688 406 310 4.162 1.257 -1.711 -3.323 -1.662 -2.470 -1.714 408 319 5.304 3.643 -1.816 -2.961 -1.462 -2.360 -1.650 209 423 3.978 1.404 -1.787 -2.785 -1.611 -1.964 -1.853 178 445 3.376 3.209 -1.646 -2.693 -1.580 -2.031 -1.825 185 459 4.135 2.018 -1.307 -2.154 -1.570 -1.766 -1.656 122 511 3.078 3.322 -1.436 -1.970 -1.210 -1.524 -1.419 125 514 4.027 2.148 -1.600 -2.694 -1.160 -2.051 -1.414 132 531 4.462 2.929 -1.622 -2.246 -1.248 -1.760 -1.494 130 532 4.122 1.971 -1.328 -1.882 -1.436 -1.601 -1.529 137 534 3.562 2.241 -1.987 -2.516 -1.444 -1.941 -1.735 Mt:2,3 4.062 2.470 -1.614 -2.557 -1.458 -1.955 -1.634 <"»:2,3 0.599 0.799 0.209 0.450 0.181 0.289 0.152 Appendix A. FEATURE DATA 145 Primary Polarimetric Data for class: ocean pixel line span l l^oo^ool l \\Shh \\SH. (dB) dt ft dt 1:2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 10:2 253 221 0.431 0.287 0.675 0.038 0.804 0.093 0.051 -0.083 0.073 0.136 260 224 0.679 0.306 0.679 0.015 0.870 0.173 0.031 -0.025 0.038 0.045 203 234 0.498 0.280 0.711 0.010 0.856 0.203 0.037 -0.042 0.051 0.079 233 235 0.680 0.348 0.638 0.014 0.885 0.287 0.020 0.006 0.020 0.002 206 243 1.020 0.292 0.695 0.013 0.865 0.341 0.044 -0.040 0.055 0.074 263 332 0.116 0.202 0.774 0.024 0.715 0.243 0.038 -0.035 0.067 0.073 Mi:2,4 0.571 0.286 0.695 0.019 0.833 0.223 0.037 -0.036 0.051 0.068 0^:2,4 0.303 0.048 0.046 0.010 0.064 0.088 0.011 0.029 0.019 0.044 Secondary Polarimetric Data for class: ocean pixel line Co-PPD Cross-PPD HH HV V V LL LR (radians) (radians) (dB) (dB) (dB) (dB) (dB) 11:2 12:2 13:2 14:2 15:2 16:2 17:2 253 221 4.824 0.996 -2.217 -3.479 -1.844 -2.780 -2.029 260 224 4.789 1.690 -1.951 -3.610 -1.595 -2.842 -1.758 203 234 5.468 1.545 -2.163 -3.986 -1.755 -2.873 -1.942 233 235 5.424 2.782 -1.892 -3.621 -1.623 -2.889 -1.754 206 243 6.090 0.771 -1.622 -3.333 -1.243 -2.358 -1.418 263 332 5.426 0.938 -2.705 -3.981 -2.101 -3.027 -2.363 A*»:2,4 5.337 1.454 -2.092 -3.668 -1.693 -2.795 -1.877 Ot:2,4 0.482 0.744 0.368 0.265 0.286 0.229 0.317 Appendix A. FEATURE DATA 146 Primary Polarimetric Data for class: building pixel line span I I W w J I IIS..s;B|| I I ^ J I \\SHH s;j \\SHHSH,\\ \\SH. s;j (dB) ft 9 ft 9 ft 9 1:2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 10:2 795 297 1.603 0.503 0.343 0.155 -0.321 -0.397 0.187 -0.015 -0.144 -0.091 464 412 2.470 0.495 0.441 0.064 -0.849 0.364 0.215 0.084 -0.181 0.125 483 416 1.407 0.325 0.599 0.076 0.497 0.289 0.074 0.013 0.004 0.051 468 425 1.978 0.551 0.379 0.070 -0.386 -0.497 0.118 0.007 -0.061 -0.027 476 433 2.100 0.356 0.596 0.049 -0.264 0.049 0.116 0.043 -0.052 0.061 272 490 2.105 0.349 0.440 0.211 -0.636 0.084 0.292 0.082 -0.296 0.111 214 505 2.602 0.791 0.143 0.066 -0.358 -0.698 0.277 0.025 -0.085 -0.035 308 505 1.581 0.654 0.252 0.094 -0.173 -0.110 0.238 -0.011 -0.024 -0.079 329 508 2.351 0.375 0.509 0.116 -0.558 0.114 0.247 0.020 -0.203 0.071 232 534 1.243 0.505 0.348 0.147 -0.108 -0.014 0.194 -0.093 -0.037 0.015 269 611 1.447 0.455 0.375 0.170 0.021 0.779 0.198 0.062 -0.054 0.058 Pi:2,6 1.899 0.487 0.402 0.111 -0.285 -0.004 0.196 0.020 -0.103 0.024 0.467 0.142 0.137 0.053 0.358 0.418 0.069 0.050 0.092 0.073 Secondary Polarimetric Data for class: building pixel line Co-PPD Cross-PPD HH HV W LL LR (radians) (radians) (dB) (dB) (dB) (dB) (dB) 11:2 12:2 13:2 14:2 15:2 16:2 17:2 795 297 2.884 2.414 -0.806 -1.637 -1.000 -1.034 -1.439 464 412 3.354 5.907 0.050 -1.156 -0.001 0.132 -1.343 483 416 5.543 3.210 -1.269 -2.236 -0.959 -1.471 -1.153 468 425 2.588 2.977 -0.412 -1.789 -0.596 -0.583 -1.052 476 433 3.229 5.224 -0.487 -1.654 -0.244 -0.438 -0.807 272 490 3.215 5.299 -0.474 -1.054 -0.382 -0.255 -1.591 214 505 2.315 2.718 0.394 -1.007 -0.416 0.027 -0.392 308 505 2.874 3.474 -0.725 -1.912 -1.328 -1.095 -1.276 329 508 2.895 4.058 -0.198 -1.016 -0.060 -0.115 -1.023 232 534 2.711 0.718 -1.204 -2.046 -1.494 -1.436 -1.594 269 611 4.849 4.470 -1.039 -1.783 -1.161 -1.161 -1.356 A*»':2,6 3.314 3.679 -0.561 -1.572 -0.695 -0.675 -1.184 0"t:2,5 0.989 1.510 0.518 0.442 0.519 0.587 0.357 Appendix A. FEATURE DATA 147 Primary Polarimetric Data for class: backstop pixel line span il-SWfc'S'fefcll 11^.11 \\SHH s;j ll5fch5hJI \\SHV s;j (dB) ft 9 ft 9 ft 9 1:2 2:2 3:2 4:2 5:2 6:2 7:2 8:2 9:2 10:2 804 300 0.971 0.449 0.357 0.194 0.032 -0.026 0.105 -0.158 -0.096 0.017 796 311 1.352 0.614 0.247 0.139 -0.164 0.024 0.292 -0.082 -0.042 0.000 805 313 1.602 0.398 0.457 0.145 -0.009 0.689 0.121 0.068 0.011 0.058 776 483 0.817 0.515 0.361 0.124 -0.467 0.268 0.122 0.076 0.001 0.041 860 501 0.863 0.530 0.353 0.117 -0.525 0.241 0.097 -0.056 -0.053 0.008 865 505 1.228 0.433 0.475 0.092 -0.301 0.220 0.086 0.028 -0.034 0.158 610 526 1.191 0.403 0.289 0.308 -0.145 0.233 0.235 0.129 -0.014 -0.038 550 632 1.080 0.386 0.429 0.185 -0.154 0.371 0.103 -0.009 -0.020 0.211 621 634 1.645 0.247 0.542 0.211 -0.295 0.483 0.126 0.040 -0.147 0.075 540 635 0.898 0.242 0.644 0.114 0.122 -0.427 0.033 -0.027 0.093 0.130 Mt:2,6 1.165 0.422 0.415 0.163 -0.191 0.208 0.132 0.001 -0.030 0.066 0»:2,6 0.296 0.117 0.119 0.064 0.210 0.304 0.075 0.085 0.064 0.078 Secondary Polarimetric Data for class: backstop pixel line Co-PPD Cross-PPD HH HV V V LL LR (radians) (radians) (dB) (dB) (dB) (dB) (dB) 11:2 12:2 13:2 14:2 15:2 16:2 17:2 804 300 2.936 1.962 -1.536 -2.210 -1.651 -1.857 -1.865 796 311 3.012 2.394 -0.981 -1.949 -1.460 -1.303 -1.556 805 313 4.644 4.170 -1.048 -1.721 -0.936 -0.969 -1.181 776 483 3.221 3.704 -1.594 -2.524 -1.760 -1.648 -2.369 860 501 3.369 2.320 -1.544 -2.522 -1.747 -1.683 -2.364 865 505 3.735 3.414 -1.291 -2.336 -1.208 -1.264 -1.769 610 526 3.920 5.014 -1.349 -1.760 -1.495 -1.330 -1.883 550 632 4.190 2.825 -1.462 -2.116 -1.478 -1.439 -1.877 621 634 3.934 4.238 -1.139 -1.574 -0.744 -0.831 -1.429 540 635 1.764 2.003 -1.845 -2.503 -1.404 -1.750 -1.818 M»":2,6 3.473 3.204 -1.379 -2.122 -1.388 -1.407 -1.811 0«:2,6 0.807 1.061 0.270 0.356 0.335 0.336 0.371 Appendix A. FEATURE DATA 148 A . 2 Summary of Polarimetric Feature Data The following tables contain a statistical summary of each spatial function for all the polari-metric features. log(span) ship car rock ocean building backstop Aggregate value # 1.810 1.355 1.427 0.838 2.663 1.726 1.749 CT 0.358 0.081 0.264 0.563 0.558 0.380 0.396 # 3 X 3 # 1.141 0.902 0.980 0.571 1.899 1.165 1.198 0.210 0.176 0.166 0.303 0.467 0.296 0.288 # 6 X 5 # 0.786 0.778 0.710 0.508 1.459 0.938 0.922 0.125 0.095 0.174 0.237 0.286 0.250 0.213 0 3 X 3 # 0.416 0.315 0.365 0.295 0.505 0.387 0.397 tT 0.136 0.048 0.073 0.048 0.126 0.076 0.089 0 5 X 5 # 0.446 0.323 0.381 0.269 0.566 0.408 0.420 0.126 0.057 0.048 0.022 0.150 0.035 0.076 V i _2 # 0.410 0.380 0.358 1.049 0.323 0.357 0.446 0.096 0.106 0.121 1.531 0.112 0.138 0.299 V i _ 3 # 0.665 0.478 0.593 1.103 0.532 0.517 0.629 0.095 0.054 0.160 1.258 0.114 0.163 0.275 V 2 - 3 # 0.435 0.213 0.406 0.198 0.307 0.272 0.319 0.114 0.091 0.136 0.083 0.129 0.158 0.127 \SHHSU ship car rock ocean building backstop Aggregate value # 0.675 0.482 0.463 0.282 0.558 0.382 0.473 tT 0.207 0.108 0.173 0.017 0.240 0.213 0.178 # 3 X 3 # 0.503 0.423 0.374 0.286 0.487 0.422 0.419 a 0.111 0.073 0.103 0.048 0.142 0.117 0.107 # 5 X 5 # 0.401 0.392 0.328 0.274 0.446 0.420 0.382 0.064 0.044 0.049 0.043 0.104 0.066 0.066 0 3 X 3 # 0.160 0.117 0.162 0.052 0.155 0.173 0.146 a 0.069 0.041 0.056 0.025 0.056 0.047 0.051 0 5 X 5 # 0.158 0.109 0.149 0.052 0.168 0.170 0.144 a 0.059 0.043 0.030 0.011 0.028 0.014 0.028 1 # 0.281 0.129 0.340 0.122 0.243 1.307 0.474 a 0.139 0.056 0.146 0.087 0.288 2.668 0.702 i V - l - 3 # 0.435 0.203 0.347 0.108 0.326 1.200 0.495 a 0.221 0.130 0.174 0.100 0.223 2.194 0.609 v" 2-3 M 0.269 0.093 0.245 0.066 0.142 0.268 0.196 a 0.116 0.082 0.107 0.066 0.092 0.228 0.123 Appendix A. FEATURE DATA 149 l |5. .5;„| | ship car rock ocean building backstop Aggregate value A* 0.251 0.505 0.445 0.698 0.358 0.501 0.448 a 0.147 0.106 0.212 0.010 0.258 0.232 0.186 1*3X3 f* 0.424 0.558 0.520 0.695 0.402 0.415 0.483 a 0.084 0.080 0.121 0.046 0.137 0.119 0.108 1*5X5 f* 0.535 0.584 0.583 0.705 0.424 0.424 0.522 a 0.068 0.046 0.064 0.043 0.092 0.043 0.063 <?3X3 f* 0.175 0.116 0.166 0.052 0.164 0.174 0.151 a 0.058 0.040 0.058 0.027 0.053 0.056 0.051 0 " 6 X 5 t* 0.169 0.107 0.166 0.052 0.169 0.174 0.151 a 0.036 0.043 0.032 0.012 0.021 0.025 0.027 Vl_2 f* 1.236 0.131 0.850 0.049 1.476 0.406 0.803 a 1.222 0.091 1.670 0.048 2.608 0.501 1.276 V i - 3 t* 1.971 0.215 1.263 0.035 1.808 0.659 1.129 a 1.425 0.215 2.429 0.048 2.859 1.031 1.659 V 2 - 3 f* 0.375 0.086 0.255 0.028 0.277 0.321 0.250 a 0.208 0.082 0.200 0.035 0.436 0.228 0.234 ship car rock ocean building backstop Aggregate value f* 0.075 0.013 0.092 0.019 0.084 0.117 0.079 o 0.072 0.002 0.072 0.021 0.052 0.071 0.056 t*3x3 f* 0.073 0.019 0.106 0.019 0.111 0.163 0.098 a 0.056 0.007 0.066 0.010 0.053 0.064 0.050 l*5xb 1* 0.064 0.024 0.089 0.021 0.130 0.156 0.097 a 0.042 0.002 0.046 0.006 0.045 0.043 0.037 03X3 f* 0.043 0.010 0.057 0.010 0.061 0.088 0.054 a 0.036 0.003 0.031 0.005 0.035 0.038 0.029 0 * 6 X 5 t* 0.055 0.017 0.061 0.013 0.080 0.096 0.063 a 0.043 0.006 0.031 0.004 0.028 0.028 0.026 V i _ 2 (* 0.684 0.649 0.609 1.264 0.788 0.855 0.799 a 0.798 0.674 0.778 1.234 0.902 1.137 0.938 V i - 3 f* 1.063 1.061 0.794 2.409 1.451 0.936 1.235 a 1.571 0.347 0.840 2.608 1.856 1.177 1.437 V 2 - 3 \* 0.297 0.633 0.325 0.438 0.508 0.221 0.376 cr 0.265 0.450 0.182 0.394 0.475 0.144 0.297 Appendix A. FEATURE DATA 150 ship car rock ocean building backstop Aggregate value P -0.301 0.869 0.577 0.834 -0.356 -0.315 0.108 a 0.576 0.085 0.276 0.076 0.451 0.439 0.352 A*3X3 A* 0.117 0.740 0.418 0.833 -0.285 -0.191 0.159 C 0.404 0.126 0.188 0.064 0.358 0.210 0.240 A*6x5 A* 0.425 0.748 0.458 0.814 -0.174 -0.125 0.246 CT 0.289 0.080 0.135 0.057 0.213 0.138 0.160 0"3X3 A* 0.421 0.165 0.292 0.057 0.226 0.305 0.258 (7 0.100 0.073 0.094 0.022 0.090 0.100 0.085 <*Sx6 A« 0.409 0.187 0.300 0.072 0.309 0.336 0.287 (7 0.103 0.045 0.075 0.024 0.068 0.049 0.063 V i _ 2 A* 1.286 0.170 0.432 0.047 1.611 1.443 0.966 <T 1.024 0.084 0.219 0.050 3.256 1.984 1.378 V i _ 3 A« 2.041 0.130 0.659 0.091 1.547 1.227 1.058 CT 1.441 0.081 0.672 0.052 2.876 1.233 1.289 V 2 - 3 A« 1.992 0.100 0.632 0.046 0.640 0.558 0.683 a 2.119 0.071 0.941 0.021 0.678 0.389 0.739 ship car rock ocean building backstop Aggregate value A» -0.378 -0.147 0.306 0.381 0.067 0.406 0.165 a 0.566 0.498 0.406 0.134 0.591 0.481 0.457 A*3X3 A» -0.106 -0.033 0.168 0.223 -0.004 0.208 0.096 CT 0.144 0.330 0.191 0.088 0.418 0.304 0.258 A»5X6 A« 0.155 0.094 0.165 0.253 0.021 0.145 0.132 a 0.155 0.109 0.089 0.073 0.289 0.162 0.159 0 3 X 3 A* 0.647 0.471 0.455 0.190 0.418 0.472 0.442 a 0.302 0.259 0.134 0.057 0.122 0.133 0.151 0^6x6 A* 0.582 0.423 0.461 0.216 0.617 0.595 0.508 a 0.109 0.194 0.074 0.019 0.118 0.071 0.089 V x _ 2 A* 2.976 0.934 0.837 0.442 2.261 0.714 1.373 a 5.220 1.001 0.674 0.313 3.703 0.404 1.881 V l _ 3 f* 9.225 1.663 1.227 0.315 1.361 1.235 2.193 CT 19.785 1.377 1.299 0.252 1.033 1.661 3.545 V 2 - 3 A* 3.364 1.388 1.412 0.676 2.727 3.166 2.247 a 3.227 0.763 1.849 1.020 3.391 4.864 2.852 Appendix A. FEATURE DATA \MSHA)\ ship car rock ocean building backstop Aggregate value # 0.160 0.072 0.120 0.040 0.259 0.169 0.155 0.132 0.022 0.171 0.033 0.132 0.131 0.121 #3X3 # 0.112 0.066 0.053 0.037 0.196 0.132 0.110 tT 0.044 0.034 0.087 0.011 0.069 0.075 0.062 #5X6 # 0.070 0.052 0.046 0.038 0.146 0.097 0.083 tT 0.028 0.012 0.048 0.013 0.046 0.045 0.038 03X3 # 0.102 0.029 0.103 0.028 0.117 0.129 0.097 tT 0.028 0.020 0.055 0.009 0.062 0.042 0.042 05X5 # 0.088 0.059 0.104 0.033 0.133 0.134 0.103 tT 0.030 0.011 0.034 0.007 0.031 0.026 0.026 Vl_2 # 0.896 0.197 0.655 0.622 0.339 1.588 0.777 tT 0.666 0.132 0.358 0.404 0.263 2.473 0.816 V X_3 # 0.706 0.339 0.921 1.094 0.502 1.854 0.979 c 0.155 0.189 0.654 1.489 0.232 2.784 1.022 V 2 - 3 # 0.544 0.345 1.720 0.475 0.396 0.506 0.755 0.384 0.210 3.233 0.436 0.197 0.354 0.996 \ ship car rock ocean building backstop Aggregate value # -0.058 -0.048 -0.085 -0.043 0.022 -0.005 -0.032 tT 0.067 0.029 0.108 0.026 0.071 0.141 0.086 #3X3 # -0.035 -0.041 -0.052 -0.036 0.020 0.001 -0.019 0.049 0.007 0.048 0.029 0.050 0.085 0.051 #5X5 # -0.044 -0.042 -0.039 -0.041 -0.003 -0.014 -0.027 tT 0.027 0.007 0.019 0.012 0.029 0.051 0.027 03X3 # 0.059 0.034 0.097 0.027 0.074 0.138 0.082 a 0.034 0.008 0.036 0.012 0.031 0.054 0.033 06X5 # 0.066 0.044 0.088 0.038 0.103 0.138 0.090 a 0.029 0.002 0.027 0.012 0.023 0.021 0.022 Vx-2 # 0.609 0.512 0.907 0.589 1.443 0.766 0.898 tT 0.409 0.388 0.925 0.636 1.781 0.524 0.903 V i _ 3 # 1.979 0.768 1.133 0.895 1.095 0.886 1.126 tT 2.602 0.671 1.196 1.017 0.729 0.544 1.071 V 2-3 # 1.343 0.195 1.289 1.297 2.602 1.798 1.643 a 1.005 0.137 1.499 2.529 4.093 1.576 2.104 Appendix A. FEATURE DATA 152 l l * ( 5 f c « 5 ; j | | skip car rock ocean building backstop Aggregate value -0.007 0.057 0.014 0.042 -0.107 -0.063 -0.027 a 0.154 0.017 0.131 0.051 0.127 0.151 0.120 M 3 X 3 M -0.010 0.050 0.031 0.051 -0.103 -0.030 -0.015 a 0.033 0.029 0.076 0.019 0.092 0.064 0.061 A»5x6 A* 0.029 0.031 0.026 0.051 -0.052 -0.025 0.001 (7 0.032 0.020 0.037 0.019 0.055 0.044 0.039 0 " 3 X 3 A« 0.086 0.030 0.121 0.044 0.088 0.143 0.098 CT 0.036 0.009 0.043 0.011 0.038 0.070 0.041 0 " 6 X 5 M 0.102 0.071 0.119 0.055 0.135 0.141 0.114 0" 0.031 0.013 0.030 0.016 0.036 0.034 0.029 V l _ 2 A« 0.959 0.259 1.751 5.542 0.666 1.629 1.759 CT 0.280 0.352 2.607 10.944 0.696 1.611 2.571 V i - 3 A« 1.648 0.633 1.018 6.505 1.053 1.797 1.948 a 0.757 0.493 0.655 12.023 0.574 2.794 2.545 V 2 - 3 P 28.988 0.550 2.140 0.652 2.712 9.884 7.057 a 31.751 0.464 2.233 0.635 5.660 28.158 12.002 l|9(s-»s;,)H ship car rock ocean building backstop Aggregate value M -0.064 0.031 0.058 0.083 0.029 0.140 0.055 a 0.047 0.064 0.068 0.053 0.109 0.115 0.083 A*3x3 A* -0.045 0.042 0.080 0.068 0.024 0.066 0.044 a 0.071 0.027 0.063 0.044 0.073 0.078 0.065 A«5x5 A» 0.014 0.058 0.070 0.075 0.037 0.039 0.048 a 0.045 0.016 0.030 0.020 0.045 0.055 0.039 0 " 3 X 3 A> 0.075 0.051 0.104 0.042 0.091 0.119 0.089 0 0.037 0.004 0.040 0.015 0.037 0.030 0.031 <""6x6 M 0.100 0.060 0.105 0.060 0.123 0.128 0.105 CT 0.047 0.005 0.028 0.016 0.025 0.025 0.026 V i _ 2 A» 0.780 0.779 1.036 0.382 3.594 0.777 1.447 CT 0.440 0.921 0.829 0.342 9.257 0.473 2.620 V x _ 3 A* 2.123 1.380 0.575 1.036 1.618 1.007 1.219 0 1.335 1.992 0.642 1.516 3.502 0.452 1.557 V 2 - 3 A» 5.363 0.969 2.790 39.760 0.769 0.997 6.867 a 5.502 1.056 6.417 96.724 0.682 0.828 14.955 Appendix A. FEATURE DATA Co-PPD ship car rock ocean building backstop Aggregate value # 2.200 2.190 4.209 6.056 3.368 3.692 3.753 a 1.233 3.422 2.642 0.089 1.143 1.340 1.558 A » 3 x 3 A* 3.168 3.767 4.062 5.337 3.314 3.473 3.791 a 0.741 2.070 0.599 0.482 0.989 0.807 0.832 # 6 X 5 # 4.069 4.235 3.878 5.330 3.223 3.395 3.854 0.760 1.076 0.688 0.541 0.618 0.528 0.653 0 3 X 3 # 1.944 1.934 2.423 1.912 0.834 1.170 1.627 tT 0.842 1.598 0.387 0.959 0.440 0.529 0.638 0 6 X 5 # 2.170 2.537 2.462 1.902 1.345 1.532 1.899 0.552 0.462 0.377 0.606 0.400 0.248 0.412 V i _ 2 # 1.640 47.315 15.102 0.134 0.182 0.323 6.893 tT 2.571 75.955 31.958 0.090 0.236 0.525 12.834 V i - 3 # 3.831 75.609 12.452 0.122 0.161 0.454 8.379 C 7.091 122.260 25.368 0.123 0.114 0.815 14.862 V 2 - 3 # 0.507 0.384 0.216 0.148 0.196 0.237 0.255 CT 0.518 0.211 0.153 0.107 0.153 0.157 0.199 Cross-PPD ship car rock ocean building backstop Aggregate value # 1.473 0.568 2.006 0.954 3.943 3.771 2.541 tT 2.382 0.334 2.132 0.462 2.864 2.527 2.091 # 3 X 3 # 2.476 1.255 2.470 1.454 3.679 3.204 2.703 a 1.188 0.143 0.799 0.744 1.510 1.061 1.022 # 5 X 6 # 2.206 1.622 2.399 1.395 3.301 2.776 2.488 a 0.713 0.241 0.385 0.281 0.591 0.740 0.528 0 3 X 3 # 2.316 1.844 1.712 1.132 2.166 2.213 1.936 a 0.345 0.034 0.750 0.831 0.929 0.374 0.625 0 6 X 6 # 2.126 1.867 1.918 1.359 2.486 2.128 2.047 a 0.356 0.293 0.346 0.387 0.198 0.293 0.303 # 14.844 1.954 1.951 1.334 6.706 1.996 4.641 23.543 1.713 2.083 1.653 14.695 4.086 8.122 V i - 3 # 11.646 2.959 1.934 1.692 8.728 1.881 4.787 a 16.960 1.936 1.831 3.341 15.805 3.445 7.576 V 2 - 3 # 0.373 0.403 0.350 0.475 0.493 0.215 0.377 a 0.207 0.527 0.218 0.278 0.878 0.181 0.391 Appendix A. FEATURE DATA \og[HHHH*) ship car rock ocean building backstop Aggregate value # 1.613 1.031 1.062 0.289 2.365 1.216 1.369 tT 0.448 0.098 0.308 0.587 0.611 0.514 0.463 # 3 X 3 # 0.810 0.509 0.489 0.012 1.543 0.725 0.767 tT 0.256 0.137 0.209 0.368 0.518 0.270 0.316 # 5 X 5 # 0.346 0.353 0.177 -0.068 1.062 0.512 0.457 cr 0.115 0.093 0.182 0.310 0.337 0.229 0.231 0 3 X 3 # 0.505 0.356 0.466 0.298 0.566 0.435 0.459 tT 0.155 0.073 0.092 0.061 0.141 0.123 0.113 0 5 X 5 # 0.557 0.369 0.473 0.295 0.624 0.424 0.479 a 0.178 0.072 0.054 0.029 0.142 0.059 0.089 V i - 2 # 0.555 0.571 0.635 0.670 0.396 0.605 0.563 0.109 0.125 0.179 0.411 0.132 0.262 0.203 V X _ 3 # 0.941 0.747 1.022 0.930 0.660 0.654 0.819 tT 0.065 0.072 0.188 0.506 0.114 0.274 0.206 V 2 - 3 # 0.878 0.367 1.829 0.848 0.437 0.467 0.873 tT 0.126 0.298 2.497 0.495 0.135 0.319 0.782 \og(HVHV*) ship car rock ocean building backstop Aggregate value # 0.244 -0.832 -0.056 -1.417 1.196 0.429 0.156 tT 0.347 0.099 0.569 0.232 0.606 0.352 0.430 A*3X3 # -0.482 -1.214 -0.454 -1.564 0.532 -0.018 -0.324 tT 0.488 0.087 0.450 0.265 0.442 0.356 0.386 # 6 X 5 # -0.937 -1.250 -0.807 -1.564 0.155 -0.266 -0.608 a 0.386 0.050 0.374 0.262 0.223 0.301 0.290 0 3 X 3 # 0.565 0.264 0.441 0.236 0.477 0.376 0.414 0.065 0.036 0.114 0.096 0.107 0.096 0.095 0 5 x 5 # 0.595 0.229 0.517 0.261 0.552 0.469 0.474 tT 0.110 0.020 0.061 0.035 0.149 0.087 0.087 Vl-2 # 2.861 0.531 2.331 0.167 0.663 1.427 1.425 a 1.751 0.228 2.386 0.144 0.380 1.221 1.163 V i _ 3 # 5.254 0.542 4.383 0.153 1.088 2.093 2.451 tT 3.473 0.196 4.044 0.131 0.264 1.248 1.746 V 2 - 3 # 2.778 0.039 2.498 0.020 0.990 2.018 1.605 t r 2.777 0.034 3.821 0.023 0.319 2.491 1.859 Appendix A. FEATURE DATA \og(VVVV) ship car rock ocean building backstop Aggregate value # 1.156 1.052 0.995 0.682 2.055 1.365 1.306 CT 0.443 0.154 0.402 0.565 0.747 0.405 0.494 # 3 X 3 # 0.708 0.629 0.645 0.410 1.409 0.715 0.816 CT 0.177 0.248 0.181 0.286 0.519 0.335 0.310 # 5 X 5 # 0.477 0.532 0.444 0.354 1.023 0.518 0.594 CT 0.134 0.124 0.154 0.216 0.309 0.257 0.216 0 3 X 3 # 0.341 0.325 0.377 0.310 0.512 0.482 0.415 CT 0.186 0.028 0.105 0.060 0.149 0.113 0.117 0 6 X 6 # 0.380 0.333 0.365 0.273 0.573 0.480 0.426 a 0.116 0.039 0.061 0.026 0.180 0.065 0.091 V i _ 2 # 0.380 0.470 0.654 0.652 0.345 0.546 0.512 0.219 0.184 0.773 0.268 0.140 0.154 0.320 V x _ 3 # 0.641 0.542 0.815 0.850 0.572 0.690 0.696 CT 0.225 0.087 0.459 0.350 0.150 0.194 0.263 V 2 - 3 # 0.454 0.424 0.463 0.320 0.346 0.400 0.401 CT 0.209 0.179 0.216 0.189 0.201 0.371 0.239 log (LLLL*) ship car rock ocean building backstop Aggregate value # 1.206 -0.210 0.414 -0.390 2.221 1.321 0.989 CT 0.546 0.040 0.477 0.408 0.650 0.424 0.478 # 3 X 3 # 0.340 -0.337 0.149 -0.691 1.429 0.697 0.451 CT 0.343 0.131 0.289 0.229 0.587 0.336 0.358 # 5 X 5 # -0.193 -0.440 -0.148 -0.699 0.953 0.416 0.135 CT 0.258 0.039 0.271 0.164 0.347 0.293 0.263 0 3 X 3 # 0.614 0.248 0.407 0.321 0.547 0.428 0.449 CT 0.125 0.069 0.092 0.123 0.127 0.108 0.110 0 6 X 6 # 0.639 0.283 0.447 0.269 0.632 0.486 0.490 CT 0.128 0.052 0.077 0.060 0.169 0.038 0.093 V ! _ 2 # 0.861 0.635 0.755 6.433 0.418 0.542 1.361 CT 0.222 0.393 0.447 9.508 0.162 0.166 1.445 V i _ 3 # 1.541 1.448 1.603 6.962 0.683 0.830 1.889 CT 0.384 0.589 0.787 9.679 0.135 0.233 1.588 V 2 - 3 # 4.338 0.660 13.668 0.198 0.452 0.751 4.085 CT 4.683 0.784 32.002 0.253 0.193 0.507 8.323 Appendix A. FEATURE DATA log(LRLR*) ship car rock ocean building backstop Aggregate value A« 0.869 1.021 0.986 0.495 1.576 0.735 0.995 <T 0.262 0.093 0.259 0.584 0.493 0.542 0.405 A*3x3 P 0.461 0.532 0.469 0.226 0.920 0.293 0.509 a 0.213 0.201 0.152 0.317 0.357 0.371 0.279 M 5 X 6 A* 0.261 0.409 0.222 0.159 0.608 0.126 0.301 a 0.170 0.116 0.161 0.248 0.237 0.261 0.210 <*3X3 A* 0.322 0.334 0.418 0.303 0.482 0.393 0.395 CT 0.082 0.055 0.084 0.048 0.159 0.122 0.103 <*6X5 A« 0.367 0.355 0.406 0.281 0.513 0.410 0.408 CT 0.091 0.062 0.062 0.024 0.124 0.068 0.077 V i _ 2 A« 0.522 0.548 0.568 0.989 0.464 1.286 0.743 a 0.202 0.174 0.209 0.978 0.167 1.571 0.584 V i - 3 A* 0.833 0.670 0.909 1.624 0.717 0.987 0.947 CT 0.148 0.071 0.193 2.250 0.118 0.563 0.503 V 3 - 3 A« 0.678 0.398 0.935 0.632 0.469 1.167 0.770 a 0.325 0.083 0.520 0.585 0.177 1.016 0.501 Appendix B S P A N P R O F I L E S A N D P O L A R I Z A T I O N S I G N A T U R E S O F T A R G E T S 157 Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS 162 Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS 163 building(795,297) Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS 164 6u.'Wt'n5i(476,433) Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS 165 Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS 166 Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS Appendix B. SPAN PROFILES AND POLARIZATION SIGNATURES OF TARGETS 168 6aeJbsfop(540,635) Appendix C I N T E R C L A S S D I S T A N C E M E A S U R E M E N T S The following tables contain the class conditional interclass distances for the spatial functions of the polarimetric features. log (span) Inter Class Distance Z>i:i,*,» value ship car rock ocean building backstop Pl:l,fc ship 0.000 0.889 0.576 1.000 0.848 0.067 0.638 car 0.000 0.636 0.778 1.000 0.800 0.819 rock 0.000 0.848 1.000 0.655 0.757 ocean 0.000 1.000 0.867 0.906 building 0.000 0.945 0.959 backstop 0.000 0.677 Pv.i 0.79 log (span) Inter Class Distance P 1 : 2 A*3X3 ship cor rock ocean building backstop ship 0.000 0.778 0.424 0.944 0.909 0.033 0.585 car 0.000 0.273 0.778 1.000 0.533 0.657 rock 0.000 0.818 1.000 0.382 0.604 ocean 0.000 1.000 0.867 0.889 building 0.000 0.818 0.941 backstop 0.000 0.535 Pl:2 0.70 log (span) Inter Class Distance Pi :3, k,l A»6X5 ship car rock ocean building backstop Pl:3,fc ship 0.000 0.000 0.273 0.833 1.000 0.400 0.536 car 0.000 0.212 0.778 1.000 0.467 0.509 rock 0.000 0.455 1.000 0.509 0.528 ocean 0.000 1.000 0.800 0.768 building 0.000 0.836 0.962 backstop 0.000 0.615 Pl:3 0.65 169 Appendix C. INTERCLASS DISTANCE MEASUREMENTS log (span) Inter Class Distance P\-.A,k,i C3X3 ship car rock ocean building backstop Pl-A,k ship 0.000 0.444 0.182 0.611 0.364 0.033 0.298 car 0.000 0.515 0.333 0.818 0.533 0.553 rock 0.000 0.545 0.719 0.145 0.423 ocean 0.000 0.879 0.700 0.644 building 0.000 0.618 0.674 backstop 0.000 0.399 PlA 0.50 log (span) Inter Class Distance Pi:c\,k,i 0 6 X 5 ship car rock ocean building backstop Pl:&,k ship 0.000 0.778 0.303 1.000 0.485 0.133 0.486 car 0.000 0.515 0.778 1.000 0.733 0.758 rock 0.000 0.970 0.835 0.327 0.594 ocean 0.000 1.000 1.000 0.965 building 0.000 0.764 0.809 backstop 0.000 0.581 Pi* 0.70 log (span) Inter Class Distance Di:6,k,i ship car rock ocean building backstop Pi*,k ship 0.000 0.111 0.212 0.278 0.515 0.267 0.295 car 0.000 0.091 0.333 0.333 0.000 0.168 rock 0.000 0.364 0.157 0.036 0.168 ocean 0.000 0.576 0.367 0.397 building 0.000 0.236 0.348 backstop 0.000 0.182 Dl* 0.26 log (span) Inter Class Distance Pi:rtk,i V X _ 3 ship car rock ocean building backstop Pl:7,k ship 0.000 1.000 0.212 0.222 0.606 0.633 0.503 car 0.000 0.455 0.889 0.212 0.267 0.505 rock 0.000 0.152 0.256 0.309 0.271 ocean 0.000 0.455 0.467 0.401 building 0.000 0.091 0.314 backstop 0.000 0.339 Pl:7 0.39 log (span) Inter Class Distance Pi-.s,k,i V 2 _ 3 ship car rock ocean building backstop Pl:S,k ship 0.000 0.889 0.091 0.944 0.515 0.633 0.560 car 0.000 0.758 0.111 0.455 0.200 0.484 rock 0.000 0.758 0.421 0.527 0.499 ocean 0.000 0.545 0.267 0.546 building 0.000 0.127 0.399 backstop 0.000 0.353 Di* 0.47 Appendix C. INTERCLASS DISTANCE MEASUREMENTS Inter Class Distance #2:1,*,/ value ship car rock ocean building backstop 02:l,k ship 0.000 0.667 0.636 0.778 0.212 0.700 0.577 car 0.000 0.091 1.000 0.091 0.400 0.386 rock 0.000 0.636 0.207 0.164 0.339 ocean 0.000 0.818 0.133 0.637 building 0.000 0.473 0.366 backstop 0.000 0.367 02:1 0.45 ll-Wfcfcll Inter Class Distance 02:2,k,i A»3X3 ship car rock ocean building backstop 02:2,k ship 0.000 0.556 0.697 0.944 0.182 0.467 0.546 car 0.000 0.152 1.000 0.152 0.067 0.324 rock 0.000 0.455 0.471 0.164 0.390 ocean 0.000 0.970 0.667 0.778 building 0.000 0.182 0.394 backstop 0.000 0.302 02:2 0.46 Inter Class Distance 02:3,*,! A»5X5 ship car roc A; ocean building backstop 02:3,k ship 0.000 0.333 0.697 0.778 0.152 0.167 0.414 car 0.000 0.636 1.000 0.273 0.200 0.463 rock 0.000 0.515 0.752 0.709 0.670 ocean 0.000 1.000 1.000 0.846 building 0.000 0.091 0.460 backstop 0.000 0.438 P2:3 0.55 Inter Class Distance #2:4,*,/ ° 3 X 3 ship car roc A; ocean building backstop 02:4,fc ship 0.000 0.444 0.030 0.833 0.121 0.000 0.233 car 0.000 0.394 0.778 0.333 0.733 0.521 rock 0.000 0.939 0.140 0.073 0.293 ocean 0.000 1.000 1.000 0.929 building 0.000 0.109 0.320 backstop 0.000 0.337 02:4 0.44 Inter Class Distance 02:5,fc,j ° " 6 X 5 ship car roc A: ocean building backstop 02:5,k ship 0.000 0.556 0.061 0.944 0.091 0.267 0.326 car 0.000 0.636 1.000 0.697 0.933 0.759 rock 0.000 1.000 0.372 0.455 0.491 ocean 0.000 1.000 1.000 0.991 building 0.000 0.036 0.409 backstop 0.000 0.491 02:5 0.58 Appendix C. INTERCLASS DISTANCE MEASUREMENTS l l w h f c l l Inter Class Distance 2^ 2:6,fc,i Vl_2 ship car rocJfc ocean building backstop #2:6,fc ship 0.000 0.778 0.303 0.667 0.364 0.133 0.401 car 0.000 0.818 0.111 0.152 0.533 0.483 rock 0.000 0.818 0.455 0.000 0.445 ocean 0.000 0.182 0.533 0.486 building 0.000 0.418 0.332 backstop 0.000 0.303 0.41 Inter ( Dlass Distance Z>2:7,*,J ship car rock ocean building backstop #2:7,k ship 0.000 0.667 0.364 0.667 0.455 0.200 0.438 car 0.000 0.515 0.556 0.455 0.600 0.549 rock 0.000 0.727 0.207 0.345 0.413 ocean 0.000 0.727 0.867 0.727 building 0.000 0.382 0.429 backstop 0.000 0.461 P2.7 0.50 Inter Class Distance P2-.&,k,i V2-3 ship car rock ocean building backstop ^2:8,fc ship 0.000 0.889 0.061 0.889 0.636 0.167 0.467 car 0.000 0.818 0.111 0.273 0.600 0.544 rock 0.000 0.848 0.521 0.127 0.451 ocean 0.000 0.636 0.667 0.670 building 0.000 0.327 0.481 backstop 0.000 0.353 P2.8 0.49 \\SVVS*V || Inter Class Distance Pza,k,i value ship car rock ocean building backstop D»:l,k ship 0.000 0.889 0.515 1.000 0.212 0.633 0.599 car 0.000 0.091 1.000 0.394 0.000 0.403 rock 0.000 0.818 0.223 0.164 0.355 ocean 0.000 0.818 0.767 0.860 building 0.000 0.364 0.391 backstop 0.000 0.385 P3.1 0.50 \\SvvS*v\\ Inter Class Distance Z?3 : 2, M # 3 X 3 ship car rock ocean building backstop Ps.2,k ship 0.000 0.889 0.515 1.000 0.182 0.133 0.479 car 0.000 0.091 1.000 0.697 0.667 0.622 rock 0.000 0.848 0.488 0.491 0.500 ocean 0.000 1.000 0.967 0.956 building 0.000 0.036 0.454 backstop 0.000 0.429 P3.2 0.57 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 173 ||5,,„S*,,|| Inter Class Distance 03:3,*,i #6X5 ship car rock ocean building backstop 03:3,fc ship 0.000 0.444 0.545 1.000 0.697 0.833 0.711 car 0.000 0.030 1.000 0.939 1.000 0.671 rock 0.000 1.000 0.851 0.927 0.713 ocean 0.000 1.000 1.000 1.000 building 0.000 0.036 0.676 backstop 0.000 0.719 03:3 0.75 114 4* II || "till'-'tin || Inter C /lass Distance 03:4,k,/ 03X3 ship car rock ocean building backstop 03:4,fc ship 0.000 0.556 0.030 1.000 0.091 0.067 0.283 car 0.000 0.515 0.778 0.576 0.600 0.594 rock 0.000 0.970 0.058 0.055 0.293 ocean 0.000 0.970 1.000 0.957 building 0.000 0.109 0.326 backstop 0.000 0.325 03:4 0.46 ||5V V5*„ || Inter Class Distance 03,5, 06X5 ship car rock ocean building backstop 03:5,fc ship 0.000 0.889 0.000 1.000 0.030 0.133 0.319 car 0.000 0.758 0.889 0.818 0.933 0.851 rock 0.000 1.000 0.107 0.036 0.338 ocean 0.000 1.000 1.000 0.986 building 0.000 0.018 0.349 backstop 0.000 0.361 03:5 0.53 Inter Class Distance 03:6,fc,J ship car rock ocean building backstop 03:6,fc ship 0.000 0.667 0.333 0.722 0.182 0.367 0.413 car 0.000 0.576 0.667 0.636 0.400 0.579 rock 0.000 0.909 0.058 0.127 0.364 ocean 0.000 0.879 0.667 0.785 building 0.000 0.218 0.360 backstop 0.000 0.333 03:6 0.47 Inter Class Distance Ds-.7,k,i ship car rock ocean building backstop 03:7,fc ship 0.000 0.889 0.515 1.000 0.364 0.633 0.635 car 0.000 0.333 0.667 0.455 0.467 0.527 rock 0.000 0.939 0.174 0.109 0.390 ocean 0.000 0.879 0.900 0.892 building 0.000 0.182 0.386 backstop 0.000 0.422 03:7 0.54 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 174 \\s..s;j Inter Class Distance Dz-.s,k,i v2-s ship car roc A: ocean building backstop 03:8,k ship 0.000 1.000 0.333 1.000 0.667 0.200 0.580 car 0.000 0.576 0.778 0.394 0.800 0.678 rock 0.000 0.909 0.174 0.182 0.405 ocean 0.000 0.848 0.933 0.899 building 0.000 0.400 0.478 backstop 0.000 0.468 03:8 0.58 \\SKA\\ Inter C "lass Distance 04:i,jb,j value ship car rock ocean building backstop 04:l.k ship 0.000 1.000 0.121 0.667 0.182 0.533 0.432 car 0.000 1.000 0.333 1.000 1.000 0.898 rock 0.000 0.788 0.008 0.218 0.376 ocean 0.000 0.758 0.967 0.743 building 0.000 0.273 0.394 backstop 0.000 0.544 04:1 0.56 Inter Class Distance 04:2,fc,i ^3X3 ship car roc A: ocean building backstop 04:2,fc ship 0.000 0.778 0.273 0.833 0.364 0.800 0.572 car 0.000 0.939 0.222 1.000 1.000 0.833 rock 0.000 0.970 0.124 0.473 0.516 ocean 0.000 1.000 1.000 0.866 building 0.000 0.455 0.543 backstop 0.000 0.704 04:2 0.67 Inter Class Distance 04:3,k,i A*5X5 ship car roc A: ocean building backstop 04:3,k ship 0.000 0.556 0.364 0.722 0.727 0.967 0.672 car 0.000 1.000 0.444 1.000 1.000 0.847 rock 0.000 1.000 0.521 0.673 0.690 ocean 0.000 1.000 1.000 0.883 building 0.000 0.200 0.649 backstop 0.000 0.722 04:3 0.74 llsfcusfcJI Inter Class Distance 04:4,k,i ° 3 x 3 ship car rocA: ocean building backstop 04:4,k ship 0.000 0.778 0.212 0.778 0.333 0.700 0.518 car 0.000 1.000 0.111 1.000 1.000 0.831 rock 0.000 1.000 0.025 0.527 0.508 , ocean 0.000 1.000 1.000 0.850 building 0.000 0.455 0.514 backstop 0.000 0.699 04:4 0.65 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 175 Inter Class Distance 04:6,k,j C 5 X 6 ship car rock ocean building backstop 04:6,fc ship 0.000 0.333 0.091 0.500 0.364 0.500 0.348 car 0.000 1.000 0.333 1.000 1.000 0.797 rock 0.000 1.000 0.355 0.600 0.582 ocean 0.000 1.000 1.000 0.831 building 0.000 0.236 0.549 backstop 0.000 0.627 04:5 0.62 Inter Class Distance 04:e,fc,j V i - 2 ship car rock ocean building backstop 04:6,fc ship 0.000 0.111 0.121 0.278 0.000 0.033 0.098 car 0.000 0.152 0.222 0.030 0.133 0.123 rock 0.000 0.576 0.157 0.036 0.200 ocean 0.000 0.242 0.333 0.346 building 0.000 0.127 0.117 backstop 0.000 0.127 04:6 0.17 \\SH.S-V\\ Inter Class Distance 04:7,*,/ ship car rock ocean building backstop 04:7,fc ship 0.000 0.667 0.061 0.556 0.182 0.033 0.244 car 0.000 0.455 0.111 0.333 0.467 0.408 rock 0.000 0.485 0.157 0.036 0.218 ocean 0.000 0.364 0.533 0.431 building 0.000 0.182 0.233 backstop 0.000 0.227 04:7 0.29 l|5fc»5/*J| Inter Class Distance 04:g, k,l V 2 - 3 ship car rock ocean building backstop 04:8,fc ship 0.000 0.556 0.182 0.222 0.273 0.100 0.239 car 0.000 0.455 0.333 0.030 0.600 0.383 rock 0.000 0.091 0.074 0.327 0.214 ocean 0.000 0.000 0.300 0.169 building 0.000 0.345 0.153 backstop 0.000 0.326 04:8 0.25 ys^ hfcS*,, value ship car Inter Class Distance Pb-.i,k,i rock ocean building backstop ship car rock ocean building backstop 06:1 0.000 1.000 0.000 0.848 0.758 0.000 1.000 0.222 0.636 0.000 0.030 1.000 0.934 1.000 0.000 0.067 1.000 0.855 1.000 0.073 0.000 0.521 0.824 0.817 0.814 0.589 0.569 0.69 Appendix C. INTERCLASS DISTANCE MEASUREMENTS \MSHHS;V)\\ Inter Class Distance 05:2,jt,( A*3x3 ship car rock ocean building backstop 05:2,fc ship 0.000 1.000 0.455 1.000 0.545 0.467 0.640 car 0.000 0.879 0.444 1.000 1.000 0.886 rock 0.000 0.970 0.884 0.982 0.842 ocean 0.000 1.000 1.000 0.922 building 0.000 0.291 0.723 backstop 0.000 0.726 05:2 0.79 Inter Class Distance 06:3,k,( P 5 X 5 ship cor roc A: ocean building backstop 05:3,fc ship 0.000 1.000 0.061 1.000 0.879 0.900 0.724 car 0.000 1.000 0.444 1.000 1.000 0.915 rock 0.000 1.000 0.950 1.000 0.813 ocean 0.000 1.000 1.000 0.930 building 0.000 . 0.364 0.819 backstop 0.000 0.828 06:3 0.84 Inter Class Distance 0e:4,fc,( 0"3X3 ship car roc Jb ocean building backstop 05:4,fc ship 0.000 1.000 0.727 1.000 0.909 0.567 0.815 car 0.000 0.818 0.889 0.455 0.733 0.752 rock 0.000 1.000 0.471 0.109 0.588 ocean 0.000 0.939 1.000 0.971 building 0.000 0.400 0.621 backstop 0.000 0.521 06:4 0.71 Inter Class Distance 06:6,fc,< 0"5X5 ship car rock ocean building backstop 06:6,fc 8 hip 0.000 1.000 0.636 1.000 0.576 0.400 0.676 car 0.000 0.818 1.000 1.000 1.000 0.957 rock 0.000 1.000 0.140 0.327 0.541 ocean 0.000 1.000 1.000 1.000 building 0.000 0.436 0.583 backstop 0.000 0.591 05:6 0.72 ll«(5fc*5.%)| | Inter Class Distance 06:6,fc,( V X _ 2 ship car rock ocean building backstop 06:6,fc ship 0.000 1.000 0.455 1.000 0.333 0.067 0.499 car 0.000 0.818 0.889 0.697 1.000 0.868 rock 0.000 0.848 0.157 0.618 0.550 ocean 0.000 0.939 1.000 0.935 building 0.000 0.309 0.454 backstop 0.000 0.569 05:6 0.65 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 177 IW-SW^JII Inter Class Distance P&:7,k,i V l _ 3 ship car rock ocean building backstop 06:7,fe ship 0.000 0.778 0.606 0.944 0.545 0.467 0.639 car 0.000 0.879 0.333 0.879 1.000 0.807 rock 0.000 0.939 0.521 0.636 0.697 ocean 0.000 0.970 1.000 0.884 building 0.000 0.327 0.620 backstop 0.000 0.652 05:7 0.72 Inter Class Distance 06:8, V2-3 ship car rock ocean building backstop 05:8 , f c ship 0.000 1.000 0.667 1.000 0.455 0.567 0.692 car 0.000 0.636 0.556 0.758 0.933 0.774 rock 0.000 1.000 0.240 0.382 0.555 ocean 0.000 0.939 1.000 0.929 building 0.000 0.055 0.447 backstop 0.000 0.533 05:8 0.66 l | 3 ( W 0 - J | | Inter Class Distance 06:i,jt,j value ship car rock ocean building backstop 06:l,fc ship 0.000 0.333 0.636 0.722 0.455 0.700 0.583 car 0.000 0.515 0.889 0.152 0.733 0.506 rock 0.000 0.091 0.322 0.164 0.331 ocean 0.000 0.485 0.100 0.395 building 0.000 0.418 0.373 backstop 0.000 0.397 Pea 0.43 \Mshhs;v)\\ Inter Class Distance De-.2 fc,; A*3x3 ship car rock ocean building backstop 06:2,fc ship 0.000 0.111 0.788 1.000 0.303 0.667 0.595 car 0.000 0.455 0.444 0.030 0.400 0.288 rock 0.000 0.212 0.355 0.145 0.376 ocean 0.000 0.455 0.033 0.392 building 0.000 0.327 0.308 backstop 0.000 0.303 06:2 0.38 Inter Class Distance Pe-.s,k,i A»6x5 ship cor rock ocean building backstop 06:3,fc ship 0.000 0.333 0.152 0.389 0.394 0.067 0.254 car 0.000 0.394 0.667 0.212 0.467 0.399 rock 0.000 0.606 0.421 0.000 0.304 ocean 0.000 0.636 0.500 0.560 building 0.000 0.345 0.407 backstop 0.000 0.257 06:3 0.36 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 178 l |9(w;ju Inter Class Distance 06:4, 0 3 X 3 ship car rock ocean building backstop 06:4,fc ship 0.000 0.222 0.455 0.944 0.545 0.367 0.510 car 0.000 0.212 0.556 0.333 0.133 0.277 rock 0.000 0.939 0.107 0.018 0.323 ocean 0.000 0.909 1.000 0.898 building 0.000 0.200 0.395 backstop 0.000 0.324 06:4 0.45 Inter Class Distance 0a:6,k,i 05X5 ship car rock ocean building backstop 06:5,fc ship 0.000 0.556 0.636 1.000 0.212 0.067 0.458 car 0.000 0.212 0.444 0.636 0.733 0.515 rock 0.000 1.000 0.702 0.764 0.684 ocean 0.000 1.000 1.000 0.930 building 0.000 0.036 0.503 backstop 0.000 0.499 06:5 0.60 Inter Class Distance De-.6,k,t ship car rock ocean building backstop 06:6,fc ship 0.000 0.222 0.273 0.667 0.030 0.467 0.319 car 0.000 0.030 0.333 0.212 0.133 0.172 rock 0.000 0.364 0.273 0.036 0.198 ocean 0.000 0.545 0.400 0.463 building 0.000 0.255 0.265 backstop 0.000 0.250 06:6 0.28 l|9(w;,)H Inter Class Distance 06:7,fc,i V x _ 3 ship car rock ocean building backstop 06:7,fc ship 0.000 0.111 0.394 0.778 0.273 0.433 0.403 car 0.000 0.273 0.778 0.212 0.400 0.339 rock 0.000 0.788 0.306 0.091 0.358 ocean 0.000 0.909 0.800 0.817 building 0.000 0.364 0.411 backstop 0.000 0.396 06:7 0.45 Inter Class Distance 06:8,fc,i V 2 _ 3 ship car rock ocean building backstop 06:8,fc ship 0.000 0.444 0.485 0.667 0.212 0.333 0.411 car 0.000 0.394 0.556 0.152 0.200 0.326 rock 0.000 0.485 0.306 0.255 0.374 ocean 0.000 0.576 0.533 0.557 building 0.000 0.018 0.249 backstop 0.000 0.255 06:8 0.36 Appendix C. INTERCLASS DISTANCE MEASUREMENTS Inter Class Distance Pra^i value ship car rock ocean building backstop 07:l,fc ship 0.000 0.556 0.303 0.611 0.394 0.033 0.348 car 0.000 0.152 0.667 0.818 0.533 0.534 rock 0.000 0.515 0.554 0.255 0.369 ocean 0.000 0.939 0.600 0.671 building 0.000 0.400 0.601 backstop 0.000 0.354 PT.I 0.48 \mshhsu\\ Inter C /lass Distance 07:2lk,j # 3 X 3 ship car rock ocean building backstop 07:2,fc ship 0.000 0.667 0.545 0.944 0.727 0.133 0.579 car 0.000 0.273 0.667 0.939 0.533 0.609 rock 0.000 0.061 0.818 0.618 0.496 ocean 0.000 1.000 0.867 0.693 building 0.000 0.455 0.770 backstop 0.000 0.523 07:2 0.61 u » ( w , ; j i i Inter Class Distance Pj.^kj # 5 X 5 ship car rock ocean building backstop 07:3,* ship 0.000 0.667 0.333 0.667 0.848 0.500 0.593 car 0.000 0.212 0.778 1.000 0.667 0.655 rock 0.000 0.152 0.884 0.545 0.463 ocean 0.000 1.000 0.900 0.690 building 0.000 0.545 0.839 backstop 0.000 0.620 07:3 0.64 Inter Class Distance Pr-Atk,i 0 3 X 3 ship car rock ocean building backstop 07:4,fc ship 0.000 1.000 0.212 1.000 0.091 0.300 0.436 car 0.000 1.000 0.222 0.939 1.000 0.867 rock 0.000 1.000 0.190 0.309 0.498 ocean 0.000 1.000 1.000 0.901 building 0.000 0.236 0.449 backstop 0.000 0.520 Pr-A 0.61 MSHHSUW Inter Class Distance 07:6,fc,i 0 5 x 5 ship car rock ocean building backstop 07:5.fc ship 0.000 0.667 0.333 1.000 0.667 0.800 0.673 car 0.000 0.939 1.000 1.000 1.000 0.935 rock 0.000 1.000 0.504 0.582 0.650 ocean 0.000 1.000 1.000 1.000 building 0.000 0.055 0.600 backstop 0.000 0.634 PT.S 0.75 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 180 Inter Class Distance VT-Z, fc,i V i - 2 ship car rock ocean building backstop 07:6,fc ship 0.000 1.000 0.030 0.056 0.697 0.333 0.385 car 0.000 0.758 0.667 0.273 0.733 0.660 rock 0.000 0.061 0.554 0.109 0.293 ocean 0.000 0.485 0.033 0.232 building 0.000 0.455 0.502 backstop 0.000 0.313 07:6 0.40 I W ^ J I I Inter Class Distance 07:7, k,l V l _ 3 ship car rock ocean building backstop 07;7,fc ship 0.000 0.889 0.121 0.000 0.576 0.067 0.295 car 0.000 0.879 0.222 0.455 0.733 0.647 rock 0.000 0.182 0.521 0.000 0.318 ocean 0.000 0.091 0.300 0.161 building 0.000 0.436 0.421 backstop 0.000 0.282 07:7 0.35 P(WjU)|| Inter Class Distance Dji8, fc,( V 2 - 3 ship car rock ocean building backstop 07:8,fc ship 0.000 0.444 0.182 0.167 0.273 0.067 0.208 car 0.000 0.394 0.222 0.152 0.200 0.275 rock 0.000 0.303 0.306 0.236 0.280 ocean 0.000 0.000 0.133 0.159 building 0.000 0.127 0.178 backstop 0.000 0.154 07:8 0.21 l | 9 ( ^ h ^ J | | Inter Class Distance 08:i,fc,j value ship car rock ocean building backstop 08:l.fc ship 0.000 0.111 0.121 0.056 0.545 0.367 0.266 car 0.000 0.212 0.222 0.636 0.267 0.311 rock 0.000 0.333 0.620 0.418 0.364 ocean 0.000 0.515 0.233 0.293 building 0.000 0.036 0.454 backstop 0.000 0.260 08:1 0.32 Inter Class Distance 08:2,Jt,( M3X3 ship car rock ocean building backstop 08:2jfc ship 0.000 0.333 0.242 0.111 0.636 0.300 0.339 car 0.000 0.091 0.222 0.818 0.400 0.389 rock 0.000 0.182 0.736 0.455 0.376 ocean 0.000 0.758 0.367 0.354 building 0.000 0.109 0.589 backstop 0.000 0.318 08:2 0.39 Appendix C. INTERCLASS DISTANCE MEASUREMENTS Inter Class Distance Pg:3, fc,/ A*5X5 ship car roc A; ocean building backstop 08:3,fc skip 0.000 0.111 0.061 0.056 0.727 0.333 0.287 car 0.000 0.152 0.111 0.818 0.333 0.337 rock 0.000 0.182 0.702 0.182 0.280 ocean 0.000 0.818 0.233 0.315 building 0.000 0.182 0.626 backstop 0.000 0.242 08:3 0.35 Inter Class Distance 08:4,*,/ 0"3X3 ship car rock ocean building backstop 08:4,fc skip 0.000 0.333 0.515 0.667 0.242 0.867 0.532 car 0.000 1.000 0.333 0.818 1.000 0.753 rock 0.000 1.000 0.388 0.491 0.644 ocean 0.000 0.909 1.000 0.837 building 0.000 0.709 0.599 backstop 0.000 0.782 08:4 0.78 l | 9 ( ^ h ^ J | | Inter Class Distance 08:6,*,/ ° " 5 X 5 ship car roc A; ocean building backstop 08:6,fc skip 0.000 0.444 0.364 0.667 0.727 1.000 0.656 car 0.000 1.000 0.333 1.000 1.000 0.814 rock 0.000 1.000 0.339 0.818 0.679 ocean 0.000 1.000 1.000 0.859 building 0.000 0.818 0.747 backstop 0.000 0.912 08:5 0.78 Inter Class Distance 08:6,fc,( ship car roe A: ocean building backstop 08:6,fc ship 0.000 0.222 0.091 0.222 0.333 0.100 0.190 car 0.000 0.212 0.000 0.455 0.267 0.251 rock 0.000 0.242 0.190 0.055 0.153 ocean 0.000 0.394 0.267 0.250 building 0.000 0.164 0.290 backstop 0.000 0.160 08:6 0.22 Inter Class Distance 08:7,fc,; V ! _ 3 ship car rock ocean building backstop 08:7,fc ship 0.000 0.111 0.061 0.278 0.061 0.033 0.098 car 0.000 0.091 0.111 0.152 0.133 0.121 rock 0.000 0.182 0.124 0.036 0.098 ocean 0.000 0.212 0.133 0.185 building 0.000 0.218 0.155 backstop 0.000 0.112 08:7 0.13 Appendix C. INTERCLASS DISTANCE MEASUREMENTS \mshhsh\)\\ Inter Class Distance 08:8,k,/ v 2 - 3 ship car rock ocean building backstop 08:8,* ship 0.000 1.000 0.182 0.611 0.182 0.133 0.347 car 0.000 0.758 0.222 1.000 0.867 0.794 rock 0.000 0.394 0.405 0.273 0.385 ocean 0.000 0.667 0.533 0.506 building 0.000 0.055 0.423 backstop 0.000 0.331 08:8 0.46 ll«(S h.S„%)|| Inter Class Distance 0g:ii k,i value ship car rock ocean building backstop 08:1,* ship 0.000 0.667 0.273 0.333 0.333 0.300 0.354 car 0.000 0.091 0.444 0.758 0.600 0.503 rock 0.000 0.030 0.554 0.364 0.288 ocean 0.000 0.727 0.567 0.422 building 0.000 0.164 0.486 backstop 0.000 0.376 09:1 0.40 ll*(sfc,s;,)|| Inter Class Distance 0g:2,fc,( # 3 X 3 ship car rock ocean building backstop 09:2,fc ship 0.000 0.889 0.364 0.944 0.697 0.300 0.594 car 0.000 0.273 0.000 1.000 0.800 0.614 rock 0.000 0.273 0.785 0.491 0.464 ocean 0.000 1.000 0.800 0.645 building 0.000 0.509 0.778 backstop 0.000 0.563 09:2 0.61 ll*(sfc.s;.)|| Inter Class Distance 09:3,*,1 # 5 X 5 ship car rock ocean building backstop 09:3,fe ship 0.000 0.111 0.152 0.278 0.788 0.700 0.444 car 0.000 0.212 0.444 0.818 0.733 0.491 rock 0.000 0.606 0.802 0.655 0.519 ocean 0.000 0.848 0.933 0.662 building 0.000 0.327 0.701 backstop 0.000 0.647 09:3 0.58 ||R(sk„s;.)H Inter Class Distance 09:4,*,1 0 3 x 3 ship car rock ocean building backstop 09:4,fc ship 0.000 1.000 0.515 0.833 0.091 0.533 0.533 car 0.000 0.939 0.667 0.939 1.000 0.920 rock 0.000 0.909 0.504 0.164 0.570 ocean 0.000 0.758 0.967 0.842 building 0.000 0.473 0.534 backstop 0.000 0.579 09:4 0.66 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 183 MSH.S;.)\\ Inter Class Distance 0 " 6 X 5 ship car rock ocean building backstop A>:6,fc ship 0.000 0.778 0.303 0.889 0.515 0.667 0.595 car 0.000 0.758 0.667 0.939 1.000 0.843 rock 0.000 0.909 0.306 0.418 0.513 ocean 0.000 1.000 1.000 0.917 building 0.000 0.109 0.527 backstop 0.000 0.583 A>:6 0.66 Inter Class Distance Po:6,fc,( Vi_2 ship car rock ocean building backstop Po-.6,k ship 0.000 0.889 0.152 0.056 0.455 0.100 0.290 car 0.000 0.818 0.667 0.455 1.000 0.760 rock 0.000 0.030 0.455 0.036 0.278 ocean 0.000 0.333 0.133 0.211 building 0.000 0.564 0.457 backstop 0.000 0.337 A>:6 0.39 ll*(5i..5;,j|| Inter C "lass Distance A> :7,k,i V ! _ 3 ship car rock ocean building backstop 09 :7 , f c ship 0.000 0.778 0.545 0.222 0.455 0.467 0.481 car 0.000 0.273 0.333 0.394 0.333 0.401 rock 0.000 0.030 0.025 0.055 0.168 ocean 0.000 0.091 0.067 0.124 building 0.000 0.018 0.173 backstop 0.000 0.165 A>:7 0.25 | | » ( S f c . 5 ; . ) | | Inter Class Distance A>:8, fc,i V 2 _ 3 ship cor rock ocean building backstop #9:8,fc ship 0.000 0.778 0.515 0.778 0.606 0.467 0.604 car 0.000 0.697 0.111 0.455 0.333 0.482 rock 0.000 0.576 0.174 0.255 0.412 ocean 0.000 0.364 0.333 0.446 building 0.000 0.091 0.314 backstop 0.000 0.280 A>:8 0.42 Inter Class Distance Pio:i,fc,; value ship car rock ocean building backstop #10:1^  ship 0.000 0.889 0.848 1.000 0.545 0.800 0.796 car 0.000 0.273 0.444 0.030 0.800 0.452 rock 0.000 0.091 0.190 0.673 0.419 ocean 0.000 0.303 0.633 0.462 building 0.000 0.618 0.352 backstop 0.000 0.695 0.53 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 184 Inter Class Distance 0io:2,fc,J A*3x3 ship car rock ocean building backstop 01O:2,fc ship 0.000 1.000 0.970 0.889 0.515 0.767 0.805 car 0.000 0.394 0.444 0.152 0.067 0.364 rock 0.000 0.061 0.322 0.145 0.365 ocean 0.000 0.424 0.133 0.353 building 0.000 0.236 0.331 backstop 0.000 0.268 1^0:2 0.41 l|9(sfc,s;B)|| Inter Class Distance 0io;3,fc,j # 5 x 5 ship car rock ocean building backstop 01O:3,fc ship 0.000 0.667 0.758 0.889 0.273 0.300 0.549 car 0.000 0.152 0.444 0.273 0.133 0.300 rock 0.000 0.182 0.388 0.236 0.347 ocean 0.000 0.576 0.333 0.463 building 0.000 0.018 0.299 backstop 0.000 0.198 010:3 0.36 Inter Class Distance 0io:4,fc,i 0 3 X 3 ship car rock ocean building backstop 010:4,* ship 0.000 0.444 0.545 0.778 0.394 0.700 0.570 car 0.000 0.818 0.333 0.758 1.000 0.713 rock 0.000 0.909 0.207 0.255 0.506 ocean 0.000 0.818 1.000 0.813 building 0.000 0.418 0.489 backstop 0.000 0.624 010:4 0.62 Inter Class Distance 0io:5,*,f 0 6 X 5 ship car rock ocean building backstop 01O:5,fc ship 0.000 0.667 0.212 0.611 0.394 0.400 0.423 car 0.000 1.000 0.111 1.000 1.000 0.814 rock 0.000 0.939 0.405 0.491 0.580 ocean 0.000 1.000 1.000 0.807 building 0.000 0.127 0.542 backstop 0.000 0.556 010:6 0.62 Inter Class Distance 0io:6,fc,< Vl-2 ship car rock ocean building backstop 01O:6,fc ship 0.000 0.111 0.061 0.611 0.000 0.033 0.139 car 0.000 0.394 0.222 0.333 0.200 0.267 rock 0.000 0.606 0.041 0.091 0.216 ocean 0.000 0.667 0.567 0.564 building 0.000 0.036 0.194 backstop 0.000 0.171 010:6 0.26 Appendix C. INTERCLASS DISTANCE MEASUREMENTS \\*{Si*s;v)\\ Inter Class Distance 0io:7,k,j V l _ 3 ship car rock ocean building backstop 01O:7,fc ship 0.000 0.444 0.818 0.667 0.667 0.600 0.660 car 0.000 0.152 0.111 0.212 0.267 0.230 rock 0.000 0.152 0.256 0.527 0.388 ocean 0.000 0.091 0.467 0.290 building 0.000 0.400 0.328 backstop 0.000 0.460 010:7 0.39 Inter Class Distance 0io:8,fc,f V 2 _ 3 ship car rock ocean building backstop 01O:8,fc ship 0.000 0.667 0.697 0.667 0.788 0.700 0.710 car 0.000 0.091 0.222 0.091 0.067 0.193 rock 0.000 0.212 0.058 0.091 0.219 ocean 0.000 0.182 0.267 0.295 building 0.000 0.164 0.247 backstop 0.000 0.249 010:8 0.32 Co-PPD Inter Class Distance 0n:i,fc,i value ship car rock ocean building backstop 011:l,k ship 0.000 0.222 0.455 1.000 0.485 0.733 0.587 car 0.000 0.212 0.333 0.333 0.333 0.288 rock 0.000 0.636 0.372 0.345 0.406 ocean 0.000 0.970 0.967 0.814 building 0.000 0.400 0.505 backstop 0.000 0.542 011:1 0.52 Co-PPD Inter Class Distance 0ii:2,fc,i # 3 x 3 ship car rock ocean building backstop 011:2,k skip 0.000 0.000 0.697 1.000 0.091 0.300 0.425 car 0.000 0.333 0.333 0.030 0.067 0.152 rock 0.000 0.939 0.603 0.473 0.612 ocean 0.000 0.788 1.000 0.850 building 0.000 0.327 0.390 backstop 0.000 0.442 011:2 0.48 Co-PPD Inter Class Distance 0ii:3,k,j # 6 x 6 ship car rock ocean building backstop 011:3,k ship 0.000 0.111 0.303 0.833 0.576 0.567 0.493 car 0.000 0.152 0.667 0.636 0.467 0.408 rock 0.000 0.879 0.570 0.455 0.487 ocean 0.000 0.970 1.000 0.893 building 0.000 0.291 0.592 backstop 0.000 0.538 011:3 0.57 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 186 Co-PPD Inter Class Distance Div.i,k,t 0"3X3 ship car roc A: ocean building backstop 011:4,* ship 0.000 0.333 0.394 0.056 0.758 0.600 0.463 car 0.000 0.152 0.444 0.333 0.333 0.307 rock 0.000 0.394 1.000 0.927 0.626 ocean 0.000 0.667 0.567 0.447 building 0.000 0.364 0.643 backstop 0.000 0.576 Din* 0.51 Co-PPD Inter Class Distance V\ uz,k,i 0"6X6 ship cor rock ocean building backstop Pi 1:6,* ship 0.000 0.556 0.333 0.278 0.848 0.600 0.536 car 0.000 0.273 0.556 0.939 1.000 0.677 rock 0.000 0.636 0.901 0.873 0.642 ocean 0.000 0.606 0.433 0.513 building 0.000 0.509 0.752 backstop 0.000 0.673 #11:6 0.63 Co-PPD Inter C 31as8 Distance Pn-.e,k,i Vi_ 2 ship car rock ocean building backstop Pll:6,fc ship 0.000 0.333 0.121 0.556 0.636 0.467 0.423 car 0.000 0.152 0.444 0.333 0.333 0.307 rock 0.000 0.848 0.769 0.582 0.525 ocean 0.000 0.091 0.100 0.398 building 0.000 0.200 0.419 backstop 0.000 0.343 Pll:6 0.40 Co-PPD Inter Class Distance Pii:7,*,i Vl_3 ship car roc A; ocean building backstop Pll:7,fc ship 0.000 0.333 0.182 0.889 0.879 0.600 0.582 car 0.000 0.273 0.889 0.818 0.667 0.592 rock 0.000 0.879 0.802 0.491 0.547 ocean 0.000 0.394 0.267 0.628 building 0.000 0.055 0.570 backstop 0.000 0.391 Pi 1:7 0.55 Co-PPD Inter Class Distance Pn-.s,k,i V 2_ 3 ship car rock ocean building backstop Pll:8,fc ship 0.000 0.000 0.394 0.556 0.545 0.367 0.401 car 0.000 0.455 0.778 0.636 0.400 0.466 rock 0.000 0.303 0.074 0.073 0.235 ocean 0.000 0.152 0.333 0.376 building 0.000 0.164 0.284 backstop 0.000 0.246 Pll-.S 0.33 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 187 Cross-PPD Inter Class Distance 0i2:i,fc,j value ship car rock ocean building backstop 012:l,fc ship 0.000 0.222 0.424 0.222 0.394 0.467 0.367 car 0.000 0.636 0.444 0.455 0.733 0.522 rock 0.000 0.182 0.256 0.345 0.353 ocean 0.000 0.364 0.567 0.352 building 0.000 0.164 0.311 backstop 0.000 0.423 012:1 0.39 Cross-PPD Inter C Jlass Distance 0i2:2,fc,« A*3X3 ship car rock ocean building backstop 012:2,fc ship 0.000 0.778 0.091 0.611 0.545 0.367 0.437 car 0.000 0.879 0.000 0.818 1.000 0.742 rock 0.000 0.697 0.537 0.400 0.505 ocean 0.000 0.758 0.867 0.647 building 0.000 0.273 0.562 backstop 0.000 0.539 012:2 0.57 Cross-PPD Inter Class Distance 0i2:3,fc,j A*5X5 ship car rock ocean building backstop 012:3,fc ship 0.000 0.556 0.242 0.667 0.758 0.400 0.513 car 0.000 0.939 0.444 1.000 0.933 0.818 rock 0.000 1.000 0.802 0.327 0.646 ocean 0.000 1.000 1.000 0.873 building 0.000 0.473 0.785 backstop 0.000 0.590 012:3 0.70 Cross-PPD Inter Class Distance 0i2:4,fc,i 03x3 ship car rock ocean building backstop 012:4,fc ship 0.000 0.667 0.515 0.889 0.061 0.200 0.418 car 0.000 0.030 0.667 0.455 0.667 0.465 rock 0.000 0.515 0.388 0.382 0.379 ocean 0.000 0.667 0.833 0.706 building 0.000 0.182 0.342 backstop 0.000 0.426 012:4 0.46 Cross-PPD Inter Class Distance 0i2:5,*,j 06X5 ship car rock ocean building backstop 012:5,fc ship 0.000 0.333 0.303 0.889 0.636 0.067 0.432 car 0.000 0.152 0.667 1.000 0.467 0.529 rock 0.000 0.727 0.868 0.400 0.518 ocean 0.000 1.000 0.867 0.844 building 0.000 0.727 0.837 backstop 0.000 0.513 012:5 0.61 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 188 Cross-PPD Inter Class Distance 0i2:6,fc,i V l - 2 ship car roe A: ocean building backstop 012:6,k ship 0.000 0.333 0.394 0.444 0.424 0.467 0.418 car 0.000 0.091 0.333 0.273 0.467 0.291 rock 0.000 0.091 0.140 0.182 0.180 ocean 0.000 0.121 0.133 0.198 building 0.000 0.036 0.186 backstop 0.000 0.233 012:6 0.25 Cross-PPD Inter Class Distance V\2:7,k,i Vx-3 ship car roc A: ocean building backstop 012:7,fc ship 0.000 0.111 0.364 0.667 0.424 0.600 0.451 car 0.000 0.394 0.556 0.394 0.667 0.436 rock 0.000 0.545 0.174 0.382 0.361 ocean 0.000 0.545 0.367 0.527 building 0.000 0.018 0.288 backstop 0.000 0.374 012:7 0.41 Cross-PPD Inter Class Distance 0i2:8,fc,» V 2 - s ship car rock ocean building backstop 012:8,fc ship 0.000 0.222 0.152 0.333 0.303 0.433 0.291 car 0.000 0.212 0.111 0.030 0.133 0.138 rock 0.000 0.242 0.157 0.436 0.245 ocean 0.000 0.394 0.567 0.350 building 0.000 0.182 0.215 backstop 0.000 0.353 012:8 0.27 \oz(HHHH") Inter Class Distance 0i3 : i ,k,j value ship car rock ocean building backstop 013:l,fc ship 0.000 0.778 0.697 1.000 0.727 0.467 0.714 car 0.000 0.333 1.000 1.000 0.200 0.632 rock 0.000 0.879 1.000 0.200 0.634 ocean 0.000 1.000 0.833 0.933 building 0.000 0.891 0.924 backstop 0.000 0.532 013:1 0.73 \og(HHHH') Inter Class Distance Pis-.2,k,i A*3x3 ship car rock ocean building backstop 013:2,fc ship 0.000 0.778 0.727 1.000 0.848 0.167 0.682 car 0.000 0.030 0.778 1.000 0.533 0.599 rock 0.000 0.758 1.000 0.509 0.641 ocean 0.000 1.000 0.967 0.906 building 0.000 0.873 0.942 backstop 0.000 0.622 013:2 0.73 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 189 \og(HHHH*) Inter Class Distance 013:3,*,! M 6 X 6 ship car rock ocean building backstop 013:3,fc ship 0.000 0.000 0.545 0.944 0.970 0.600 0.658 car 0.000 0.576 0.889 1.000 0.600 0.642 rock 0.000 0.485 1.000 0.764 0.699 ocean 0.000 1.000 0.933 0.838 building 0.000 0.855 0.961 backstop 0.000 0.762 013:3 0.76 \oZ(HHHH*) Inter Class Distance 013:4,*,/ 0 3 X 3 ship car rock ocean building backstop 013:4,* ship 0.000 0.667 0.364 0.667 0.152 0.300 0.388 car 0.000 0.636 0.556 0.879 0.333 0.619 rock 0.000 0.909 0.521 0.145 0.495 ocean 0.000 0.970 0.800 0.813 building 0.000 0.527 0.592 backstop 0.000 0.414 013:4 0.55 l0g(HHHH*) Inter Class Distance 0i3:6,*,i 0 6 X 6 ship car rock ocean building backstop 013:5,* ship 0.000 0.778 0.424 1.000 0.212 0.600 0.555 car 0.000 0.818 0.667 1.000 0.400 0.740 rock 0.000 1.000 0.686 0.473 0.667 ocean 0.000 1.000 0.900 0.935 building 0.000 0.873 0.748 backstop 0.000 0.660 013:5 0.72 \o%(HHHH<) Inter Class Distance 0i3:6,*,i V i _ 2 ship car rock ocean building backstop 013:6,* ship 0.000 0.667 0.667 0.889 0.061 0.500 0.522 car 0.000 0.394 1.000 1.000 0.333 0.658 rock 0.000 0.848 0.901 0.182 0.603 ocean 0.000 1.000 0.600 0.855 building 0.000 0.545 0.696 backstop 0.000 0.428 013:6 0.63 \og(HHHH*) Inter Class Distance 0i3:7,*,i V l _ 3 ship car rock ocean building backstop 013:7jfc ship 0.000 0.778 0.667 1.000 0.030 0.533 0.555 car 0.000 0.758 1.000 1.000 0.400 0.776 rock 0.000 0.879 0.967 0.073 0.656 ocean 0.000 1.000 0.533 0.866 building 0.000 0.673 0.735 backstop 0.000 0.436 013:7 0.67 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 190 log Inter Class Distance Pis-.s,k,i V2-3 ship car rock ocean building backstop 013:8,fc ship 0.000 1.000 0.576 1.000 0.424 0.733 0.700 car 0.000 0.758 0.222 0.939 0.467 0.692 rock 0.000 0.818 0.769 0.145 0.596 ocean 0.000 0.970 0.667 0.775 building 0.000 0.855 0.788 backstop 0.000 0.569 013:8 0.69 \o%[HVHV*) Inter Class Distance Du;i,k,i value ship car rock ocean building backstop 014:l,fc ship 0.000 1.000 0.333 1.000 0.818 0.333 0.647 car 0.000 0.818 1.000 1.000 1.000 0.957 rock 0.000 0.879 0.901 0.527 0.692 ocean 0.000 1.000 1.000 0.971 building 0.000 0.727 0.879 backstop 0.000 0.697 014:1 0.81 lot(HVHV*) Inter Class Distance 0i4:2,k,i A*3X3 ship car rock ocean building backstop 014:2,fe ship 0.000 0.778 0.000 0.944 0.939 0.467 0.588 car 0.000 0.939 0.889 1.000 1.000 0.935 rock 0.000 1.000 0.901 0.564 0.679 ocean 0.000 1.000 1.000 0.977 building 0.000 0.618 0.877 backstop 0.000 0.704 014:2 0.79 \og{HVHV*) Inter Class Distance 014:3,*,; ship car rock ocean building backstop 014:3,fc ship 0.000 0.444 0.121 0.833 1.000 0.900 0.668 car 0.000 0.697 0.778 1.000 1.000 0.809 rock 0.000 0.939 0.983 0.727 0.711 ocean 0.000 1.000 1.000 0.929 building 0.000 0.691 0.925 backstop 0.000 0.841 014:3 0.81 \og(HVHV) Inter Class Distance 0i4:4,fc,( <""3x3 ship car rock ocean building backstop 014:4,fc ship 0.000 1.000 0.606 1.000 0.485 0.967 0.775 car 0.000 0.879 0.000 1.000 0.667 0.745 rock 0.000 0.788 0.174 0.345 0.517 ocean 0.000 0.970 0.767 0.763 building 0.000 0.527 0.589 backstop 0.000 0.629 014:4 0.67 Appendix C. INTERCLASS DISTANCE MEASUREMENTS log{HVHV*) Inter Class Distance 0i4:5,k,< 0 5 x 6 ship car rock ocean building backstop 014:6,fc ship 0.000 1.000 0.515 1.000 0.182 0.600 0.598 car 0.000 1.000 0.556 1.000 1.000 0.932 rock 0.000 1.000 0.107 0.436 0.564 ocean, 0.000 1.000 1.000 0.944 building 0.000 0.309 0.472 backstop 0.000 0.624 014:6 0.69 log(HVHV*) Inter Class Distance 014:6,*,; ship car rock ocean building backstop 014:6,fc ship 0.000 0.889 0.697 1.000 0.364 0.367 0.618 car 0.000 0.455 0.667 0.758 0.400 0.613 rock 0.000 0.818 0.653 0.309 0.582 ocean 0.000 0.879 0.800 0.840 building 0.000 0.545 0.632 backstop 0.000 0.481 014:6 0.63 log(HVHV*) Inter Class Distance VH-T^,I V i - 3 ship car roc* ocean building backstop 014:7,fc ship 0.000 1.000 0.606 1.000 0.394 0.467 0.640 car 0.000 0.758 0.778 0.939 0.867 0.865 rock 0.000 0.848 0.587 0.018 0.534 ocean 0.000 1.000 0.933 0.921 building 0.000 0.600 0.685 backstop 0.000 0.536 014:7 0.70 \og(HVHV) Inter Class Distance 0i4:8,fc,i V 2 - 3 ship car rock ocean building backstop 014:8,fc ship 0.000 1.000 0.242 1.000 0.061 0.367 0.451 car 0.000 1.000 0.111 1.000 0.867 0.835 rock 0.000 1.000 0.107 0.200 0.458 ocean 0.000 0.970 0.900 0.858 building 0.000 0.255 0.431 backstop 0.000 0.472 014:8 0.58 log(vvyv) Inter Class Distance 0is:i,fc,( value ship car rock ocean building backstop 015:l,fc ship 0.000 0.222 0.242 0.500 0.697 0.300 0.405 car 0.000 0.030 0.444 0.818 0.600 0.435 rock 0.000 0.394 0.736 0.473 0.410 ocean 0.000 0.879 0.767 0.618 building 0.000 0.527 0.720 backstop 0.000 0.527 015:1 0.52 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 192 log{VVVV) Inter Class Distance 0i6:2,fc,f A*3X3 ship car rock ocean building backstop 016:2,fc ship 0.000 0.222 0.152 0.611 0.758 0.167 0.387 car 0.000 0.030 0.333 0.818 0.000 0.286 rock 0.000 0.576 0.835 0.055 0.355 ocean 0.000 0.939 0.533 0.629 building 0.000 0.727 0.813 backstop 0.000 0.317 0 1 5 : 2 0.46 log(VWV') Inter Class Distance 0i6 :3,fc ,j A«6X6 ship car rock ocean building backstop 015:3,* ship 0.000 0.333 0.242 0.333 0.909 0.067 0.389 car 0.000 0.394 0.667 0.879 0.133 0.484 rock 0.000 0.152 0.917 0.091 0.377 ocean 0.000 0.939 0.233 0.455 building 0.000 0.818 0.891 backstop 0.000 0.295 015:3 0.48 l o g ( V V W ) Inter Class Distance 0i5;4,k,i 0 " 3 x 3 ship car roc A; ocean building backstop 015:4,fc ship 0.000 0.333 0.303 0.111 0.545 0.467 0.369 car 0.000 0.333 0.222 0.697 0.867 0.520 rock 0.000 0.303 0.521 0.545 0.416 ocean 0.000 0.788 0.800 0.488 building 0.000 0.127 0.512 backstop 0.000 0.525 016:4 0.47 log[VVVV*) Inter Class Distance 0 i s : 5,k, i 0 " 5 x 5 ship car rock ocean building backstop 015:5,k ship 0.000 0.111 0.091 0.722 0.727 0.567 0.460 car 0.000 0.455 1.000 0.879 0.933 0.692 rock 0.000 0.909 0.719 0.818 0.619 ocean 0.000 1.000 1.000 0.931 building 0.000 0.327 0.707 backstop 0.000 0.704 015:6 0.68 Inter Class Distance 0i5:6 ,k,i V i _ 2 ship car roc A; ocean building backstop 015:6,k ship 0.000 0.111 0.030 0.111 0.364 0.367 0.210 car 0.000 0.091 0.111 0.515 0.667 0.325 rock 0.000 0.273 0.504 0.545 0.318 ocean 0.000 0.606 0.733 0.409 building 0.000 0.073 0.399 backstop 0.000 0.451 016:6 0.35 Appendix C. INTERCLASS DISTANCE MEASUREMENTS log(VVVV) Inter Class Distance 015:7,*,/ V i _ 3 ship car rock ocean building backstop 015:7,fc ship 0.000 0.333 0.121 0.500 0.485 0.233 0.324 car 0.000 0.212 0.222 0.758 0.600 0.447 rock 0.000 0.455 0.603 0.418 0.383 ocean 0.000 0.818 0.667 0.568 building 0.000 0.218 0.556 backstop 0.000 0.409 015:7 0.45 log(VVW) Inter Class Distance 0i6 ;8 ,* ,/ V 2 - 3 ship car rock ocean building backstop 016:8,fe ship 0.000 0.667 0.273 0.667 0.333 0.100 0.365 car 0.000 0.273 0.667 0.455 0.067 0.391 rock 0.000 0.667 0.405 0.109 0.341 ocean 0.000 0.636 0.267 0.569 building 0.000 0.509 0.466 backstop 0.000 0.227 015:8 0.39 log[LLLL*) Inter CI ass Distance 0i6:i,fc,i value ship car rock ocean building backstop 016:l,fc ship 0.000 1.000 0.818 1.000 0.758 0.133 0.703 car 0.000 0.818 0.333 1.000 1.000 0.855 rock 0.000 0.848 1.000 0.909 0.889 ocean 0.000 1.000 1.000 0.879 building 0.000 0.782 0.904 backstop 0.000 0.766 016:1 0.83 \og(LLLL*) Inter Class Dist ance 0i6:2,fc,J #3X3 ship car rock ocean building backstop 016:2,fc ship 0.000 1.000 0.333 1.000 0.939 0.500 0.713 car 0.000 0.818 0.778 1.000 1.000 0.923 rock 0.000 0.939 1.000 0.818 0.794 ocean 0.000 1.000 1.000 0.957 building 0.000 0.691 0.917 backstop 0.000 0.790 016:2 0.85 \og(LLLL*) Inter Class Distance 0i6:3,fc,j #5X5 ship car rock ocean building backstop 016:3,fc ship 0.000 0.667 0.121 0.944 1.000 0.933 0.723 car 0.000 0.697 0.778 1.000 1.000 0.843 rock 0.000 0.939 1.000 0.855 0.744 ocean 0.000 1.000 1.000 0.948 building 0.000 0.764 0.945 backstop 0.000 0.896 016:3 0.85 Appendix C. INTERCLASS DISTANCE MEASUREMENTS log (LLLL*) Inter Class Distance 016:4,*,. 0 " 3 X 3 ship car rock ocean building backstop 016:4,* ship 0.000 1.000 0.848 0.944 0.303 0.733 0.727 car 0.000 1.000 0.556 1.000 0.867 0.903 rock 0.000 0.485 0.636 0.109 0.582 ocean 0.000 0.788 0.533 0.655 building 0.000 0.564 0.642 backstop 0.000 0.525 016:4 0.67 log (IIII*) Inter Class Distance 0i6:&,*,< 0"6X5 ship car rock ocean building backstop 016:5,* ship 0.000 1.000 0.848 1.000 0.061 0.700 0.671 car 0.000 0.939 0.000 1.000 1.000 0.833 rock 0.000 0.939 0.669 0.327 0.716 ocean 0.000 1.000 1.000 0.859 building 0.000 0.509 0.631 backstop 0.000 0.664 016:6 0.73 log(IIII*) Inter C "lass Distance 016:6,*,. V i _ 2 ship car roc A; ocean building backstop 016:6,* ship 0.000 1.000 0.879 0.944 0.273 0.133 0.592 car 0.000 0.818 0.556 1.000 1.000 0.889 rock 0.000 0.303 1.000 0.764 0.765 ocean 0.000 1.000 0.900 0.745 building 0.000 0.364 0.717 backstop 0.000 0.612 016:6 0.72 log (LLLL*) Inter Class Distance 0ie:7,*,i V i _ 3 ship car rock ocean building backstop 016:7,* ship 0.000 1.000 0.848 1.000 0.242 0.200 0.602 car 0.000 0.879 0.111 1.000 1.000 0.836 rock 0.000 0.788 1.000 0.745 0.855 ocean 0.000 1.000 0.967 0.829 building 0.000 0.564 0.758 backstop 0.000 0.680 016:7 0.76 log(IIII*) Inter Class Distance 0i6:8,*,( V 2 - 3 ship car rock ocean building backstop 016:8,* ship 0.000 1.000 0.636 1.000 0.152 0.300 0.552 car 0.000 0.879 0.222 0.939 0.800 0.794 rock 0.000 1.000 0.504 0.273 0.626 ocean 0.000 1.000 0.967 0.894 building 0.000 0.364 0.565 backstop 0.000 0.506 016:8 0.66 Appendix C. INTERCLASS DISTANCE MEASUREMENTS log(LRLR*) Inter Class Distance Dna,k,i value ship car rock ocean building backstop 017:l,fc ship 0.000 0.222 0.303 0.389 0.879 0.133 0.407 car 0.000 0.091 0.778 0.879 0.267 0.441 rock 0.000 0.636 0.736 0.273 0.430 ocean 0.000 0.970 0.200 0.594 building 0.000 0.782 0.839 backstop 0.000 0.356 017:1 0.51 \og(LRLR*) Inter Class Distance Vn-2,k,i #3x3 ship car roc A; ocean building backstop 017:2,fc ship 0.000 0.222 0.061 0.389 0.788 0.367 0.380 car 0.000 0.212 0.556 0.697 0.467 0.437 rock 0.000 0.576 0.835 0.436 0.454 ocean 0.000 0.909 0.033 0.499 building 0.000 0.800 0.811 backstop 0.000 0.442 017:2 0.50 log(LRLR*) Inter Class Distance Dn-.^kj # 5 X 5 ship car rock ocean building backstop 017:3,* ship 0.000 0.556 0.182 0.167 0.727 0.433 0.414 car 0.000 0.636 0.778 0.455 0.733 0.624 rock 0.000 0.091 0.868 0.327 0.434 ocean 0.000 0.909 0.133 0.396 building 0.000 0.836 0.778 backstop 0.000 0.495 017:3 0.52 \og{LRLR*) Inter Class Distance 0i7:4,*,j 03x3 ship car rock ocean building backstop 017:4,fc ship 0.000 0.222 0.606 0.167 0.606 0.333 0.422 car 0.000 0.636 0.444 0.697 0.400 0.506 rock 0.000 0.758 0.240 0.164 0.448 ocean 0.000 0.758 0.433 0.545 building 0.000 0.345 0.500 backstop 0.000 0.324 PITA 0.46 log(LRLR*) Inter Class Distance Vn-.b,k,i 0 5 X 5 ship car rock ocean building backstop 017:5,fc ship 0.000 0.000 0.333 0.722 0.697 0.467 0.474 car 0.000 0.515 0.889 0.818 0.400 0.540 rock 0.000 0.909 0.587 0.000 0.453 ocean 0.000 1.000 1.000 0.917 building 0.000 0.545 0.711 backstop 0.000 0.461 017:6 0.59 Appendix C. INTERCLASS DISTANCE MEASUREMENTS 196 \o%(LRLR*) Inter Class Distance Dn-.e,k,i Vl_2 ship cor rock ocean building backstop 017:6,fc ship 0.000 0.000 0.485 0.389 0.606 0.000 0.327 car 0.000 0.333 0.778 0.394 0.200 0.335 rock 0.000 0.606 0.405 0.182 0.395 ocean 0.000 0.697 0.133 0.506 building 0.000 0.618 0.545 backstop 0.000 0.248 017:6 0.39 \oZ(LRLR*) Inter Class Distance Vn-Ttk,i Vi-3 ship car rock ocean building backstop 017:7,fc ship 0.000 0.111 0.394 0.500 0.758 0.000 0.374 car 0.000 0.394 0.778 0.879 0.133 0.467 rock 0.000 0.667 0.521 0.164 0.422 ocean 0.000 0.939 0.200 0.613 building 0.000 0.636 0.725 backstop 0.000 0.250 017:7 0.48 \og{LRLR*) Inter Class Distance Dn-.z,k,i V 2 _ 3 ship cor rock ocean building backstop 017:8,*: ship 0.000 0.444 0.212 0.556 0.485 0.000 0.317 car 0.000 0.576 0.333 0.636 0.200 0.450 rock 0.000 0.758 0.339 0.127 0.381 ocean 0.000 0.848 0.567 0.648 building 0.000 0.400 0.521 backstop 0.000 0.261 017:8 0.43 Appendix D D E S C R I P T I O N O F S O F T W A R E Software Overview This appendix is intended to provide a brief description of the software developed in the perfor-mance of this research. Each program is menu driven and "should" require little documentation. In addition to these descriptions, each program and subroutine is documented in the source code. All programs are written in VAX Fortran and run on the VAX 11/750 computer in the Department of Electrical Engineering. The source code has not been included here because of it huge size. The following is a list of the main packages used. 1. RIP - polarimetric image analysis 2. RTP - polarimetric feature extraction and classification 3. PFA - feature correlation analysis 4. FSA - feature selection algorithm 5. P T M - polarimetric target model and analysis RD? This package is a collection of polarimetric analysis tools for analyzing images in the JPL compressed data format [35]. The original package was developed by Dr. Mark Scivier [14] at MDA and then adapted and expanded by the author. RIP has the following functions: • synthesize an image 197 Appendix D. DESCRIPTION OF SOFTWARE 198 • compute polarization signature • compute ratio of signatures • minimize recieved power using SIMPLX method • maximize ratio of signatures using SIMPLX method • examine stokes matrix • synthesize span image • compute target span profile • compute target to clutter ratio • maximize target to clutter ratio These routines are located in VAX save set [RUSSH.MDASOFT]*.* The three dimensional signatures and span profiles are output in .XYZ format or .GRD format for input into SURFER (Golden Software) on a PC AT. R T P RTP is a re-written and expanded version of software developed by Sanjay Singhal for his MSc Thesis [3]. It is used to display synthesised byte array images on the RAMTEK, extract polarimetric features of compressed JPL data, calculate statistical summaries of the data for training, and conduct target classification. RTP has the following main functions: 1. enter image for display and evalutation 2. select targets manually by inputing target coordinates 3. detect targets automatically (intensity threshold detector) Appendix D. DESCRIPTION OF SOFTWARE 199 4. train recognition system: (a) enter a priori classes (b) extract polarimetric features (c) calculate a statistical summary 5. classify targets (a) get training data from a file (b) get feature vector from user (c) classify targets using minimum distance classifier 6. output classification results 7. highlight targets in the image 8. compare classification results with training data 9. evaluate classifier 10. highlight points in an image 11. enter target classes 12. reset image highlights 13. calculate target feature profile for a point RTP creates several files to store specific information. These file are defined below: • .FTR - FeaTuRe data file • .SUM - feature data SUMmary file • .CLS - CLaSsifcation output file Appendix D. DESCRIPTION OF SOFTWARE 200 • .IMG - image file (input only) • .XYZ - 3-D data file • .GRD - 3-D data file The subroutines for this package are contained in VAX file [RUSSH.SAR.PROGSjRTP*.*. RTP is a package of research utilities and not meant as a complete target recognition system. P F A PFA (Polarimetric Feature Analysis) calculates the aggregate correlation coefficient matrix from a feature data file (.FTR) and a feature vector specified by the user. The output is a .COR file and is in L A T E X format. • .COR - CORrelation matrix in L A T E X format F S A FSA (Feature Selection Algorithm) is a rewritten, internally documented and adapted version of the feature selection program written by Sanjay Singhal [3]. It is used to calculate the spread, interclass distance, performance, orthogonality and marginal utility of features as presented in Singhal et al. [45]. It takes as input a feature data file (.FTR) and a feature vector specified by the user. The outputs are: • .NET - distance, spread and orthogonality • .MU - marginal utility and performance data A program was written to convert the distance and orthogonality data in a .NET file to .TEX format. It is called NET2TEX. Appendix D. DESCRIPTION OF SOFTWARE 201 P T M P T M produces a Polarimetric Target and Clutter Model as presented in Chapter 5. Once the user has specified the input parameters: • target scattering matrix • clutter scattering matrix • target gain factor • receiver noise factor • transmit and receive phase distortion a number of functions may be done to the data. The P T M functions are: 1. enter model parameters - manually 2. read model parameters from a file (.DAT) 3. produce power profiles (.GRD, .XYZ) 4. extract target to clutter ratio 5. compute polarization signature (.GRD) 6. extract polarimetric features (.FTR) 7. compute classifier distances (.CLS) and output to (.DAT) file The model covers an area of 15 X 15 pixels. The three dimensional profile and signatures are output in a format compatible with a plotting program SURFER from Golden Software Inc. Miscellaneous SUM2TEX Converts .SUM files to .TEX files and calculates spread. Appendix D. DESCRIPTION OF SOFTWARE 202 N E T 2 T E X Converts .NET files to .TEX format. P T M _ N O I S E _ M A P Produces noise maps for P T M program. P D E T E C T I O N Calculates probabiility of detection given TCR, Weibull parameter 0, and the probability of false alarm. F T R 2 P R T Converts .FTR files to .PRT files that are suitable for output. 

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