A Comparative Study of Non-separable Wavelet and Tensor-product. Wavelet; Image Compression

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1 Copyright c 007 Tech Science Press CMES, vol., no., pp.91-96, 007 A Comparative Study o Non-separable Wavelet and Tensor-product Wavelet in Image Compression Jun Zhang 1 Abstract: The most commonly used wavelets or image processing are the tensor-product o univariate wavelets, which have a disadvantage o giving a particular importance to the horizontal and vertical directions. In this paper, a new class o wavelet, non-separable wavelet, is investigated or image compression applications. The comparative results o image compression preprocessed with two dierent kinds o wavelet transorm are presented: (1) non-separable wavelet transorm; () tensor-product wavelet transorm. The results o our experiments show that in the same vanishing moment, the non-separable wavelets perorm better than the tensor-product wavelets in dealing with still images. Keyword: Non-separable Wavelet; Tensorproduct Wavelet; Image Compression 1 Introduction Univariate wavelets have played an important role in signal processing since wavelet expansions became more appropriate than conventional Fourier series to characterize the local behavior o non-stationary signals [Antonini M, Barlaud M, and MathieuP, et al (199), Mira Mitra, S. Gopalakrishnan (006), Duddeck, Fabian M.E (006)]. To apply wavelet methods to digital image processing, bivariate wavelets have to be constructed. The most commonly used method is the tensor product o univariate wavelets. This construction leads to a separable wavelet, which has a disadvantage o giving a particular importance to the horizontal and vertical directions [W. He and M. J. Lai (000)]. There 1 Inormation Engineering College, Guangdong University o Technology, Guangzhou, , China, zhangjun7907@hotmail.com has been a growing research interest in the area o construction o non-separable wavelets over the past ew years. Much eort has been made on constructing non-separable bivariate wavelet. For example, J.Kovacevic and Vetterli studied properties o multidimensional nonseparable wavelets and numerically constructed examples o continuous non-separable compactly supported bivariate wavelets [J.Kovacevic and M.Vetterli (199)]. Cohen and Daubechies generalized the method [I.Daubechies (1988)] to construct non-separable bidimensional (discontinuous) compactly supported wavelets [A. Cohen, I.Daubechies (1993)]. Both Cohen-Daubechies s wavelets and J.Kovacevic -Vetterli s examples are based on the dilation matrix [ 1 1. Also, Ming- Jun Lai constructed bivariate non-separable compactly supported orthonormal wavelets based on the commonly used uniorm dilation matrix [ 0, which have 1-order vanishing moment and linear phase [W. He, M. J. Lai (000)]. David stanhill and Yehoshua Y.Zeevi [D. Stanhill and Y.Zeevi (1996)] constructed the non-separable orthonormal wavelets based on the commonly used uniorm dilation matrix [ 0, which holds higherorder vanishing moment. In this paper, our classes o non-separable wavelet ilter banks with dierent properties are studied or image compression. The irst one is Lai s non-separable wavelet based on the dilation [ 0 that holds 1-order vanishing moment and linear phase. The second one is David stanhill s non-separable orthonormal wavelet based on the dilation [ 0 that holds higher-order vanishing moment, but does no have the linear phase. The third one is the non-separable orthonormal wavelet based on the dilation [ 1 1 that holds 1-order vanishing moment. The ourth one is the non-separable orthonormal wavelet based on

2 9 Copyright c 007 Tech Science Press CMES, vol., no., pp.91-96, 007 the dilation [ 1 1 that holds higher-order vanishing moment. For the purpose o comparison, we also implemented the image compression method using tensor-product o univariate wavelets (CDF53, DB97). Wavelet Transorm in Two Dimensions: Non-separable Wavelet and Tensor Product o Univariate Wavelet There are various ways to extend one-dimensional (1-D) wavelet transorm to two dimensions. The simplest way to generate a -D wavelet transorm is to apply two 1-D transorms separately. Thus, image decomposition can be computed with separable iltering along the abscissa and ordinate, by using the same pyramidal algorithm as in the 1- D case. As shown in Figure. 1, this separable transorm (ST) called tensor-product decomposes images with a multi-resolution scale actor o two, providing one low-resolution subimage and three spatially oriented wavelet coeicient subimages at each resolution level. Another way o extending wavelet transorm to higher dimensions is to use non-separable ilters. The -D MRA (multiresolution analysis) with a dilation { } matrix D is a ladder o closed subspaces Vj j Z, which approximates L (R ) and satisies {0}...V 1 V 0 V 1... L (R ) (x) V j 1 (Mx) V j φ V 0 s.t. Where, the set {φ(x k)} k Z is an orthonormal basis or V 0. The unction φ(x) is called scaling unction and since V 0 V 1, φ(x) has to be the solution o a dilation equation o the orm φ(x)= k Z Hφ(Dx k) The associated wavelet is then derived rom the scaling unction by the ormula ψ i (x)= k Z G i φ(dx k) i = 1,...,det(D) 1 As shown in Figure., the commonly transorms o non-separable wavelet are quincunx transorm (QT) and separable sampling wavelets transorm (FST) with our bands. The ormer bases on the dilation matrix D = [ 1 1, and the latter bases on the dilation D = [ 0 and uses non-separable and nonoriented ilters (M = det(d)). 3 The Image Compression Scheme The standard process o image compression is as shown in Figure 3. Where, Y denotes the wavelet coeicients that are obtained by perorming wavelet transorm to image data. A compression scheme can be designed as below [Wang Ling and Song Guo-xiang (001)]: 1. Perorm 8-bit Scalar quantization o Y. Let D e j denote the wavelet coeicients o part e(e = H denotes horizontal; e = V denotes vertical; e = D denotes diagonal). δ j e denotes the threshold. { D e j,δ = D e j, when abs(de j ) > δ j e 0, otherwise 3. Compute the entropy od e j, which is denoted ast e j. t e j = k p(d e j,δ (k,l))log p(de j,δ (k,l)) l p(d e j,δ (k,l)) is the probability o pixel (k,l) in D e j,δ. Compute the compression rate as ollowing: /{ 1 CR = 8 det(d) (a 1t1 H +a t1 V +a 3 t1 D )+... } 1 + (det(d)) j (a 1t H j +a t V j +a 3 t D 1 j )+ (det(d)) j tl j a 1, a, a 3 all can set 1. 4 Experiment In our experiment, the non-separable wavelet method is used to implement the image compression scheme. As a comparison, the tensor-product method o well-known biorthogonal wavelets CDF53 and DB97 (the best wavelet or image compression so ar) is also implemented. The non-separable wavelet ilter banks used in the experiments are as ollowing:

3 Non-separable Wavelet and Tensor-product Wavelet 93 Figure 1: Tensor-product wavelet decomposition o S j S j + 1 S j W j + 1, 1 Figure : Non-separable wavelet decomposition o S j W j + 1, M 1 Figure 3: The processing o image compression Quincunx Wavelet: Orthogonal, 1-order vanishing moment (-M1): H =[ 1 1, ], G =[ 1, 1 ] Orthogonal, -order vanishing moment (-M): H = G = Four-band, Separable Sampling Wavelet Orthogonal, Linear phase, 1-order vanishing moment (4-M1): H = G 1 = G = [ G 1 G ]

4 94 Copyright c 007 Tech Science Press CMES, vol., no., pp.91-96, 007 Figure 4: Lena Figure 5: Barbara Figure 6: Barbara Figure 7: Pepper

5 Non-separable Wavelet and Tensor-product Wavelet 95 Figure 8: Mandrill where G 1 = G = G 3 = [ G 31 G 3 ] where G 31 = G 3 = Orthogonal, -order vanishing moment (4-M): H = (3 3 1) G 1 = (1 + 3) G 3 = 1 [ ] G = (5 3 +7) The peak signal to noise ratio (PSNR) or a grayscale image xand its compressed reconstruction x is given by (x i, j x i, j ) i M, j N RMSE = NM PSNR = 0log 10 ( 55 RMSE ) Because this application is image dependent, we have chosen ive images: Lena, Barbara, Goldhill, Peppers and mandrill o size The results are shown in Figure.4 Figure.8: As shown in the Figure.4 Figure.8, DB97 has the best ascendant perormance. Although the perormance o the non-separable wavelet in these experiments is not as good as DB97, which is the best wavelet in image compression applications, the perormance o the our-band, -order vanishing moment non-separable wavelet is close to that o DB97 and better than that o CDF53

6 96 Copyright c 007 Tech Science Press CMES, vol., no., pp.91-96, 007 that is also an excellent wavelet in image compression applications. Multidimensional nonseparable wavelet is ar away rom being well understood so ar. We believe that the non-separable wavelet will perorm better and better in image compression applications as the advance o nonseparable wavelet theory. 5 Conclusion In this paper, we present the comparative results o image compression preprocessed with two dierent wavelet transorms: the non-separable wavelet transorms (quincunx and our-band) and the Tensor-product wavelet transorm. The results show that: 1) the vanishing moment is an important property in still image compression ) the our-band, -order vanishing moment nonseparable wavelet has better perormance in still image compression than Quincunx non-separable wavelet 3) the non-separable wavelet has better ascendant perormance than the tensor-product o univariate wavelet. As the uture work, we will investigate the application o non-separable wavelets in image compression. tion or Wave Propagation Analysis in Isotropic Plates. CMES: Computer Modeling in Engineering & Sciences, vol.15, pp Kovacevic, J.; Vetterli, M. (199): Nonseparable multi- dimensional perect reconstruction ilter banks and wavelet bases or r n. IEEE Trans.Ino.Theory, vol.38, pp Stanhill, D.; Zeevi, Y. (1996): Two-Dimensional Orthogonal Wavelets with Vanishing Moments. IEEE Trans. Signal Processing, vol.10, pp Wang, L.; Song, G. (001): The Pre-Processing o Multiwavelet and Its application in Image Compression. Journal o Electron, vol.10, pp Reerences Antonini, M.; Barlaud, M.; MathieuP et al. (199): Image coding using wavelet transorm. IEEE Trans.Image Processing, vol.1, pp Cohen, A.; Daubechies, I. (1993): Nonseparable dimensional wavelet bases. Revista Mat. Iberoamericana, Vol.9, pp Daubechies, I. (1988): Orthonormal bases o compactly supported wavelets. Comm. Pure Appl.Math, vol.41, pp Duddeck; Fabian, M.E. (006): An alternative approach to boundary element methods via the Fourier transorm. CMES: Computer Modeling in Engineering & Sciences, vol.16, pp He, W.; Lai, M.J.(000): Examples o bivariate nonseparable compactly supported orthonormal continuous wavelets. IEEE Trans. Image Processing, vol.9, pp Mitra, M.; Gopalakrishnan, S. (006): Wavelet Based -D Spectral Finite Element Formula-