2009 Seventh International Conference on Advances in Pattern Recognition
Best Basis Selection using Singular Value Decomposition
S. Esakkirajan
T. Veerakumar,
P. Navaneethan,
Lecturer, Department of I&CE, PSG College of Technology, Coimbatore, India.
[email protected]
Lecturer, Department of ECE, PSG College of Technology, Coimbatore, India.
[email protected]
Professor, Department of EEE, PSG College of Technology, Coimbatore, India.
[email protected]
Abstract— This paper presents a new idea of best basis selection through singular value decomposition. Wavelet and Wavelet Packet Transform are efficient tools to represent the image. Wavelet Packet Transform is a generalization of wavelet transform which is more adaptive than the wavelet transform because it offers a rich library of bases from which the best one can be chosen for a certain class of images with a specified cost function. Wavelet packet decomposition yields a redundant representation of the image. The problem of wavelet packet image coding consists of considering all possible wavelet packet bases in the library, and choosing the one that gives the best coding performance. In this work, Singular Value Decomposition is used as a tool to select the best basis. Experimental results have demonstrated the validity of the approach.
II.
Wavelet packet transforms are more adaptive than the wavelet transform because they offer a rich library of bases from which the best one can be chosen for a certain class of images. The extra adaptivity of the wavelet packet is obtained at a price of added computation in searching for the best wavelet packet basis. The cost function is an essential part of the best basis algorithm. Initially Coifman and Wickerhauser [3] suggested using a version of entropy as the cost function for the best basis selection. Their algorithm requires the key restriction of additivity for the information cost function. Ramachandran and Vetterli proposed ratedistortion optimization technique [4] for the best basis selection. The entropy as well as the rate-distortion method of best basis selection work by fully decomposing all subbands to a predefined depth. Then the best basis is found by pruning this decomposition tree in a recursive bottom-up fashion. The entropy-based technique prunes the tree to minimize the overall estimated entropy of the wavelet packet structure. The rate-distortion method is given a particular target bit rate for the image and prunes the tree to minimize the distortion of the image. In this work, singular value of the subband is used as a tool to select the best basis.
Keywords- Wavelet Packet Transform; Singular Value Decomposition; Best basis;
I.
INTRODUCTION
Wavelet packet transform (WPT) is a generalization of the dyadic wavelet transform (DWT) that offers a rich set of decomposition structures. WPT was first introduced by Coifman et al. [1] for dealing with non-stationarities of the data. Wavelet transform recursively decomposes sub-images belonging to the low frequency channels. For images that contain rich texture information, wavelet transform will not perform well, and improvements can be found if the search is made over all possible binary sub trees instead of using the pyramid wavelet tree. Textures are composed of a large number of more or less ordered similar patters, giving rise to a perception of homogeneity. Texture is typically a composition of mid and high frequency components; hence wavelet decomposition does not capture its typical structure. Fingerprint images can be seen as texture patterns of flow orientations with sharp discontinuities [2]. For fingerprints, the analysis by the wavelet transform is inefficient because it only partitions the frequency axis toward the low frequency, whereas the wavelet packet transform decomposes even the high-frequency bands which are kept intact in the wavelet transform. This paper is organized as follows: Section II deals with best basis selection, Section III gives an introduction to Singular Value Decomposition, the proposed algorithm is given in Section IV, experimental results are given in Section V and finally conclusions are drawn in Section VI. 978-0-7695-3520-3/09 $25.00 © 2009 IEEE DOI 10.1109/ICAPR.2009.13
BEST BASIS SELECTION
III.
SINGULAR VALUE DECOMPOSITION
The Singular Value Decomposition (SVD) is a powerful technique in many matrix computations and analyses. Using the SVD of a matrix in computations, rather than the original matrix, has the advantage of being more robust to numerical error. The SVD exposes the geometric structure of a matrix, an important aspect of many matrix calculations. The SVD of a M × N matrix A is given in (1) A = UΣV T (1) where U is an M × M orthogonal matrix, V is an N × N orthogonal matrix and Σ is an M × N diagonal matrix. The columns of U are eigen vectors of AA T , the columns of V are eigen vectors of A T A [5]. The diagonal entries of Σ are the normalized singular values which are the square roots of nonzero eigenvalues of both AA T and A T A . The matrix A is given in (2) A = σ1 u 1 v1T + σ 2 u 2 v T2 + .......... + σ n u n v Tn (2)
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Among the number of bands, the best bands are selected through singular value of the subband. Thus, the singular value is used to select the best basis. Here, we have not taken the Peak Signal to Noise Ratio (PSNR) into consideration, because to determine the PSNR, the image has to be reconstructed each time, which is time consuming. The flowchart of the proposed algorithm is shown in Fig. 1. In the figure, IWPT stands for Inverse Wavelet Packet Transform, CR stands for Compression Ratio and Level refers to level of decomposition.
The singular values of σ i are ordered as σ1 ≥ σ 2 ≥ ........... ≥ σ n . The number of singular values is equal to the rank of the matrix A . The major advantage is that the first singular value σ1 is greater than the remaining singular values. In the diagonal matrix Σ , the singular values are arranged in descending order along the diagonal. Thus, by observing the first singular value, it is possible to find the energy contained in the subband. If the energy is more, then the band is taken into consideration for further decomposition. Thus SVD can be used as a tool for selecting the basis. IV.
Lena, Barbara and Finger images of size 256 x 256 were used for testing our algorithm. Lena image do not contain large amounts of high-frequency or oscillating patterns [6]. Barbara image exhibits large amounts of high-frequency and oscillating patterns. Finger print contains a large amount of texture. Thus, the algorithm is tested for low-frequency, midfrequency and high-frequency images. The different wavelet filters taken into consideration are Haar, Db4, La8, biorthogonal 9/7 [7] and 5/3. Test images are subjected to two level of wavelet packet decomposition, because at the first level of decomposition, there is no difference between wavelet and wavelet packet.
PROPOSED ALGORITHM
The objective is to select the best basis using SVD. First the SVD of each subband of the WPT is taken. By comparing the first singular value of each subband it is possible to find which subband has high energy content. The first singular values of the subbands are arranged in descending order. Test Image
WPT of test image
TABLE I. Compression Ratio 1:16 1:8 1:4 1:2.67 1:2 1:1.45 1:1.23
Apply SVD to each subband of WPT Arrange the first singular values of all subbands in descending order
RESULTS FOR LENA IMAGE
PSNR in dB 24.15 26.37 28.61 31.44 33.37 37.44 41.77
Best Tree [1] [1,2] [1,2,3,5] [1,2,3,5,6,4] [1,2,3,5,6,4,8,9] [1,2,3,5,6,4,8,9,12,11,16] [1,2,3,5,6,4,8,9,12,11,16,7,10]
The experimental result of Lena image for different compression ratio using Haar wavelet is shown in Table I. Table I shows the PSNR for the desired compression ratio and the best tree used for the reconstruction of the image. The number in the best tree column indicates the bands used for reconstruction. From the table it is obvious that as the number of bands used for reconstruction increases, the quality of the reconstructed image increases which is reflected in the PSNR value.
Fix the compression ratio (CR)
No. of subbands =
EXPERIMENTAL RESULTS
V.
4 Level CR
Selection of best subbands through singular value
TABLE II.
Take IWPT Reconstructed image Figure 1. Flowchart of the proposed algorithm.
The compression ratio is used to decide the number of subbands to be taken into consideration for reconstruction.
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RESULTS FOR BARBARA IMAGE
Compression Ratio 1:16
PSNR in dB 21.24
[1]
1:8
22.16
[1,2]
1:4
23.46
[1,2,3,7]
1:2.67
24.57
[1,2,3,7,5,13]
1:2
25.19
[1,2,3,7,5,13,6,16]
1:1.45 1:1.23
29.17 31.64
[1,2,3,7,5,13,6,16,4,8,15] [1,2,3,7,5,13,6,16,4,8,15,12,11]
Best Tree
TABLE III.
RESULTS FOR FINGERPRINT IMAGE
Compression Ratio 1:16
PSNR in dB 17.77
[1]
1:8
19.98
[1,3]
1:4
22.09
[1,3,11,2]
1:2.67
24.66
[1,3,11,2,9,4]
value increases. Table III shows the performance of the proposed algorithm for finger print using Haar wavelet. The number of level of decomposition performed on the test image is two.
Best Tree
1:2
26.02
[1,3,11,2,9,4,5,6]
1:1.45
29.29
[1,3,11,2,9,4,5,6,12,10,7]
1:1.23
32.44
[1,3,11,2,9,4,5,6,12,10,7,8,16]
TABLE IV. Wavelets Haar
Table II shows the performance of the proposed algorithm for Barbara image using Haar wavelet. The number of level of decomposition is two. From the table, it is obvious that with decrease in compression ratio, the PSNR
BEST TREE FOR DIFFERENT WAVELETS PSNR in dB 26.02
Best Tree [1,3,11,2,9,4,5,6]
Bi9
28.48
[1,3,11,2,4,9,12,6]
Bi5
27.30
[1,3,2,9,5,11,4,10]
La8
28.44
[1,3,11,2,4,9,12,6]
Db4
27.86
[1,3,11,2,9,4,6,12]
PSNR = 28.61
PSNR = 31.64
PSNR = 26.02 Figure 2. Original, reconstructed test images and the corresponding best basis
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The performance of the proposed algorithm for different wavelet for fingerprint image is given in Table IV. From the table it is clear that the quality of the reconstructed image depends on the choice of the wavelet. By observing the best tree of fingerprint, it is possible to conclude that fingerprint contains information in both low and high frequency region. The original and the reconstructed test images for a specified PSNR value is shown in Fig. 2. The best tree is illustrated through a red line. Two level of decomposition is performed on the test images. From the figure it is obvious that for Lena image, the best basis is towards left which indicates that maximum information is in low frequency region, where as in Barbara image, the best basis is equally spread in left and right side of the tree which implies that Barbara image contains high frequency oscillatory patterns. For fingerprint image, the best basis is spread on the left and right side of the tree which implies that fingerprint contains both low frequency and high frequency information.
ACKNOWLEDGMENT The authors wish to thank the management and principal of PSG College of Technology for their encouragement. The authors wish to thank Dr. N. Malmurugan and Dr. R. Sudhakar for their valuable suggestions. REFERENCES [1]
[2]
[3]
[4]
VI. CONCLUSION The best basis selection based on SVD is given in this paper. The algorithm is tested on different images. The algorithm is simple and only comparison of singular values between subbands is required to select the best basis for a specified compression ratio. The SVD approach to best basis selection is robust, simple, and easy to implement.
[5] [6]
[7]
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