Chapter 3 OPTIMAL DECORRELATION AND THE KLT 3.1 Introduction Karhunen-Loeve transform (KLT) is a unitary transform that diagonalizes the covariance or the correlation matrix of a discrete random sequence. [3.5] This decorrelation property is desirable because processing (quantization, coding etc) of any one coefficient in the KLT domain has no direct bearing on the others. Also, as will be shown later, it is considered as an optimal transform among all discrete transforms based on a number of criteria. It is, however, used infrequently as it is dependent on the statistics of the sequence i.e. when the statistics change so also the KLT. Because of this signal dependence, generally it has no fast algorithm. Other discrete transforms such as cosine transform (DCT), (see chapter 5) even though suboptimal, have been extremely popular in video coding. The principal reasons for the heavy usage of DCT are 1) it is signal independent 2) it has fast algorithms resulting in efficient implementation and 3) its performance approaches that of KLT for a Markov-1 signal with large adjacent correlation coefficient. In spite of this, KLT has been used as a bench mark in evaluating the performance of other transforms. It has also provided an incentive for the researchers to develop signal independent (fixed) transforms that not only have fast algorithms, but also approach KLT in terms of performance. This chapter defines and develops the KLT and also lists the performance criteria. It is also extended to two dimensional random signals. It concludes with applications which illustrate the decorrelation property and its significance in image compression. 3.2 Karhunen-Loeve Transform Let R be the N N correlation matrix of a random complex sequence x = x1 , x2 , x3 ,, x N T given by x1 x 2 x3 R = E [xxH ] =E . . . x N x1 x1* * x 2 x1 R = E x3 x1* .... x x* N 1 * * * x1 x 2 x3 x *N x1 x 2* x1 x3* .... .... .... x 2 x 2* x3 x 2* x 2 x3* x3 x3* .... .... .... .... .... .... .... .... x N x 2* x N x3* .... .... .... .... .... .... * * x1 x *N E x1 x1 E x1 x N x 2 x *N * = x3 x N .... x N x N * E x N x1* E x N x *N 1 where E is the expectation operator and E x j x *j is the autocorrelation of x j and E x j xk* is the crosscorrelation between x j and xk , j k., Note that R is Hermitian. Let the unitary matrix which diagonalizes R be defined as such that 1 = H , H = , H R = 1 R = , (3.1) = Diag. 1 , 2 , k ,, N . Here, i , i 1,2,3,...N are the eigenvalues of R. is called the KLT matrix and it decorrelates the random sequence x. This can be seen when the forward and inverse KLT are considered. Let y be the forward transform of x y= 1 x = H x and let the inverse transform of y be x = y (3.2) where y = y1, y 2 ,..., y N T represents the random sequence in the transform domain. The correlation matrix for y is then E yy H = E H xx H (3.3) = H E xx H = H R = It is clear from (3.3) that the random sequence y has no cross- correlation. In other words, x has been decorrelated by the KLT matrix . Such a transform is also called principal component or Hotelling transform. It is a statistically optimal transform. All other transforms (suboptimal) are evaluated against this benchmark, KLT. Eigenvalues and eigenvectors of R To show the signal dependence of KLT, we consider the eigenvalues and eigenvectors of R. From (3.1) H R = or R = (3.4) where 1 2 3 ... N , 2 and i i1 i 2 i 3 ... iN T i 1,2,3, , N being the ith column of .Writing the right hand side of (3.4) in full, we have 11 12 13 1 N 21 22 23 ... ... 2N ... ... ... 1 N 1 N 2 N 3 NN 11 21 22 12 = 1 13 , 2 23 1N 2 N 2 3 . . 31 32 , 3 33 3 N . N (3.5) N 1 N 2 , , N N 3 NN or R i = i i , i 1,2,3, , N (3.6) It is clear form (3.6) that i ’s are real and positive the eigenvalues of R and i ’s are the corresponding eigenvectors. When the eigenvalues i ’s are arranged in descending order so that 1 2 N , the auto-correlations of transformed signal vector are arranged in descending order. Similar considerations based on the diagonalization of the covariance matrix of a random sequence x produce the KLT based on the covariance matrix. Let the covariance matrix in KLT domain be E y y y y H 3 y1 y1 y2 y2 = E y N yN * * * y1 y1 , y 2 y 2 , , y N yN (3.7) It is desired that this covariance matrix be diagonal, i.e. E y - y y y H = = diag. 1 , 2 ,, N Here y = E y= E y1 , E y 2 ,, E y N = y1 , y 2 ,, yN is the mean of T the random vector y. This diagonalization can be achieved if the vector y y is related to the signal vector x x by a unitary tansformation such that y x 1 x y where x E x and is the unitary transform matrix. It can be seen that if is made up of the eigenvectors for the data covariance matrix E x x x x R ' , or R i i i , H = R , so that i 1,2,3, , N where ' 1 R' then, E y y y y =E 1 x x x x H = 1 E x x x x H H 1 R (3.8) It is noted that although and play the same role of diagonalization, they are in general different unless x is a zero mean random vector. 3.3 Application of KLT in data compression As KLT diagonalizes a correlation matrix or covariance matrix, it is possible to represent an N-dimensional random vector x by only some of its coefficients in the KLT domain with negligible error. It is only logical to select m out of the N KLT coefficients that represent the first m largest eigenvalues. By quantizing and coding these m coefficients, x can be reconstructed with minimal error. This is the essence of data compression or bandwidth reduction i.e., the m KLT coefficients require less number of bits compared to that required for x. This role of the KLT in data compression can be illustrated in a general maximum variance zonal filter as shown in Figure 3.1. 4 x ŷ m y Î m A B x̂ Fig. 3.1 Maximum variance zonal filter The data vector x is transformed by the operation block A into the transform domain vector y which undergoes compression by the operator block Î m and then transformed by B back into the data domain as x̂ . The operators A, Î m and B are selected to minimize the mean square error(mse) of x x̂ . In the case of simple compression Î m is a N N 2 diagonal matrix 1 m N with the first m diagonal elements as ones and remaining diagonal elements, as zeros i.e. Î m Diag. 1,1,,1,0,0,,0 . The random vector x = x1 , x2 , x3 ,, x N is mapped into y by the orthogonal matrix T A i.e., y Ax y1 , y2 , y3 ,, y N T T yˆ m Iˆm y y1 , y2 , y3 ,, ym ,0,0,0,,0 T xˆ Byˆ m xˆ1 , xˆ2 , xˆ3 ,, xˆ N (3.9a) (3.9b) (3.9c) where B is another orthogonal matrix. It can be shown that the mse between x and x̂ is minimized for a given m when A = H and B A 1 provided the columns (eigenvectors) of are arranged, so that the corresponding eigenvalues are in descending order 1 2 N . This implies that A and B correspond to KLT and inverse KLT respectively. Hence the mean square error (mse) is given by mse = 1 N E x k x€k 2 N k 1 N 2 = E yk = k m1 (3.10) N k m 1 k 5 since mse is invariant under unitary transformation. This implies that of all the discrete orthogonal transforms, KLT achieves the minimum mse when only a subset of the m KLT coefficients are retained. The remaining N-m coefficients representing small eigenvalues are set to zero. This is the key to data compression. In general, KLT is considered to: 1. pack the most energy in the least number of KLT coefficients; 2. minimize the mse between the original and reconstructed signal for a given number of coefficients; 3. achieve the minimum rate for rate-distortion function, among all unitary transforms; 4. decorrelate the signal in the transform domain. In view of these properties, KLT is applicable to pattern recognition, classification and bit rate reduction (compression) by retaining only the first m KLT coefficients i.e. y1 , y 2 ,, y m that correspond to the m largest eigenvalues out of the N coefficients. The KLT basis functions (eigenfunctions) for a Markov-1 process with 0.95 and N=16 are shown in Fig.3.2 (See Prob. 3.5) Fig. 3.2 KLT basis functions for N=16 and 0.95 for a Markov-1 signal 6 3.4 KLT for 2D random field KLT developed for 1D random field can be extended to the 2D case. This has applications in processing of 2D signals such as multispectral imagery. For simplicity, assume the square random field to be real and represented by x11 x 21 x31 x N 1 x12 x13 x 22 x 23 x32 x33 xN 2 xN 3 x1N x 2 N x3 N . x NN (3.11) Define the N 2 N 2 correlation matrix as follows: E uuT , (3.12) where u is the N 2 1 column vector obtained from the lexicographic ordering of i.e., u = x11 x12 x1N x21 x22 x2 N x N1 x N 2 x NN T . By using u, a one–dimensional random field, to represent the N N image, the diagonalization of its correlation matrix defined in (3.12) can be considered as in Section 3.2. The problem, however, quickly becomes formidable for N of moderate size, since the diagonalization problem to produce the eigenvectors for the KLT is of dimension N 2 N 2 . The relevant equation is k k k k 1,2,, N 2 (3.13) where k and k are respectively the eigenvalues and eigenvectors. k ’s form the columns of the KLT matrix for this 2-dimensional random field. In terms of the pixels x m ,n in the 2-dimensional image, (3.13) becomes E x N N m 1 n 1 m ,n , xm,n k ,l m, n (3.14) k , l , m, n 1,2,, N . = k ,l k ,l m, n , By assuming separable statistics the development of 2D KLT can be considerably simplified. Under this assumption, the row and column statistics are considered completely independent 7 and identically distributed. Hence the correlation matrix and the eigenfunctions become separable as shown in the following, E xm,n ; xm,n E x m,n ; x m,n E x m,n ; x m,n (3.15) and kl m, n 1 mk 2 nl m, m, n, n, k , l 1,2,, N . 1 and 2 are factor matrices in the KLT matrix due to the separable statistics of rows and columns. In fact, the diagonalization problem of the N 2 N 2 correlation matrix can now be separated so that we have: and 1 2 , (3.16a) 1 2 , (3.16b) 11 1 H 1 , (3.17a) where and 2 2 2H 2 . (3.17b) Equations (3.16a) through (3.17b) represent two N N diagonalization problems. 1 and 2 are N N diagonal matrices whose diagonal elements are the eigenvalues of the N N correlation matrices 1 (based on row statistics) and 2 ( based on column statistics) respectively. The symbol stands for Kronecker product of two matrices i.e. A B C a 11 B a 12 B . . . a 1n B a 21 B a 22 B . . . a 2n B --- a m1 B --- --- a m2 B . . . (3.18) --- a mn B C is a matrix of size mp nq when the matrices A and B are of sizes m n and p q respectively. The KLT of u is i.e. H u 1H 2H u 8 (3.19) In conclusion, by modeling the image autocorrelation by a separable function ( independent row and column statistics), diagonalization of an N 2 N 2 matrix is considerably simplified into diagonalization of two N N matrices 1 and 2 . Obtain basis images for KLT, p=0.9543, N=8. Assuming that the statistical properties along row and along column are independent. A first order Markov process is defined by 1 p R p2 p N 1 p p p2 p3 1 p p2 p 1 p N 2 p N 1 p N 2 p N 3 1 Obtain eigenvectors of the correlation matrix R and sketch the basis images based on the eigenvectors. Results: KLT basis images when p=0.9543 9 rou=0.95; N=8; for i=1:N for j=1:N R(i,j)=rou^(abs(j-i)); end end [V,D]=eig(R); for i=1:N for j=1:N B=V(:,i)*V(:,j)' subplot(N,N,(j-1)*8+i); imshow(B); end end 3.4 Applications Although the generation of KLT involves estimating correlation/covariance matrices with their diagonalization leading to eigenvalues and eigenvectors, a perusal of the references under KLT indicates that it has found applications in image compression [3.10, 3.19], multispectral image compression [3.26, 3.27, 3.28], image segmentation and indexing [3.46], recursive filtering [3.12, 3.30], image restoration [3.24], image representation, recovery and analysis [3.42], multi layer image coding [3.28], neural clustering [3.3], speech recognition [3.41], speaker recognition [3.47], speaker verification [3.4], feature selection [3.45], texture classification [3.60], image and video retrieval[3.50, 3.59, 3.61, 3.62, 3.63, 3.64] etc. Of particular significance is the paper by Saghri, Tescher and Reagan [3.26] wherein the KLT is applied in decorrelating across spectral bands followed by JPEG (Joint Photographic Experts Group) algorithm. They were able to produce a range of compression ratios (CR) starting with near lossless result at 5:1 CR to visually lossy results beginning at 50:1 CR. An adaptive approach in which the covariance matrix is periodically updated based on the terrain (water, forest, cloud, ice, desert, etc.) is utilized in achieving these high compression ratios. It is only appropriate at this stage to describe this application in detail. 10 Multispectral images (both satellite and airborne) exhibit a high degree of spatial and spectral correlations. The proposed scheme (Fig. 3.3) involves 1D KLT to decorrelate across spectral bands followed by the JPEG algorithm (This involves 2D DCT of spectrally decorrelated images for spatial decorrelation.) Fig. 3.3 Terrain-adaptive compression block diagram [3.26] IEEE 1995. In this experiment 16 unequal multispectral bands (images) covering the visible through infrared regions (0.36 to 12.11 micron wavelength) acquired by a multispectral scanner (M7 sensor) (16 band Airfield test image set) are partitioned into sets of nonoverlaping images i.e., sub-block sets. The multispectral sub-block sets are used to obtain the covariance matrices, (Fig.3.4) eigenvalues and the eigenvectors. The eigenplanes (spectrally decorrelated images) are formed by matrix multiplication of the sub- block set and basis functions (Fig. 3.5) 11 Fig. 3.4 Correlation coefficient matrix [3.26] IEEE 1995 Fig. 3.5 Removing spectral correlation via a Karhunen-Loeve transformation [3.26] IEEE1995 The effectiveness of KLT can be observed in Fig 3.6 wherein the first nine spectrally decorrelated eigen-planes of the 16 bands are shown. The first two and three eigen-planes have more than 80% of the energy of the test set. Hence the remaining eigen-planes can be coarsely quantized (some of them can even be dropped) resulting in bit rate reduction. Another measure of compression capability is the rapid decrease in the eigen-values (variances of the eigen-planes). This is evident from the variance distribution of the test images (Fig. 3.7) [3.26]. In fact, the superiority of KLT can be further observed in Fig 3.8 12 wherein the variance distributions for the KLT and DCT are compared . Large variances for DCT imply more bits to code those images. Further gains in spectral decorrelation via KLT have been achieved by using a terrain adaptive approach. As the multispectral images exhibit a number of different terrains (water, forest, cloud, ice, desert, etc.) the covariance matrix and eigen-planes (hence the KLT) are updated frequently. This, of course, involves additional complexity and increased overhead. Overall bit-rate reduction is a cumulative result of decorrelation of spectral bands by KLT followed by implementing the JPEG algorithm [3.54] on the spectrally decorrelated eigen-images. Fig. 3.6 Eigen images of the test image set (first of the total 16) [3.26] IEEE 1995 13 Fig. 3.7 Ordered variances of the eigen images [3.26] IEEE 1995 Fig. 3.8 KLT versus DCT for spectral decorrelation [3.26] IEEE 1995 Another application of KLT is in video segmentation, classification and indexing [3.46]. This is useful in random retrievals of video clips from large data bases. This approach for automatic video scene segmentation and content-based indexing is very robust and reduces the potential for fast scene change detection. The principal component analysis extracts effective discriminating features from the reduced data set that can be reliably used in video scene change detection, segmentation and indexing tasks. Summary Both 1D and 2D Karhunen-Loeve transforms (KLT) are defined and developed. Their properties are outlined. By assuming row and column independent statistics, generation and implementation of 2D-KLT are simplified into two 1D-KLTs. In spite of its computational complexity it is utilized in specific fields such as multispectral imaging. Also, KLT serves as a benchmark in evaluating other discrete transforms. A specific application wherein multispectral imagery has been decorrelated is illustrated. 14 Exercises 3.1 Show that when eigen-vectors (columns) of are rearranged, the corresponding eigen-values rearrange accordingly. 3.2 See Fig. 3.1 It is stated that the mse between x and x€ is minimized when A and B N correspond to KLT. Prove this. Show that this mse is k m 1 k . 3.3 Show that the sum of variances of an N-point signal under orthogonal transformation is invariant. Hint: Given y = Ax where A* A 1 , x = x1 , x2 ,, x N T data vector and y T = y1 , y 2, , y N transform vector. Show that T N k 1 ( 2x kk 2x kk N 2y k 1 kk and y2kk are variances of x k and y k respectively.) 3.4 The rate-distortion function RD in bits/sample for a specified distortion D is defined as RD = D= 1 N 1 2 max 0, log 2 ~kk / N k 1 2 1 N 2 min , ~kk . N k 1 ~ 2 is the variance of the kth coefficient The parameter is determined from D. kk in any orthonormal transform domain. Show that among all the unitary transforms, KLT yields the minimum rate R D . 3.5 For a first order Markov process defined by 1 R = 2 N 1 3 N 1 2 N 2 N 3 1 2 1 1 15 where is the adjacent correlation coefficient, show that the eigenvalues are 1 2 k 1 2 cos k 2 k = 1, 2, ..., N where k are the real positive roots of the transcendental equation tan N = 1 sin 2 cos 2 2 cos for N = even. (Similar result is valid for N odd.) Show that the m k th element of the KLT matrix is mk 2 N k 1/ 2 N 1 k , sin k m 2 2 m, k 1,2,, N 3.6 Obtain the eigenvalues and sketch the eigenvectors for an order-1 Markov process with = 0.9 and N = 16 3.7 Repeat Prob. 3.6 for =0.85. 3.8 Assume separable statistics, obtain and sketch eigenimages based on Probs. 3.6 and 3.7. 3.9 Derive (3.19). 3.10 Simulate the automatic video scene segmentation, classification, indexing and retrieval schemes based on the techniques presented in [3.46] using some test sequences. 16 References on KLT 3.1. D.B. Ponceleon el al, “Transform coding for low bit rate applications,” IS&T/SPIE Symposium on Electronic Imaging: Science & Technology, Vol. 2187, pp. San Jose, CA, Feb.1994. 3.2. I.S. Reed and L.S. Lan, “ A fast KLT for data compression,” IEEE Trans. SP, (under review). Also published in SPIE/VCIP, Vol.2094, pp. Cambridge, MA, Nov. 1993. 3.3. G. Martinelli, L.P. 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