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UBC Theses and Dissertations

Image processing 1995

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I M A G E P R O C E S S I N G by A N D Y B I N G - B I L L C H A N B . S c , The University of Br i t i sh Columbia , 1993 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Statistics We accept this thesis as conforming to ^he required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 1995 © A n d y B i n g - B i l l Chan, 1995 In p resen t ing this thesis in partial fu l f i lment of t h e r e q u i r e m e n t s fo r an advanced d e g r e e at the Univers i ty o f British C o l u m b i a , I agree tha t t h e Library shall make it f reely available f o r re ference and s tudy. I fu r ther agree that pe rmiss ion f o r ex tens ive c o p y i n g of this thesis f o r scholar ly pu rposes may be g r a n t e d by the head o f m y d e p a r t m e n t or by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on o f this thesis fo r f inancial gain shall n o t be a l l o w e d w i t h o u t m y w r i t t e n pe rmiss ion . D e p a r t m e n t o f The Univers i ty o f Brit ish C o l u m b i a Vancouver , Canada Date DE-6 (2788) ABSTRACT In this thesis, we consider the restoration of multiple grey levels image. The problem is to clean up or restore the dirty picture, that is, to construct an estimate of the true image from a noisy picture of that true image. Following a method proposed by Meloche and Zamar (1994), we estimate the colour at each site by a function of the data available in a neighbourhood of that site. In this approach, the local characteristics of that image, that is, the frequency with which each pattern appears in the true unobserved image are particularly important. We will propose a family of unbiased estimates of the pattern distribution and the noise level which are used in the restoration process. We will use our estimates of the pattern distribution in an attempt to select the best neighbourhood shape for the restoration process. ii CONTENTS Abstract " Table of Contents iii List of Tables iv List of Figures v Acknowledgements vii 1 Introduction 1 2 Estimations of Q 10 2.1 Introduction 10 2.2 Unbiased Estimations of Q 10 2.3 A Simple Representation for Q 21 3 Joint Estimations of Q and a 29 3.1 Introduction 29 3.2 Estimation of a using Estimating Equation derived from E{\ £" = 1 *i2) 2 9 3.3 Estimation of a using Estimating Equation derived from E(\ J2i=i 3 3 4 Neighbourhood Shapes 35 5 Conclusions 3§ References 40 i n LIST OF TABLES Table 1.1 Frequency distribution (r = 1) 5 Table 1.2 Frequency distribution (r = 2) 5 Table 1.3 Frequency distribution (r = 5) 7 Table 2.1 Pattern distribution with r = 1, <p(8,y) = Indicator functions 25 Table 2.2 Pattern distribution with r = 1, <p(8,y) = Power functions 25 Table 2.3 Pattern distribution with r = 2, ip(6, y) — Indicator functions 26 Table 2.4 Pattern distribution with r = 2, o?(<5, y) = Power functions 27 Table 4.1 AMSE for various neighbourhood sizes and shapes .36 iv LIST OF FIGURES Figure 2.1 True Image (Unobserved) 41 Figure 2.2 Noisy Image (Observed) 41 Figure 2.3 Pattern distribution with r = 3 42 (i) <p(8, y) = Indicator functions 42 (a) Q3(8) vs«5 42 (b) Q3(8) vs8 42 (c) Q3{8)vs8 : . . 42 (ii) <p(8, y) = Power functions 43 (d) Q3(6) vs8 43 (e) Q3(8) vs8 43 (f) Q3(<5)vsc5 43 Figure 2.4 Pattern distribution with r = 5 44 (i) tp(6, y) = Indicator functions 44 (a) Q3(8) vs8 44 (b) Q3(8) vs8 44 (c) Q3(8) vsS 44 (ii) (p(8, y) = Power functions 45 (d) Q3{8) vs8 45 (e) Q3(8) vsc5 45 v (f) Q3(6)vs6 45 Figure 2.5 Restored Image 46 (i) with (f(8, y) = Indicator functions 46 (a) r = 1 46 (b) r = 2 46 (c) r = 3 .' 46 (d) r = 5 46 (ii) with (f(6, y) — Power functions 47 (e) r = 1 ^ 47 (f) r = 2 47 (g) r = 3 47 (h) r = 5 47 Figure 3.1 True Image (0-1-2 strips) 48 Figure 3.2 Noisy Image (0-1-2 strips) 48 vi ACKNOWLEDGEMENTS I would like to thank my supervisor, Jean Meloche, for all his support, encouragement, valuable time, and excellent supervision he has provided for the development of my thesis and my thesis presentation on April 4th 1995. I am honoured to be the first graduate student to work on research and thesis under his supervision. I was introduced to the field of Image Processing by Jean Meloche in the fall of 1993. I am amazed by the fact that statistics can be widely applied to Image Processing. It has always been fun working with Jean, and I have learned a lot from him. He has been one of the greatest teachers whom I have ever met. Without him, this thesis may never be made possible. I would also like to thank Ruben Zamar for all his wonderful and helpful comments, and the time he spent with me on the directed study course. Special thanks to my two special friends, Grace Chiu and Andy Ho, for their help in my C programming, and all the fellow graduate students for making my life in the Department of Statistics exciting. vii 1 Introduction. In this thesis, we consider multiple grey levels image restoration. One kind of grey levels is grey levels from white to black. Some other kinds are grey levels are from white to red, white to green, white to blue, etc. If we combine the three latter kinds, we will get a coloured image. The problem is to clean up or restore the dirty picture, that is to construct an estimate of the true image from a noisy picture of that true image. The true image is not observed, while the noisy picture is. Figure 2.1 is the true image while Figure 2.2 is a degraded version of true image, the noisy picture. We want to recover the true image (Figure 2.1) from the noisy picture (Figure 2.2) which is the observed data. The estimate of the colour at each site proposed by Meloche and Zamar (1994) is a function of the data available in a neighbourhood of that site. The local characteristics of the underlying image is central to their approach. The notion of local characteristics is formalized in what follows. We assume that there are n sites or pixels on a plane (or a line), numbered 1 to n. The vector {zi, z^,..., zn} represents the true image, and 2,- € C where C = {ci, C 2 , . . . , cjt} is a set of k grey levels. The observations Yi, Y2,..., Yn (called the records) form the degraded image and are modelled as Yi = zi + aei (1) where e\,..., e„ i.i.d. N(0,1), Yi,..., Yn are the records, and cre.'s are the noise. The ob- jective of image restoration is to work out estimates z\,..., zn of the true image z\,..., zn from the records Y\,..., Yn. Following Meloche and Zamar (1994), we define a system of neighbourhoods, Ni,..., Nn. 1 The neighbourhood Ni, centred at i, consists of r sites called the neighbours of site i and includes the site i itself. In general, the neighbourhood JV,- is the set of all sites j at a small distance from site i. The neighbourhoods N\,..., /Vn have the same shape but different centres. For examples, if /Y, = {l,...,n} for all i, then r equals n and we have just one neighbourhood, the whole image. On the other hand, if N{ — {i} for all i, then r equals 1 and then we have n single-site neighbourhoods. Neighbourhoods can be 1-dimensional (linear), 2-dimensional (planar), 3-dimensional, or even higher than 3-dimensional. A pattern is a particular colouring of the sites (or pixels) in a neigh- bourhood. Some examples of neighbourhoods are graphically shown in the following examples. Example 1.1 : If the true image consists of 0 — 1 bits sent over a line, then the following is a typical linear 3-site neighbourhood: The true image is one-dimensional, and the colour set C is {0,1}, where 0 = black, and 1 = white. In this example of 3-site neighbourhood, there will be 23 = 8 possible patterns: Mathematically, a pattern here is a vector of 3 coordinates of black or white that represent the colour of neighbourhood sites in some fixed order.* As we will notice from the next example, the size of the pattern space increases rapidly with the size of the neighbourhood. Example 1.2 : Suppose we have a colour set C = {0,1,2} (0 = black, 1 = middle grey, 2 2 = white), and a cross-shaped neighbourhood There will then be 3s = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 2 Again, mathematically, a pattern here is a vector of 5 coordinates of black, middle grey or white that represents the colour of neighbourhood sites in some fixed order.* There is a need for a systematic convention for the ordering of the components of a neighbourhood. Firstly, we denote a pattern by 8. We use the bold face 6 as a vector of <5's in a neighbourhood. In Example 1.1, 6 is a vector of 3 coordinates, that is, 6 — (6"i, <52, 83), and in Example 1.2, 6 is a vector of 5 coordinates, that is, 6 = (Si, 82, S3,84,6*5). In the case of 3-site neighbourhoods, we use 6 = (Si, <52, 83) in the following order: <5i And then in the case of cross-shaped neighbourhoods, we use 6 = (81,62,83,84,85) in the following order: 82 83 84 Ss Bold face notation is used to represent a vector of variables in a neighbourhood. The coordinates are ordered in the same way as the coordinates of the patterns are. For example, for 3-site neighbourhoods, Y,- = ( ^ , 1 , 5 ^ , 2 , ^ , 3 ) is the vector of records, Yj, for j in the neighbourhood Ni of site i. Yij is the record at the jth neighbour of site i. By equation (1), Yi.i Yi,2 = zi,l zi,2 Zi,3 + creit2 o-ei<3 so that the distribution of Yt- is multivariate normal with mean z,- and covariance matrix <7 2/ 3. Throughout this thesis, we look at the distribution of all the possible patterns in the true image. The true distribution of the patterns is usually unknown, but it can be estimated from the records (the noisy picture) in many different ways. As stated before, the pattern distribution plays a useful role in the process of image restoration. The estimates z\,...,zn of the true image proposed by Meloche and Zamar (1994) are based on the knowledge of the pattern distribution and cr. Example 1.3 : In this example, we use the Figure 2.1 with 3 grey levels as the true image. Let r denote the neighbourhood size, and C = {0,1,2}. When r = 1, there are only three possible patterns, namely, black, middle grey, and white. Table 1.1 below shows how frequently each pattern appears in the true image. 4 Table 1.1: Frequency distribution (r = 1) pattern frequency black 0.48340 grey 0.42505 white 0.09155 Total 1.00000 This frequency distribution reflects the fact that 48% of all pixels are black, 43% of them are middle grey, and 9% are white. When r = 2, a horizontal 2-site neighbourhood, there is a total of 32 = 9 possible patterns, namely, 00, 01, 02, 10, 11, 12, 20, 21, and 22. Table 1.2 below shows the frequency with which each pattern appears in the true image. Table 1.2: Frequency distribution (r = 2) pattern frequency 0 0 0.47192 0 1 0.00842 0 2 0.00305 1 0 0.00842 1 1 0.41541 1 2 0.00122 2 0 0.00305 2 1 0.00122 2 2 0.08728 Total 1.00000 5 The high frequency of the patterns 00, 11, and 22 in this frequency distribution reflects the fact that neighbouring pixels typically have the same colour. When r = 5, we consider the cross-shaped neighbourhood, there are 35 = 243 possible patterns. We use 6 = (6"i, 82,83,84, 85) in the following order: Table 1.3 below is the corresponding frequency table. 6 Table 1.3: Frequency distribution (r = 3) pattern frequency pattern frequency pattern frequency 0 0 0 0 0 0.43811 1 0 0 0 0 0.00824 2 0 0 0  0.00293 0 0 0 0 1 0.00824 1 0 0 1 0 0.00006 2 0 0 1 0 0.00012 0 0 0 0 2 0.00293 1 0 0 20 0.00012 2 0 2 2 0 0.00018 0 0 0 1 0 0.00824 1 0 1 1 0 0.00024 2 0 2 2 0.00281 0 0 0 2 0 0.00293 1 0 1 1 1 0.00806 2 111 1 0.00110 0 0 1 1 1 0.00012 1 1 1 0 0 0.00012 2 12 2 2 0.00110 0 0 2 2 2 0.00006 1 1 1 0 1 0.00793 2 2 11 1 0.00012 0 1 0 0 0 0.00824 1 1 1 0 2 0.00012 2 2 2 0  0.00006 0 1 0 0 1 0.00006 1 1 1 1 0 0.00793 2 2 2 0 1 0.00012 0 1 0 0 2 0.00012 1 1 1 1 1 0.38757 2 2 2 0 2 0.00269 0 1 1 0 1 0.00024 1 1 1 1 2 0.00110 2 2 2 1 0 0.00012 0 1 1 1 1 0.00806 1 1 1 2 0 0.00012 2 2 2 1 2 0.00110 0 2 0 0 0 0.00293 1 1 1 2 1 0.00110 2 2 2 2 0 0.00269 0 2 0 0 1 0.00012 1 1 2 2 2 0.00012 2 2 2 2 1 0.00110 0 2 0 0 2 0.00018 1 2 1 1 1 0.00110 2 2 2 2  0.07532 0 2 2 2 2 0.00281 1 2 2 2 2 0.00110 Total 1.00000 7 Note that Table 1.3 has fewer than 243 lines. It is because some patterns do not appear in the image. The high frequency of the patterns 00000, 11111, and 22222 reflects the fact that cross-shaped neighbourhoods of 5 pixels typically have the same colour. In general, the pattern distribution is unknown, and Meloche and Zamar (1994) pro- pose to estimate it directly from the records Y\,..., Yn, for binary (black/white) images. In this thesis, we extend this to the case of images with an arbitrary, but finite, number of grey levels, and we develop a family of estimates of the pattern distribution. In the case of single-site neighbourhoods, the above frequency distribution can be written as <?i(*) = -x; i{*=*}, (2) n ,=i where l{z,- = 6} = 1 when the true colour at site i is equal to <5, otherwise, it is equal to zero. The subscript 1 is used to indicate that we are dealing with single-site neigh- bourhoods. More generally, for an arbitrary neighbourhood size r, the above frequency distribution can be written as Qr(6) = " £ !Ki = • • •. *i,T = M, (3) n i=i and the 6 in equation (3) is r-dimensional. The subscript r in the above equation is used to indicate that we are dealing with r-site neighbourhoods. For binary images, Meloche and Zamar (1994) propose to use where Qr defined by equation (3) depends on the Zi, z2,.. •, zn. Note that z,- depends 8 on the typically unknown parameters QT and a. For this reason, Meloche and Zamar (1994) propose methods to estimate QT and cr in the case of C = {0,1}, that is, binary images. While equation (4) defines an estimate no matter what the colour set C is, the estimates of QT and a proposed by Meloche and Zamar (1994) are only valid in the case of C = {0,1}. In this thesis, we propose a family of simple estimates for QT and a in the case of C = {ci, c2,..., cjt}, that is, images with an arbitrary, but finite, number of grey levels. Chapter 2 will focus on the estimation of Qr, assuming a is known, while Chapter 3 will deal with the joint estimation of Qr and cr. 9 2 Estimations of Q. 2.1 Introduction Firstly, let Q r, which will be used frequently, be the |C| r vector of all possible Qr(6ys, where |C| r is the total number of patterns. Meloche and Zamar (1994) propose a biased estimate of Q r based on indicator functions for binary images, and a general formula for obtaining an unbiased estimate from the biased one. In this chapter, we extend their idea to images with an arbitrary, but finite, number of grey levels. And we also discover a simple representation which makes the computations of the estimates easier. In this chapter we assume that a is known. The joint estimation of Q r and a will be discussed in Chapter 3. 2.2 Unbiased Estimations of Q r We start by reviewing the biased and unbiased estimates of Q r based on indicator functions for binary images proposed by Meloche and Zamar (1994). Later on, we will extend to image with more than two grey levels. Consider a case of 2 grey levels (eg. black/white), that is, C = {0,1}, with single-site neighbourhoods (r = 1). Let Now, define <?i(0) £ £ > ( o , t f ) 1 n 1 n . = i 10 and Qi to be the vector (Qi(0), Qi(l))T. This Qi is a biased estimate of Qi because, for example, EQl{0) = £ ± E 1 { * ; - < n I = -EPf>,- + ere,- < -) n 2 = ^E{P(e,- < ±-)\{zi = 0 } + P(c- < £)l{zi = 1 } } = IE1{Z,- = 0}$(i-) + ± E 1 { * = 1 } * ( ^ ) n la n la Similarly, EQx(l) = $(̂ -)<2i(0) + $(±)Qi(l), where $(•) is the cumulative distribution function of a N(0,1) random variable. Now we let a = b — $ ( 5 7 ) , then we can write the above in matrix notation. £Qi = AiQi where Ax = t a b b a So, an unbiased estimate can be obtained by inverting Ai and defining Qi = Ax 1 Q 1 , so that EQi = Af^Qi = A^AiQi = Qx. That is, and Qi is unbiased for Qi. 11 Using the same colour set C, we now consider linear 3-site neighbourhoods, that is, 3 pixels in a row, either vertically or horizontally, with C = {0,1}. As we have just mentioned in Example 1.1, there are 23 = 8 possible patterns, namely 000, 001, . . . , 111. Recall from equations (1) and (2) in Chapter 1 that 1 n Qsi^l, <$2, fa) = - 53 Hzi,l ~ fa,zi,2 = fa, Z,,3 = fa}, where <5,- = 0,1, i — 1,2,3. Following the same idea as above, define 03(000) = ±Y:i{Yi,i<\}HYi,2<\}l{Yi>3<\} $3(001) = I^i{y;.1<I}i{y;,2<I}i{y;.3>I} W i l ) = ^HYi^lMY^^^Wt^^}. Again, we write Q3 for the vector (g3(000), $3(001), $3(010),..., <23(111))T. In the case of 2 grey levels, we use two indicator functions, \{Yi < |} for black pixel, and l{Yi > |} for white pixel (for single-site neighbourhoods). That is, if Y{ < | , then we say that the colour at site i is black. And if Yi > | then we say that the colour at site i is white. Observe that there is a relationship between Qi and Q3. In the case of linear 3-site neighbourhoods, we use products of 3 indicators chosen from l{Yi < |}, l{Yi > |}. So Qi and Q3 have similar structure. Since Qi is biased for Qi, we would expect that Q 3 is also biased for Q3. Now, we can compute the expectations of Q3(6i, 62, S3)'s. For example, EQ3(000) = E^l{Yitl < \}\{Yia < l-}l{Yi<3 < \) 12 = < \)P(Yia < \)P(Yii3 < i ) = ±p{[P{eitl < =0} + Pfa < = 1}] x [ ^ , 3 < ^ ) l k 3 = 0} + P ( e ^ < ^ ) l h 3 = l}]}. Performing similar calculations for the other seven expectations, and letting a be and 6 be $ ( 5 7 ) , we get E ' 03(000) > 3 a26 a2b ab2 a2b ab2 ab2 b3 > ( Q3(000) > <? 3(ooi) a26 a3 ab2 a2b ab2 a2b b3 ab2 <?3(001) Qs(OlO) a26 ab2 a3 a2b ab2 b3 a2b ab2 Qs(OlO) <33(011) a62 a2b a2b a3 b3 ab2 ab2 a2b Q3(0ll) Q 3 ( i o o ) a26 ab2 ab2 b3 a3 a?b a?b ab2 Q 3 ( i o o ) g 3 ( i o i ) ab2 a2b b 3 ab2 a2b a3 ab2 a2b <? 3 ( ioi) Q 3 ( n o ) ab2 b3 a2b ab2 a2b ab2 a3 a2b Q 3 ( n o ) ^ $ 3 ( 1 1 1 ) , v 6 3 ab 2 ab2 a2b ab2 a2b a2b , 03(111) J In matrix notation, we can write the above as £Q 3 = A 3 Q 3 As before, we obtain an unbiased estimate of Q3 by defining Q3 = A 3 1 Q 3 . As shown in Meloche and Zamar (1994), the above matrices Ai, A 3 (as well as the corresponding ones for all r) can be analytically inverted. Meloche and Zamar (1994) show that, in general, EQT — A r Q r with A R ( 6 \ 7 ) H«) # (* , 7W~ # (* , 7 ), 13 (5) where A r ( £ , 7 ) is the ( £ , 7 ) entry of A r , and # ( £ , 7 ) = 9 Sk = 7*:) is the number of k such that 6k = "fk, for k = 1,2,..., r, and that Ar-1(^,7) = (-1) (62 - a2)r (6) where a = $ ( 5 7 ) , 6 = $ ( 2 7 ) - Equation (6) is valid for all r-site neighbourhoods, where r is arbitrary and finite, with C = {0,1}. As an application of equations (5) and (6) in the case of 2-site neighbourhoods, an unbiased estimate of Q2 is f $ 2 ( 0 0 ) N ( aa ab ba u\ - 1 ' g2(oo) ^ Q2(01) ab aa bb ba Q2(oi) <? 2 (10) ba bb aa ab $ 2 ( 1 0 ) ^ 0 2 (H) j K bb ba ab aa j ^ $ 2 ( 1 1 ) j where Q2(00) = ^Wa < \}W,2 < \} g2(oi) = -vm* < hma > h n 1 & We now extend this idea for estimating Q based on indicator functions to a colour set with more than 2 grey levels. We will notice from the following examples that the bias matrix A r that needs to be inverted may not have a simple structure. Example 2.1 : Suppose C = {0,1,2} and r = 1. Now let <p(o,y) = i{y<̂ } 14 v(2,y) = ni<,<|) = H»>l) and define n t e l n . = 1 2 n«=i ",=i 2 2 Giro = ^Ev(2,«)=^i:i{«>|} n t=i n i=i ^ After carrying out some calculations, and letting a = $ ( 5 7 ) , b = $ ( 3 7 ) , c = $(=jj-),d = *(£) " = *(£) " * ( £ ) , / = - = 1" *(£)> * = 1 " = 1 — $ ( ^ 7 ) , we obtain ' & ( 0 ) ' ' a b c ̂  ( Qi(0) y E = d e f Qi(i) ^ 0l(2) , K g h i j ^ Gi(2) j or, in matrix notation, EQi = AiQx. When r = 2 (linear 2-site neighbourhoods), there are 32 = 9 possible patterns, namely, 00, 01, 02, 10, 11, 12, 20, 21, and 22. And again by carrying out some calculations, we obtain 15 E ' Q 2 ( o o ) N ^ aa ab ac ba bb be ca cb cc) ' g 2 (oo) > g 2(oi) ad ae af bd be bf cd ce cf Q 2 ( o i ) <?2(02) ag ah ai bg bh bi eg ch ci Q 2 (02) <? 2(io) da db dc ea eb ec fa fb fc Q 2 (10) Q 2 ( H ) = dd de df ed ee ef fd fe ff 0 2 ( i i ) 02(12) dg dh di eg eh ei fg fh fi Q 2 (12) 02(20) ga gb gc ha hb he ia ib ic Q 2 (20) <?2(21) gd ge 9f hd he hf id ie if g 2(2i) ^ 02(22) j \ 99 gh hg hh hi ig ih ii j < S2(22) j or EQ2 = A 2 Q 2 in matrix notation: But now, A 2 does not seem to have a simple structure.* In the case of binary images, observe that the two bias matrices, Ai, A 2 are closely related. In fact A 2 is a 2-fold Kronecker product of Ai. Definition 2.1 (Kronecker product of matrices) : Let A = (a,j) be a p x m martix and B = (bap) be a q x n matrix. The pq x mn matrix with aijbap as the element in the i,ath row and the j,/?th column is called the Kronecker or direct product of A and B and is denoted by A ® B; that is, A <g> B ^ anB aJ2B ... aimB ^ a2iB a22B ... a2mB y apiB ap2B 16 According to Anderson (1984), (A 0 B) - 1 = A - 1 0 B - 1 , (7) that is, the inverse of a Kronecker product of 2 matrices is the Kronecker product of the inverses of the 2 matrices. In what follows, A^r) denotes the r-fold Kronecker product of A, that is A ( r ) = A 0 A 0 • • • 0 A. (8) Since 6 and 7 are r-dimensional, we can write the (£,7) entry of A(r) as AV(6n) = flA(6i,7i). (9) »=i In case of binary images, we denote the matrix (bias part) of the 1-site case by Ai, the matrix of the 2-site case by A 2, and the matrix of the 3-site case by A 3 . Take a closer look at Ai, and A 2. a b b a \ A 1 = A 2 = V aa ab ba bb ab aa bb ba ba bb aa ab bb ba ab aa ' aAi bAi > ^ 6A1 aAi j By Definition 2.1, A 2 is a 2-fold Kronecker product of Ai. Similarly, A 3 is a 3-fold Kronecker product of Ai. Note that in Example 2.4, for the colour set of 3 grey levels, A 2 is a 2-fold Kronecker 17 product of A i . Also note that AT1(6,7) in equation (6) is the (6,7) entry of the r-fold Kronecker product of Aj"1 for the case of binary images. Given a function <p : C x R —• R , define = - £ > ( * , * ) ( 1 0 ) n ,=1 Qr(6) = ^f:^(61,Yi,1)(p(82,Yi,2)...<p(6r,Yi,r) (11) n t=i In the examples so far, we have been using indicator function as the choice of <p(8, y). Proposition 2.1 : If Qr(8) = " E ,̂lM ,̂ ̂ ,2) • • • V(*r, X>) then £ Q r = A * r ) Q r (12) where A\(8,7) = Eip(8,~f + ere), and A ^ is the r-fold Kronecker product of A i . A ^ is a |C| r x |C| r matrix, where |C| is the size of the colour set. Furthermore, if | A i | > 0, then Q r = (Af 1 ) ^ Q r is an unbiased estimate of Q r .» Proof : We start the proof by expressing the equation EQr = A r Q r . We need to show . EQT{6) = Y,M6,l)Qr{l) (13) 7 for all 6. Note that 18 EQT{6) = E-J2v(^YiA)---V(SriYitr) n i^i = - £ %>(*!, E 1{*M = 7l} • • • K>) E 1{*> = 7r} 7 1 i = l 71 7r 1 " = -EEH^.l = 7l}̂ V(<5l,7l + ̂ 0---E1{^,r = -yr}E<p(8r,*fr + (TeT) n 1=1 Ti 7 r 1 n = - E E HZi,l = 7 l } • • • E H^r = 7r}^V(*l»7l + °"eO • • • £V(<*r,7r + " t=l 71 7r = X:^(7)Ai(cSa,70-"^i(^,70 7 = E n U ^ i ( A , 7 , ) a ( 7 ) 7 = E^(«,7)Qr(7) 7 = E4 r ) (« ,7 )Qr (7) 7 which is just equation (12). Therefore, £ ( A i r ) ) " 1 Q r = ^ ( A r ^ W Q r = ( A < r ^ A ^ Q , = Q r , and Q r = (Ax a ) ^ Q r is unbiased for Q r . » Proposition 2.1 shows how to obtain an unbiased and consistent estimate of Q r for any function <p(6,y) provided |Ai| ^ 0. We therefore have a family of unbiased and consistent estimates. In order to avoid the problem of inverting a large matrix, we can invert Ai first for the case of single-site neighbourhood, then we can use Definition 2.1 to build the A; 1 = (A"1)^. Example 2.2 : Suppose C = {0,1,2}, r = 3, and define ¥>(0,y) = 1 <p(hy) = y v>(2,y) = y2 19 and as usual 1 ^ •=i For instance, $ 3 ( 0 , 0 , 0 ) Q 3 ( i , i , i ) $ 3 ( 2 , 2 , 2 ) 1 n - V l - 1 - 1 = 1 1 n — £ *»M ' *«'.2 ' ^.3 7 t «=1 Obviously, Q3 is biased for Q 3 . Just notice, for instance, that $ 3 ( 0 , 0 , 0 ) = 1 indepen- dently from the records. By definition, Ax ( 6 , 7 ) = E(p(8,~f + ere), we have Ax(0, 7) = £ l = l Ai(l,7) = J5(7 + ae) = 7 Ax(2, 7) = £ ( 7 + at) 2 = 7 2 + cr2 By Proposition 2 . 1 , Q3 = (A_i)^C»3 is unbiased for Q 3 , where A 1 = 1 Ax(0,0) Ax(0,l) Ax(0,2) N Ax(l,0) Ax(l,l) Ai(l,2) ^ ( 2 , 0 ) Ax(2,l) Ax(2,2) J 1 0 1 2 and Ax"^ 1 _ 2 i =3 I 1 2 2 2 CT 2 2 - 1 —a 2 -1 2 1 / 20 Now, we express Q3 = (A_i)(3)Q3 in matrix form. ' $ 3 ( 0 , 0 , 0 ) > t $ 3 ( 0 , 0 , 1 ) $ 3 ( 0 , 0 , 2 ) ( 1 - £)2(f) 2 > V 2 ( l - - d ) V ) ( i - ^ ) 2 ( 2 ) / $ 3 ( 0 , 0 , 0 ) ^ $ 3 ( 0 , 0 , 1 ) $ 3 ( 0 , 0 , 2 ) \ 2.3 A Simple Representation for Qr Proposition 2 . 1 provides a class of unbiased estimates Qr of Qr. The proposed esti- mates have the form Qr = (A-1)(r)Qr, where Qr(6) = -J2^{6i,Yi^{S2,Yia)...9{6r,Yi,r). 7 1 ,=1 The matrix (A-1)(r) is the r-fold Kronecker product of Ai, and when we have a large colour set, C, and a big r, (A-1)(r) is a large matrix. In this section, we show that Qr can be expressed as 1 ^ Qr(6) = - £ m, >5.l)*(«2. Y*) • • • * ( « r . * > ) ni=l (14) for some \&(t\ y)'s. Proposition 2.2 : If we define Qr(S) = -Ev&,YM63,Yu)---tp(6T,Yi,r), then EQr = A(r)Qr, where  r^ is the r-fold Kronecker product of the bias matrix Ai, and by definition, Ai(£,7) = Etp{6,i + at) (Af0 is |C| r x |C]r). If |Ai| > 0, then 2 1 Q r = (A 1)^r^Qr is unbiased and consistent for Q r, and 7 '' can be expressed as Qr(6) = -it*(6uYi,1)---y(6r,Yi,r) n i=i where *(*,y) = E^V(7,y).« Proof: 7 '' 71 7r = £ • • • E f t • • • A i \ r ^ £ V ( T I , no • • • noi 71 7r  11  1=1 = ^E{E---Eft^(7l,ni)-"^r>(7r,nr)]} U 1=1 71 7 r = ^E{(E^7^(7i5ni))-:-(E^^(7r,nr))} n i = l 71 7 r = -fyil{8uYi,i)...^{8T,Yi<r)]. n i = i Note that wi th this representation, Q r is easily seen to be unbiased. Indeed, EV(6,-y + ae) = £ £ A j T V ( r , 7 ) T = E ^ f i T - ^ T . - y T = 1{* = 7}. 22 And so 1 ^ EQr{6) = — E D^(8i, Yiti)^(82, Yi>2) • • • \P(cy, Yi>r) 1 A n = - E H** = ^ } i { ^ . , 2 = s2}--. i{Zi<r = tsr> t=i i n = Qr(6). Note that since Qr(6) is an average of an r-dependent sequence, it converges to its mean. Therefore, Qr{6) is an unbiased and consistent estimate of Qr(6).» Example 2.2 (Continued) : Recall that ip(0,y) = l,c?(l,y) = y,ip(2,y) = y2. By equation (17), f *(o,y)' ( 1 " 2 - 3 1 2 2 1 ^ 2 '¥>(0,y) ^ * ( i , y ) = CT 2 2 -1 v ( i > y ) < * ( 2 > y ) > -<r» - 1 \ 2 2 1 2 y ^ v(2,y); Therefore, we can write mv) = ( i - y ) i + (^)y + (|)ya, ¥(0,y) = (cr2)l4,(2)y + (-l)y2, *(i,y) = (^H-Hy)y + (;;)y2, and so 1 A 03(0,0,0) = -E*(0,ni)*(0,na)*(0,^), n »=i $3(0,0,1) = -E*(0,ni)*(0,na)*(l.^). n i=l 23 $3(2,2,2) = -y£^(2,Yi,1)^!(2,Yii2)^(2,Yi,3). As stated in Proposition 2.2, such a new representation can be obtain generally.* In summary, we have the following general procedure for obtaining unbiased and consistent estimates for Qr: Let C = {ci, c2,..., ĉ }, and r > 0. Take any (p : C x R —• R, and then define A as the |C| x |C|(= k x k) bias matrix with elements As*, = E<p(6,i + ere). If |A| > 0, then define Qr(6) = if; *(*i,ni)*(fc.na) •••*(*, no where * ( W = £ ^ V ( W i Then QT(S) is an unbiased and consistent estimate of Qr(S). Example 2.3 : We use Figure 2.1 as the true and unobserved image, and Figure 2.2 as the noisy and observed image. The colour set C is {0,1,2}, r is the neighbourhood size, and cr = 0.50. The following tables shows how far away the biased Q r is from the true Q r, and how good and close the unbiased Q r is to the true Q r. 24 Table 2.1: Pattern distribution with r = 1, <p(6,y) = Indicator functions <5 QiV) Qi(S) Qi(S) 0 0.48340 0.47498 0.48505 1 0.42505 0.37836 0.42074 2 0.09155 0.14667 0.09421 Total 1.00000 1.00000 1.00000 Table 2.2: Pattern distribution with r = 1, <p(6,y) — Power functions 8 QM Qi(8) Qi(S) 0 0.48340 0.37644 0.48306 1 0.42505 0.22966 0.42380 2 0.09155 0.39390 0.09314 Total 1.00000 1.00000 1.00000 25 Table 2.3: Pattern distribution with r = 2, <p(8, y) =Indicator functions 6 Qi{6) W ) 0 0 0.47192 0.34601 0.46966 0 1 0.00842 0.11395 0.01315 0 2 0.00305 0.01501 0.00224 1 0 0.00842 0.11377 0.01274 1 1 0.41541 0.20496 0.40092 1 2 0.00122 0.05963 0.00709 2 0 0.00305 0.01520 0.00266 2 1 0.00122 0.05945 0.00667 2 2 0.08728 0.07202 0.08488 Total 1.00000 1.00000 1.00000 26 T a b l e 2.4: Pattern distribution wi th r = 2, <p(8,y) = Power functions 6 W) W) 0 0 0.47192 0.09996 0.46610 0 1 0.00842 0.06099 0.00452 0 2 0.00305 0.10460 0.00598 1 0 0.00842 0.06099 0.00480 1 1 0.41541 0.07745 0.42015 1 2 0.00122 0.12890 0.00000 2 0 0.00305 0.10460 0.00570 2 1 0.00122 0.12896 0.00000 2 2 0.08728 0.23355 0.09275 Total 1.00000 1.00000 1.00000 For r = 3 (linear 3-site neighbourhood), the columns 0,3(6), Qs(S), and Qs(6) are dis- played graphically as barplots in Figures 2.3a, 2.3b, 2.3c, 2.3d, 2.3e, 2.3f. Final ly , for r = 5 (cross-shaped neighbourhood), the columns Qs(6), Qs(6), and Qs(6) are also dis- played graphically as barplots in Figures 2.4a, 2.4b, 2.4c, 2.4d, 2.4e, 2.4f. We have seen that different neighbourhood shapes and sizes give different pattern distributions. They also give different estimates {z\, z2,..., zn}, because as mentioned before, the estimate of the colour at each site is a function of the data, Yi, available in a neighbourhood of that site. The estimate proposed by Meloche and Zamar (1994) requires estimates of pattern distributions, Qr(6), and a. Recal l from equation (4) that . _SgM»(Yt--g)qr(g) 27 where tf>a denotes the r-dimensional normal density with mean zero and covariance o~2IT, and 6C denotes the colour/pattern at the center of the neighbourhood. Figures 2.5a to 2.5h are the restored images based on different neighbourhood shapes and sizes, and different estimations of the pattern distributions.* 28 3 Joint Estimations of Q and a. 3.1 Introduction Meloche and Zamar (1994) propose estimating z,- by Z% Us MY,- - 6)Qr(6) ' where <5C denotes the colour/pattern at the center of the neighbourhood, and (f>„ is the normal density with mean zero and variance cr2. Furthermore, our proposed estimation of Qr involves cr, and we have been assuming that a is known. Equation (16) states that The subscript a indicates that Qr{6) involves a. But a is likely to be unknown and must be estimated from the noisy image. Ideally, we want to estimate both Qr and cr, but we will focus on the estimations of Qx and cr. Once a is estimated, Qr can also be estimated. For the rest of this thesis, we write Q instead of Qi. In this chapter, we propose some estimating equations for cr. 3.2 Estimation of a using Estimating Equation derived from E(^Yli=i Note that n t i ni=i = lj^E{Zi + cre)2 = -TE(z2 + 2ziae + a2e2) ni=i 29 S ni=l = v2+ ^62Q(8). Thus, if Q<r is any unbiased estimate of Q, E(o-2 - - E ̂ 2 + E = 0- (18) t = l a The subscript a is used to indicate that Q„ involves a. Now define A„(a) = a 2 - - X X + E 6" W) (19) and taking expection of equation (19), A(a) = EXn(a) = a2 - (a2 + £ 62Q(6)) + £ eS2£go(cT). (20) Note that according to equation (18), a is one of the root of A(a). Empirical evidence suggests that when Q is derived from indicator functions, A(a) has 2 roots, the smaller of which is a (irrespective of the colour set C). Theoretical results to that effect have not been reached yet. When Q,, is derived from power functions, and when C = {ci,c2}, equation (19) yields a closed-form estimate of a2 instead of a. 30 A(a) = a 2-(a 2 + c2g(c1) + c2g(c2)) + c2g(c1) + c2g(c2). By solving An(a) = 0, tr2 can be estimated as ra i=l C 2 - C i This is just the estimate of a2 proposed by Meloche and Zamar (1994) for the colour set {0,1}. It is unbiased and consistent. When |C| = 3, it can be shown that A(a) = 6 if is derived from power functions. then Q = A a Q, where Aa = i y) = y,<p(cs,y) Q(ci) = i n Q(C2) = 1 n n i^i Q{cz) = 1 n n , t i l 1 1 Cl c 2 c 3 ^ c{ + a2 4 +a2 4 + °' ) and A 7 J = |Aa ' (c2 - c3)(a2 - c2c3) (c2 - c3)(c2 + c3) - ( c 2 - c 3 ) > \ -(ci -c 3 ) (a 2 - c x c 3 ) -(ci - c3)(cx + c3) (cx - c3) v (ci - c2)(a2 - cic2) (ci - c2)(ci + c2) -(ci - c2) ) 31 where | A| = —(ci — c2)(ci — c3)(c2 — c3). We can write equation (20) in matrix form. Let X(a) = a2-(a2 + ,Z62Q(S)) + ^t62EQa(6) S 6 = a2-cr2 + '£62EQa(6)-Y;S2Q(S). s s Therefore, A(o) = a 2 - a 2 + ATA:1AaQ + ATQ = a2 - a2 + AT(A:1 Aa -I 3)Q, where I Vc 2 C 3 - C l C 2+cJ - C l C 3 / v C 2 C 3 - C l C 2 + c J - C l C 3 ) - ( C 2 C 3 - C l C 2 +cj - C l C 3 — C l C 3 + C l C 2 + C 2 C 3 - C ? , I / a 2 - < 7 * \ ^ - C 1 C 3 + C 1 C 2 + C 3 - C 2 C 3 - C i C 3 + C l C 2 + C 2 C 3 - C ^ / a 2 - c / 2 * - C l C 3 + C l C 2 + C ? - C 2 C 3 - C i C 3 + C l C 2 + C 2 C 3 - C ^ V - C l C 3 + C l C 2 + c i - C 2 C 3 / and -c2(a2 - a2) + c\{a2 - a2) C 2 C 3 — C aC 2 + q — CiC3 —CiC3 + CiC2 + c 2 c 3 — c| -clo*2-*2) -C1C3 + cic 2 + ci - C 2C 3 = -(a2 - a2). As a result, A(a) = a2 - a2 - (a2 - a2) = 0. Thus, when |C| = 3, A(a) defined by equation (20) is identically zero. To get an estimate of a2 when |C| = 3, one possibility is to derive a different estimating equation starting from higher moments of Y{. For example, we start from Y*. 32 3.3 Estimation of a using Estimating Equation derived from E(^YZ=\ Y4) By simple computation, E{-IZY4) = 3cr4 + 6<T2Es62Q(S) + XS64Q(6). (21) Th Thus, if Qo- is any unbiased estimate Q , E(3<r4 + 6<r2 J2 S2Qa{6) + £ 64Qa(8) - I £ Yt4) = 0. (22) S .8 7 1 t'=l Define Xn(a) = 3a4 + 6a2 £ * 2 < ? a ( < ! > ) + £ tf4^*) - - £ 1? (23) and taking expectation of equation (23), A(a) = 3a4 + 6a2 £ *2£0„(*) + £ ^.(tf) - £( - £ 1?). (24) 6 6 n i = l According to equation (22), a is one of the root of A(a). By equation (14), Oaiei) = D(c2 - C3)(a2 - c2C3) + (4 - 4)Yi - - & ( < * ) = T ^ D C c i - ^ - C ^ ^ l A l n , = l Then note that An(a) and A(a) defined by equations (23) and (24) are second degree polynomials in a2. As a result, A(a) has 2 roots. The smaller of which is a2. In general, a2 can be estimated as the smallest root of A„(a). In particular, when C = {0,1,2}, $•(0) = ^tli1 - + ( T ) Y i + ( \ ) Y A Qa(l) = iD(«2)l + (2)15 + (-1)1?] n ,=1 4.(2) = ^ D ( ^ ) i + (y)̂  + (̂ 2] 33 By solving An(a) = 0, we have 1 n 7 n «=i 6 N a2 + Q(l) + 40(2) - - ± \/(Q(l) + 4Q(2) - ^)2. We can conclude that for the colour set {0,1,2}, cr2 can be estimated as i n 7 «=i Although «T2 may not be unbiased, it is consistent. The estimate of a can be obtained by taking the square root of a2. In general, we can derive an estimating equation for cr2 with an arbitrary colour set in a similar fashion when Q r is derived from power functions. 34 4 Neighbourhood Shapes. We have seen in the previous examples that the bigger the neighbourhood size, the better the restoration performance. But this will no longer hold if the neighbourhood size is big while the image is small. Sometimes a good and small neighbourhood may result in better performance in restoration than a big and bad one. The performance of the estimates zi,...,zn can be measured by the average expected square error: 'AMSE = - JT E(Bi - Zi)2 n ,=i According to the theorem by Chan and Meloche (1995), AMSE = - Y, E(z{ - zi)2 = <T2(1 - o2h(na * Q)). (25) n,-=i <72(1 — o-2I0(r)a * Q)) can be obtained by numerical integration or Montecarlo. Iofoo- * Q) is the middle element on the diagonal of the Fisher information matrix oir]a*Q (recall that our vectors are indexed from 1 to r, so the middle element on the diagonal element is the (^2L)th element. r\a is the normal density with mean zero and variance cr2. Figure 3.1 is a 129 x 129 true image with alternating horizontal strips of 0, 1, and 2, and Figure 3.2 is the degraded version of the true image with a = 0.50. Table 3.1 provides the approximate AMSE (obtained by Montecarlo) for various neighbourhood sizes and shapes. In the table, AMSE^ = <72(1 - <72/0(77a * Q)), AMSE2 = <T2(1 - a2I0{qa * Qindicator)) AMSE3 = CT2(1 - (T2I0(Vc * Qpower)), 35 where Q is the true pattern distribution, Qindicator is the estimated pattern distribution based on the indicator functions, and Qp0wer is the estimated pattern distr ibution based on the power functions. T a b l e 4 .1 : AMSE for various neighbourhood sizes and shapes neighbourhood AMSEx AMSE2 AMSE3 linear 3-site nbhd (horizontal) 0.042 0.050 0.054 linear 3-site nbhd (vertical). 0.021 0.032 0.031 linear 5-site nbhd (horizontal) 0.013 0.042 0.080 linear 5-site nbhd (vertical) 0.0053 0.043 0.056 5-site nbhd (cross-shaped) 0.0025 0.045 0.043 linear 7-site nbhd (horizontal) 0.0042 0.092 0.125 Note that when the Qr(Sys are known, the cross-shaped neighbourhood results in a better performance wi th the lowest AMSE\ than the other neighbourhoods used here. In particular, it performs better than a larger neighbourhood which is the linear and horizontal 7-site neighbourhood. In pratice, we substitute QT(6ys for Qr(5)'s in equation (25) when we only have a noisy image. We choose the neighbourhood shape which gives the smallest AMSE. 36 When we use the estimates Qr(6~)'s in equation (25) for obtaining the AMSEi and AMSE3, the linear and vertical 3-site neighbourhood seems to be the best choice be- cause its corresponding $3(o")'s are relatively more accurate than the <5s(6)'s and <57(o")'s. When the neighbourhood size gets larger, the estimates Qr's become less accurate. There- fore, larger neighbourhood results in a worse restoration performance for our particular noisy image in Figure 3.2. If we have a larger image, the larger neighbourhoods may per- form better than the smaller ones because the estimates Qr for the larger neighbourhoods become more accurate. 37 5 Conclusions. By extending the idea of estimating Q r based on indicator functions by Meloche and Zamar (1994), we have developed a family of estimates of Q r which can be based on any arbitrary choice of a set of ip(6, y)'s. We start with any set of <p(6, y)'s which seems to have a simple structure. Then we define Qr(6) = ^EniM*2, Ik ) • • • <p(6r,YitT) n ,=1 which is biased for Qr{6). By applying the propositions stated before, we obtain a new set of \&(c5, y)'s, where *(*,y) = £A£r¥>(7,v), 1 such that Qr(6) = - £ n0*(*2, ^,2) • • • * ( * r , K > ) " ,=1 is an unbiased and consistent estimate of Qr(6). At the moment, we lack the theoretical results on judging which set of $(6, y)'s give the best and the most accurate estimate of Q r - We have addressed the problem of estimating a by proposing some estimating equa- tions for a. We have derived an estimating equation from E(^2~27=i Y?) f ° r a colour set with |C| = 2. By solving A„(a) = 0, we have obtained an estimate of a2 for any colour set with |C| = 2. But the estimating equation derived from 2~2?=i Y?) does not work for a bigger colour set when Q r is derived fron power functions. So we have derived another estimating equation from Z)"=i Y*) f ° r a colour set with |C| = 3. Again, by solving An(a) = 0, 38 we have obtained an estimate of a1 for any colour set with | C | = 3. To estimate a2 when | C | > 3, we can derive a different equation starting from a higher moment of Y{. In general, we estimate a2 by deriving an estimating equation for a2 in this fashion. The esimate may not be unbiased, but it is consistent. By taking the square root of a2, we obtain the estimate of a. When Q r is derived from indicator functions, the empirical evidence suggests that A(a) has 2 solutions for a," and the smaller of which is cr. Irrespective of the colour set C, estimating equation (18) always works. Therefore, we do not have to consider higher moments of Y,. But theoretical results have not been reached yet. 39 REFERENCES Anderson, T.W. (1984) An Introduction to Multivariate Statistical Analysis. Wiley. Chan, A., Meloche, J. (1995) Estimation of a Gaussian Mean and Image Restoration. To be submitted for publication. Meloche, J., Zamar, R. (1994) Black and White Image Restoration. The Canadian Journal of Statistics. 22, 3, 335-355. 40 Figure 2.2: Noisy Image (Observed) ^ ?, s ! * y h ' '« - < j k $ M m * * s V - ^ • 41 Figure 2.3: Pattern distribution with r = 3, and (i) (p(8, y) = Indicator functions (a) Q3(S) vs 8 (b) QZ{S) vs 8 (c) Qz{8) vs 8 42 Figure 2.3: Pattern distribution with r = 3, and (ii) <p(6, y) = Power functions SSS8888 m wwww Kawssg 888888 (e) Q3(6) vs 6 43 Figure 2.4: Pattern distribution with r = 5, and (i) y>(6, y) = Indicator functions (a) Q5(S) vs 6 LL—tn. & -—- » , ...»X.« (b) &(<!>) vs £ •*—* - -»•• t (c) vs 6 44 Figure 2.4: Pattern distribution with r = 5, and (ii) ip(6,y) = Power functions (d)Qs(6)vs6 (e) Q5(S) vs 8 t } (f) Q5(6) vs ,5 45 Figure 2.5: Restored Image, ip(6,y) = Ind ica to r Figure 2.5: Restored Image, <p{6,y) = Power Figure 3.1: True Image (0-1-2 strips) F igure 3.2: Noisy Image (0-1-2 strips) 48

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