J. Kovacevi'c and M. Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R. IEEE Trans. Inform. Theory, 38(2):533--555, March 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....addresses several questions about the potential importance of this application. II. BASIC PROPERTIES OF SEPARABLE AND QUINCUNX SAMPLING IN TWO DIMENSIONS This section presents basic concepts from the theory of lattices [13] and their connection to two dimensional (2 D) multirate systems [14], 15] that will be used through the paper. Under the original lattice in two dimensions we assume . The term multirate refers to systems living on different sublattices of . The sublattice is determined by the sampling matrix , as the set of all vectors generated by , A coset of a sublattice ....

....Fourier expression for the output of channel is (8) where and (9) are coset and modulation vectors for the case of quincunx sampling. For more general applications, such as texture synthesis, to assure the cancellation of the aliasing terms at the output of the analysis synthesis filter bank [14], the high pass filter should be designed as (10) This decomposition results in one low resolution subimage and one nonoriented wavelet subimage. Fig. 5 illustrates the idealized partition of the frequency domain after four iterations of quincunx decomposition. At each level, the input Fig. 5. ....

J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n ," IEEE Trans. Inform. Theory, vol. 38, pp. 533--555, Mar. 1992.

.... of a spiral resampling of the data, to obtain a 1 dimensional signal, where rotation invariance is reflected as translation invariance [32] Another solution is to implement non separable filter banks using nonseparable subsampling lattices in the decomposition scheme like the quincunx lattice [33, 34]. However, a sufficient angular localization is still hard to obtain in this way. Despite all these efforts, extracting features from the original data and incorporating rotation invariance in the features themselves clearly would improve results. To achieve this, we start from the 2 dimensional ....

J. Kovacevic and Martin V. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r n . IEEE Trans. Inform. Theory, 38(2):533--555, 1993.

....point out that near PR filterbanks are possible. I. INTRODUCTION T HE McCLELLAN transformation is an efficient and popular method for the design and realization of multidimensional nonseparable finite impulse response (FIR) filters [1] perfect reconstruction filterbanks (PRFB s) and wavelets [2] [4] An odd length symmetric one dimensional (1 D) prototype FIR filter is transformed into an odd length symmetric dimensional D) FIR filter using a real valued transformation function, which preserves key properties of the 1 D filter prototype. For instance, in the design of D PRFB s and ....

....in a twochannel D PRFB. They satisfy (1) and the perfect reconstruction condition (see [7] 2) for any where the symbol denotes complex conjugation and is the aliasing frequency associated with the D downsampling pattern of the filterbank (e.g. for 2 D quincunx downsampling lattice [2]) A. Design of Even Length PRFB s The frequency response of an even length symmetric 1 D FIR filter centered at can be expressed (see [5] as (3) where is a polynomial on the interval and can be viewed as the frequency response of a 1 D odd length symmetric filter. Then, the desired D ....

J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n ;" IEEE Trans. Inform. Theory, vol. 38, pp. 533--555, Mar. 1992.

....filter banks for sampling matrix 2I, holding structurally orthogonality and centrosymmetry. This family [4] is defined by polynomial matrix products. These matrices include some angles that can be chosen arbitrarily. It is well known that such filter banks may generate wavelet bases [1] [3], and a necessary condition for that is that the filters vanish at some aliasing frequencies. In addition, the resulting wavelets can be N times continuously differentiable only if the polynomials H 0 ; HM Gamma1 vanish as well as their derivatives up to order N at their aliasing ....

J. Kovacevic and M. Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases. IEEE Transactions on Information Theory, 38(2):533--555, March 1992.

....to any of the channels for a finer frequency discrimination. For example the next stage filter bank system becomes (V Gamma D;D;P ) 5. RESULTS AND CONCLUSIONS For the experiment considered, the sampling structure Gamma of the moving images analyzed by the filter bank is the non separable [5] quincunx sampling structure described by the lattice matrix (13) Defined on it, a number of 3D prototypes P (f ) with varying support regions has been designed to approximate the rhombic dodecahedral symmetry. For the downsampling matrix D = diag[2; 2; 2] the QMF (V Gamma ; D;P ) decomposition ....

J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n ," IEEE Trans. Inf. Theory, vol. 38, pp. 533--55, Mar. 1992.

....will favor horizontal, vertical and diagonal features of the original data. Other features are not that easily detected. Non separable wavelets can provide a solution to this. They allow for the construction of twodimensional wavelets on e.g. lattices (e.g. on the so called quincunx lattice [7]) or on hexagonal grids [10] All these are heavily based on Fourier tranforms, just like the classical wavelets. Second generation wavelets designed using the lifting scheme are another option. The lifting scheme is a generic method to create wavelets on intervals, irregular samplings, meshes, ....

....used, we are convinced that the two dimensional analog that we present below has the same potentials as the the CDF (2, 2) wavelet. 3.1 A Two Step Method Our approach is inspired by the well known Red Black Gauss Seidel technique for the iterative solving of linear systems. In the literature [7], the kind of lattice we use is known as a quincunx lattice. However, we prefer the name Red Black wavelet transform because it is simpler and it is more appropriate to describe the splitting step in the lifting scheme. The idea, just like in the CDF (2, 2) wavelet, is that we first split the ....

J. Kovacevic and M. Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n . IEEE Trans. Inform. Theory, 38(2):533--555, March 1992.

....rectangular divisions of the frequency spectrum. Often, symmetry axes and certain nonrectangular divisions of the frequency spectrum correspond better to the human visual system. In the early 90 s several solutions, both orthogonal and biorthogonal, and using different lattices became available [7, 10, 28, 36, 45]. These are typically concerned with two and three dimensions as the algebraic conditions in higher dimensions become increasingly cumbersome. Other work in the signal processing literature uses two techniques: either cascade structures or oneto multidimensional transformations. Although using ....

....higher dimensions become increasingly cumbersome. Other work in the signal processing literature uses two techniques: either cascade structures or oneto multidimensional transformations. Although using cascade structures it is easy to build orthogonal or biorthogonal multidimensional filter banks [27, 28], one cannot guarantee vanishing moments which Lucent Technologies, Bell Laboratories, 600 Mountain Avenue, Murray Hill NJ 07974. jelena bell labs.com, wim bell labs.com. are a necessary condition for both stability and smoothness. One to multidimensional transformations include the method of ....

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J. Kovacevic and M. Vetterli. Nonseparablemultidimensional perfect reconstruction filter banksand wavelet bases for R n . IEEE Trans. Inform. Theory, 38(2):533--555, 1992.

.... become a popular technique for multiresolution representation and analysis in a wide variety of image processing applications, including computer aided diagnosis of mammograms [13, 14, 15, 16, 17] We choose the linear phase nonseparable 2 D perfect reconstruction wavelet transform described in [18] to obtain a multiresolution representation [12] of the original mammogram. This transform does not introduce phase distortions in the decomposed images. In addition, no bias is introduced in the horizontal and vertical directions as would occur with a separable transform. The impulse response of ....

....The impulse response of the analysis low pass filter is h(n 1 , n 2 ) 0 B B B B B B B 0 0.125 0 0.125 0.5 0.125 0 0. 125 0 1 C C C C C C C A The dilation matrix used to represent the subsampling lattice is D = 2 6 6 4 1 1 1 1 3 7 7 5 which corresponds to the 2 D quincunx sublattice [18], as shown in Figure 2. Let the original mammogram have the finest resolution N N pixels. Since D expands the sampling lattice by # 2 in each direction, image resolution decreases by a factor of 1 # 2 after each decomposition. For example, the image at the second finest resolution has N ....

J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n ," IEEE Transactions on Information Theory, vol. 38, no. 2, pp. 533--555, March 1992.

....for diagonal dilation matrices one can also lift 1D wavelets to higher dimensional wavelets for dilation matrices A with det(A) 2 [4] Both types are called separable, they have an inherent 1D structure. Moreover, nonseparable quadrature mirror filters (QMFs) have been constructed in [11]; they lead to continuous 2D wavelets [18] Our aim is to develop a general construction procedure for nonseparable compactly supported orthogonal 2D wavelets for dilation matrices with det(A) 2. As usual, we exploit the connection between wavelets and multiresoluton analysis. This leads us ....

....example, # # = 1, 1) and # # = 1, 3) lead to h = 0 0 # # 1 # # 2 0 # 1 # 2 ## 2 ## 1 0 # 2 # 1 0 1 A . ORTHOGONAL TWO DIMENSIONAL WAVELETS 1479 Corollary 4.5. This set of coe#cients h is up to normalization identical to the family of QMF filters studied in [11]. In particular, the choice (#, 2 # 3, # 3) leads to a wavelet for the quincunx grid which is known to be continuous [18] Now we start iterating the procedure above. First, we choose arbitrary nonzero values for # = 1 , 2 , l and compute. Let h # 2 = h 0 2 , h 1 2 ....

<F3.735e+05> J. Kovacevic and M.<F3.806e+05> Vetterli,<F3.666e+05> Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for<F3.744e+05> IR<F3.415e+05> n<F3.806e+05> , IEEE Trans. Inform. Theory, 38 (1992), pp. 533--555.

....a decomposition of the sets of all pairs of integers Z 2 into 2 cosets: Gamma 1 and Gamma 2 : Z 2 = Gamma 1 [ Gamma 2 ; Gamma 1 = fDZ 2 g ; Gamma 2 = fDZ 2 1 0 g The number of wavelets needed is det(D) Gamma 1 = 1 in this case. Cohen, Daubechies[4] Kovacevic and Vetterli [5] have constructed several examples of nonseparable bidimensional scaling functions and multiwavelets. 4 Bidimensional multiple scaling function: We now have r bidimensional scaling functions Phi 1 ; Phi 2 ; Phi r associated to a dilation matrix D. For r = 2 we have: Phi : R 2 R 2 ....

J. Kovacevic and M.Vetterli, Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for R n , IEEE, Transactions on Information Theory 38(2), 533--555 (1992).

....a m;n are real or complex depending on whether f( Delta) is a real or a complex valued signal. Here, we restrict ourselves to the case where all the eigenvalues of the matrix D have magnitudes strictly larger than unity. This is necessary to ensure that dilation occurs in every direction [8] [19]. In addition, we also assume that X n ja m;n j 1 and X n k n k 2 1; 1:2) where k n k 2 is the 2 norm of vector n . These conditions are automatically satisfied when the sum over n is finite. An MSDE is a generalization of the two scale difference equation [8] 19] which ....

....[8] 19] In addition, we also assume that X n ja m;n j 1 and X n k n k 2 1; 1:2) where k n k 2 is the 2 norm of vector n . These conditions are automatically satisfied when the sum over n is finite. An MSDE is a generalization of the two scale difference equation [8] [19] which has the form f( x) X n c n f(D x Gamma n ) 1:3) As we shall see later in Section 6 (see also [3] MSDEs provide more compact and more flexible models for wider classes of arbitrary signals than two scale difference equations. This is the main motivation for studying such ....

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Kovacevic, J. and Vetterli, M., "Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for R n ," IEEE Trans. on Information Theory, Vol. 38, No. 2, pp. 533-555, March 1992.

.... wavelet transforms on the quincunx lattice and compare results under simple data reduction methods using separable and nonseparable symmetrical (linear phase) biorthogonal filters, where the two dimensional filters for use on the quincunx lattice are obtained from the one dimensional filters [3]. We attempt to use the most general coding schemes possible which do not disadvantage either separable or nonseparable techniques. For a comparison of separable wavelets, quantizers and coders the reader is directed to [2] We attempt to use separable and quincunx wavelet filters of similar size ....

....subsampling and convolution processes. We also constrain ourselves to filters for which there are well defined one and twodimensional versions. We concentrate on two wavelet filter sets here. We use a linear phase (i.e. symmetrical) bi orthogonal set as proposed by Kovacevi c and Vetterli [3] and its onedimensional version (from which the two dimesnional filter is derived) The one dimensional filter set derived from the generalized filter [3] using a 1 = 2 and a 2 = Gamma6 is: h 0 = 1 2 1 (2.1) g 0 = 1 2 Gamma6 2 1 (2.2) The two dimensional filter set derived from the above ....

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Jelena Kovacevi'c and Martin Vetterli "Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for R n .", IEEE Trans. Info. Theory. Vol. 38, No. 2 March 1992

....information does not necessarily reflect the position or the policy of the Government and no official endorsement should be inferred. 1 Introduction Wavelet theory provides a unified framework for a number of techniques developed in multiresolution analysis [21, 22, 25, 31] and subband coding [16, 31]. Although wavelets have been studied by mathematicians for many years [4] wavelet transforms have recently generated a great deal of interest as a new form of multiresolution representation for 1 D signals and 2 D images [23] Multiresolution representations are commonly used for the analysis of ....

J. Kovacevic and M. Vetterli. Non-separable multidimensional perfect reconstruction filter banks and wavelet bases for R. IEEE Transactions on Information Theory, Special Issue on Wavelet Transforms and Multiresolution Signal Analysis, 38(2):533--555, 1992. I. Daubechies, S. Mallat, and A. S. Willsky, editors.

....product wavelets for diagonal dilation matrices one can also lift 1d wavelets to higher dimensional wavelets for dilation matrices A with jdet(A)j = 2 [4] Both types are called separable, they have an inherent 1d structure. Moreover non separable quadrature mirror filters have been constructed in [11], they lead to continuous 2d wavelets [18] Our aim is to develop a general construction procedure for non separable compactly supported orthogonal 2d wavelets for dilation matrices with jdet(A)j = 2. As usual we exploit the connection between wavelets and multiresoluton analysis. This leads ....

.... (2; 0) e.g. fi = 1; Gamma1) and fl = Gamma1; Gamma3) leads to h = 0 0 Gamma ff 1 Gamma ff 2 0 ff 1 ff 2 Gammaff 2 ff 1 0 Gamma ff 2 ff 1 0 1 A : Corollary 4. 5 This set of coefficients h is up to normalization identical to the family of QMF filters studied in [11]. In particular the choice ( Gamma2 p 3; Gamma p 3) leads to a wavelet for the quincunx grid which is known to be continuous [18] Now we start iterating the procedure above. First we choose arbitrary nonzero values for = 1 ; 2 ; l and we compute Let h ff 2 = h 0 2 ; ....

J. Kovacevic, M. Vetterli, Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for IR n , IEEE Trans. on Inform. Theory 38 (1992), p. 556-568.

....its eigenvalues are strictly larger than one in magnitude. Several authors have studied the lattice structure and the accompanying decimation interpolation process in multidimensions that corresponds to the matrix D because of their importance in multidimensional multi rate filters and wavelets [4, 7]. In 1 D, eqn. 1) takes a simpler form f(x) M X m=1 X n a m;n f(fi m x Gamma n) 2) where fi is an integer greater than unity. Eqns. 1) and (2) are called (M 1) scale difference equation as there is a total of (M 1) scales on both sides of these equations. Therefore, the special ....

.... 1) scales on both sides of these equations. Therefore, the special case of two scale difference equations is obtained by setting M = 1. All our modeling is done under the conditions X l a m;D m l Gamma p = 1 M ; l 2 U m ; 3) where U m represents the fundamental parallelepiped [4, 7] for the decimation matrix D m . The fundamental parallelepiped corresponding to D m is the unit cell of the lattice defined by the columns of D m . In 1 D these conditions become X l a m;fi m l Gammap = 1 M ; p = 0; 1; 2; fi m Gamma 1: 4) As shown in [1] These conditions, ....

Kovacevic, J. and Vetterli, M., "Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for R n ," IEEE Trans. on Information Theory, Vol. 38, No. 2, pp. 533-555, March 1992.

....approach is discussed in Section 4. Finally we present our experimental results and conclusions in Section 5. 2. MULTIRESOLUTION DECOMPOSITION A multiresolution representation [4] of the original mammogram is obtained using the linear phase nonseparable 2 D wavelet transform described in [5]. This is chosen for two reasons. First, it does not introduce phase distortions in the decomposed images; second, no bias is introduced in the horizontal and vertical directions as a separable transform would. The coe#cients of the analysis low pass filter are h(n 1 , n 2 ) 0 0 0.125 0 ....

J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n ," IEEE Transactions on Information Theory, vol. 38, no. 2, pp. 533--555, March 1992.

.... The theory of uniform band (or integer sampling rate) filter banks in the one dimensional case is fairly well understood [20, 17] However, not much in the form of a well developed theory is known in the multidimensional case of uniform band (or integer matrix sampling rate) filter banks [21, 13]. Moreover, in the multidimensional case, there has not been any coherent attempt to develop a complete set of tools for the analysis of arbitrary combinations of upsamplers, downsamplers and filters. The real difficultly has been that in multidimensions the operations of upsampling and ....

J. Kovacevic and M. J. Vetterli. Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for IR n . IEEE Trans. on Information Theory, 38:533--555, March 1991.

....some of these tilings are charactristic functions of fractal sets. It is still an unsolved problem to determine in the smoothness of nonseparable orthogonal 2 d wavelets for the dilation matrix R. The only positive result in this direction was reported in [83] it shows that the filter given in [46] leads to a continuous 2 d wavelet for R. This example is identical with the shifted ff wavelet for parameters ( Gamma2 Gamma p 3; p 3) described in [53] 4.4 Algorithms Algorithms for the wavelet transform split in two groups: discretizations of the continuous transform and ....

J. Kovacevic, M. Vetterli; Non-separable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for IR n , IEEE Trans. on Info. Theory, Special Issue on Wavelets, 1992

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J. Kovacevic and M. Vetterli, "Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for # n ," IEEE Trans. Inform. Theory, pp. 533--555, Mar. 1992.

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J. Kovacevi'c and M. Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R. IEEE Trans. Inform. Theory, 38(2):533--555, March 1992.

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J. Kovacevic and M. Vetterli. Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for  . IEEE Transactions on Information Theory, 38(2):533--555, 1992.

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Jelena Kovacevic, Martin Vetterli, "Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for R n

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Kovacevic, J., Vetterli, M.: Nonseparable Multidimensional Perfect Reconstruction Filter Banks and Wavelet Bases for R , IEEE Trans. Information Theory, Vol. 38, No. 2, pp. 533-555, March 1992

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J. Kovacevic and M. Vetterli, Non-separable Multidimensional Perfect Reconstruction Filter banks, IEEE Trans. Inform. Theory 38(1992), pp. 558-568.

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J. Kovacevic, M. Vetterli. Nonseparable multidimensional perfect reconstruction banks and wavelet bases for IR n . IEEE Trans. Inform. Theory 38 (1992), 533 -- 555.

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