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Information content of polarimetric synthetic aperture radar data Small , David L. 1991

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Information Content of Polarimetric Synthetic Aperture Radar Data by David L. Small BA.Sc. University of Waterloo, Waterloo, 1988. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1991 © David L. Small, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference 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 E l e c t r i c a l E n g i n e e r i n g The University of British Columbia Vancouver, Canada _ 21 J u n e 1991 Date DE-6 (2/88) Abstract Research into the analysis of polarimetric synthetic aperture radar (SAR) data continues to reveal new applications and data extraction techniques. The objective of this thesis is to examine the information content of a quad-polarization SAR, and determine which polarimetric variables are most useful for classification purposes. The four complex polarimetric radar channels (HH, HV, VH, and W) are expressed as nine scattering matrix cross-product "features" (with the loss of only absolute phase), and the relative utility of each for terrain classification is examined. Feature utility is examined in two ways — by measuring how each feature separates classes of terrain in an image, and by measuring how well a classifier performs with and without each feature. The features are then ranked in order of utility to the classifier, or in order of information content. A sharp distinction is found between those features that provide information useful to the classifier, and those that do not. It is found that those features that are defined as the product of a co-polarized and a cross-polarized term can be relatively safely ignored, with little loss of classification accuracy. This would be useful for reducing data transmission, storage, and processing requirements, and for designing future simplified radar systems. There is qualitative evidence that classification performance can actually be improved when these features are ignored. Of three simplified radar systems considered, the co-polarized design (returning only the complex HH and W channels) in general produced classifications closest to that of a fully polarimetric SAR. ii Table of Contents Abstract u List of Tables vii List of Figures ix List of Illustrations xi Acknowledgements xiii 1 Properties of Polarimetric Radar Data 1 1.1 General Properties 1 1.1.1 Coordinate System l 1.1.2 Polarization of Electromagnetic Waves 2 1.1.3 Elliptical Polarization 4 1.1.4 Poincare" Sphere 5 1.1.5 Stokes' Vectors 6 1.1.6 Partially Polarized Waves 7 1.1.7 Scattering Matrix 7 1.1.8 Voltage at Antenna 8 1.1.9 Antenna Stokes' Vectors 9 l.l.lOStokes' Matrix 9 l.l.lO.lDerivation 10 1.1.10.2Advantages/Disadvantages 12 1.1.10.3Relation to Scattering Matrix 13 1.2 Properties of JPL AIRSAR Data n 1.2.1 Introduction to JPL AIRSAR 13 1.2.2 Stokes' Matrix Compression 14 1.2.2.1 Compression 15 1.2.2.2 Decompression 15 1.3 Polarimetric Features 16 1.3.1 General Observations 16 1.3.2 Total Power 17 1.3.3 Number of Bounces During Scattering 17 1.3.4 Polarization Phase Difference 17 1.3.4.1 Definition 17 1.3.4.2 Use as Discriminator 18 1.3.4.3 Relation to Stokes' Matrix 18 1.3.5 Scattering Matrix Cross Products 18 1.3.6 Polarization Signature 19 1.3.6.1 Coefficient of Variation 20 1.3.6.2 Fractional Polarization 21 iii 2 Data Calibration 2 2 2.1 Background 2 2 2.2 Radar System Model 2 2 2.3 Phase Calibration 2 4 2.3.1 Measured Scattering Matrix 24 2.3.2 Symmetrization of Scattering Matrix 25 2.3.3 Solving for the Phase Distortions 26 2.3.3.1 Equation 1: Reciprocity 26 2.3.3.2 Equation 2: "Known" target 27 2.3.4 Phase Corrected Measurements 27 2.3.5 Kronecker Delta Form 28 2.4 Cross-Talk Calibration 30 2.4.1 Assumptions 30 2.4.2 Parameters 30 2.4.3 Cross-Talk Correction 31 2.4.4 Iteration 32 2.5 Co-polarized Channel Imbalance 32 2.6 Radiometric Calibration 32 3 Classification 3 4 3.1 Introduction 3 4 3.2 Unsupervised Classification 35 3.2.1 Introduction 35 3.2.2 Algorithm 37 3.2.2.1 Importance 37 3.2.2.2 Algorithm Overview 37 3.2.2.3 Scatterer Signature 38 Odd Number of Bounces 38 Even Number of Bounces 38 Diffuse Scattering 38 3.2.2.4 Consistency 38 3.2.2.5 Forest Scattering 40 3.2.3 Implementation 41 3.2.3.1 Verification 41 3.2.3.2 Calibration 43 3.2.3.3 Results 44 3.2.3.4 Discussion 45 3.3 Supervised Classification 46 3.3.1 Theory 46 3.3.1.1 Introduction 47 3.3.1.2 Euclidean Distance 47 3.3.1.3 Intra-Class Distance 48 3.3.1.4 MAP Classifier 49 3.3.1.5 Multivariate Gaussian Classes 50 3.3.1.6 MED vs. ML 50 3.3.1.7 Hybrid 51 3.4 Supervised vs. Unsupervised 5i iv 4 Class Separation by Feature 53 4.1 Motivation 53 4.2 Feature Definitions 53 4.2.1 Feature Vector 53 4.2.2 Polarization Ratio 55 4.2.3 Linear Depolarization Ratio 55 4.2.4 Polarization Phase Difference 55 4.3 Feature Distributions 56 4.3.1 Training Set Definitions 56 4.3.2 Distributions 58 4.4 Class Separation 58 4.4.1 Introduction 58 4.4.2 Results 60 4.4.2.1 Forestry 60 Bonanza Creek - L-band (2 Scenes) 61 Fairbanks - L-band (2 Scenes) 64 Punta Cacao - P, L, and C bands 68 Mt. Shasta - P, L, and C bands 71 Traverse City - L-band 75 Weeks Lake - P, L , and C bands 77 4.4.2.2 Agricultural 80 Flevoland - P, L , and C bands 80 4.4.2.3 Urban 84 San Francisco - L-band 84 4.4.2.4 Geological 86 Pisgah - P, L, and C bands 86 4.4.3 Summary of Feature Performance 90 SR{HHVV*} 90 Span 90 HH, HV, VV 90 HV 91 W 91 Frequency 92 5 Feature Utility in Classification 94 5.1 Introduction 94 5.1.1 Confusion Matrix 94 5.1.2 Relative Confusion 94 5.2 Results 95 5.2.1 One Feature Ignored 95 5.2.2 One Feature Used 97 5.2.3 Features Dropped in Reverse Rank Order 100 Calibrated EDO Scenes 1 0 3 All Scenes 103 5.3 Feature Ranking ios 5.3.1 Introduction 105 5.3.2 Varying Scene-types 107 Calibrated EDO vs. Uncalibrated EDO 107 v 6 Simplified Radar Systems ios 6.1 Introduction 1 0 8 6.1.1 Copolarized Radar , 108 6.1.2 Amplitude Radar 109 6.1.3 Single Transmit Radars 109 6.2 Results no 6.2.1 Relative Confusion H2 6.2.2 Discussion 1 1 2 7 Conclusions n 5 7.1 Possible Areas for Future Work H5 A Tabular Feature Statistics and Class Separabilities in A.1 Bonanza Creek n 7 A.2 Fairbanks 1 1 8 A.3 Flevoland n9 A.4 Pisgah 1 2 0 A.5 Punta Cacao 1 2 2 A.6 San Francisco i 2 3 A.7 Mt. Shasta 1 2 4 A.8 Traverse City 1 2 5 A.9 Weeks Lake 1 2 6 References 1 2 8 vi L i s t o f Tab l e s 3.1 Decision Process for Van Zyl's Unsupervised Classification Algorithm 39 3.2 Results of Unsupervised Classification of Forests 40 3.3 Percentages from Van Zyl Unsupervised Classification of San Francisco L-Band Data 42 3.4 Percentages from Van Zyl Unsupervised Classification of Traverse City L-Band Data 43 3.5 Percentages from Van Zyl Unsupervised Classification of Weeks Lake L-Band Data 44 3.6 Percentages from Van Zyl Unsupervised Classification of Calibrated and Uncalibrated Weeks Lake L-Band Data (near range excluded) 45 4.7 20x 20 Training Set Locations 58 4.8 Span of Classes in Pisgah Scene for P, L, and C-bands 88 4.9 Mean and Standard Deviation of P-band Normalized HH and VV Cross Products for Classes in Pisgah Scene 89 4.10 Average Class Separations of L-band Features (All Scenes) 92 5.11 Relative Confusion for Single Normalized Cross-Products Ignored with L-band Full Feature Set as Reference 96 5.12 Relative Confusion for Single Normalized Cross-Products Ignored with P, L, and C-band Full Feature Set as Reference 97 5.13 Relative Confusion for Single Normalized Cross-Products Used with Full L-band Feature Set as Reference 98 5.14 Relative Confusion for Single Normalized Cross-Products Used with Full P, L, and C-band Feature Set as Reference 100 5.15 Basis for Feature Utility Ranking 106 5.16 Feature Utility Rankings for various scene types 107 6.17 Relative Confusion of Simplified Radars for Four-Class Scenes 114 A. 18 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Bonanza Creek L-band data) 117 A. 19 Class Separations for Normalized Cross Products (Bonanza Creek L-band data) 117 A.20 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Fairbanks L-band data) 118 A.21 Class Separations for Normalized Cross Products (Fairbanks L-band data). . . . 118 A.22 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Flevoland P, L, and C-band data) 119 A.23 Class Separations for Normalized Cross Products (Flevoland P, L, and C-band data) 119 A.24 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Pisgah P, L, and C-band data) 120 vii A.25 Class Separations for Normalized Cross Products (Pisgah P, L, and C-band data) 122 A.26 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Punta Cacao P, L, and C-band data) 122 A.27 Class Separations for Normalized Cross Products (Punta Cacao P, L, and C-band data) 123 A.28 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (San Francisco L-band data) 123 A.29 Class Separations for Normalized Cross Products (San Francisco L-band data) 124 A.30 Values for Span, HH, VV, HV and Rhhvv Normalized Cross Products (Mt. Shasta P, L, and C-band data) 124 A.31 Class Separations for Normalized Cross Products (Mt. Shasta P, L, and C-band data) 125 A.32 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Traverse City L-band data) 125 A.33 Class Separations for Normalized Cross Products (Traverse City L-band data) 126 A.34 Values for Span, HH, VV, HV and Rhhvv Normalized Cross Products (Weeks Lake P, L, and C-band data). . . 126 A.35 Class Separations for Normalized Cross Products (Weeks Lake P, L and C-band data) 127 viii L i s t o f F i g u r e s 1.1 Coordinate System 2 1.2 Polarization Ellipse 4 1.3 The Poincar6 Sphere 5 1.4 The Unwrapped Poincare" Sphere 6 1.5 Transmit/Receive Timing Diagram 14 1.6 Polarization Signature 20 3.7 Scattering Mechanisms (a) Slightly rough surface (odd bounce) (b) dihedral corner reflector (even bounce) (c) Forested area (diffuse) {1: direct canopy backscatter; 2: double bounce scattering; 3: direct ground backscatter; 4: direct tree trunk backscatter} 36 4.8 Feature Distribution in Weeks Lake L-band scene 59 4.9 Feature Distribution in Bonanza Creek L-band scene (13 March 1988) 62 4.10 Feature Distribution in Bonanza Creek L-band scene (20 March 1988) 63 4.11 Feature Distribution in Fairbanks L-band scene (13 March 1988) 66 4.12 Feature Distribution in Fairbanks L-band scene (20 March 1988) 66 4.13 Feature Distribution in Punta Cacao P-band scene 70 4.14 Feature Distribution in Punta Cacao L-band scene 70 4.15 Feature Distribution in Punta Cacao C-band scene 71 4.16 Feature Distribution in Mt. Shasta P-band scene 73 4.17 Feature Distribution in Mt. Shasta L-band scene 73 4.18 Feature Distribution in Mt. Shasta C-band scene 74 4.19 Feature Distribution in Traverse City L-band scene 76 4.20 Feature Distribution in Weeks Lake P-band scene 78 4.21 Feature Distribution in Weeks Lake L-band scene 78 4.22 Feature Distribution in Weeks Lake C-band scene 79 4.23 Feature Distribution in Flevoland P-band scene 81 4.24 Feature Distribution in Flevoland L-band scene 81 4.25 Feature Distribution in Flevoland C-band scene 82 4.26 Feature Distribution in San Francisco L-band scene 85 4.27 Feature Distribution in Pisgah P-band scene 87 4.28 Feature Distribution in Pisgah L-band scene 87 4.29 Feature Distribution in Pisgah C-band scene 88 4.30 Average Class Separations for L-band Scenes 91 5.31 Average Relative Confusion for Single Feature Ignored (Reference is L-band Full Feature Set Classification) 95 ix 5.32 Relative Confusion when Features Dropped in Rank Order in EDO Class Scenes - Full Feature Reference 104 5.33 Relative Confusion when Features Dropped in Rank Order - Full Feature Reference 105 6.34 Relative Confusion of Simple Radars in All L-band Scenes 113 6.35 Relative Confusion of Simple Radars in L-band EDO Scenes 114 x L i s t o f I l l u s t r a t i o n s Illustration 1 Unsupervised Classification of 3 x 3 Averaged San Francisco L-band Data (white=odd-bounce, light-gray=even-bounce, dark-gray=diffuse scatterer, black=unclassified) 42 Illustration 2 Unsupervised Classification of 2x2 Averaged Phase Calibrated Weeks Lake L-band Data (mid-gray=odd-bounce, white=even-bounce, black=diffuse scatterer, dark-gray=unclassified) 45 Illustration 3 Supervised Classification of 2x2 Averaged 13 March 1988 Bonanza Creek L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 61 Illustration 4 Supervised Classification of 2x2 Averaged 20 March 1988 Bonanza Creek L-band Data (black=odd-bounce, gray=diffuse, white=even-bounce) 62 Illustration 5 Supervised Classification of 2x2 Averaged 13 March 1988 Fairbanks L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified (very little in this scene)) 64 Illustration 6 Supervised Classification of 2x2 Averaged 20 March 1988 Fairbanks L-band Data (black=odd-bounce, gray=diffuse, white=even-bounce) . . . . 65 Illustration 7 Supervised Classification of 2x2 Averaged Punta Cacao L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 69 Illustration 8 Supervised Classification of 2x2 Averaged Mt. Shasta L-band Data (black=Tree2, light-gray=Treel, white=Clear-Cut, dark-gray=unclassified) 72 Illustration 9 Supervised Classification of 2x2 Averaged Traverse City L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 75 Illustration 10 Supervised Classification of 2x2 Averaged Weeks Lake L-band Data (black=lake, light-gray=forest, white=clear-cut, dark-gray=unclassified) 77 Illustration 11 Supervised Classification of 2x2 Averaged Flevoland L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 80 Illustration 12 Supervised Classification of 2x2 Averaged San Francisco L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 84 Illustration 13 Supervised Classification of 2x2 Averaged Pisgah L-band Data; black(I)=Phase I Lava, dark gray(IH)=Phase HI Lava, medium gray(II)=Phase II Lava, light gray(A)=Alluvial Surface, white(LB)=Lakebed (Playa), gray(seen in rectangular interference regions)=unclassified 86 Illustration 14 Sequence of Features Dropped (9 colour images) in Weeks Lake, BC Scene (23 March 1988). Features were dropped in reverse rank order of Table 5.15. The last feature not dropped labels each image 101 xi Illustration 15 Air Photograph (August 1988) of Area Close to Weeks Lake AIRSAR Scene 102 Illustration 16 Visual Classification Comparison (colour) of Simplified Radar Systems — Flevoland, NL Scene (16 August 1989) I l l xii Acknowledgements I owe many thanks to my supervisors Dr. Ian G. dimming, and Dr. M. R. Ito, for their guidance throughout my studies at UBC. I especially appreciate Dr. Cumming's patience in explaining SAR processing, and Dr. Ito's assistance in administrative matters. Dr. Jakob van Zyl of the Jet Propulsion Laboratory (JPL) in Pasadena, California deserves credit for many of the ideas explored within this thesis. I wish to thank the B.C. Science council, who, through their G.R.E.A.T. award program, supported me financially during the past two years. I am thankful for the help given by the staff within the research group at MacDonald Dettwiler and Associates (MDA), especially from Melanie Dutkiewicz. MDA also graciously provided their computer facilities and film recorders for the production of colour hardcopy. Rob Ross and Dave Gagne also deserve thanks for (usually) keeping the Electrical Engineering Sun computer network up, and for continually improving the suite of available software tools. The radar data librarians at JPL were also most helpful in helping us obtain AIRSAR data. But most of all, many thanks to Marina RuB for her assistance in ending my days as a "Bachelor". xiii Chapter 1: Properties of Polarimetric Radar Data Chapter 1 Properties of Polarimetric Radar Data Polarimetric Synthetic Aperture Radar (SAR) differs from conventional SAR in that backscatter returns are recorded from more than one combination of transmit and receive polarizations. Whereas conventional SARs have a fixed transmit and receive polarization (eg. horizontal transmit horizontal receive, or HH), polarimetric SARs measure other polarization combinations as well. A fully polarimetric SAR records both the amplitude and phase of four combinations, that is horizontal transmit horizontal receive (HH), horizontal transmit vertical receive (HV), vertical transmit horizontal receive (VH), and vertical transmit vertical receive (VV). Knowledge of these four combinations enables the calculation of backscattered power for any combination of transmit and receive antenna polarizations. 1.1 General Properties A general introduction to polarimetric SAR follows. The coordinate system is outlined, followed by a description of various representational forms of elliptically polarized electromagnetic waves. 1.1.1 Coordinate System The horizontal (H) and vertical (V) basis vectors are shown in figure 1.1. Antenna boresight (the direction of propagation) is along the vector N. The plane of incidence is defined as containing both N and the surface normal at the point of incidence [63, p. 819]. V is defined as perpendicular to N within the plane of incidence and H is parallel to the horizon. The axes are mutually orthogonal: B_=VxE (1) 1 Chapter 1: Properties of Polarimetric Radar Data SAR Antenna Figure 1.1: Coordinate System 1.1.2 Polarization of Electromagnetic Waves Electromagnetic waves are denned in terms of their electric and magnetic field vectors. The orientation of the electric field vector defines the polarization of a plane harmonic wave (physicists use the magnetic field vector in their definition) [47, p. 3-2]. The electric field vector can be separated into horizontally and vertically polarized components. The path traced out by the electric field vector is then defined (after [29, p. 247]) by: (EH{z,t)\ \Ev(z,t)J (2) = hexpJT with T = u>t — kz the variable part of the phase factor (z being the distance along the direction of propagation N, w the angular frequency, k the wavenumber of the radiating wave). The complex 2 Chapter 1: Properties of Polarimetric Radar Data vector h represents the polarization of the plane wave, and is time-invariant: H A (hff\ /anexpjS"\ \hv) \avexpjSvJ ( 3 ) = ( " ) expJ'5" \ ay expJC / with e = 6y — f>H die phase difference between the EH and Ey components. Note that when the antenna radiates a field defined by equation (3) then the vector h defines the antenna's polarization [29, p. 248]. The electric field vector may also be stated [4] as: E_ = E-H + Kv (4) = an cos (r + 8H)2L + «v cos (r + 6H + c)V * where H and V are the unit vectors in the H and V directions, and an and ay the magnitudes of the horizontal and vertical components respectively (from [4, p. 24], [47, p. 3-2]). Expanding the second cosine term: cos (r + 6fj + e) = cos (r + 6H) cos e — sin (r + sin e (5) Substituting | £ = cos (r + fa) and = cos (r + % + e) into (5) yields: cost — \\\— [ J sine (6) ay ajj Rearranging terms and squaring both sides leads to: 2 / i p \ 2 ( * V + ( « E V cos*. - 2 ^ c o 5 £ = 1 - (SL)) sin* £ (7) \ av) \ aH) aHav \ \aH) J Simplifying results in: (Ey\2 . (EH\2 EjjEy . 2 / o \ — + - 2 cos e = sin € (8) V ay) \aH J aHay This is the equation of a conic. Investigating the determinant [4, p. 25] 1 cos t det cose 1 > 0 sin2 e (9) a\a2H shows that it is always positive, implying that (8) is the equation of an ellipse. The electric field vector therefore traces out an ellipse as the wave propagates forward along N. 3 Chapter 1: Properties of Polarimetric Radar Data 1.1.3 Elliptical Polarization The section above introduced the interpretation of the polarization state as an ellipse like that seen in figure 1.2. The ellipse can be specified by its ellipticity angle x (measuring the "roundness" of the wave), and its orientation angle tp (measuring the angular deviation from horizontal polarization of its semi-major axis). The wave's amplitude is represented by the length of the semimajor axis of the ellipse. Waves receding from the observer whose electric field vector rotates clockwise (negative x) ®K denoted as right-handed. Those whose electric field vector rotates counter-clockwise (positive x) are left-handed. The ellipticity and orientation angles are related to the amplitudes and phase difference as follows V *• H Figure 1.2: Polarization Ellipse [4, p. 27]: sin(2x) = 2a#ay sine (10) tan (2*) = 2anay cos e (11) Linear polarizations correspond to x=0o> *=0° or 180° being horizontal, and $=90° being vertical. For circular polarizations x=45° Geft handed) or -45° (right handed). 4 Chapter 1: Properties of Polarimetric Radar Data Note that -45° < x ^  45° and 0° < * < 180° are sufficient for the representation of all polarizations. 1.1.4 Poincare Sphere Z *- Y Figure 1.3: The Poincard Sphere 5 Chapter 1: Properties of Polarimetric Radar Data +45 X Left Handed Circular Linear Right Handed Circular 0 180 ¥ Figure 1.4: The Unwrapped Poincani Sphere There is a one-to-one mapping [4, p. 31] between any possible polarization state and a corresponding point P o n a Poincare sphere of radius So, with 2\ and 2ip being the spherical angular coordinates of the point. Points in the northern hemisphere are left handed, while points in the southern hemisphere are right handed. Linear polarizations lie on the equator (Sj=0) while circular polarizations lie at the poles. The sphere may also be "unwrapped" and displayed as a two dimensional rectangle, much in the same way as one sees with world maps (see figure 1.4). Linear polarizations run along the equator through the center of the rectangle. 1.1.5 Stokes' Vectors The Stokes' vector of a plane monochromatic wave (after [4, p. 30] and [29]) is defined as: /S0\ /\hH\2 + \hv\2\ S = Si s2 \s3J \h„\2-\hy\2 2$t(h*Hhv) ( *H + a2v \ aH ~ aV 2anav cos e (12) V 2^s(h*Hhv) ' \2a#aysine/ So is proportional to the wave's intensity [4, p. 30]. The other three parameters are functions of the ellipticity and orientation angles: Si = So cos 2x cos 2V> •S-j = So cos 2x sin 2V> 53 = S0 sin 2% (13) 6 Chapter 1: Properties of Polarimetric Radar Data Only three of the Stokes' vector elements (Stokes' parameters) are independent. For polarized waves they are related by: 502 > S 2 + Si + S32 (14) with the equality holding for fully polarized waves. The three parameters Si, S2 and S3 define the Cartesian coordinates of the polarization state on the Poincare" sphere. So is the radius of the sphere. For partially polarized waves the point falls within the sphere while fully polarized waves lie on the surface. 1.1.6 Partially Polarized Waves The Stokes' vector of a partially polarized wave can be decomposed [29] into a fully polarized component and an unpolarized component: /{S0)\ /S0-(S0)\ Si S2 V s3 ) \ 0 0 0 (15) with (So)p = yjS\ + S\ + S% ranging between zero and one indicating the portion of the wave that is fully polarized. The degree of polarization p of the wave is defined [29, p. 249] as: (S0)p _ y/S'j + SI + Sj (16) So So representing the ratio of the intensity of the polarized component of the wave.to the wave's total intensity. 1.1.7 Scattering Matrix Each scatterer on the ground produces a scattered wave with a polarization different from that of the original incident wave. The relationship between the incident and scattered waves is codified in the scattering matrix, which is stated as: ( Shh S^v \ s s <17) 1 Chapter I: Properties of Polarimetric Radar Data with Sij denoting the relation between the i transmit and j receive polarizations. The relationship between the transmitted (incident) and scattered electric field vectors is then: Ea = SEt (18) where Es and Et are the scattered and transmitted electric field vectors respectively. Each of the four parameters in the scattering matrix is complex, requiring eight floating point elements for storage. However, the absolute phase does not influence the received power and may be neglected, resulting in only seven independent parameters. Additionally, in the monostatic case (transmitter and receiver co-located) reciprocity dictates that Shv = S v / i . resulting in only five independent parameters. 1.1.8 Voltage at Antenna Each of the four recorded echoes (HH, HV, VH, VV) that make up the measured scattering matrix is a complex quantity, containing both amplitude and phase information. The voltage measurement V at the antenna for each resolution element can be stated [29, p. 249] as: V = CjSCt (19) with S the scattering matrix, and Ct and Cr complex vectors defined similarly to h denoting the polarization states of the antenna at transmission and reception respectively: C, = ( £ ) (20, Note that Ct,h is a phasor representing [74, p. 685] "the complex (amplitude and phase) wave amplitude" in the H direction when "a unit voltage signal is applied to the antenna feed line". A similar definition holds for Cr,h-Expanding equation (19) yields: V = CTthShhCt,h. + CT>h.ShvCtiV + Cr,vSvh.Ct,h + CriVSvvCtiV (21) The resulting received power P is then: P = VV* (22) Note that for an HH radar system, Cf = (1 0) and CJ = (1 0), only the first term of equation (21) contributes [74, p. 688], and Shh is easily extracted. 8 Chapter 1: Properties of Polarimetric Radar Data 1.1.9 Antenna Stokes' Vectors The polarization states of the transmit and receive antennas can also be represented by their Stokes' vectors, denned as follows [74, p. 685]: (\Ct,h\2 + \Ct,v\\ \CUH\2 - \Ci,v\2 2»(C«lfcC7tW) V 23(CaQ)„) / (23) and G r /\Cr,h\2 + \CTtV\2\ \Cr,h\ ~ \Gr,v\ \ 2%{cr,hc;tV) ) (24) where Gt and Gr are the transmit and receive antenna Stokes' vectors respectively. 1.1.10 Stokes' Matrix A 4x4 real matrix analogous to the scattering matrix can be derived [35, p. 362], [47, p. 3-10], [64, p. 532], [63, pp. 819-820], that relates the Stokes' parameters of the transmitted pulse to those of the received pulse. This 4x4 real matrix is known as the Stokes' matrix. This section derives the elements of the Stokes' matrix from the scattering matrix, and then notes some of the advantages and disadvantages of the Stokes' matrix representation. 9 Chapter 1: Properties of Polarimetric Radar Data 1.1.10.1 Derivation Substituting equation (21) in equation (22) the received power can be expressed as: P ={cT,hc*th) (Cr,vC*h (cr,vc*v (cr,vc*h (shhS*.h){Ct,hCth) + (Cr,hC*yh)(ShhShv)(Ct,hClv) + (Shh.S*h)(CtihC*h) + (CTthC*tV)(ShhSyV)(Ct<hClv) + (ShvS*h)(Ct>vClh) + {Cr,hC?,h,)(shhS*.k)(ct,hClh)-\-(5'/,„5*/l)(Ct)l,C*^) + (Cr,hC*tV)(ShvSyV)(Ct,vC*tV) + (SvhSlh)(Ct,hCtth) + (Cr,vC*h){SvhSlv){Ct,hC*iV) + (SvhSvh){Ct,hC*h) + (Cr,vC*v)(SvhS*v)(CtthC*v) + {SvvS^h)(Ct,vC*ih) + (CT,vC*h)(SvvShv)(Ct,vC*tV) + (SvvS*h)(Ct,vC*h) + {Cr,vC*tV)(SvvS*v)(CttVC*v) (25) Rearranging into matrix form, the expression simplifies to: P = (Cr,h.C*%h\ \ Cr,vC*h I = YTTWYt (shhSlh shvsiv shhsiv ShvS*hh\ /Ct,hC*h\ SvhSyh. SVySvv Svh,Svv SvvSvfl G O * G C * G C* C G* 'Jhh'Jyh J h v > - > v v J h h J V v Jhv<Jvfl SvvS^v Svh,S^v SvvS^h/ \Ct,vC*h I (26) Yr describes the polarization state of the receive antenna while Yt describes the polarization state of the transmit antenna. The matrix W is independent of the polarization state of either antenna, and is descriptive only of the properties of the surface scatterer. We wish to obtain an equation relating the received power to the Stokes' vectors of the antennas and a 4x4 real matrix, otherwise known as the Stokes' matrix. We can write the antenna Stokes' vectors Gr and G, in terms of Yr and Yt (after [35, p. 362]): Gr — KYr (27) Gt = RYt where / l l 0 0\ 1 - 1 0 0 0 0 1 1 VO 0 -j jJ (28) 10 Chapter 1: Properties of Polarimetric Radar Data Solving for YT and Yt: Yt = R~ Gt = ~ Yr — R 1 Gr — — where I 0 ° \ /\ctM2 + \Ct,v\ 2\ 1 1 - l 0 0 \ct,h\2 - \ctiV\2 2 0 0 1 3 l o 0 1 -j) \ - 2 9 ( c a q g / 1 0 o \ /ic r iAr+ic r i„i a\ 1 1 - 1 0 0 \Cr,h\2 - \Cr,v\2 2 0 0 1 3 Vo 0 1 -j) V -2%{cr,hc;<v) / Z 1 1 0 1 - 1 0 0 0 0 1 j Vo o i -j/ and (R-'f = (RTyl = \ (29) (30) (31) (32) (33) / l 1 0 0 \ 1 - 1 0 0 0 0 1 1 \0 0 j -jJ Substituting equations (29) and (30) in (26) produces: P = k(R-1Gr)TW{R-lGt) = kGJiR-^WR^Gt where k is a scalar constant. Combining the three 4x4 matrices (R'^WR-1 into a single 4x4 matrix creates the real Stokes' matrix M: M = (R-ifWRT1 / l 1 0 0 \ /ShhS£ h Sh.vSlv ShhSlv Sh,vSlh\ / l SvhSyh SVVSVV Svh,Svv SvvSvfl ShhSyh Shv^vv Shh,S*v ^hvS*^ \SvhShh Svv Slv SvhSlv 1 0 Vo - 1 0 0 0 1 1 o j -j I 1 0 0 \ 1 - 1 0 0 0 0 1 j V0 0 1 -j I (34) 11 Chapter 1: Properties of Polarimetric Radar Data One can then calculate the power simply as the product of a constant, the receive antenna Stokes' vector, the Stokes' matrix and the transmit antenna Stokes' vector [14, p. 299]: P = kGjMGt (35) with M the Stokes' matrix, G> the receive antenna Stokes' vector, and G, the transmit antenna Stokes' vector. The Stokes' vector of the scattered wave S5' is related [14, p. 299] to the Stokes' vector of the incident wave S'r via: Ssc = RRTMStr (36) In the bistatic case, the scattering matrix S has only seven independent parameters, implying that there are nine relations between the M's sixteen elements (see [64, pp. 532-533]). In the monostatic case, both S and M are symmetric, S has five independent parameters, and M nine. 1.1.10.2 Advantages/Disadvantages The Stokes' matrix direcdy relates the antenna Stokes* vectors to the received power. The principal advantage of the Stokes' matrix over the scattering matrix representation is that a set of resolution elements' Stokes' matrices may be averaged [63, p. 533] to produce a Stokes' matrix representative of a coarser resolution element. Such an operation is valid for a scatterer composed of several incoherent scattering centers. For N waves with no bias in their phase relationship (incoherendy related), their superposition can be represented by the sum of their Stokes' vectors: /(So)i\ 1 N S A V ~ JV ^ 1=1 (SO,-(37) No analogous "average" scattering matrix exists. When Stokes' matrix averaging is done, the one-to-one relationship between the Stokes' matrix and a corresponding 2x2 scattering matrix is lost. 12 Chapter 1: Properties of Polarimetric Radar Data A reverse transformation to a scattering matrix representation from an averaged Stokes' matrix Sav would be invalid. An averaged Stokes' matrix must be stored as a set of nine (not seven) independent parameters for the monostatic case (sixteen for the bistatic case). 1.1.10.3 Relation to Scattering Matrix From equation (34) one can derive that the elements of the Stokes' matrix M are related [73, p. 252] to the scattering matrix cross-products via the following equations: = -^.Shh • S^h + • Syy + 2SfiV • S^y] Mi ) 2 = ~^[Shh ' Sfrh — Svv ' Svv] Mx ,3 = -R[Shh • Sly] -T ^[Shv • Syy] MiA M2<2 -- — ^ S[Shh ' Sfrh + Syy • Syy — 2Shv ' S^y M2,3 = ^ [Shh • S^] - ^[Shv • Syy] M2,A = - ^ [ s h h - s i v ] + ^[shy-s:v] = -Shv • Sly -T - R[Shh • Syy] -•^[Shh • Syy] M4,4 = —Shv ' S^y ~ 2^-^llfl' ^vv^ The other Stokes' matrix elements are known from symmetry in the monostatic case. 1.2 Properties of JPL AIRSAR Data 1.2.1 Introduction to JPL AIRSAR The Jet Propulsion Laboratory in Pasadena, California operates a fully polarimetric SAR aboard a DC-8 aircraft. Known as the AIRSAR, the system collects data simultaneously at the C (5.3 cm), L (24 cm), and P-band (67 cm) wavelengths. The radar is a test-bed for the SIR-C space shuttle mission scheduled for 1993 [61, p. 337] and for the Earth Observing System (EOS) SAR mission [17], [66] planned for launch in the late 1990's. 1 3 Chapter 1: Properties of Polarimetric Radar Data The NASA/JPL DC-8 radar transmits alternately at H and V polarizations, and receives the H and V components of each echo simultaneously. Figure 1.5 shows a transmit/receive timing diagram. Horizontal Transmit Event I I I I I 1 1 Vertical Transmit Event I | | | | | | HH Echo r\ r> r\ r> d o CX. HV Echo r>, r\ r\ r~\ VH Echo r~\ r\ r\ r-\  W Echo Q dk Q Q Ch Q Figure 1.5: Transmit/Receive Timing Diagram [74, p. 685] A complete synthetic aperture consists of approximately 1500 echoes. The interpulse period is approximately 1.3 milliseconds [74, p. 685]. 1.2.2 Stokes' Matrix Compression A polarimetric radar returns a veritable flood of data. In the future, JPL plans [62] on collecting the following amount of data for every data take: Data Volume = (#Az.pixels) X (#Range pixels) x (#Matrix Elem.) X (#Bytes/Complex Number) x (^Frequencies) (39) = 16384 x 1200 x 4 x 8 X 3 « 1800 Megabytes This is well beyond the current data processing abilities of affordable computers. Currently only a subset of the data acquired is ever processed and analysed. To maximize the size of that subset it is desirable to compress the data returned into a more manageable size. A compression algorithm [10] is used by JPL to reduce the original single look scattering matrix data into four look Stokes' matrix data. A compression factor of 12.8 is achieved. Single look (hi res) data is also available in a compressed scattering matrix format [62], at a compression factor of 3.2. 14 Chapter 1: Properties of Polarimetric Radar Data 1.2.2.1 Compression For a Stokes' matrix M, the compression operation used by JPL proceeds as follows [73, p. 251]. First a scale factor is stored: / / M \ \ ( 4 0 ) 6y«e(2) = / » < ( 2 5 4 ^ ^ _ - 1 . 5 J j To minimize quantization errors during normalization, a factor x approximately equal to M i , i is introduced. X = ( ^ + 1 . 5 ) . 2 * « M (41) The remaining elements are then stored as: byte(3) = 127-Mia/x (42) byte(4) = 127 • sign(Mlfl/x) • y/\M1<3/x\ byte(5) = 127 • sign(M1A/x) • ^\MiA/x\ byte{6) = 127 • sign(M2,3/x) • y/\M2,3/x\ byte(7) = 127 • sign(M2A/x) • ^\M2A/x\ byte(8) = 127 • M3,3/x byte(9) = 127 -M3A/x byte(\0) = 127-M4A/x where Int( ) is the integer part of the operand, sign(a) returns +1 for positive a, 0 for a=0, and -1 for negative a. Note that nine independent parameters are stored, as noted in section 1.1.10.2. 1.2.2.2 Decompression Data stored in JPL's 4-look format [10] is decompressed via the following equations [73, p. 15 Chapter 1: Properties of Polarimetric Radar Data 251]: *.x = ( ^ + 1.5)-2**> = byte{Z) • ^ M1>3 = sign(byte(4)) • ( ^ ^ ) • M M M M = «*n(&yte(5)) • {^^f • M u M 2 , 3 = «*n(6yte(6)) • (^9) 2 ' M M ( 4 3 ) M 2 ) 4 = sign(byte(7)) • [ J L ^ r L ) • Mh M3,3 = byte(8)-^ M 3 , 4 = W 9 ) ~ M4A = byte(10)-^ M 2 , 2 = Mhi - M 3 , 3 - M 4 , 4 Note that there are nine independent matrix elements recovered. A tenth is a linear combination of three of the nine, while the other elements of M are known from symmetry. 1.3 Polarimetric Features 1.3.1 General Observations In general, heavily vegetated areas will have a strong cross polarized (HV) return, while water surfaces have their strongest return at VV polarization. This suggests a method for the viewing of polarimetric radar data whereby the HH return is fed to the red gun of the video display, the HV to the green, and the VV to the blue. Vegetation then appears green and water blue. For a Stokes' matrix Af: "Mi,! Mi, 2 Ml ,3 M a , 4 -M 2 ,i M 2 , 2 M 2 , 3 M 2 , 4 M3,i M3,2 M 3 , 3 M 3 ) 4 -M 4 , i M 4 ) 2 M 4 , 3 M 4 , 4 . (44) 16 Chapter 1: Properties of Polarimetric Radar Data this section investigates the physical significance of some of the matrix elements and describes some quantities that can be derived from them. 1.3.2 Total Power The first Stokes* matrix element M\Y\ is known as the total power, or span. It represents the backscattered power when the incident (transmitted) pulse is completely unpolarized, containing equal amounts of all polarizations, (ie. SL = ( SQ 0 0 0 ) r ) . From equation (38) one can see that the span is the sum of the HH, HV, VH, and V V returns. Being a combination of multiple channels, a span image has less speckle (and contrast) than any individual channel. 1.3.3 Number of Bounces During Scattering Different scattering mechanisms result in different polarization signatures. Reflections from a flat surface will have a single bounce, while buildings, being dihedral corner reflectors, exhibit a strong double-bounce scattering return. Reflections from forests typically undergo multiple bounces in the canopy before reaching the radar receive antenna. Figure 3.7 illustrates each of these scattering mechanisms. The use of simple scattering models such as these removes the need for training for some classifications, and leads to an unsupervised classification algorithm [60]. 1.3.4 Polarization Phase Difference 1.3.4.1 Definition The polarization phase difference (PPD) is defined as the phase difference between the HH and VV signals. For signals: SHH = AHH exp*'*" (45) the polarization phase difference is then: PPD = A<fi = 4>hh - <f>vv (46) 17 Chapter J: Properties of Polarimetric Radar Data The PPD can be calculated by evaluating ShhS^v and extracting the phase term. 1.3.4.2 Use as Discriminator Ulaby [57] and Boerner [2] have reported that the radar backscatter from some types of vegetation has different path lengths depending on the polarization, suggesting that the PPD can be used to discriminate between different types of ground cover. 1.3.4.3 Relation to Stokes' Matrix The PPD is the phase of ShhSyV. The real and imaginary parts of this quantity can be extracted from the Stokes' matrix via: K(ShhS:v) = M 3 i 3 - M4,4 (47) and 3(S f c f cS;„) = - 2 M 3 i 4 (48) The PPD is therefore calculated as; PPD = A<fi = arctan ( ~ 2 M \ \ ) (49) \Af 3,3 - M 4 ) 4 / 1.3.5 Scattering Matrix Cross Products The scattering matrix cross products are often computed as the inputs to a classifier. The cross products and the manner in which they can be computed from the scattering matrix are shown below: 1. ShhS*hh = Mltl + M2t2 + 2 M i l 2 (HH image) 2. SVVS*VV = + M2>2 - 2Mh2 (VV image) 3. ShvS*hv = Mi,i - M 2 , 2 (HV image) 4. U(SkhS:v) = M 3 ) 3 - M 4 , 4 5. %{ShhS:v) = - 2 M 3 ) 4 6. ®(ShvS*vv) = M l i 3 - M2i3 18 Chapter 1: Properties of Polarimetric Radar Data 7. $s(ShvS;v) = M2A - M M 8. K(ShhS*hv) = M 1 ) 3 + M 2 ) 3 9. %(ShhS*hv) = -(Mi,4 + M2A) 1.3.6 Polarization Signature Given a Stokes' matrix representation of a ground resolution element, from equation (35) one can calculate the received power P over a wide range of transmit and receive antenna polarizations. The received power for each combination of antenna polarizations can then be mapped over the two dimensional grid (x=-45°..45°, ^=0°..180°) known as the unwrapped Poincare sphere and displayed as a surface plot, as in figure 1.6. Such a representation is known as a polarization signature and is useful for the visual comparison of scattering properties. Figure 1.6 shows the polarization signature of a trihedral corner reflector. The case where one sets the transmit and receive polarizations equal: Xr = Xt (50) is known as a co-polarized antenna. A cross-polarized antenna results when the receive polarization is equal to the transmit polarization shifted by 90° in the orientation angle: (51) = (V>t + 90°) mod 180° 19 Chapter 1: Properties of Polarimetric Radar Data Figure 1.6: Polarization Signature The quantities Pmin and Pmax can be computed as the minimum and maximum received power respectively over all polarization states (x.VO- This allows the calculation of two additional quantities: the coefficient of variation and the fractional polarization. 1.3.6.1 Coefficient of Variation The coefficient of variation is the ratio of minimum to maximum received power [63, p. 540]: CoV = (52) M H O I One can see that the smaller the coefficient of variation, the more pronounced is the effect of varying the polarization of the antennas, and the more information that is likely to be found in the polarization signature. A coefficient of variation close to unity indicates that changing the polarizations of the antennas will have minimal effect on the backscattered power, and that there is little useful information content in the polarization-specific data. 20 Chapter 1: Properties of Polarimetric Radar Data 1.3.6.2 Fractional Polarization The fractional polarization is defined [74, p. 696] by: P - P r t •*• 771 a x -* 771171 f ^ 1 \ = P +P~ C ' •* max i J mm One can see that a high fractional polarization shows a high degree of polarized return from the scatterer, and indicates that varying the antenna polarizations will strongly influence the received power. The fractional polarization is therefore useful as an indicator of the polarization "purity" of the backscatter return, and of the possibilities for enhancing the contrast between a given target and other parts of the image. 21 Chapter 2: Data Calibration Chapter 2 Data Calibration 2.1 Background Calibration of polarimetric radar data is a necessary first step if one wishes to avoid coming to false conclusions about the nature of scatterers on the ground. There are four steps [61, p. 342] in the calibration of a polarimetric SAR image: 1. Relative Phase Calibration 2. Cross-Talk Calibration 3. Co-Polarized Channel Imbalance Calibration 4. Radiometric Calibration The needs and resources of each user determine his or her required degree of calibration. The first two steps can be performed on any image, with no ground truth devices required in the scene at the time of data acquisition. Steps three and four can not be performed without auxiliary ground truth data. Returns from trihedral corner reflectors and/or polarimetric active radar calibrator (PARCs) in the scene allow one to perform the last two steps, enabling the complete calibration of the data set. 2.2 Radar System Model The scattering matrix measured by the radar is distorted by the transmit and receive subsystems within the radar. Data calibration aims to remove the distortions from the data, by first devising a model that explains how the corrupted measured scattering matrix is derived from the true scattering matrix, and then inverting the model to arrive at the best estimate of the true scattering matrix given the measurement. The radar system architecture can be modelled by the following formula (after [61, 49, 74]): (54) 22 Chapter 2: Data Calibration where A is an absolute amplitude factor, <f> the absolute phase (dependent on slant range from radar to target), R and T are the receive and transmit system distortion matrices, S is the true scattering matrix, and N the noise in each channel. The object of Data Calibration is the determination of the values of A, R, T, and N. Since <j> has no effect on the received power, it can safely be ignored. Requiring only the four calibration steps listed above results in the following system model: Z = A exp* RTST = w r H)(SHH ^ v 1 h) < 5 5 ) V 1^ A / V Svh svv J V 84 h ) where, for the receive system,/; is the co-polarized channel imbalance in both amplitude and phase, while bi is the cross talk when vertically polarized waves are received and 82 is the cross-talk when horizontally polarized waves are received. Similar definitions hold for the parameters of the transmit system (fc, 83, and 84). Alternatively, the model can be stated in Kronecker delta format as: ( 1 &t 82 8284 \ /Shh\ 83 h 8283 84 j2 #1 #1#4 f l 84 f l \8183 81 f 2 83 j1 / 1 / 2 / Shv Svh \SvvS (56) From equation (55) one should note that the cross-talk and channel imbalance calibrations can be performed independendy, as: / 1 83 \ (1 0 \ and similarly, T = 1 * 3 .84 h ) Vo h)\hlf2 1 — TCTX (57) R ' 1 62 T 0 1 Vo /1/W/1 1 — RQRX (58) where TC and RC describe the co-polarized channel imbalance in the transmitting and receiving systems and TX and RX express the cross talk. 23 Chapter 2: Data Calibration Substituting equations (57) and (58) in (55) results in: Z = A expJ> Rx RCSTCTX (59) Note that this implies that the cross-talk and co-polarized channel imbalances may be performed separately. This is important, as the cross-talk calibration can be done on all data, whereas the co-polarized channel imbalance calibration requires the presence of in-scene trihedral corner reflectors or PARCs. Neglecting absolute amplitude and phase, we can rewrite the above in the form: Z = RXWTX (60) where W is known as the intermediate scattering matrix [61, p. 338]: (Whh Whv\ W = I = Aextf+RcSTc V W«« / (61) ., / Sh.h hShv \ \f\Shv hh Both the relative phase calibration and the cross-talk calibration (determination of Tx and Rx) can be performed without the use of external calibration devices. In the case of no corner reflectors or PARCs, W is then the best possible estimate of 5. 2.3 Phase Calibration Phase calibration is important, since if it is not performed, improper conclusions may be made about the nature of the scatterers being imaged. Phase calibration corrects for the initial ignorance of the lengths of the phase paths of the different channels (HH, HV, VH, VV) in the radar. This section describes the theoretical basis of the phase calibration operation. Literature on the phase calibration of polarimetric SARs can be found in [25], [27], [49], [73], and [74]. 2.3.1 Measured Scattering Matrix The measured scattering matrix R is related to the true scattering matrix S (neglecting noise for the moment) by the unknown transmit and receive phase paths (after [73, p. 247] and [74, p. 694]): (Rhh Rhv\ /SfcfcexpW'.»+*-'> Shv expW«"+*-*) \ R = [ = (62) \Rvh RwJ \Svhexp^+^ Svv expW'.'+*") / 24 Chapter 2: Data Calibration where the phase factor <j>t,h denotes the phase distance travelled by horizontally polarized signals within the radar system (from the transmit amplifier to the antenna) and the other phase factors fcj are defined similarly. For simplification we introduce: <t>t = 4>t,h - <t>t,v (63) <t>r = <t>T,h ~ <I>T,V resulting in: R = exp J ^ < - + *-" ) (64) V S^exp*' Svv J The leading phase term (dependent on the slant range from the target to the aircraft) in the above equation can be ignored, as only the relative phase terms between the polarimetric channels affect the received power. 2.3.2 Symmetrization of Scattering Matrix During the transformation from scattering matrix to Stokes* matrix representation, the first step is the symmetrization of the scattering matrix. This section describes that operation. Neglecting the absolute phase, a symmetrized measured scattering matrix Z is formed by applying a phase shift of -(<j}t — <t>r) to the SVA element [24, p. 774]. From equation (64): Z = (65) V Svh exp*' Svv j This eliminates the hv-vh phase difference. The off diagonal elements of Z are then in phase and may be averaged [73, p. 248]: w = ±(z + zT) / Shh expJ'(0l+*') 5ft"+s"" exp*- \ (66) V s i > " + s " h exp*' Svv J Note that: Wvh = \ (Zhv + Zvh exp-**'-*')) (67) 25 Chapter 2: Data Calibration Once the measured scattering matrix R is transformed into a symmetric scattering matrix W, it is next converted to a Stokes' matrix via equation (34). The cross products of the symmetric scattering matrix W are related to those of the true scattering matrix 5 via: whhw*hh - ShhShh w w* ''VV '' VV — c c* — I J v v k J v v whvw*hv — Shv^hv whhw*vv whhw*hv = ShhS*hvexp>*< = SvhSvv expJ^>'' 2.3.3 Solving for the Phase Distortions Two equations are required to solve for the 4>t and <f>r of equation (63). They can be supplied by: 1. Assuming reciprocity (Shv = Svh) 2. Selecting a "known" target or point in the image where the polarimetric phase difference (PPD) is known 2.3.3.1 Equation 1: Reciprocity Reciprocity dictates that ShV=Svh- Neglecting the absolute phase, from equation (64) we can therefore take the average value of R*hvRvh = SlvSvh exp-7^1-^-) over the entire image to estimate <t>dif j = 4>t-4>r = arg {{RlvRvh)) (69) for (R*hvRvh) coherently averaged over the entire scene [74, p. 694]. For images stored in the Stokes' matrix format [10], this calculation must be done before conversion and stored in the image header for later use during calibration. 26 Chapter 2: Data Calibration 2.3.3.2 Equation 2: "Known" target If one can predict the phase difference between the hh and vv signals (PPD) at one or more parts of a scene, then from equation (64) one may observe that RhhR*v = ShhS*v exp J '^ , +*') and form an estimate of 4>t + <j>r via: <f>sum = <t>t + <t>r = Kg(RhhKv) ~ &I^(ShhS*v) (70) = a,Ig(Rhh.Rlv) - PPDexpected for RhhR*v in the "known" regions [74, p. 694]. Only one point is required but the use of more points will decrease the effect of noise on the estimate. The quantity <f>t + <f>r is most easily estimated by examining the polarization phase difference (PPD) via equation (49) of an area in the uncalibrated image where the theoretical scattering phase difference is near zero (eg. open water). For such a case, the above simplifies to: <f>t + 4>r = ™g(RhhR*vv) (71) 2.3.4 Phase Corrected Measurements Solving the two equations (69) and (70) in two unknowns gives us estimates for the relative phases <j>t and <pr. Inverting equation (64) and omitting the leading phase term lets us estimate 5 from R via: S = V Rvh exp-*' Rvv to a relative phase. This estimate of the scattering matrix has had the effects of the phase path lengths in the receive and transmit systems of the radar removed. Once translated into Stokes' matrix 27 Chapter 2: Data Calibration form, this becomes: shhs*hh = whhw*hh Uvv^vv — ,,vv'rvv ShhS*vv = WhhW:vexp-^+^ ShhS*hv = WhhW*hvexp-i*< svhs:v = wvhw:vexP-^ representative of a Stokes' matrix M that has been properly phase calibrated. (73) 2.3.5 Kronecker Delta Form The phase calibration operation can neglect absolute amplitude, absolute phase, cross-talk, and channel amplitude imbalance distortions. This results in a simplified radar system model: R = 1 0 : ° h Shh ( Shh Shv hShv \ 1 0 ,0 h (74) where /} and h are pure phase terms (no amplitude imbalance), R the measured scattering matrix, and 5 the true scattering matrix. Note that the scattering matrix is in standard format, unlike that seen in [24, p. 774]. In Kronecker delta format equation (74) can be restated as: /l 0 0 0 \ /Shh\ R = 0 f2 0 0 0 0 / i 0 Vo 0 0 hhl Shv Svh V Svv ' = DS (75) This can be compared to the representation of equation (64) (derived from [73]) which becomes the following after normalizing to the absolute phase of Shh' Shh Shv exp-**' \ R = Aexp j ( ^ k + < ^ h ) .S^exp"^ Svvexp~^'+^) (76) 28 Chapter 2: Data Calibration with <f>, and <f>r as defined in equation (63). In Kronecker delta format we have: 0 \ (Shh\ R 0 0 0 e x p - * ' 0 0 0 e x p " * -Vo 0 0 0 0 Shv Svh V Svv ' = DS Comparing equations (74) and (76), we conclude: / i = e x p - * ' h = e x p " * ' (77) (78) During the symmetrization operation that precedes the conversion to Stokes' matrix format, the cross-polarized element is calculated as: Zhv = \(Rhv + fl^exp-^'-*')) (79) Assuming that Shv — Svh we can modify equation (77) to model the phase distortion equivalendy as: / i 0 z! = 0 I {h + fi e x p " Vo 0 0 o \ - 0 e x p " * ' 0 Vo 0 e x p " -j(<t>t+<t>r) J /Shh\ D'S' (80) This suggests a phase calibration step of. / I S = 0 0 \ 0 e x p J ^ ' 0 VO 0 expW«+*')/ (Z'hh\ Jhv (81) Kz'vvJ Note that this phase calibration step conflicts with that seen in Freeman [24, p. 774]. The discrepancy may be due to a failure by Freeman to normalize in terms of Shh (as in equation (76)) rather than Svv (as in equation (64)). 29 Chapter 2: Data Calibration 2.4 Cross-Talk Calibration Cross-talk calibration consists of the determination of the matrices Tx and Rx (see equation (60)). This can be done without comer reflectors or PARCs in the scene. The method is outlined in Van Zyl [61, pp. 339-342]. 2.4.1 Assumptions Assuming that the system is reasonably well isolated (6\62 ^ 0), and that the co- and cross-polarized components of S are uncorrelated ((S^S^) = 0 and {S*vShv) = 0), one can expand equation (60) to show that: {ZhhZ*hh) » (WhhW*hh) (82) (ZVVZ*VV) « (WVVW:V) (83) {ZvvZ*hh) « {WvvW*hh) (84) {ZhvZ*hh) - (WkvW*hh) *h{WhhWh\) + S2/f(WvvW*hh) (85) + 2(62/fT{WhvW*hv) (zhvz*vv) - (whvw;v) *h{whhw;v) + S2/f{WvvW:v) (86) + 26*{WhvW*hv) 2.4.2 Parameters With the above assumptions, one can combine the above equations to solve for the cross-talk parameters: S &P + 2P*(WhvWZv)(ZvvZ*hh) - 4(WhvW*J2(ZvvZ*hh){ZhhZ*vv) 30 Chapter 2: Data Calibration and where s if = &Q + 2Q*(whvw*hv)(zhhz:v) m 2 A* - 4(WhvW*hv)2(ZvvZ*hh)(ZhhZ*vv) A = (ZhhZ*hh)(ZvvZ:v) - (ZvvZU)(ZhhZ:v) - (WhvW*hvY (89) P = (ZhvZ*hh){ZvvZ:v) - (ZvvZih)(ZhvZ;v) - (WhvW*hv)(ZvvZ*hv) (90) Q = (ZhhZ*hh)(ZhvZ;v) - {ZhvZlh)(ZhhZlv) - {WhvW*hv)(ZhhZ*hv) <91) Note that we require knowledge of {WhvW^v) when it is W in fact that we are attempting to calculate. The solution is to iterate. However, first the calibration process will be explained. 2.4.3 Cross-Talk Correction The cross-talk calibration is performed by inverting equation (60): W = R-1ZT~1 (92) to produce the following estimates of the intermediate scattering matrix cross products: (WhhW*hh) « {ZhhZlh) (93) (WVVW:V) * {ZVVZ*VV) (94) (WhhW:v) « (ZhhZ:v) (95) {WhhW*hv) *(ZhvZ*hh) - h(ZhhZlh) - (62/f){ZvvZ*hh) - 26l(ZhvZtv) 31 (96) Chapter 2: Data Calibration (whvw:v) *{ZvvZ*hv) - t>l{ZvvZ*hh) (97) - {hin(zvvz*m ) - 2(62/f)(ZhvZ*hv) (WhvW*hv) *(ZhvZt ,v + \61\2(ZhhZih) + \62/f\2(ZvvZ:, VV - 8i(ZhhZlv) - 6\(ZlhZhv) (98) -(62/f)(ZvvZ*hv)-(62/f)'(Z:vZhv) + h(62/fr{ZhhZ:v) + 6l{62/f){ZthZvv) 2.4.4 Iteration Equation (98) allows one to form an estimate of (W/i«W£„) given an initial guess of say 6j=0 and #2//=0. The estimate of {WhvW^v) can then be used to iteratively improve the estimates of 6i and 62/f via equations (87) and (88). The iteration process is complete once all three estimates (WhvW^), 6j, and 62/f reach stable solutions. The inclusion of noise in the radar system model complicates the estimation process further. Van Zyl's paper [61], from which the preceding explanations are derived, extends the process to cover the case of noise in the system. 2.5 Co-polarized Channel Imbalance The amplitude portion of the channel imbalance results from the differing treatments of horizon-tally and vertically polarized waves within the radar. The internal gain factors for each polarization are different, resulting in system bias. The extent of the imbalance cannot be determined without at least one corner reflector in the scene. It is therefore not dealt with in detail here. See [61, p. 341] for more information. 2.6 Radiometric Calibration The absolute amplitude factor .4 can be determined by comparing the absolute radar cross section measured from a trihedral corner reflector in the scene with what is theoretically expected from a 32 Chapter 2: Data Calibration corner reflector of that size and then performing the appropriate relative amplitude scaling to the rest of the image. Note that an in-scene corner reflector is required. Radiometric calibration is therefore not investigated in detail here. For further information, see [61, p. 342]. 33 Chapter 3: Classification Chapter 3 Classification 3.1 Introduction One important aim of SAR polarimetry is improved accuracy in SAR-based classification of land cover. Rather than returning just a single scalar radar backscatter quantity, as with conventional SARs, polarimetric SARs return the complete scattering matrix (see equation (17)). The additional information can be used to improve classification accuracy. The sections that follow describe the theoretical basis for some classification algorithms, and discuss their effectiveness. There are two broad categories of classification. Unsupervised classification is the classification of land cover into types without any a-priori information required from the operator. Supervised classification requires the investigator to select training areas that are then used as class prototypes by the computer in the classification of the land cover outside the training areas. Both types of classification take as input a set of features for each pixel upon which the classification decisions are based. The features input to a classifier could be [47, p. 3-18] for example: 1. The Stokes' matrix elements 2. Three amplitude (HH, HV, VV) and two phase difference images 3. The scattering matrix cross-products 4. Specially defined combinations of Stokes' matrix elements The following sections describe both the supervised and unsupervised classification algorithms as they apply to polarimetric SAR images. Chapter 3: Classification 3.2 Unsupervised Classification 3.2.1 Introduction Van Zyl's unsupervised classification algorithm [60] assigns each pixel to a class based on its polarization properties. It prescribes three scattering classes, namely: 1. Odd number of reflections (eg. ocean, clear-cut logged area) 2. Even number of reflections (eg. building) 3. Diffuse scattering (eg. forest) 35 Chapter 3: Classification 4 2 2 2 3 1 Forested Areas Figure 3.7: Scattering Mechanisms [60, p. 39] (a) Slightly rough surface (odd bounce) (b) dihedral comer reflector (even bounce) (c) Forested area (diffuse) [1: direct canopy backscatter; 2: double bounce scattering; 3: direct ground backscatter; 4: direct tree trunk backscatter] See figure 3.7 for a physical interpretation of the scattering mechanisms of each of the three classes. 36 Chapter 3: Classification Heavily vegetated areas are found to exhibit a mixture of the properties of all three scattering classes, with the diffuse component increasing at high incidence angles due to greater interaction with the canopy. 3.2.2 Algorithm This section describes the Van Zyl unsupervised classification algorithm. The theoretical basis of the algorithm is explained, its operation outlined, and its utility for classifying forest scenes explored. 3.2.2.1 Importance An unsupervised classification algorithm requires no a-priori knowledge of the study area, and is therefore simpler to operate than supervised techniques. Van Zyl's unsupervised classification algorithm by no means exhausts the information content of the Stokes' matrices. Supervised classifiers, or even unsupervised classifiers using more detailed models of the scattering mechanisms at work in a scene will produce more complete and more \ informative classifications. Van Zyl's algorithm does however provide a quick glimpse of the scattering mechanisms in a scene that may later be expanded upon by the investigator. 3.2.2.2 Algorithm Overview The unsupervised classification algorithm of Van Zyl [60, p. 37] works in the following way. From the Stokes' matrix data for each pixel, simulations (or experiments) of various transmit and receive polarizations are conducted: 1. Linear polarizations of varying orientation angles are transmitted and the orientation of the received (scattered) pulse is observed. 2. Circularly polarized pulses are transmitted, and the handedness of the received (scattered) wave is compared to that of the transmitted pulse. 37 Chapter 3: Classification The results from these two experiments are used to determine the classification of each pixel. Handedness is defined here from the point of view of the wave receding from the observer [64, p. 531). 3.2.2.3 Scatterer Signature The scattering mechanisms of the three classes chosen by Van Zyl are developed in [60, pp. 37-38] to produce the following decision rules: Odd Number of Bounces 1. The orientation angle ip of the scattered wave increases as the orientation angle of the transmitted wave increases. 2. The handedness of the scattered wave is the opposite of the transmitted wave. Even Number of Bounces 1. The orientation angle tp of the scattered wave increases as the orientation angle of the transmitted wave decreases. 2. The handedness of the scattered wave is the same as the transmitted wave. Diffuse Scattering 1. The orientation angle ip of the scattered wave increases as the orientation angle of the transmitted wave increases. 2. The handedness of the scattered wave is the same as the transmitted wave. 3.2.2.4 Consistency To counter the effects of noise, each experiment is checked for consistency. The average behaviour of the scattered wave in the first experiment Oinear polarizations) is checked for the ranges of transmitted wave orientation angles 0° < ip < 90° and 90° < ip < 180°. 38 Chapter 3: Classification If the scattered wave's orientation angle does not consistendy either increase or decrease for both cases, then the pixel is set aside as unclassified. Also, in the second experiment (circular polarizations) first a left-circular transmitted pulse is simulated and the handedness of the scattered wave is noted. Then a right-circular transmitted pulse is simulated and the handedness of the scattered wave is noted. If the handedness of the scattered wave does not have the same relation to that of the transmitted wave for both cases, then the pixel is set aside as unclassified. The decision process, including consistency checks, for the unsupervised classification algorithm is summarized in table 3.1. Effect on scattered wave orientation angle when transmitted wave orientation angle increases between 0° < V < 90° 90° < V < 180° Handedness of scattered wave when transmitted wave is Right Handed Left Handed Classification ++ ++ Left Handed Right Handed Odd Right Handed Left Handed Even ++ ++ Right Handed Left Handed Diffuse ++ XX XX not classified ++ XX XX not classified XX XX Right Handed Right Handed not classified XX XX Left Handed Left Handed not classified xx don't care ++ increases — decreases Table 3.1 Decision Process [60, p. 39] for Van Zyl's Unsupervised Classification Algorithm 39 Chapter 3: Classification 3.2.2.5 Forest Scattering Literature on the classification of forest scenes based on polarimetric radar data may be found in [14], [15], [21, p. 784], and [60, p. 40]. The unsupervised classification method of Van Zyl does not unambiguously classify forest, as demonstrated in table 3.2. Model Classification Very sparse forest Odd Sparse forest Even Moderately thick forest Diffuse Very thick forest Odd Table 3.2 Results of Unsupervised Classification of Forests [60, p. 40] The seemingly chaotic results can be explained by the scattering models developed in [15]. Trees are modelled as large vertical dielectric cylinders (the trunk) oriented statistically about the surface normal, with smaller cylinders (branches) radiating at random orientations governed by a probability density function (pdf)- Leaves can be ignored, as they do not have a large effect on backscatter except at short radar wavelengths. A very sparse forest is essentially an open plain, producing an odd-bounce classification. In a sparse forest, double bounce return off the tree trunk/ground dominates, yielding an even-bounce classification. In a moderately thick forest, volume scattering within the canopy dominates, and a diffuse classification results. In a very thick forest the canopy is closed, and scattering off the tops of the trees dominates, producing an odd-bounce classification. The transition points between the scattering mechanisms shown in table 3.2 are a function of radar frequency, forest density, and forest water content (which influences the dielectric constant of the cylinders) [60, p. 40], and are the subject of current research. 40 Chapter 3: Classification 3.2.3 Implementation Van Zyl's unsupervised classification algorithm was implemented by the author in C within the Sunview/Xview graphic user interface as a program called "POLVIEW". This section describes the verification of the implementation, the effect of calibration on classification results, and discusses the utility of the algorithm for the classification of a scene acquired over the Weeks Lake region of Vancouver Island, Canada. Results are presented in the form of tables and gray-scale classification maps. All four types of unclassified pixels are represented by a single gray tone in the images to increase the contrast between the odd, even, and diffuse classifications. 3.2.3.1 Verification The software implementation of the algorithm was first verified by testing it on data of San Francisco acquired by the original JPL CV-990 SAR in the summer of 1985. Van Zyl reported averaging his data before running it through the classifier, so this step was necessary within POLVIEW as well. Averaging is necessary to reduce statistical variations, but if one selects too large an area then the target area's scattering properties lose homogeneity. Van Zyl reported that averaging 36 samples yielded the best compromise between these two considerations [60, p. 42]. Although Van Zyl indicated that he used an average of 36 resolution elements in obtaining his results, he did not report the form of resolution elements (whether they were in single look [62] or four look [10] format). From table 3.3 we see that Van Zyl's reported results match those obtained from POLVIEW when 3x3 averaging is used. Noting that the data supplied to POLVIEW is 4-look, this implies that Van Zyl used single look data, as the POLVIEW 9-element averaging produces the same results as Van Zyl 36-element averaging. 41 Chapter 3: Classification Illustration 1 Unsupervised Classification of 3 x 3 Averaged San Francisco L-band Data (white=odd-bounce, light-gray=even-bounce, dark-gray=diffuse scatterer, black=unclassified) The unsupervised classification results for 3x3 averaged San Francisco data are shown in Illustration 1. The results are very similar to those reported by Van Zyl [60, p. 41]. Source Avg Odd Even Diffuse Unclassified 11 w Both O Both O POLVIEW l x l 50.6 22.8 1.5 10.8 6.9 3.8 3.3 POLVIEW 2x2 48.9 21.7 6.4 8.7 5.7 4.4 3.9 POLVIEW 3x3 47.7 22.2 10.1 7.3 5.4 3.6 3.3 POLVIEW 6x6 46.1 25.0 13.3 5.8 6.0 1.6 1.9 Van Zyl 36 47.7 22.3 10.1 6.0 5.5 4.6 3.8 Table 3.3 Percentages from Van Zyl Unsupervised Classification of San Francisco L-Band Data 42 Chapter 3: Classification For additional certainty, the POLVEEW implementation was then tested on JPL CV-990 data acquired from a forested area near Traverse City, Michigan. The results, in comparison with those reported by Van Zyl [60, p. 43] are listed in table 3.4. Source Avg Odd Even Diffuse Unclassified 11 w Both O Both O POLVIEW l x l 36.2 12.6 3.5 22.1 9.2 8.9 7.2 POLVIEW 2x2 36.8 3.7 17.5 13.2 3.2 14.6 10.5 POLVIEW 3x3 35.3 1.5 28.9 5.9 1.0 16.4 10.6 POLVIEW 6x6 30.5 0.7 41.4 0.8 0.2 17.0 8.9 Van Zyl 36 35.4 1.5 28.8 5.1 1.1 17.1 10.9 Table 3.4 Percentages from Van Zyl Unsupervised Classification of Traverse City L-Band Data There is considerable agreement here as well. The POLVIEW classified images of both the San Francisco and Traverse City scenes appear similar to those published by Van Zyl [60], indicating that the implementation appears to be sound. 3.2.3.2 Calibration Calibration is necessary for the accurate estimation of the scattered wave's polarization from the incident (transmitted) wave's polarization. Uncalibrated Stokes' matrices will introduce errors into the estimation of the scattered wave's polarization, and diminish the success of the unsupervised classification algorithm. Having verified the POLVIEW implementation of Van Zyl's unsupervised classification algo-rithm, POLVIEW was then used to classify the Weeks Lake L-band data. The data were classified both before and after calibration with POLTOOL v6.3 [31]. Note that POLTOOL can perform only phase calibration and no cross-talk calibration. However, this is not a serious deficiency, as the JPL radar exhibited good cross-talk isolation [25] during the season that the data was acquired. Further calibration would not have been possible, as no external calibration devices were deployed in the scene at the time of imaging. Absolute radiometric calibration was therefore not available, but 43 Chapter 3: Classification would not have improved the algorithm's accuracy in any case, as only the handedness and orientation angle of the scattered wave, and not its amplitude, are used by the algorithm. However, the lack of external calibration devices also made a co-polarized channel imbalance calibration impossible, a deficiency that might affect classification accuracy. 3.2.3.3 Results The unsupervised classification results from the Weeks Lake L-band scene, both for uncalibrated and POLTOOL phase calibrated data, are reported in table 3.5. The most important results are those from three-by-three averaging (the Weeks Lake data, as with the San Francisco data, is in 4-look averaged Stokes' matrix format). Classification results using other averaging box sizes are, however, included for comparison. Data Set Avg Odd Even Diffuse Unclassified 11 j r Both O Both O Uncalibrated l x l 24.6 10.1 2.7 33.8 11.0 14.7 2.7 Uncalibrated 2x2 19.7 1.7 10.0 30.8 3.5 32.3 1.8 Uncalibrated 3x3 15.2 0.3 13.2 24.3 0.8 45.0 0.8 Uncalibrated 6x6 9.3 0.0 14.1 11.8 0.1 64.2 0.1 Phase Calibrated l x l 35.5 7.1 2.2 31.5 8.2 8.9 6.2 Phase Calibrated 2x2 39.1 0.7 8.6 24.9 1.5 15.4 9.3 Phase Calibrated 3x3 41.4 0.1 12.2 18.2 0.3 17.5 9.9 Phase Calibrated 6x6 44.5 0.0 14.3 12.6 0.1 18.8 9.3 Table 3.5 Percentages from Van Zyl Unsupervised Classification of Weeks Lake L-Band Data 44 Chapter 3: Classification 3.2.3.4 Discussion Data Set Avg Odd Even Diffuse Unclassified 11 \\ Both 0 Both 0 Uncalibrated 3x3 10.4 0.4 16.2 22.8 0.9 48.4 0.9 Phase Calibrated 3x3 47.8 0.2 15.0 8.3 0.3 21.9 6.6 Table 3.6 Percentages from Van Zyl Unsupervised Classification of Calibrated and Uncalibrated Weeks Lake L-Band Data (near range excluded) Illustration 2 Unsupervised Classification of 2x2 Averaged Phase Calibrated Weeks Lake L-band Data (mid-gray=odd-bounce, white=even-bounce, black=diffuse scatterer, dark-gray=unclassified) Visual inspection of the classified Weeks Lake image (Illustration 2) indicates that most of the unclassified pixels occur in the extreme near-range. More informative tabular results might therefore be obtained by omitting the extreme near-range from the study area. Excluding the nearest range 150 pixels (of 750) in the Weeks Lake scene yields the results seen in table 3.6. The classified calibrated Weeks Lake image (Illustration 2) shows that, in general, the odd-bounce class corresponds to open water or clear cuts (identified from aerial photos), while the diffuse scattering class corresponds to forested areas. 45 Chapter 3: Classification Table 3.6 shows that the clear cut (odd bounce) areas are not properly identified unless the data is calibrated. Attempting classification based on uncalibrated data leads to wrong classifications, as well as many more unclassified pixels than when calibrated data is used. 3.3 Supervised Classification Supervised classification requires the investigator to specify a training set made up of areas with known ground cover types. Class prototypes are then computed from the training set Other pixels in the image are then compared to these class prototypes and, using a minimum euclidean distance (MED) or minimum intra-class distance (MICD, which scales the distance by the measured intra-class standard deviation) metric, are assigned to one of the given classes. The class features used in the determination of class distance could be any of those listed in section 3.1. The supervised classification algorithms described here use the features listed in section 1.3.5 as the basis for classification. This section describes the theory behind supervised classification, an implementation of the algorithm in C as the program "POLVIEW", and results obtained from classifications of the JPL San Francisco and Weeks Lake L-band SAR datasets. 3.3.1 Theory This section describes the theory of the supervised classification algorithm. The main task is the determination of the "distance" between the feature set of the pixel that we wish to classify and the feature set of each class prototype. 46 Chapter 3: Classification 3.3.1.1 Introduction As noted in section 1.3.5, each feature set can be specified by a feature vector of the form: / / o \ ( ShhS*hh \ / a n + a 2 2 + 2a i 2 \ fx c c* « 1 1 + 2^2 _ 2 d i 2 h « 1 1 - «22 h k = - 2 a 3 4 h «13 - 0 2 3 h «24 - « 1 4 h a i 3 + a23 \ h ) \ - ( a i 4 + a 2 4 ) / (99) where A is the Stokes' matrix of the resolution element being considered, and fo ... fg are the scattering matrix cross-products. The class C,- can then be represented by the mean of all feature vectors X that are its members: Ui = E X where XeCi. The class mean is best estimated by: (100) (101) where there are N, members of class C,-. 3.3.1.2 Euclidean Distance The Euclidean distance between a feature vector X and the class prototype rhi of class C; is defined as: = yj'(x - mi)T (X - rfti) (102) Qassifiers based on this distance metric are known as Minimum Euclidean Distance (MED). 47 Chapter 3: Classification This distance is easily calculated but does not take into account the variance within each class. Some classes have compact feature spaces while others are more spread out. Comparing Euclidean distances to the means of (a) a compact class, and (b) a distributed class may therefore lead to suboptimal classifications. By computing the covariance matrix of each class, the within-class variability can become a part of the algorithm. 3.3.1.3 Intra-Class Distance If in addition to the mean, one makes use of the covariance matrix of each class, a more useful distance metric results. The covariance matrix of class C; is defined as: X S,- = E = E (x - [x - /?,•) XXT] - fctf (103) Its estimate is then: * = i:E(*-i*)(*-'*)3 (104) The intra-class distance between a feature vector X and a feature vector TO,- is defined as: dMiCD (X, mi) ^ [W (X - mt) ] w(x-fh^ = (X- rhlfwTw(x - rhi) (105) = (X - rhi)TS-'(X - mi) where (106) wTw = (A-*$T) (A-*$T) = ($A$T)_1 W represents the orthonormal whitening transformation, and 5, the measured co-variance matrix of class Ci. The transformation W operates by first removing correlations through a rotation of the 48 Chapter 3: Classification feature vectors onto orthogonal axes (via the eigenvector matrix $), and then scaling each feature to a unit variance (using the diagonal matrix A). Note that: = $A (107) In summary, the orthonormal whitening transformation enables the Minimum Intra-Class Distance (MICD) metric to measure distance in standard deviation units along uncorrelated feature axes. 3.3.1.4 MAP Classifier The a-posteriori probability of class C, given the observed feature vector X is p(Ci\x). The maximum a-posteriori (MAP) classifier is then: P^dlX^P^CAx) (108) 3 where the most probable class of C, and Cj is chosen given the observation X. Using Baye's theorem we have: P^XlC^PiCi^pfxicijPiCj) (109) j with P(Ci) the a-priori class probability of class C, and p(^X\C,the class-conditional PDF for X. For the JPL SAR images under study, we cannot assume one class more probable than another, so we set P(C\) = P(C2) = • • • = P(C,) = • • • = P(Cm) = ± where there are m classes. This leaves: i pfxic^p^xiCj) (110) j as our classifier. 49 Chapter 3: Classification 3.3.1.5 Multivariate Gaussian Classes For multivariate Gaussian classes, the class-conditional PDF for X is p(x\d)= l -—exp-i( ; ? -' r 0 T sr 1 (^-«0 (111) where there are n classes, pi is the true (not measured) mean of class C,-, and £,• is the true (not measured) covariance matrix of class C«. Taking logarithms and substituting in classifier (110) results in a classifier of the form: (x-lTj) £- 1(x-M;)+21n|£ j|>(x-/r i) S f 1 (x - #) + 21n|£; | j (X - rifL? (X - £•) -(it- / i - ) r s r 1 (X - £•) >21n^j (112) If we lack knowledge of the true class means and variances £,, we can substitute our estimates of those quantities (m, and Si). The resulting maximum likelihood (ML) classifier is: t (x - nij)1S-1 (x - rnj) + 2ln\Sj\*(x~ ™ . ) T S t _ 1 (x - m\) + 2In|5,| (113) i Defining the maximum likelihood distance as: dML(X,m\) = (*- n ^ S - 1 (X - m t) + 2In \S{\ (114) the class chosen as best corresponding to feature vector X is then C,- where dML (x, m) < dML (x, rnj) Vi / j (X - m l ) r S i - 1 (x - Mi) + 21n\Si\ < (x - m ^ S " 1 (x - rnj) + 21n Vi ^ j (115) 3.3.1.6 MED vs. ML In summary, although the MED classifier is simpler (and consequently operates much more quickly), the M L classifier uses more information to describe each class, and should therefore be more accurate. 50 Chapter 3: Classification 3.3.1.7 Hybrid This section describes a classification algorithm that compromises between the speed of MED classifier and the accuracy of the ML classifier. The algorithm avoids the computational price of orthonormalization, but has many of the advantages of the maximum likelihood classifier. For a given class C,-, the standard deviation of each feature j can be calculated from the training set: \ ml, (116) where there are Ni members in the training set of class C,. The average standard deviation of each feature over all classes is then: i N '3 = N Vi  JrY,0-i,J (117) where there are N classes. Once the above quantity has been computed from the training sets for each feature, a distance metric can be formed from the following simple computation: dH(x,rnt)=^X<-mi^ (118) where / denotes the feature currendy being summed, and there are Nf features. This hybrid classification technique is used in chapter 4, mainly for speed. 3.4 Supervised vs. Unsupervised This section describes the pros and cons of the supervised and unsupervised classification techniques. An unsupervised classifier does not require a-priori knowledge of the ground cover within a scene, while supervised classifiers require the investigator to specify a training set. Unsupervised classifiers are therefore, other considerations being equal, preferable, as they are simpler to operate, needing no instruction prior to application. 51 Chapter 3: Classification For JPL AIRSAR data, a supervised classifier using an in-scene training set is less sensitive to poor calibration than Van Zyl's unsupervised classifier. An explanation for this follows. The supervised classifier bases its decisions on distance measures between pixels within the scene and class prototypes derived from training sets (also assumed to be within the scene). The training sets are themselves miscalibrated in the same way as the other pixels, so distance measures between them are not as affected. Van Zyl's unsupervised classifier relies on estimates of the scattered wave's polarization. The estimate is less accurate in a poorly calibrated image, resulting in incorrect classifications. Both classifiers should however be more equally subject to poor calibration if the supervised classifier is restricted to training data from outside the scene. Supervised classifiers can have their accuracy improved by adding more detailed training areas. Improvements in the accuracy of unsupervised classifiers are more complex. Supervised classifiers also have the advantage that each class corresponds direcdy to a ground cover type. In an unsupervised classifier, each class need not have a corresponding ground cover type. Although a class may correspond to a particular behaviour in a scattering model (as with Van Zyl's algorithm) or to a cluster in feature space (as with an unsupervised classifier consisting of a clustering algorithm followed by the normal distance-based classification) there need not be a ground cover analog to each class. 52 Chapter 4: Class Separation by Feature Chapter 4 Class Separation by Feature This chapter explores the relative utility of each of the polarimetric features when attempting to discriminate between various terrain classes using a JPL AIRSAR image. 4.1 Motivation Knowledge of the utility of each feature in discriminating between different ground cover types is a necessary first step in the construction of knowledge-based classifiers. For polarimetric AIRSAR images, a model of the scattering mechanism can be used to explain and/or derive a feature's utility for a given set of tasks. The model can then be integrated into a classifier. Knowledge of feature utility is also vital when choosing (a) the best partially polarimetric radar for a given task or (b) those features that must be preserved and those that might be dropped in a data compression operation. 4.2 Feature Definitions The following section describes features that can be calculated from polarimetric SAR data. The feature set described here is used as the basis for classifications later in this chapter. 4.2.1 Feature Vector Given the scattering matrix S, defined as: (119) and the Stokes' matrix A: /an Oi2 ai3 « i 4 \ a21 «22 a23 a 2 4 A = (120) «31 «32 a33 «34 53 Chapter 4: Class Separation by Feature the cross products (e.g. ShhSlh) are the product of the complex return from one channel with the complex conjugate of the return from another channel. The cross products themselves are generally complex, unless the two channels are the same, in which case their cross product is real. As a short explanation of the nomenclature used in the following sections, Shh and Svv are in amplitude units, and are complex numbers. ShhS*v is unnormalized, and is in power units. HH VV* = SkhSyv/an is normalized, and is a power ratio. The polarimetric feature set F was selected to be the span plus the nine normalized cross products: /FQ\ / span \ / -su x an \ * i HH HH* [2(a u + a12) - (a 3 3 + a 4 4 ) ] /a n F2 VV VV* [2(ou - a12) - (a 3 3 + a 4 4 ) ] /o n F3 HV HV* (a 3 3 + a 4 4 ) / « n F4 U{HH VV*} (a 3 3 - a 4 4 ) /an F5 %{HH VV*} - 2 a 3 4 / a n Fe ${HV VV*} (a 1 3 - o 2 3 ) / o n F7 %{HV VV*} (a 2 4 - a i 4 ) / a u Fs ${HH HV*} (ai3 + a 2 3 ) /an \FJ \%{HH HV*} ) V -(«24 + Ol4)/Oll / Note that span is unnormalized, and is in power units. su is a weighting constant used to bring the span into a range comparable with the normalized scattering matrix cross products. Note that the span is the only unnormalized member of the feature set. Al l of the normalized cross products represent information unavailable to conventional non-polarimetric radars, while the span feature is representative of the brightness collected by conventional SARs. The span is affected by the roughness of the objects illuminated relative to the radar wavelength and can therefore be used to discriminate between terrains of different roughnesses. The normalized cross products are affected by the structure of the scatterers, and can sometimes discriminate between terrain types that have similar backscatter strength. 5 4 Chapter 4: Class Separation by Feature 4.2.2 Polarization Ratio The polarization ratio p is denned as the ratio of the HH cross section to the V V cross section, and is affected by the shape and orientation of the scatterers. P = f * f k (122) Durden [13] notes that this feature decreases with increasing frequency in a forested area, as branches are thin cylinders at P-band (strong polarization dependence) while at higher frequencies the branches are effectively thicker cylinders, with a return influenced less by the branch orientation and microwave polarization. 4.2.3 Linear Depolarization Ratio The linear depolarization ratio LDR, often used in radar meteorology, is defined as the ratio of the cross-polarized HV cross section to the like-polarized VV cross section. LDR = (123) In forested scenes, high values of LDR indicate canopy scattering, while lower values indicate surface and tree-surface scattering [13]. 4.2.4 Polarization Phase Difference The polarization phase difference was defined previously as the difference between the HH and V V phase terms: PPD 4 tan"1 (111^ 14) (124) Ulaby [57] and Boemer [2] have reported that the radar backscatter from some types of vegetation has different path lengths depending on the polarization, suggesting that the PPD can be used to discriminate between different types of ground cover. 55 Chapter 4: Class Separation by Feature 4.3 Feature Distributions 4.3.1 Training Set Definitions This chapter considers eleven polarimetric SAR scenes. Two were collected by the first JPL fully polarimetric L-band SAR (San Francisco, CA and Traverse City, MI), and have been reported on extensively in the literature [64]. The remainder were acquired with the current JPL DC-8 AIRSAR. The AIRSAR has a multifrequency capability to simultaneously acquire P, L, and C-band data. Within the eleven scenes, the similar class-types were often used for training: many scenes contained large odd-bounce areas, many had multipath scattering areas (usually forest), and some contained small even-bounce areas dispersed about the scene. These class-types were prototyped using training areas selected based on the results of a van Zyl [60] unsupervised classification. The Pisgah, Mt. Shasta, and Weeks Lake scenes contain no sizeable even-bounce scatterers; alternate class-types were improvised. Results from these alternate training sets provide a basis for judging the impact of the class-dependence on the general conclusions. The Pisgah, Mt. Shasta, and Weeks Lake scenes are differentiated from the others during some totals in this chapter. Where such differentiation is performed, these three scenes are referred to as non-EDO (Even/Diffuse/Odd). For the record, the locations of the training sets used in the following sections are listed in Table 4.7. A l l coordinates reference the top-left pixel of a 20x20 bounding box. The 4-look JPL compressed file format coordinate system is used, with 1,1 being the coordinate of the first pixel in the file (top left corner). The acquistion date, tape ID, and frequencies (P, L, and/or C) analysed are also listed for each dataset. Scene Location Tape ID, Acquisition Date Class y Bonanza Creek, Alaska CC0117L, 88.03.13 1 Even 660 493 2 Diffuse 852 501 3 Odd 808 399 Table 4.7 20x 20 Training Set Locations (Continued . . . ) 56 Chapter 4: Class Separation by Feature Scene Location Tape ID, Acquisition Date Class X y Bonanza Creek, Alaska CC0045L, 88.03.20 1 Even 564 534 2 Diffuse 764 535 3 Odd 718 437 Fairbanks, Alaska CC0181L, 88.03.13 1 Even 554 634 2 Diffuse 132 274 3 Odd 592 524 Fairbanks, Alaska CC1252L, 88.03.20 1 Even 674 682 2 Diffuse 258 336 3 Odd 710 576 1 Even 688 662 Flevoland, The CC1267PLC, 2 Diffuse 968 622 Netherlands 89.08.16 3 Odd (Ocean) 960 66 4 Odd (Field) 262 294 1 Phase I Lava 303 375 Pisgah Lava Flow, California CC0089PLC, 88.06.02 2 Phase II Lava 4 89 3 Phase III Lava 80 235 4 Alluvial Surface 629 664 5 Dry Lakebed 289 604 Punta Cacao, Costa Rica CC1213PLC, 90.03.15 1 Even 434 1068 2 Diffuse 784 790 3 Odd 580 474 San Francisco, California ?L, 85.05.21 1 Even (City) 366 454 2 Diffuse (Park) 424 346 3 Odd (Ocean) 288 324 Mt. Shasta, California CC1199PLC, 89.08.06 1 Clear Cut 879 234 2 Tree 1 746 481 3 Tree 2 788 118 Table 4.7 20 x 20 Training Set Locations (Continued . . . ) 57 Chapter 4: Class Separation by Feature Scene Location Tape ID, Acquisition Date Class X y 1 Even (Dead Trees) 750 644 Traverse City, ?L, 2 Diffuse (Forest) 796 580 Michigan 85.07.10 3 Odd (Lake) 840 548 4 Odd (Field) 660 790 . Weeks Lake, British Columbia CM1234PLC, 88.03.23 1 Clear Cut 910 401 2 Forest 316 425 3 Lake 356 352 Table 4.7 20x20 Training Set Locations 4.3.2 Distributions A graphical representation of the feature distributions within the training areas of the Weeks Lake L-band scene is shown as an example in Figure 4.8. Each bar is centered on the feature's mean value with a length twice its standard deviation. If the class bars are well separated for a certain feature, then that feature helps in discrimination between those classes. For instance, it can be seen from Figure 4.8 that both the span and HV features discriminate well between lake and forest. 4.4 Class Separation 4.4.1 Introduction If two terrain classes A and B are considered, a large difference between a feature's mean values in the two classes relative to its standard deviations in A and B indicates that the feature is useful for discriminating between them. In Figure 4.8, when the class bars of a feature have a distinct range of values (are separate), then the feature is a useful discriminator. The concept is formalized in the class separability [13] and [71], defined as the ratio of the difference in the feature's mean values to the sum of its standard deviations. For feature vectors A and B as defined in equation (121), the class separation is defined as: ^'^--i&rfk  (,25) 58 Chapter 4: Class Separation by Feature Value Weeks Lake - L-band 2.50 2.00 — Clear Cut Forest Lake 1.50 -1.00 Feature Span HH W HV Re Im Re Im Re Im HHVV* HVVV* HHHV* Figure 4.8: Feature Distribution in Weeks Lake L-band scene where A, is the mean value of feature i in class A, and (Ai)a is the standard deviation, with similar definitions holding for class B. The standard deviations are calculated for 2 x 2 averaged pixels and all classifications discussed in this chapter are based on such averaging. The single pixel variation was found to be too large for reliable classification. The definition of class separability shown in equation (125) is different than the inter-class distance measures used by Singhal [51, p. 85] and Heal [32, p. 91], as the training areas used in these studies are uniformly large enough (20x20) that the second order statistics of each feature for each training area can be collected. 59 0.50 o.oo -0.50 Chapter 4: Class Separation by Feature A class separability above unity indicates that the feature is useful for discrimination; above 2, the feature provides close to perfect discrimination on its own. Other definitions of class separation consider the influence of "third party" classes on the separation of class pairs. For example, the difference in class means can be scaled by the mean standard deviation over all classes in the scene, rather than by the sum of the standard deviations of the class pair being considered. Under such definitions, the "separation" between two classes is influenced by the presence or absence of other classes. Classifiers must operate in such an environment, so the approach has some merit. However it is not clear that the modified definitions of class separability accurately model the effect of "other" classes on class pair discrimination. For simplicity, the class separation definition shown in equation (125) is used throughout this chapter. 4.4.2 Results The following sections consider the class separation results for the class pairs within each scene. Each scene is introduced with the classification map resulting from a classification using the ten L-band features of equation (121). Look direction is always to the bottom of the page. The training areas are demarcated and labelled. For each scene, the feature distributions are illustrated for each feature at all available frequencies. Numerical values of the feature means and standard deviations, and the class-pair separations are available in Appendix A. Note that a quantity called the span scale (su) is listed with the class separation results for each scene that required a normalization. This is the quantity used to scale the span to put it in a range comparable with the normalized cross products. Four different scene-types were studied. Most were forestry scenes (9 datasets), with single agricultural, urban, and geological scenes also included for comparison. 4.4.2.1 Forestry A variety of forest scenes were studied, ranging from tropical (Costa Rica) to subarctic (Alaska). Even-bounce, odd-bounce, diffuse, clear-cut, water, and different tree-type classes were investigated. 60 Chapter 4: Class Separation by Feature Bonanza Creek - L-band (2 Scenes) This scene was acquired during a campaign in Alaska to study the effect of changing environmental conditions on microwave signatures. Preliminary analysis of the data was reported in [67]. The training areas for the Bonanza Creek scene were chosen as the best homogenous 20x20 representatives of the even-bounce, diffuse, and odd-bounce scattering classes (from the Van Zyl [60] unsupervised classification algorithm). Scenes from two different dates (one with a temperature above freezing, one below) are considered. Although the two sets of data are not co-registered, the same patches of river/forest were used for training in each dataset. Note that both datasets are L-band only. Even-bounce scattering is predominandy found along the edges of river banks opposite the aircraft's track. An even-bounce scattering geometry is formed by the flat water against the vertical trunks of trees along the river bank. There are also a few tree stands away from river banks that exhibit even-bounce scattering behaviour. Such stands must have a relatively stronger ground-trunk interaction term in their backscatter, possibly caused by dead branches or the loss of branch needles. Either would reduce multipath backscatter, and the ground-trunk even-bounce scatter would dominate. Illustration 3 Supervised Classification of 2x2 Averaged 13 March 1988 Bonanza Creek L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 61 Chapter 4: Class Separation by Feature Illustration 4 Supervised Classification of 2x2 Averaged 20 March 1988 Bonanza Creek L-band Data (black=odd-bounce, gray=diffuse, white=even-bounce) Bonanza Creek - L-band - 13 March 1988 T 2.00 \ -1.50h T Even Diffuse Odd"'" Spin HH Re bn H H W * Re Im H V W » Re Im HHHV* Figure 4.9: Feature Distribution in Bonanza Creek L-band scene (13 March 1988) The Bonanza Creek results suggest the following comments: 62 Chapter 4: Class Separation by Feature V a l u e B o n a n z a C r e e k - L - b a n d - 20 M a r c h 1988 2J0 Spin HH W HV Rc 1m Rc Im Re toi H H W - H V W » HHHV* Figure 4.10: Feature Distribution in Bonanza Creek L-band scene (20 March 1988) 1. The span does well at discriminating the odd-bounce (river) scatterer from the diffuse and even-bounce terrain, due to the low specular backscatter from a relatively flat surface (for incidence angles above 20° [16, p. 25]). 2. The %1{HH VV*} normalized cross-product performs the even/odd separation well in both datasets, due to its characteristically high positive return from odd-bounce scatterers. 3. The W normalized cross product discriminates the odd-bounce (river) terrain from the even-bounce scatterers on both days, and is helpful in distinguishing the diffuse (forest) scatterers from the odd-bounce targets on the frozen day. 4. The HV normalized cross product distinguishes odd-bounce (river) from diffuse (forest) on both days, as vegetation has a strong depolarizing effect while water has a much weaker one. 5. The last five normalized cross products have standard deviations that are very large compared to the difference between their mean values in each class, and therefore provide poor class separation. 63 Chapter 4: Class Separation by Feature Fairbanks - L-band (2 Scenes) The training areas for the Fairbanks scene were chosen as the best homogenous 20x20 representatives of the even-bounce, diffuse, and odd-bounce scattering classes. Scenes from two different dates are considered. Although the two sets of data are not co-registered, the same areas of river/forest were used for training in each dataset. Note that both datasets are L-band only. Illustration 5 Supervised Classification of 2x2 Averaged 13 March 1988 Fairbanks L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified (very litde in this scene)) As can be seen from the mean values of span in Figure 4.11 (and Table A.20), the even-bounce scatterer (man-made structure) is much brighter in the 20 March 1988 data set This may have been caused by the aircraft's track aligning itself with the building more closely. It should also be noted that in the areas that overlap in the two datasets the data was collected at a higher incidence angle on 20 March 1988. 64 Chapter 4: Class Separation by Feature Illustration 6 Supervised Classification of 2x2 Averaged 20 March 1988 Fairbanks L-band Data (black=odd-bounce, gray=diffuse, white=even-bounce) Much the same pattern of discrimination is observed in the data from both dates, with the largest difference being an improvement in the performance of the HV feature in the 20 March 1988 data. The results from the Fairbanks scene suggest the following comments: 1. The span discriminates well between all three classes, with the even-bounce being extremely bright, the odd-bounce very dark, and the diffuse scatterers in between. The low class separations for class pairs including the even-bounce class are due to the extremely bright even-bounce returns (and standard deviations) observed in this scene. Despite the low class separabilities and poor prima facie results evident in Figures 4.11 and 4.12, the classes are well separated, as setting a high threshold on span easily identifies the even-bounce scatterers. For the span, the assumption of a Gaussian distribution within the even-bounce class appears not to hold. 65 Chapter 4: Class Separation by Feature F a i r b a n k s - L - b a n d - 13 M a r c h 1988 3.00 1.50-0.50 --0.50 — Span HH W HV Re Im Re Im H H W * H V W Re Im HHHV* Figure 4.11: Feature Distribution in Fairbanks L-band scene (13 March 1988) vdue F a i r b a n k s - L - b a n d - 20 M a r c h 1988 1.50-Figure 4.12: Feature Distribution in Fairbanks L-band scene (20 March 1988) 66 Chapter 4: Class Separation by Feature 2. The U{HH VV*} feature shows its strength in discriminating odd-bounce (river/runway) from even-bounce (buildings) and diffuse (forest) targets. Even-bounce targets produce a negative normalized U{HH VV*} return, while odd-bounce targets produce a high positive normalized ${HH VV*} return. 3. The HV feature distinguishes diffuse (forest) from odd-bounce (river/runway), as vegetation has a strong depolarizing effect while water/smooth surfaces have almost none. 4. The five last normalized cross products have standard deviations that are very large compared to the difference between their mean values, and therefore provide poor class separation. 67 Chapter 4: Class Separation by Feature Punta Cacao - P, L , and C bands There are three distinct classes within this scene. Although the area boundaries differ slighdy with frequency, at L-band, three areas are distinct after a van Zyl unsupervised EDO classification. It is noted that the scattering mechanism of the even-bounce terrain class in the Punta Cacao scene is presendy unknown to the author. It is worthy of note however, that the area is atypical of most even-bounce scatterers, in that it returns very little backscatter (less than even the odd-bounce clear-cut areas), though it is significandy brighter at C-band than in P and L-bands (see Figures 4.13, 4.14, and 4.15). The results from the Punta Cacao scene suggest the following comments: 1. The $t{HH VV*} feature shows its strength in discriminating odd-bounce (clear-cut) from even-bounce (unknown ground terrain) targets. It does best in P-band, but succeeds almost as well in L and C bands. The feature's characteristic high positive value for odd-bounce targets and negative value for even-bounce targets are evident in Figures 4.13, 4.14, and 4.15. 2. In P and L bands, the HV normalized cross product distinguishes diffuse (forest) from odd-bounce (clear-cut), as vegetation has a strong depolarizing effect while flat land is a less powerful depolarizer. As can be seen in Figure 4.15 however, the HV return from the clear cut is larger at C-band, decreasing the effective class separation. This effect is probably due to increased multipath backscatter due to a higher effective surface roughness at the shorter wavelength. 3. The effect noted for HV above is also evident in the W normalized cross products, as P and L-band returns are able to distinguish the forest (diffuse scatterer) from both the even and odd scattering terrain, while at C-band their returns are almost indistinguishable. 4. The four last normalized cross products have standard deviations that are large compared to the difference between their mean values. They therefore provide poor class separation. 5. In general, of the three frequencies studied, P-band features provide the best discrimination between forest (diffuse) and clear-cut (odd-bounce). 6. The polarization phase difference (PPD) succeeds in providing useful discrimination between the three terrain classes. In all three bands it does very well at discriminating even-bounce targets (PPD near 180°) from odd-bounce scatterers (PPD near 0°). 68 Chapter 4: Class Separation by Feature Illustration 7 Supervised Classification of 2x2 Averaged Punta Cacao L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) 69 Chapter 4: Class Separation by Feature Punta Cacao - P-band Span HH W HV Re Im Re Im Re Im H H W » H V W * HHHV* Figure 4.13: Feature Distribution in Punta Cacao P-band scene v«iue Punta Cacao - L-band ISO -1.00 T T T I 1 Even Difnue Odd"" Span HH W HV Re Im Re Im Re Im H H W » H V W ' HHHV* Figure 4.14: Feature Distribution in Punta Cacao L-band scene 70 Chapter 4: Class Separation by Feature P u n t a C a c a o - C - b a n d Span HH W HV Re lm Re Im Re Im H H W » H V W * HHHV* Figure 4.15: Feature Distribution in Punta Cacao C-band scene Mt. Shasta - P, L, and C bands This scene is of a forested area in northern California near ML Shasta collected by the AIRSAR in September 1989. The area contains coniferous tree stands dominated by ponderosa pine and white fir [13], and clear cuts. A van Zyl unsupervised classification revealed no even-bounce scatterers, so the most homogenous 20x20 boxes representative of three classes visible in an HV image were selected as training areas. No normalized L-band cross products discriminate well between the classes within this scene. The clear cut class is an odd-bounce scattering class while both tree classes are diffuse scattering classes, although the Treel class is "purer" in this respect than the Tree2 class. Lack of ground truth prevents a more detailed description of each of the three training areas. The L-band results suggest the following comments: 1. The $1{HH VV*} normalized cross product distinguishes well between clear cut and the first tree type. The clear-cut area is an odd-bounce scatterer resulting in a high positive value for $1{HH VV*} while the tree is a diffuse scatterer, resulting in a lower Qjut still positive) value for the cross product. 71 Chapter 4: Class Separation by Feature Illustration 8 Supervised Classification of 2x2 Averaged Mt. Shasta L-band Data (black=Tree2, light-gray=Treel, white=Clear-Cut, dark-gray=unclassified) 2. The HV normalized cross product provides some discrimination between the clear-cut (poor depolarizer) and the first tree type (relatively strong depolarizer). 3. The span discriminates well between the clear cut and "tree 1" terrain types, and also between the two tree types. The clear cut is dark Cow backscatter due to specular reflection), while the "tree 1" terrain class is relatively bright. The "tree 2" terrain class has an intermediate brightness at P and L-bands, and is comparable to the clear cut at C-band. A multifrequency comparison of the class separations seen in Figures 4.16,4.17, and 4.18 reveals that in P-band the span discriminates between all three classes, in L-band less well between two out of three class pairs, and in C-band less well between the same two class pairs. Also, for discriminating the 'Treel" terrain class from "Clear Cut", in P-band the U{HH VV*}, HV, and W features all 72 Chapter 4: Class Separation by Feature M t Shas ta - P-band Re lm mrw* Re lm H V W Figure 4.16: Feature Distribution in Mt. Shasta P-band scene M t . Shas ta - L - b a n d T T aoa TTO*1 Tniei Span Re Im H H W * Re Im H V W ' Re Im HHHV" Figure 4.17: Feature Distribution in Mt. Shasta L-band scene 73 Chapter 4: Class Separation by Feature M t . Shasta - C - b a n d Span HH W HV Re Im HHHV* Figure 4.18: Feature Distribution in Mt. Shasta C-band scene provide a class separation above unity, while in L-band only the ${HH VV*} and HV features discriminate so well, and in C-band only the R{HH VV*} feature. P-band is revealed to be the best wavelength for Forest/Clear Cut discrimination. This finding is consistent with the results from the Punta Cacao scene. 74 Chapter 4: Class Separation by Feature Traverse City - L-band The Traverse City scene is not completely calibrated, and was acquired with the original JPL airborne polarimetric SAR over central Michigan. Illustration 9 Supervised Classification of 2x2 Averaged Traverse City L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) The following comments are notable after considering the Traverse City scene. 1. The U{HH VV*} feature discriminates well between odd-bounce (lake, field) and even-bounce (dead trees) terrain, but not between odd-bounce classes (lake and field). 2. The W feature again works well in distinguishing water from land, due to water's Bragg scattering. Notably, the W feature provides good separation even when both land and water classes are odd-bounce (eg. lake and field). 75 Chapter 4: Class Separation by Feature Traverse City - L-band 0.50 Span HH W HV Re Im Re Im Rc lm H H W * H V W * HHHV* Figure 4.19: Feature Distribution in Traverse City L-band scene 3. The HV feature is good for discriminating forest (strong depolarizer) from lake (extremely weak depolarizer), but not much else. 76 Chapter 4: Class Separation by Feature Weeks Lake - P, L, and C bands The Weeks Lake scene was acquired by the AIRSAR over a forested area northwest of Victoria, British Columbia, Canada. The area is hilly forest timberland, with predominantly fir, and some hemlock mixed in places. Open clear-cuts recendy pared by forest companies fill about half of the scene. Tree stands range in age up to eighty years. The L-band scene was phase-calibrated using the lake as a zero PPD reference. At P and C bands, the histogram of PPD values within the Lake training area was uniform, and no calibration could be done. »\ •- '. -v-..... \J $%\ Illustration 10 Supervised Classification of 2x2 Averaged Weeks Lake L-band Data (black=lake, light-gray=forest, white=clear-cut, dark-gray=unclassified) The Weeks Lake scene results shown in Figures 4.20, 4.21, and 4.22 (as well as Tables A.34 and A.35) suggest the following comments: 1. In L and C bands, the W feature does well in distinguishing water from land classes, due to the water's Bragg scattering. At P-band, no such effect is evident, and the lake's normalized W return is actually less than that from the Clear-cut and Forest classes. Bragg scattering appears to drop off markedly between L-band (24cm) and P-band (68cm) wavelengths, probably due to light winds in the tree-protected small lake. Light winds imply water waves with a small period, and little Bragg scattering at low frequencies (eg. P-band). In the Flevoland scene, 77 Chapter 4: Class Separation by Feature W e e k s L a k e - P - b a n d 150 0.50 T dear Cut Pbicst Feature Spin HH W HV Re Im Re Im Re Im H H W * H V W HHHV* Figure 4.20: Feature Distribution in Weeks Lake P-band scene Value W e e k s L a k e - L - b a n d 1.00 0.50 -0J0J-..I T T T dear Cut Rarest Late'" Span HH W HV Re Im Re Im Re Im H H W - H V W ' HHHV* Figure 4.21: Feature Distribution in Weeks Lake L-band scene 78 Chapter 4: Class Separation by Feature W e e k s L a k e - C-band 3.00 — 0.50 T T If Ii' dear Cm Feature Span HH W HV Re Im Re Im Re Im HHW* HVW* HHHV* Figure 4.22: Feature Distribution in Weeks Lake C-band scene stronger winds would be expected on the more open ocean, causing longer period waves. Bragg scattering is strong at P-band in that scene (see the normalized W return in Figures 4.23, 4.24, and 4.25, and Table A.22). The HV feature distinguishes forest (strong depolarizer) from lake (extremely weak depolarizer) well in L and C bands. At P-band however, the HV normalized cross product varies little between the three terrain classes. The forest's lower HV return at P-band may be a result of less multipath scattering at the longer wavelength. This effect is also noticable in the Punta Cacao data. The Weeks Lake L-band data was only phase calibrated, and did not undergo cross-talk, channel gain balance, or absolute amplitude calibration. The P and C-band data was not even phase calibrated, as the histogram of polarization phase difference values within the lake training area had a near-uniform distribution. The lack of calibration may explain the abnormally high values of U{HV VV*} and $s{HV VV*} in the Lake class, and their higher than usual class separations. This may be caused by an SNR problem due to the very low return from the Lake. 7 9 Chapter 4: Class Separation by Feature 4.4.2.2 Agricultural Flevoland - P, L, and C bands The Flevoland scene was collected by the AIRSAR in 1989 over an agricultural area close to the ocean in the Netherlands. The training areas in the Flevoland scene were selected as the most homogenous 20x20 representatives of the diffuse, odd and even-bounce scattering classes (from the Van Zyl [60] unsupervised classification algorithm). Illustration 11 Supervised Classification of 2x2 Averaged Flevoland L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) The results from the Flevoland scene suggest the following comments: 1. At L and C bands the span provides discrimination of odd-bounce terrain from both diffuse and even-bounce scatterers, due to the low specular return from flat areas (for incidence angles 80 Chapter 4: Class Separation by Feature Value Flevoland - P-band Span HH W HV Re Im Re Im Re Im H H W * H V W * HHHV* Figure 4.23: Feature Distribution in Flevoland P-band scene V"Jue Flevoland - L-band ll'l Ii'"-Even Diffuse Odd Field Span H H W H V R e l m R e l m R e l m H H W * H V W * HHHV* Figure 4.24: Feature Distribution in Flevoland L-band scene 81 Chapter 4: Class Separation by Feature Value 150-100-Flevoland - C-band • Even Diffuse Odd Field 1.50-0.50 -0.50 Feature Span H H W H V R e l m R e l m R e l m HHW> H V W ' HHHV* Figure 4.25: Feature Distribution in Flevoland C-band scene above 20°). At P-band, the return is relatively higher from the odd-bounce (ocean) training area. Bragg scattering is also notably higher (visible in the greater normalized W return). The higher Bragg scattering return at P-band (due to the ocean wave size) is largely responsible for the overall higher span of the odd-bounce training area at P-band, and the subsequent poor separation between the odd and even-bounce classes provided by the span at that frequency. For all three frequencies, Bragg scattering produces a higher normalized W return in the odd-bounce (ocean) class than in the other classes. The variance of normalized W return is also low, especially at P and C-bands (see Figures 4.23 and 4.25). This result stands in contrast to the low P-band normalized W return in the lake in the Weeks Lake scene. The low Bragg scattering seen at P-band in that scene is most likely explained by a much lower level of wind on the tree-protected small lake than on the open ocean of Flevoland. The U{HH VV*} feature also shows its strength in discriminating odd-bounce (ocean, field) from diffuse (vegetation with multipath scatter) and even-bounce (vegetation with strong ground-trunk interaction term and man-made structures). The feature's characteristic high positive value 82 Chapter 4: Class Separation by Feature for odd-bounce targets and negative value for even-bounce targets is evident in Figures 4.23 and 4.25. The mean value of the feature is not negative in the even-bounce training area at P and C bands, as a different scattering mechanism seems to dominate at these frequencies. 4. As expected from the %1{HH VV*} results at L-band, the PPD does well in distinguishing even from odd-bounce targets. At L-band the mean PPD is close to 0° for odd-bounce scatterers, but closer to 180° for even-bounce terrain. 5. At P and C bands, the odd-bounce HV normalized cross product is distinctive from the diffuse and even-bounce returns due to the extremely low depolarization occuring bn the ocean. At L-band the normalized HV return is distinctively high for the diffuse class (due to the multipath backscatter), allowing it to be discriminated from the odd and even-bounce classes. The relatively lower normalized HV return at L-band is currently unexplained. 6. The linear depolarization ratio (LDR, HVIW) discriminates even or diffuse terrain from odd-bounce pixels in P and L bands, due to the very low HV return from flat odd-bounce scatterers. One might conclude from this that the even-bounce area has a lot of multipatning. 83 Chapter 4: Class Separation by Feature 4.4.2.3 Urban San Francisco - L-band This scene was one of the first to be acquired by the original JPL fully polarimetric SAR, and has been extensively studied [64]. Illustration 12 Supervised Classification of 2x2 Averaged San Francisco L-band Data (black=odd-bounce, light-gray=diffuse, white=even-bounce, dark-gray=unclassified) The results from the San Francisco scene suggest the following comments: 1. The U{HH VV*} feature shows its strength in discriminating odd-bounce (ocean) versus even-bounce (city) targets. 84 Chapter 4: Class Separation by Feature San Francisco - L-band 0.00 --OJO feature Span HH W HV Re Im Re Im Re Im H H W * H V W » HHHV* Figure 4.26: Feature Distribution in San Francisco L-band scene 2. The W feature discriminates water from land well, due to water's relatively high W to power ratio, attributable to Bragg scattering [64, p. 538]. 3. The HV feature distinguishes park (diffuse) from ocean (odd-bounce), as vegetation has a strong depolarizing effect while water has almost none. 4. The five last normalized cross products normally have standard deviations that are very large compared to their mean values. One deviates from this pattern in the San Francisco scene. The city (even-bounce) $t{HH HV*} cross product takes on a mean value larger than the sum of its standard deviation and that of either of the other classes, which leads to high class separation values. One proposed explanation for this behaviour (unseen in other scenes) is that it may be due to rows of streets with significant relief angled at 45° to azimuth. This hypothesis has yet to be fully investigated. 85 Chapter 4: Class Separation by Feature 4.4.2.4 Geological Pisgah - P, L, and C bands The Pisgah dataset was acquired over the Pisgah lava flow in California, and has been reported on previously in [21], and [56]. The Pisgah scene was affected by ground interference in both P and L-bands [56]. The L-band interference is of unknown origin, but interference at P-band is more common. Since it is at such a longer wavelength (68cm), it is susceptible to ground-based UHF interference [25, p. 14, p. 37] in populated areas. Illustration 13 Supervised Classification of 2x2 Averaged Pisgah L-band Data; black(I)=Phase I Lava, dark gray(III)=Phase III Lava, medium gray(II)=Phase II Lava, light gray(A)=Alluvial Surface, white(LB)=Lakebed (Playa), gray(seen in rectangular interference regions)=unclassified 86 Chapter 4: Class Separation by Feature Pisgah - P-band -0.50 Spm HH V V HV Re Im Re lm Re Im HHVV* H V V V * HHHV* Figure 4.27: Feature Distribution in Pisgah P-band scene Pisgah - L-band Spm HH V V HV Re Re Im Re Im H V W * HHHV* Figure 4.28: Feature Distribution in Pisgah L-band scene 87 Chapter 4: Class Separation by Feature Value Pisgah - C-band 2.00 Feature Span HH V V HV Re Im Re Im Re Im H H V V * H V V V * HHHV* Figure 4.29: Feature Distribution in Pisgah C-band scene Class Span P-band L-band C-band Phase I Lava .259 .238 .524 Phase II Lava 1.75 1.06 1.13 Phase III Lava .571 .442 .622 Alluvial Surface .021 .016 .069 Dry Lakebed (Playa) .026 .0084 .042 P: s„=5xl0 7 ; L: s„=2xl0 6 ; C:su=\.0 Table 4.8 Span of Classes in Pisgah Scene for P, L, and C-bands The L-band results from the Pisgah scene suggest the following comments: 1. The L-band V V normalized cross-product was able to discriminate dry lakebed from phase II or III lava (with a class separation above one). 88 Chapter 4: Class Separation by Feature 2. The L-band HV normalized cross-product could also distinguish dry lakebed from phase II lava, due to the low depolarizing effect of the smooth lakebed contrasted with the stronger depolarizing effect of the relatively rougher phase II lava terrain. 3. No other L-band normalized cross products provided class separations above unity for any of the other class pairs in this scene. Unnormalized feature vectors are reported to have more success in this scene [56, p. 351]. 4. The L-band span provides good discrimination (class separabilities were approximately 2.8) of phase II lava from dry lakebed and alluvial surfaces. The dry lakebed and alluvial surface are both relatively smooth flat surfaces, resulting in specular scattering and negligible backscatter (for incidence angles above 20° [16, p. 25]). The phase II lava in contrast is relatively rough (for L-band), resulting in approximately 100 times as much backscatter. For this scene, the C-band span was approximately six orders of magnitude brighter than the return in either L or P bands (see Table 4.8). Such a large difference in brightness cannot be due to relative roughness changes, but might have been caused by a miscalibration or misprocessing of one or more of the images. Class HH W Mean S.D. Mean S.D. Phase I Lava 1.74 .36 2.12 .37 Phase II Lava .57 .15 3.32 .17 Phase III Lava 1.22 .22 2.70 .23 Alluvial Surface 2.55 .27 1.22 .24 Dry Lakebed (Playa) 2.63 .41 1.10 .39 Table 4.9 Mean and Standard Deviation of P-band Normalized HH and VV Cross Products for Classes in Pisgah Scene The P-band HH and W normalized cross products discriminate well between almost all five classes (class separations above two). The other normalized cross products (at all three frequencies) fail to provide separabilities greater than unity (with the exception of R{HH VV*} discriminating phase II from phase III lava). 89 Chapter 4: Class Separation by Feature The means and standard deviations of the P-band HH and W normalized cross-products are shown in Table 4.9. The smooth areas (alluvial surface and dry lakebed) have a relatively strong normalized HH return, while the the rougher areas (lavas) have a relatively strong normalized V V return. The normalized HH and W cross products are therefore able to discriminate between all class pairs except alluvial surface vs. playa and phase I lava vs. phase III lava, which are too alike in roughness to be distinguished from one another. 4.4.3 Summary of Feature Performance Class separations for each scene were computed between all classes shown in Table 4.7. The class separations were averaged over class pairs and scenes for each feature, to give a measure of the overall separation ability of each feature. The class separation averages are displayed in Figure 4.30 and Table 4.10. Note that class separabilities vary considerably from scene-type to scene-type, and from class-set definition to class-set definition. This variation will be discussed later. The following sections discuss the results shown in Table 4.10 and Figure 4.30. 3ft{HH V V * } For the scenes studied, the U{HH VV*} feature is the best discriminator overall. Its success is based on the fact that its sign can be used to distinguish between odd and even bounce scattering mechanisms. Odd-bounce terrain classes have a PPD of near zero [64], from equation (124) we see that the sign of U{HH VV*} must then be positive. Even-bounce targets have a PPD near 180°; the sign of $t{HH VV*} is therefore negative. Span The span also works well on average, notably better than in the scenes studied in [8]. In the Pisgah and Mt. Shasta scenes, we see that an unnormalized feature set may be more appropriate [56], as the span offers the best discrimination. HH, HV, VV The normalized radar channel amplitudes (HH, HV, and W) are also good discrim-inators, while the remaining features are generally not useful, with rare exceptions. The exceptions are discussed in the sections dealing with each individual scene. 9 0 Chapter 4: Class Separation by Feature Class Separation Average Class Separation 2.50 2.00 1.50 1.00 0.50 0.00 Bonanza Creek 13.03 bonanza Creek 20.03 Fairbanks i3.03 Fairbanks 20.03 Flevoland Punta Cacao San Francisco Traverse City ML Shasta Weeks Lake ; i : Span VV HV Re Im HHVV* Re Im H V W * Re Im HHHV* Feature Figure 4.30: Average Class Separations for L-band Scenes HV The normalized HV cross product is often able to discriminate between water and vegetation, as the vegetation's multipath backscatter has a strong depolarizing effect, while water is an extremely weak depolarizer. VV The normalized W cross product can often distinguish land terrain from water. Water produces a relatively high normalized W return, due to Bragg scattering [64, p. 538]. The theoretical WIHH ratio for pure Bragg scattering [16, pp. 14-15] is: cos( O~HH = (e(l + sin20) - s i n 2 0 ) -50+ y/t - sin29 e cos 9 + y/e — sin2 9 (126) where e is the water's dielectric constant at the frequency of interest, and 9 is the incidence angle. 91 Chapter 4: Class Separation by Feature Scene Feature span HH W HV HH W* HVW* HHHV* ft & Bonanza Creek1 1.74 .50 .76 1.49 1.90 .19 .12 .10 .14 .31 Bonanza Creek2 1.55 .72 1.23 1.03 1.77 .67 .16 .06 .02 .05 Fairbanks1 .74 .74 .72 .95 1.41 .28 .13 .19 .26 .07 Fairbanks2 .64 .92 .69 1.25 1.43 .43 .14 .13 .26 .01 Punta Cacao 1.39 .86 1.53 1.20 1.92 .89 .13 .38 .40 .33 M L Shasta* 1.38 .14 .40 .76 .95 .20 .03 .06 .11 .05 Traverse City 1.62 1.25 1.37 1.01 1.06 .41 .14 .31 .33 .18 Weeks Lake* 1.74 .85 1.88 1.28 1.56 .20 .66 .59 .19 .13 Mean Forested 1.42 .69 1.04 .97 1.27 .35 .25 .26 .24 .14 Flevoland 1.07 1.47 1.99 1.59 2.70 .46 .12 .30 .17 .20 Pisgah* 1.43 .49 .65 .44 .32 .22 .12 .08 .08 .09 San Francisco .83 1.48 2.09 1.62 2.24 .19 .50 .32 .98 .29 Mean EDO 1.20 .99 1.30 1.27 1.80 .44 .18 .23 .32 .18 Mean non-EDO 1.51 .49 .98 .83 .94 .21 .27 .24 .13 .09 Mean AU 1.28 .86 1.21 1.15 1.57 .38 .20 .23 .27 .16 ! 1 3 March 1988; 2 2 0 March 1988; *Non Even/Diffuse/Odd (EDO) Training Set Table 4.10 Average Class Separations of L-band Features (All Scenes) Frequency In general, P-band features appear to provide the best discrimination between forest (diffuse) and clear-cut (odd-bounce) terrain. This was confirmed in the Punta Cacao, and Mt. Shasta, and Weeks Lake scenes. The reason for P-band's better performance is probably that the clear-cut is smooth compared to that wavelength, while it is rougher (and hence more similar to diffuse multipath scatterers) at L and C bands. P-band HH and W normalized cross-products were also able to discriminate many of the classes in the geological Pisgah scene where features from all other frequencies failed. The class separations calculated from the C-band data are often the worst of the three frequencies available. This may be partially due to the band's poorer calibration [25], but the principal cause is probably that many terrain surfaces in the scenes have roughnesses greater than 6.3 cm (the C-band 92 Chapter 4: Class Separation by Feature wavelength), resulting in poorer contrast than in P or L bands, where a greater variety of relative roughnesses manifest themselves. The C-band radar also had by far the lowest SNR. The following exceptions were noted. In the Pisgah scene, the C-band span discriminates between the five classes better than the span from P or L-band (see Table 4.8) due to a better relevant roughness sensitivity. In the Weeks Lake scene, the C-band span discriminates better between lake and clear-cut or forest. In these cases, C-band's advantage is due to the higher backscatter from the clear-cut class, and the lower variation in brightness within the forest class. 93 Chapter 5: Feature Utility in Classification Chapter 5 Feature Utility in Classification 5.1 Introduction Although class separation measures are useful for determining the ability of single features to discriminate between classes, they ignore the covariance between features. The amount of new information that a feature adds to that already provided by the others can be more important than what it supplies in isolation. This section introduces a measure of the information that each feature adds. If one assumes that a classification performed with the full feature set is "correct", then counting the number of pixels that change classification between that "reference" run and one using an incomplete feature set gives an indication of the importance (for the purposes of classification) of the features missing in the incomplete feature set. 5.1.1 Confusion Matrix A confusion matrix is constructed by counting the number of pixels that are class Cj in the reference run, and class Ci in the run using an incomplete feature set. Totals are computed for all d and Cj and arranged in a square matrix with the row indicating the classification in the reference run, and the column the classification in the run using the limited feature set. The diagonal elements contain the number of pixels that do not change classifications between runs. 5.1.2 Relative Confusion The total confusion is computed by adding the non-diagonal elements, with the exception of transitions to and from the unclassified "class". The relative confusion is then the total confusion expressed as a percentage of the total number of pixels considered. 94 Chapter 5: Feature Utility in Classification 5.2 Results 5.2.1 One Feature Ignored The relative confusion resulting from removal of a single feature from the full feature set was computed for each feature in each scene. The results are shown in Figure 5.31 and Table 5.11. Two by two averaging was used throughout. Average Relative Confusion - Single Feature Ignored RC 35.00 30.00 -25.00 -20.00 -15.00-10.00 -5.00 0.00 till Bonanza Creek 13.03 Bonanza Creek 20.03 Fairbanks i3.03 Fairbanks 20.03 Flevoland Punta Cacao San Francisco Traverse City Mt. Shasta Weeks Lake \h\\ i: i.n hi \1 Span HH VV HV Re Im HHVV* Re Im H V W * Re Im HHHV* Feature Figure 5.31: Average Relative Confusion for Single Feature Ignored (Reference is L-band Full Feature Set Classification) 95 Chapter 5: Feature Utility in Classification Scene Relative Confusion when Single Feature Ignored span HH VV HV HH W HVW HHHV ft ft ft Bonanza Creek1 15.1 5.4 3.7 11.0 13.1 1.2 .4 .5 .7 1.0 Bonanza Creek2 10.9 3.1 7.0 7.3 12.2 3.6 1.1 .4 .1 .3 Fairbanks1 .9 4.2 4.6 10.8 13.5 2.0 1.5 1.1 1.6 .6 Fairbanks2 .5 4.3 4.3 10.8 12.5 1.9 .4 .9 1.5 .2 Punta Cacao 8.6 1.4 2.8 5.3 4.0 1.1 .0 1.1 .8 .4 Mt Shasta* 36.6 1.2 3.1 6.4 10.5 1.4 .4 1.0 1.2 .9 Traverse City 5.7 6.9 5.5 9.5 13.3 4.1 1.1 1.2 2.3 1.8 Weeks Lake* 23.2 1.9 5.6 9.1 9.5 2.4 1.0 2.7 2.0 1.9 Mean Forested 12.7 3.6 4.6 8.8 11.1 2.2 .74 1.1 1.3 .89 Flevoland 2.1 2.1 7.6 11.7 14.6 1.6 .3 1.0 .6 .3 Pisgah* 34.2 12.2 13.8 9.6 12.2 11.9 7.4 3.3 2.7 2.8 San Francisco 1.6 3.1 1.2 6.2 8.5 .1 2.2 .5 5.6 1.3 Mean EDO 5.7 3.8 4.6 9.1 11.5 2.0 .9 .8 1.7 .7 Mean non-EDO 31.3 5.1 7.5 8.4 10.7 5.2 2.9 2.3 2.0 1.9 Mean AU 12.7 4.2 5.4 8.9 113 2.8 1.4 1.2 1.7 1.0 x\3 March 1988; 220 March 1988; *Non Even/Diffuse/Odd (EDO) Training Set Table 5.11 Relative Confusion for Single Normalized Cross-Products Ignored with L-band Full Feature Set as Reference Note how the presence or absence of HV HV", $1{HH VV*}, or the span can strongly influence classification decisions. Knowledge of only the sign of the U{HH VV*} normalized cross product is often enough (for non volume scatterers) to determine that the polarization phase difference (PPD) is either in the neighbourhood of 0° or 180°, indicating odd or even-bounce scatterers respectively. Features that produce a low RC in Table 5.11 relative to a high class separation (in Table 4.10) are correlated with other features. They do discriminate between different classes, but add little information useful for improving the classifier to the sum of information already provided by other features. 96 Chapter 5: Feature Utility in Classification Scene Relative Confusion when Single Feature Ignored span HH VV HV HH W HVW HHHV » » Flevoland 8.6 6.8 7.9 8.9 16.9 4.9 .6 1.2 .6 1.0 Pisgah* 30.2 12.9 13.9 5.5 7.3 2.7 1.7 1.1 1.6 1.9 Punta Cacao 2.7 .5 .7 2.3 1.9 .3 .1 .2 .2 .2 M L Shasta* 29.5 1.0 2.0 4.1 6.3 .6 .8 .8 .7 .6 Weeks Lake* 30.2 2.5 3.4 4.2 4.5 1.6 1.0 2.5 1.6 .9 Mean Forested 20.8 1.3 2.0 3.5 4.2 .83 .63 1.2 .83 .57 Mean EDO 5.7 3.7 4.3 5.6 9.4 2.6 .4 .7 .4 .6 Mean non-EDO 30.0 5.5 6.4 4.6 6.0 1.6 1.2 1.5 1.3 1.1 Mean AU 20.2 4.7 5.6 5.0 7.4 2.0 .8 1.2 .9 .9 *Non Even/Diffuse/Odd (EDO) Training Set Table 5.12 Relative Confusion for Single Normalized Cross-Products Ignored with P, L , and C-band Full Feature Set as Reference Table 5.12 shows the relative confusion results when the reference scene is a classification using the full P, L, and C-band feature set, rather than just the L-band features. The full feature set classification uses all 30 features, while each "single feature ignored" run uses 27, as the single feature is ignored at all three frequencies simultaneously. The three-frequency reference might be judged to be a more "correct" classification, as its decisions are less "noisy" [56], and make use of more information. One should keep in mind that in this case, the relative confusion results apply to each feature averaged across three frequencies. 5.2.2 One Feature Used In order to evaluate how much relevant classification information each feature alone possessed, the relative confusion resulting from the comparison of a classification using the full feature set with a classification using only a single feature was computed. Table 5.13 shows the relative confusion of each single-feature classification when compared against a baseline classification that used all ten L-band features. Note that RC values are often very high, as single feature classification is 97 Chapter 5: Feature Utility in Classification frequently unable to distinguish between terrain classes. The table illustrates the percentage of the scene's pixels that change classification when 9 of the feature vector's components are ignored and only one is retained. Scene Relative Confusion when Single Feature Used span HH VV HV HH W HVW HHHV ft ft ft Bonanza Creek1 34.0 47.4 45.5 37.6 26.5 67.7 67.0 53.2 66.7 62.2 Bonanza Creek2 38.0 49.8 31.1 45.9 23.1 52.4 52.1 51.8 50.5 72.6 Fairbanks1 52.3 42.9 46.1 37.0 21.4 65.9 65.1 65.3 62.9 48.0 Fairbanks2 32.9 37.2 50.1 25.3 19.1 46.2 49.4 77.0 71.1 41.8 Punta Cacao 14.9 46.1 20.1 15.9 23.1 43.3 70.3 41.4 42.3 53.9 M L Shasta* 19.8 62.1 48.9 45.0 39.2 70.3 51.6 49.2 62.2 49.0 Traverse City 54.0 42.3 30.5 22.4 39.5 40.2 40.4 38.2 59.4 43.5 Weeks Lake* 37.8 49.4 33.6 33.9 33.7 46.1 58.8 55.7 53.0 46.4 Mean Forested 35.5 47.2 38.2 32.9 28.2 54.0 56.8 54.0 58.5 52.2 Pisgah* 50.3 67.9 60.3 63.8 67.8 73.1 71.3 73.1 72.7 80.0 Flevoland 44.8 43.4 31.9 59.5 22.0 51.1 72.8 68.1 73.1 71.1 San Francisco 40.1 38.8 26.9 21.7 16.7 48.5 47.2 45.7 44.7 49.3 Mean EDO 38.9 43.5 35.3 33.2 23.9 51.9 58.0 55.1 58.8 55.3 Mean non-EDO 36.0 59.8 47.6 47.6 46.9 63.2 60.6 59.3 62.6 58.5 Mean AU 38.1 47.9 38.6 37.1 30.2 55.0 58.7 56.2 59.9 56.2 '13 March 1988; 220 March 1988; *Non Even/Diffuse/Odd (EDO) Training Set Table 5.13 Relative Confusion for Single Normalized Cross-Products Used with Full L-band Feature Set as Reference As the accuracy of a classification using only a single feature is not high, these results should not be taken too seriously. They serve only to illustrate that the $t{HH VV*} and HV features provide better discrimination than the other single features. 98 Chapter 5: Feature Utility in Classification The following observations are listed for the record: 1. Results are impressive for the U{HH VV*} feature in the Fairbanks, Bonanza Creek, Flevoland, Punta Cacao, and San Francisco datasets, with only approximately 20% of the pixels changing classification. These three scenes have even, odd, and diffuse scattering class definitions. The poor results in the Pisgah scene are due to the fact that all five classes in that scene are odd-bounce scatterers, reducing the effectiveness of the U{HH VV*} feature's capability to differentiate odd-bounce from non-odd-bounce scatterers. 2. The HV feature does well in the Punta Cacao, Traverse City, and San Francisco scenes. In San Francisco and Traverse City, the water class has a distintively low HV return, while the diffuse (park/forest) class has a distinctively high one, and the even-bounce (city/dead trees) class is in between. All have sufficiendy low standard deviations that the HV feature can do reasonable discrimination on its own. 3. The span feature exhibits excellent maintenance of the "correct" classification in the Punta Cacao and Mt. Shasta scenes. In the Mt. Shasta scene, the classes were chosen based on HV brightness so it is not surprising that a brightness measure discriminates the three classes well. In the Punta Cacao scene, the forest is bright, the odd-bounce dark, and the even-bounce mysteriously darker still, allowing the span to discriminate between the three on its own. 4. The W feature maintains a close-to-"correct" classification single-handedly in the Punta Cacao, and San Francisco scenes, for reasons similar to those oudined above. As with the "single feature ignored" relative confusion runs, multifrequency "single feature used" relative confusion results were also computed. Results are less noisy, but apply to each feature "averaged across the three frequencies". With reference to Table 5.14: 1. In the Punta Cacao scene, the span, as well as the VV VV*, HV HV*, and U{HH VV*} normalized cross products are extremely good at maintaining the "correct" combined PLC full feature set reference classification when used in isolation. 99 Chapter 5: Feature Utility in Classification 2. In the ML Shasta scene, the span also does a good job of maintaining the reference PLC full feature classification. Scene Relative Confusion when Single Feature Used span HH V V HV HH W HVW HHHV » & Flevoland 64.2 37.1 28.3 45.7 23.8 49.1 65.7 65.0 70.5 63.5 Pisgah 55.6 45.2 37.3 59.6 69.7 73.4 75.2 79.6 74.9 75.3 Punta Cacao 9.5 40.9 10.8 6.7 10.6 38.2 52.7 49.0 47.3 64.8 Mt. Shasta 14.7 54.7 39.2 36.2 32.3 60.4 63.0 53.3 56.0 57.8 Weeks Lake 31.8 41.6 35.7 39.2 35.4 58.4 49.8 46.3 48.2 58.1 Mean Forested 18.7 45.7 28.6 27.4 26.1 52.3 55.2 49.5 50.5 60.2 Mean EDO 36.9 39.0 20.0 26.2 17.2 43.7 59.2 57.0 58.9 64.2 Mean non-EDO 34.0 47.2 37.4 45.0 45.8 64.1 62.7 59.7 59.7 63.7 Mean All 35.2 43.9 30.3 37.5 34.4 55.9 61.3 58.6 59.4 63.9 Table 5.14 Relative Confusion for Single Normalized Cross-Products Used with Full P, L , and C-band Feature Set as Reference 5.2.3 Features Dropped in Reverse Rank Order In order to determine whether or not the classification differences resulting when sets of features were dropped were due to classification mistakes or noise, the following study was performed. Increasing numbers of features were dropped from the classifier, and the resulting classification maps were compared to a full feature set classification to determine if the classification changes were (a) randomly distributed, indicating that the changes were predominandy due to noise, or (b) systematically distributed, indicating that real classification mistakes were occurring. Comparisons were done both visually (contrasting the classification maps), and via the relative confusion resulting from each feature subset. An example of the classifications resulting as increasing numbers of features are dropped is shown in Illustration 14. Red pixels are classified as clear-cut, green as forest, and blue as water. Black pixels are assigned to the "unclassified" class. The intensity of each classified pixel is proportional to the magnitude of the HH return within its resolution element. 100 Illustration 14 Sequence of Features Dropped (9 colour images) in Weeks Lake, BC Scene (23 March 1988). Features were dropped in reverse rank order of Table 5.15. The last feature not dropped labels each image. Chapter 5: Feature Utility in Classification Chapter 5: Feature Utility in Classification The feature labelling each classification image denotes the lowest ranked feature present in the feature subset responsible for the given classification map. Note that no systematic changes are apparent until the the W feature is dropped; the lower-ranked features do not appear to have a systematic influence on classification results for the chosen classes in this scene. A qualitative visual comparison of Illustrations 14 and 15 provides some evidence that dropping the four bottom-ranked features may actually improve classification accuracy. Figure 5.32 illustrates the relative confusion results for classifiers using progressively fewer features. For a classification using j features, RCorig was computed with respect to the full 10 feature set. Figure 5.32 shows the upward trend in RCorig as more features are dropped. Features were dropped in the feature utility rank order shown in Table 5.15. The features labelled on the abcissa of Figure 5.32 refer to the feature dropped in that run. Calibrated EDO Scenes The results indicate that, for calibrated EDO scenes, no sizeable (>5%) changes in pixel classifications occurred until the sixth feature (HH) was dropped. That is, the set of features {span, &{HH VV*}, HV HV*, VV VV*, HH HH*} manage to reproduce the full-feature classification with at least 95% accuracy. In general, RC increases monotonically as more features are dropped, indicating that no feature is significantly more important for classification than those with higher rank. All Scenes Results from incompletely calibrated or non-EDO scenes vary more widely, as can be seen in Figure 5.33. An examination of Figure 5.33 reveals the following: 1. Figures 5.32 and 5.33 show that the results from all scenes but Pisgah and Traverse City indicate that the [${HH VV*}, HV HV*, VV VV*, span] feature subset is enough to closely approximate a full-feature classification. 2. The generally high relative confusion in the Pisgah scene is probably due to the higher number of classes used (5 rather than 3) and the poor performance of normalized features in the scene. 103 Chapter 5: Feature Utility in Classification RC 60.00 50.00 40.00 30.00 20.00 10.00 0.00 Relat ive Confus ion — Sequence of Features D r o p p e d Bonanza Creek 13.03 Bonanza Creek 20.03 Fairbaiikl'l3"03 Fairbanks 20.03 Flevoland Punta Cacao San Francisco Traverse City Feature Ihhhv Dwvv Rhvvv Rhhhv ihhvv H H H V Rhhw Figure 5.32: Relative Confusion when Features Dropped in Rank Order in EDO Class Scenes - Full Feature Reference In the incompletely calibrated Traverse City scene, relative confusion is actually higher for the three-member {%t{HH VV*}, HV HV*, span} feature subset than for the two-member [?fc{HH VV*}, HV HV*} subset. The addition of span decreases the classification accuracy. This is the only case where such non-monotonic behaviour is observed. In the incompletely calibrated San Francisco scene, the loss of the $t{HH HV*} feature, with its relatively high class separation for this scene, significantly increases relative confusion The feature is relatively unimportant in the other scenes (ranking fourth from the bottom). 104 Chapter 5: Feature Utility in Classification Relat ive Con fus ion — Sequence of Features D r o p p e d RC 60.00 50.00 40.00 30.00 20.00 10.00 0.00 Dihhv Ihvw Rhvw Rhhhv Dihw HH W HV Rhhw Figure 5.33: Relative Confusion when Features Dropped in Rank Order - Full Feature Reference 5.3 Feature Ranking 5.3.1 Introduction Based on the overall class separation results, together with the relative confusion experiments in the previous sections, the ten features were "ranked" in a feature utility order. As the "single feature ignored" experiment revealed the impact of each feature on classification accuracy, features were ranked based on the relative confusion caused when the feature was removed from a full (10 feature) classification, close ties being broken by the class separation. For consistency, only L-band data was used. 1 0 5 Bonanza Creek 13.03 Bonanza Creek 20?03 Fairbanks 13"03 Fairbanks 20.03 Flevoland Pisgah Punta Cacao San Francisco Mt. Shasta Traverse City Weeks Lake Chapter 5: Feature Utility in Classification A ranking of the ten feature vector elements based on the above criterion is shown in Table 5.15 for the average of all scenes. The average L-band "single feature ignored" relative confusion results are shown together with the average class separations for each feature. The averages were applied over all scenes with EDO (even/diffuse/odd) training sets. The following points should be kept in mind: 1. The rank of the span is not important when evaluating the utility of the normalized polarimetric features, although it useful when evaluating the relative advantage over conventional radars that the extra features returned by a polarimetric radar provide. Al l polarimetric radars provide brightness measurement for classification purposes. The feature utility ranking is important insofar as it (a) illustrates the order in which features may be dropped (during data compression), and (b) calls attention to those information-laden features that will be useful in constructing improved classifiers. 2. The polarimetric data should be well calibrated for the ranking to be valid. For example, if the data is not phase calibrated, some information will migrate from the %t{HH VV*} feature to the %{HH VV*} feature. Rank Feature Mean RC Mean Class Separation 1 span 12.7 1.28 2 ?&{HH VV*} 11.3 1.57 3 HV 8.9 1.15 4 W 5.4 1.21 5 HH 4.2 .86 6 %{HH VV*} 2.8 .38 7 &{HH HV*} 1.7 .27 8 ft{HV VV*} 1.4 .20 9 %{HV VV*} 1.2 .23 10 %{HH HV*} 1.0 .16 Table 5.15 Basis for Feature Utility Ranking 106 Chapter 5: Feature Utility in Classification 5.3.2 Varying Scene-types Different scene-types produce slighdy different rankings, as can be seen in Table 5.16. The table shows that for a variety of scene-types, the discriminatory information resides mainly in the {&{HH VV*}, HV HV*, span, VV VV*, HH HH*} feature subset Investigation of partially polarimetric radars that acquire most of that set might therefore be warranted. Three such radars are investigated in the next chapter. Rank All EDO non-EDO Forest Agricultural (Flevoland) Comments 1 span %t{HH VV*} span span $t{HH VV*} Most distinctive 2 %t{HH VV*} HV •$t{HH VV*} %{HH VV*} HV 3 HV span HV HV W Provide some useful separation 4 W W W W span 5 HH HH HH HH HH 6 3{HH VV*} Q{HH VV*} ^{HH VV*} %{HH VV*} %{HH VV*} Mainly noise 7 $t{HH HV*} ${HH HV*} $t{HV VV*} $t{HH HV*} Q{HV VV*} 8 tH{HV VV*} %{HV VV*} Q{HV VV*} %{HV VV*} $t{HH HV*} 9 ${HV VV*} $t{HV VV*} $t{HH HV*} %{HH HV*} ^{HV VV*} 10 %{HH HV*} %{HH HV*} %{HH HV*} $t{HV VV*} %{HH HV*} Table 5.16 Feature Utility Rankings for various scene types Calibrated EDO vs. Uncalibrated EDO In comparison with the uncalibrated San Francisco and Traverse City, the calibrated EDO scenes behave more uniformly. The following differences are apparent. 1. Many of the calibrated EDO scenes attribute greater importance to the span. 2. For the two uncalibrated scenes, the results of this study do not differ widely from those of [8] (although a different definition of class separation, different scaling, and smaller training areas were used). The extra eleven scenes (nine forestry, one agricultural, and one geological) show the low value of span evident in the San Francisco scene to be the anomaly that it is. 107 Chapter 6: Simplified Radar Systems Chapter 6 Simplified Radar Systems Polarimetric radars are different than conventional single-channel radars in many ways [8]. A polarimetric radar requires more complex hardware, including waveguide switches, improved timing control, extra power, a dual polarized antenna, and a second receiver. Calibration is also more difficult, and data collection, transmission, processing, and archiving all require more resources. This chapter studies the classification accuracy attained using simplified radar systems to gauge how many of the advantages of fully polarimetric radars can be obtained from simplified partially-polarimetric radars. 6.1 Introduction Fully polarimetric radars pay a price in complexity and data storage/processing requirements for the extra radar channels that they make available. Fully polarimetric radars return four channels (HH, HV, VH, W), while conventional single-channel radars return only one. Dual-channel radars are intermediate between conventional single-channel and fully polarimetric radars. Only one of the two most distinctive features ($t{HH VV} and HV HV*) can be included in a two-channel radar simultaneously, as three channels are required to obtain both. A copolarized radar is the simplest radar that can acquire the U{HH VV} feature, and the single transmit radar is the simplest one that includes the HV HV* feature. The following sections study these dual-channel radars, together with a three-channel amplitude-only radar. 6.1.1 Copolarized Radar A CO-POL radar returns only the HH and W channels. Only one receiver is required, and less transmitted power is needed for a given SNR, as the comparatively weak HV channel is omitted. The data transmission, processing, and storage requirements are halved compared to the fully polarimetric radar. In comparison to a fully polarimetric SAR (see equation (121)), a CO-POL radar returns the following features: 108 Chapter 6: Simplified Radar Systems /F0\ / span' \ FT HH HH* F2 = VV VV* FA ${HH VV*} \FJ \%{HH VV*} J (127) The following points should be noted: 1. The definition of span is modified in this case to half the sum of the unnormalized HH and W channels recognizing that the HV HV* feature is not available. 2. The polarization phase difference (PPD) is preserved, as it can be computed from the tft{HH VV*} and %{HH VV*} features. 6.1.2 Amplitude Radar An ampltitude (AMPL) radar measures the amplitude of the HH, HV, VH, and W channels, but not the phase. No inter-channel phase calibration, and less storage are its principal advantages. Such a radar collects the following features: Fi F2 \F3J I span \ HH HH* VV VV* \HV HV* J (128) Note that the important HV HV* feature is present, but that the $1{HH VV*} feature is missing. 6.1.3 Single Transmit Radars Single transmit radars transmit on only a single polarization, but receive on two polarizations. They require no high-speed, high-power waveguide switch, less transmit power and less complex timing control than a fully polarimetric radar. Two receivers are needed, but there is no receive-path waveguide switch, and the receive data does not need to be de-multiplexed. Transmission, processing, and archive storage requirements are halved. 109 Chapter 6: Simplified Radar Systems A horizontal transmit radar returns the following features: /F0\ / span" \ Fi HH HH* F3 = HV HV* Fs ${HH HV*} \%{HH HV*} ) while a vertical transmit radar returns: ( F ° \ F2 F3 Fe \Fr/ The following points should be noted: / span'" \ VV VV* HV HV* ${HV VV*} \%{HV VV*} ) (129) (130) 1. The definition of span in both equations is modified to use only those of the HH, HV, VH, and W features that are available. 2. Compared to the CO-POL radar, the HV HV* feature is available, but the co-polarized phase &{HH VV*} and %{HH VV*}) and one of the [VV VV*, HH HH*) features are not. 6.2 Results For simplified radars, features involving HV have a higher standard deviation than they would coming from fully polarimetric radars, as only one cross-pol channel is available to average. The analysis here ignores this difference, although it does take account of the modified definition of span necessary for each simplified radar. The classifications that result from three of the simplified radar subsets are visually juxtaposed in Illustration 16. Red pixels are classified as even-bounce, green as forest (diffuse), blue as ocean, and yellow as field. White pixels are unclassified (note that they are not speckle). The intensity 110 Illustration 16 Visual Classification Comparison (colour) of Simplified Radar Systems — Flevoland, NL Scene (16 August 1989) Chapter 6: Simplified Radar Systems of each pixel (with the exception of those not classified) is governed by the HH return within each resolution element. Structural differences between the classification resulting from a full feature set and that from each subset are minimal. 6.2.1 Relative Confusion Relative confusion statistics were calculated between the the fully polarimetric classification reference, and the classifications resulting from the simplified radar datasets. Figure 6.34 shows the relative confusion (in comparison to fully polarimetric-based classifica-tions) computed from classifications based on the subsets of the data that would be returned from four different simplified radars. Figure 6.35 shows the relative confusion results for the EDO scenes, grouped by radar system. The CO-POL radar performs best in most cases. 6.2.2 Discussion Of the four simplified radars considered, the CO-POL performed the best on average. Of the simplified radars, it alone returns the $t{HH VV*} and $s{HH VV*} features, and hence provides the polarization phase difference (PPD) for analysis. The distinctiveness of the PPD has been discussed in earlier chapters and by other authors as well [2], [18], [60], [57]. For the EDO scenes, the CO-POL radar always performed best. Two of the EDO scenes were studied with four classes (even, diffuse, odd (water), odd (field)). In both cases, the CO-POL radar remained the best choice (see Table 6.17), due to it having both the U{HH VV*} and W features. Within two of the non-EDO scenes, (Weeks Lake and Mt. Shasta), the AMPL radar performed better, due to its use of the HV HV* feature. However, in these non-EDO scenes where classification is dominated by span, the better performance of the AMPL radar might also be due to its unchanged definition of span. There is no question that the fully polarimetric radar enables the best classifications. If this is the top priority, then the extra complexity, data volume, and cost associated with fully polarimetric 112 Chapter 6: Simplified Radar Systems RC Simple Radars — Relative Confusion 30.00 25.00 20.00 15.00 — 10.00 5.00 0.00 13.03 20.03 13.03 20.03 Flevoland Pisgah Punta San Mt. Traverse Weeks Cacao Francisco Shasta City Lake Bonanza Creek Fairbanks Figure 6.34: Relative Confusion of Simple Radars in All L-band Scenes radars is well justified. Especially in a research environment, it is beneficial to have the full feature set available for analysis. However, if bandwidth is at a premium, and simplicity, reliability, and economy are important, then simplified radar systems offer an alternative. Their classification performance is much better 113 Chapter 6: Simplified Radar Systems Simple Radars — Relative Confusion Radar SjiUra Figure 6.35: Relative Confusion of Simple Radars in L-band EDO Scenes Flevoland Traverse City CO-POL 15.4 10.6 AMPL 16.0 13.3 H Transmit 19.4 14.1 V Transmit 18.2 12.9 Table 6.17 Relative Confusion of Simplified Radars for Four-Class Scenes than that of single-channel radars, yet they are simpler and cheaper than fully polarimetric radars. Of the four simplified radar systems considered, the CO-POL option performs the best on most of the forestry datasets, although, for other applications, the AMPL radar might be more suitable. 114 Chapter 7: Conclusions Chapter 7 Conclusions Motivated by the desire to reduce the data transmission, recording, and processing requirements attendant with polarimetric data, the information content of polarimetric SAR features has been analysed. A sharp distinction has been found between those features that carry information useful for classification and those that do not. Classification performance is good (and may be improved) when only 50 to 70% of the image data is used. Data reduction is possible, particularly if classification is the main use of the data. The U{ShvS;v}, %{ShvS:v}, ${ShkS*hv}, and Z{ShhS*hv} features may be dropped with little change to the resulting classification (the given four features were only significant in the Pisgah Ojeological) and San Francisco (urban) scenes). Of the simplified partially polarimetric SAR systems explored, the CO-POL radar (with only the HH and W channels) was found to give the best classification performance, often nearly equal to that of a fully polarimetric radar. The classification performance of the simplified radars was in general not degraded substantially between fully polarimetric and simplified radar systems; in all cases it was much better than that of single-channel radars. 7.1 Possible Areas for Future Work Many interesting avenues await new research into polarimetric SAR information content. The following is a list of topics worthy of future research: 1. Obtain more ground truth and select classes based on third-party training area choices. 2. Test conclusions in more detail at frequencies other than L-band. 3. Convert classification images to ground range, geocode, and given a "perfect" ground truth classification map, note which feature subset produces the best classification. 4. Develop a hierarchical classifier based on the utility of different features (at all available frequencies) for identifying specific terrain classes. 115 Chapter 7: Conclusions 5. Develop a fuzzy-set based classifier, allowing a pixel to maintain partial membership in more than one class, therefore not reducing the classification operation to an all-or-nothing decision. Through the concept of fractional membership in say, the "clear-cut" and "forest" classes, second growth forests of different ages might be discriminated. 6. Form a ranking of feature utility by first dropping the feature with the lowest relative confusion, then choosing the feature that produces the lowest relative confusion when it is added to the list of dropped features, dropping it, and continuing until only one feature remains. A ranking could be produced for each scene, and compared across scene-types and class-definitions. 116 Appendix A Tabular Feature Statistics and Class Separabilities A.l Bonanza Creek Class Span HH W HV ®{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. L 1 Even 2.05 .70 2.13 .44 1.11 .33 .38 .13 -.13 .39 Diffuse 1.51 .44 1.54 .41 1.32 .39 .57 .15 .29 .33 Odd .22 .087 1.91 .32 1.90 .34 .096 .058 1.52 .19 L 2 Even 1.60 .60 2.59 .49 .88 .39 .27 .12 .015 .38 Diffuse .84 .22 2.04 .39 1.18 .30 .39 .13 .52 .33 Odd .24 .12 1.75 .27 2.06 .29 .098 .054 1.52 .19 113 March 1988 (T=2°C) (su = 1.0); 220 March 1988 (frozen, T=-15°C) (su = 1.0) Table A.18 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Bonanza Creek L-band data). Class Pair Feature span HH W HV HH W* HVW* HHHV* ft ft ft L 1 Even / Diffuse .48 .69 .29 .67 .57 .12 .00 .02 .05 .02 Even/Odd 2.33 .29 1.18 1.51 2.80 .28 .19 .15 .22 .46 Diffuse / Odd 2.44 .51 .80 2.29 2.33 .16 .17 .15 .17 .44 Mean 1.75 .50 .76 1.49 1.90 .19 .12 .10 .14 .31 L 2 Even / Diffuse .93 .62 .44 .49 .71 .32 .05 .01 .01 .01 Even / Odd 1.92 1.11 1.76 .97 2.65 1.03 .21 .10 .02 .09 Diffuse / Odd 1.79 .44 1.50 1.63 1.94 .66 .21 .09 .03 .07 Mean 1.55 .72 1.23 1.03 1.77 .67 .16 .06 .02 .05 113 March 1988 (T=2°C); 220 March 1988 (frozen, T=-15°C) Table A.19 Class Separations for Normalized Cross Products (Bonanza Creek L-band data). 117 A.2 Fairbanks Class Span HH W HV U{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. L 1 Even .24 .69 2.59 .56 1.13 .48 .14 .138 -.24 .51 Diffuse .12 .070 1.61 .40 1.48 .39 .46 .18 .46 .41 Odd .0018 .0008 1.83 .30 1.98 .31 .095 .042 1.44 .30 L 2 Even .24 .88 2.64 .59 1.17 .49 .093 .10 -.29 .60 Diffuse .0092 .0065 1.44 .39 1.53 .42 .52 .16 .37 .37 Odd .0001 .00004 1.80 .29 1.98 .28 .11 .050 1.45 .29 113 March 1988, 5„=.05; 220 March 1988, su=.Q\ Table A.20 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products O a^irbanks L-band data) Class Pair Feature span HH W HV HH W* HVW* HHHV* ft ft ft L 1 Even / Diffuse .16 1.02 .40 .99 .75 .40 .18 .13 .32 .01 Even / Odd .35 .88 1.07 .24 2.08 .28 .06 .33 .44 .10 Diffuse / Odd 1.71 .32 .70 1.61 1.39 .16 .15 .11 .03 .10 Mean .74 .74 .72 .95 1.41 .28 .13 .19 .26 .07 L 2 Even / Diffuse .26 1.24 .40 1.66 .69 .62 .17 .19 .33 .01 Even / Odd .27 .97 1.04 .13 1.96 .67 .26 .11 .30 .01 Diffuse / Odd 1.40 .54 .64 1.96 1.65 .01 .00 .10 .14 .02 Mean .64 .92 .69 1.25 1.43 .43 .14 .13 .26 .01 *13 March 1988; 220 March 1988 Table A.21 Class Separations for Normalized Cross Products O'airbanks L-band data). 118 A.3 Flevoland Class Span HH W HV U{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. p Even .519 .275 .66 .26 2.65 .41 .35 .14 .22 .28 Diffuse 1.46 .622 1.25 .42 2.09 .55 .33 .15 .16 .40 Odd .572 .211 .95 .092 3.00 .099 .027 .014 1.55 .067 L Even 1.99 1.82 3.11 .63 .71 .54 .091 .083 -.31 .41 Diffuse .468 .183 1.39 .37 1.65 .40 .48 .17 .29 .38 Odd .120 .043 1.28 .12 2.68 .12 .022 .016 1.74 .08 C Even .186 .056 1.98 .44 1.19 .37 .42 .15 .44 .33 Diffuse .200 .061 1.49 .38 1.68 .40 .41 .14 .57 .31 Odd .012 .014 1.72 .25 2.01 .35 .14 .12 1.41 .40 su=5 Table A.22 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Flevoland P, L , and C-band data). Feature Class Pair span HH W HV HH W* HVW* HHHV* & ft Even / Diffuse 1.05 .86 .58 .05 .08 .71 .14 .20 .06 .01 p Even / Odd .11 .82 .70 2.11 3.90 .53 .19 .18 .37 .45 Diffuse / Odd 1.07 .58 1.40 1.89 2.99 1.38 .04 .13 .21 .30 Even / Diffuse .76 1.72 1.01 1.56 .76 .39 .05 .32 .05 .18 L Even / Odd 1.00 2.45 2.98 .70 4.16 .72 .24 .43 .33 .40 Diffuse / Odd 1.54 .24 1.97 2.52 3.18 .26 .05 .14 .15 .02 Even / Diffuse .12 .60 .64 .01 .20 .05 .01 .14 .03 .14 C Even / Odd 2.48 .37 1.13 1.03 1.32 .28 .00 .16 .03 .04 Diffuse / Odd 2.51 .36 .43 1.09 1.18 .22 .02 .03 .01 .13 Table A.23 Class Separations for Normalized Cross Products (Flevoland P, L, and C-band data). 119 A.4 Pisgah Class Span HH VV HV tt{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. Lava (I) .259 .405 1.74 .355 2.12 .371 .069 .035 1.41 .255 Lava (II) 1.75 .685 .57 .150 3.32 .166 .056 .027 1.04 .164 Lava (III) .57 .546 1.22 .222 2.70 .231 .043 .028 1.46 .200 P Alluvial Surface .021 .0078 2.54 .267 1.22 .244 .117 .063 1.31 .209 Playa .026 .018 2.63 .410 1.10 .390 .135 .081 .935 .378 Lava (I) .238 .197 1.77 .343 1.95 .415 .138 .079 1.37 .268 Lava (II) 1.06 .364 2.10 .311 1.50 .082 .199 .082 1.27 .238 Lava Gil) .442 .306 2.06 .307 1.68 .064 .127 .064 1.44 .217 L Alluvial Surface .016 .0054 1.66 .322 2.04 .069 .151 .069 1.32 .227 Playa .0084 .0041 1.56 .263 2.31 .047 .067 .047 1.57 .210 Lava (I) .524 .214 1.85 .344 1.88 .376 .137 .069 1.13 .308 Lava (II) 1.13 .384 1.73 .300 1.93 .306 .170 .084 1.06 .354 Lava (III) .622 .236 1.84 .338 1.93 .349 .117 .060 1.28 .301 C Alluvial Surface .069 .019 1.64 .353 2.07 .353 .144 .073 .986 .280 Playa .042 .015 1.37 .263 2.44 .274 .093 .037 1.36 .217 P: 5 u=5xl0 7, L: s u=2xl0 6 , C: su=5 T a b l e A . 2 4 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products O'isgah P, L, and C-band data). 120 Feature Class Pair span HH W HV HH W* HVW* HHHV* ft ft ft II / III .96 1.75 1.58 .24 1.14 .09 .08 .06 .08 .04 I I / I 1.37 2.33 2.24 .20 .88 .02 .04 .17 .00 .13 II / Alluvial 2.50 4.75 5.12 .66 .73 .10 .04 .26 .06 .26 II / Playa 2.45 3.69 4.00 .73 .20 .04 .09 .27 .19 .43 p III / I .33 .91 .96 .40 .10 .09 .03 .13 .05 .15 III / Alluvial .99 2.71 3.11 .80 .35 .17 .14 .23 .01 .27 III / Playa .97 2.23 2.57 .84 .90 .04 .02 .23 .22 .43 I / Alluvial .58 1.29 1.46 .49 .21 .07 .09 .09 .05 .13 I / Playa .55 1.16 1.34 .57 .75 .04 .04 .10 .15 .27 Alluvial / Playa .18 .12 .19 .13 .64 .11 .15 .02 .17 .12 II / III .92 .05 .32 .49 .38 .13 .14 .04 .15 .09 I I / I 1.46 .50 .64 .38 .21 .44 .02 .04 .09 .14 II / Alluvial 2.82 .69 .86 .32 .11 .22 .24 .00 .15 .09 II / Playa 2.85 .94 1.43 1.02 .69 .04 .11 .12 .11 .21 L III / I .41 .45 .40 .07 .14 .32 .11 .09 .06 .06 III / Alluvial 1.37 .64 .59 .18 .27 .10 .11 .03 .01 .00 III / Playa 1.40 .89 1.14 .54 .31 .10 .04 .10 .06 .12 I / Alluvial 1.10 .17 .11 .09 .10 .23 .21 .04 .07 .06 I / Playa 1.14 .35 .51 .56 .42 .43 .08 .18 .01 .05 Alluvial / Playa .78 .18 .44 .72 .58 .20 .15 .11 .07 .12 Table A.25 Class Separations for Normalized Cross Products O'isgah P, L, and C-band data). (Continued . . . ) 121 Feature Class Pair span HH W HV HH W* HVW* HHHV* ft ft ft II / III .83 .17 .00 .38 .33 .19 .03 .05 .03 .11 I I / I 1.02 .19 .08 .22 .10 .23 .03 .01 .03 .04 II / Alluvial 2.64 .13 .21 .17 .12 .41 .02 .02 .01 .01 II / Playa 2.74 .63 .88 .63 .53 .36 .02 .01 .03 .15 c III /I .22 .02 .07 .16 .25 .05 .07 .07 .01 .08 HI/Alluvial 2.17 .29 .20 .21 .50 .24 .02 .07 .02 .13 III / Playa 2.31 .77 .82 .24 .16 .18 .02 .04 .01 .01 I / Alluvial 1.96 .30 .27 .05 .24 .18 .05 .01 .02 .06 I / Playa 2.11 .78 .87 .41 .44 .11 .05 .03 .02 .10 Alluvial / Playa .76 .43 .59 .46 .75 .08 .00 .04 .01 .17 Table A.25 Class Separations for Normalized Cross Products O'isgah P, L , and C-band data). A.5 Punta Cacao Class Span HH W HV R{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. Even .022 .0070 .91 .31 2.73 .38 .18 .067 -.56 .31 p Diffuse 1.32 .63 1.80 .40 1.49 .45 .36 .15 .11 .37 Odd .10 .048 .90 .23 2.89 .27 .11 .052 1.11 .24 Even .091 .031 1.03 .32 2.73 .33 .12 .065 -.69 .32 L Diffuse 1.03 .48 1.93 .44 1.08 .43 .50 .15 .16 .32 Odd .22 .10 1.41 .27 2.24 .29 .17 .066 1.09 .29 Even .63 .16 2.00 .44 1.43 .47 .29 .10 -.24 .40 C Diffuse .90 .49 1.71 .47 1.42 .41 .43 .15 .50 .32 Odd .65 .28 1.80 .29 1.69 .29 .26 .11 1.17 .23 N.B. s„=200 Table A.26 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products Q^ unta Cacao P, L, and C-band data). 122 Feature Class Pair span HH W HV HH W* HVW* HHHV* ft ft ft Even / Diffuse 2.06 1.25 1.49 .81 .99 .41 .18 .49 .41 .26 p Even / Odd 1.41 .01 .24 .62 3.04 .05 .30 .45 .33 .04 Diffuse / Odd 1.82 1.41 1.91 1.25 1.63 .45 .08 .11 .24 .30 Even / Diffuse 1.83 1.19 2.16 1.73 1.32 .79 .18 .18 .60 .42 L Even / Odd .94 .66 .78 .38 2.93 1.33 .20 .39 .17 .44 Diffuse / Odd 1.40 .73 1.63 1.49 1.51 .54 .00 .55 .43 .14 Even / Diffuse .42 .31 .01 .58 1.03 .90 .63 .13 .08 .39 C Even / Odd .05 .28 .33 .13 2.21 1.14 .43 .16 .28 .47 Diffuse / Odd .33 .10 .38 .67 1.21 .04 .23 .01 .18 .07 N.B. su = 200 Table A.27 Class Separations for Normalized Cross Products fl?unta Cacao P, L , and C-band data). A.6 San Francisco Class Span HH W HV &{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. City .39 .58 1.91 .38 1.62 .39 .23 .083 -.35 .46 L Park .082 .042 1.52 .34 1.66 .34 .41 .14 .46 .33 Ocean .015 .0069 .82 .099 3.09 .10 .047 .021 1.48 .094 Table A.28 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (San Francisco L-band data) 123 Class Pair Feature span HH W HV HH W* HVW* HHHV* ft ft ft City / Park .49 .54 .05 .78 1.02 .01 .58 .11 41.04 .30 City / Ocean .63 2.30 22.97 1.81 !3.30 .27 .87 .38 41.86 .54 Park / Ocean 1.38 1.60 23.26 32.26 2.40 .29 .06 .47 .04 .02 Mean .84 1.48 2.09 1.62 2.24 .19 .50 .32 .98 .29 Table A.29 Class Separations for Normalized Cross Products (San Francisco L-band data). A.7 Mt. Shasta Class Span HH W HV &{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. Clear-Cut .12 .055 1.72 .37 1.88 .37 .20 .077 1.00 .31 P Treel .91 .27 1.87 .44 1.08 .36 .52 .18 .025 .33 Tree2 .35 .15 1.96 .39 1.38 .39 .33 .13 .60 .37 Clear-Cut .24 .10 1.76 .34 1.78 .35 .23 .11 1.14 .29 L Treel 1.57 .53 1.61 .43 1.32 .42 .54 .17 .25 .33 Tree2 .60 .31 1.67 .34 1.51 .36 .41 .13 .69 .32 Clear-Cut .34 .11 1.73 .30 1.78 .32 .24 .11 1.16 .26 C Treel .78 .32 1.51 .36 1.59 .37 .45 .15 .37 .35 Tree2 .33 .10 1.65 .34 1.61 .37 .37 .15 .71 .35 N.B. su = 500 Table A.30 Values for Span, HH, VV, HV and Rhhvv Normalized Cross Products (ML Shasta P, L, and C-band data). 124 Feature Class Pair span HH W HV HH VV* HVW* HHHV* ft ft ft Clear Cut / Treel 2.43 .19 1.09 1.26 1.53 .25 .02 .10 .03 .10 p Clear Cut / Tree2 1.10 .32 .66 .62 .59 .18 .21 .13 .10 .14 Treel / Tree2 1.33 .11 .40 .63 .82 .07 .18 .20 .04 .03 Clear Cut / Treel 2.12 .21 .59 1.11 1.44 .23 .02 .05 .17 .01 L Clear Cut / Tree2 .86 .13 .37 .75 .74 .29 .02 .05 .06 .07 Treel / Tree2 1.16 .09 .24 .44 .67 .06 .04 .09 .09 .07 Clear Cut / Treel 1.01 .34 .27 .83 1.31 .01 .02 .05 .09 .08 C Clear Cut / Tree2 .08 .13 .25 .52 .75 .02 .07 .04 .09 .09 Treel /Tree2 1.05 .20 .02 .27 .49 .00 .04 .01 .16 .16 N.B. su = 500 Table A.31 Class Separations for Normalized Cross Products (Mt. Shasta P, L, and C-band data). A.8 Traverse City Class Span HH W HV U{HH VV} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. Even 1.01 .68 1.45 .42 2.16 .45 .20 .12 -.39 .43 L Diffuse 1.13 .30 1.06 .32 2.02 .42 .46 .16 .25 .31 Odd .050 .022 .40 .15 3.35 .21 .12 .044 .60 .19 Table A.32 Values for span, and HH, VV, HV and Rhhvv Normalized Cross Products (Traverse City L-band data) 125 Class Pair Feature span HH W HV HHW* HVW* HHHV* ft ft ft Even / Diffuse .13 .52 .16 .92 .87 .54 .14 .08 .29 .20 Even / Odd 1.35 1.84 1.83 .46 1.62 .68 .06 .44 .59 .04 Diffuse / Odd 3.35 1.38 2.12 1.63 .69 .01 .21 .42 .11 .31 Mean 1.61 1.25 1.37 1.01 1.06 .41 .14 .31 .33 .18 Table A.33 Class Separations for Normalized Cross Products (Traverse City L-band data). A.9 Weeks Lake Class Span HH W HV ®{HH VV*} Mean S.D. Mean S.D. Mean S.D. Mean S.D. Mean S.D. p Clear-Cut .42 .19 2.02 .27 .65 .24 .66 .10 .027 .29 Forest 1.39 .57 1.77 .86 .69 .11 .69 .11 -.11 .34 Lake .042 .020 2.45 .18 .22 .12 .66 .074 .048 .16 L Clear-Cut .66 .32 1.88 .33 1.32 .30 .40 .13 .84 .30 Forest 1.38 .46 1.78 .35 1.01 .30 .61 .17 .36 .34 Lake .087 .030 1.23 .16 2.38 .18 .20 .044 1.49 .13 C Clear-Cut 1.01 .32 2.80 .37 .60 .20 .30 .13 .60 .22 Forest .98 .32 2.66 .32 .69 .19 .33 .12 .55 .29 Lake .051 .019 2.57 22 1.29 .22 .069 .044 1.32 .20 Table A.34 Values for Span, HH, VV, HV and Rhhvv Normalized Cross Products (Weeks Lake P, L, and C-band data). 126 Feature Gass Pair span HH VV HV HH VV* HVW* HHHV* ft ft ft Clear Cut / Forest 1.27 .45 .38 .11 .22 .14 .18 .46 .39 .05 p Clear Cut / Lake 1.81 .94 1.19 .01 .05 .03 .87 .63 .69 .17 Forest / Lake 2.28 1.45 1.53 .12 .32 .15 1.05 1.18 1.03 .11 Clear Cut / Forest .92 .15 .52 .69 .74 .27 .97 .26 .16 .16 L Clear Cut / Lake 1.64 1.32 2.24 1.22 1.51 .22 .94 .61 .35 .03 Forest / Lake 2.66 1.07 2.89 1.92 2.43 .10 .98 .91 .08 .20 Clear Cut / Forest .05 .21 .23 .10 .09 .22 .02 .01 .05 .04 C Clear Cut / Lake 2.79 .39 1.67 1.32 1.75 .98 .53 .09 .23 .26 Forest / Lake 2.74 .16 1.46 1.63 1.58 1.12 .59 .11 .16 .34 Table A.35 Class Separations for Normalized Cross Products (Weeks Lake P, L and C-band data). 127 References [I] AIRSAR Bulletin. 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