Mean Square Error for Biorthogonal M-channel Wavelet Coder

We propose a simple and efficient method to compute the weighted mean square error for a biorthogonal M-channel wavelet coder for multidimensional signals. Indeed, biorthogonal filters weight the amount of quantization error which appears on the reconstructed output.We show that the mean square error of a reconstructed signal, resulting from the quantization errors of the M cosets provided by an M-channel wavelet coder, can be expressed as only a function of the polyphase components of the synthesis filters. Hence, the weights can be computed easily, in any dimension, for any lattice, and any downsampling. As examples, we deal with the computation of the weights for the two-dimensional non separable quincunx filters and for the lifted and unlifted butterfly scheme, showing that the proposed formulation of the weights is particularly useful in case of M-channel lifting schemes. Experimental results demonstrate that the efficiency of a bit allocation process in a geometry coding of triangular meshes is increased thanks to the use of the weighted mean square error as distortion criterion: the PSNR gain reaches up to more than +3 dB at some bitrates. KEY WORDS : M-channel biorthogonal filter bank, weighted mean square error (MSE), polyphase component, lifting scheme, butterfly scheme, quincunx filter, bit allocation, geometry coding, triangular meshes Mean Square Error for Biorthogonal -Channel Wavelet Coder Frédéric Payan, Marc Antonini Laboratoire I3S UPRES-A 6070 CNRS Université de Nice Sophia Antipolis Route des Lucioles F-06903 Sophia-Antipolis FRANCE Phone: +33 (0)4 92 94 27 22, Fax: +33 (0)4 92 94 28 98 fpayan,am @i3s.unice.fr This work was supported by a grant from the Région Provence Alpes Côte d’Azur (France). October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 1 Mean Square Error for Biorthogonal -Channel Wavelet Coder Abstract In this paper we propose a simple and efficient method to compute the weighted mean square error for a biorthogonal -channel wavelet coder for multidimensional signals. Indeed, biorthogonal filters weight the amount of quantization error which appears on the reconstructed output. We show that the mean square error of a reconstructed signal, resulting from the quantization errors of the cosets provided by an -channel wavelet coder, can be expressed as only a function of the polyphase components of the synthesis filters. Hence, the weights can be computed easily, in any dimension, for any lattice, and any downsampling. As examples, we deal with the computation of the weights for the two-dimensional non separable quincunx filters and for the lifted and unlifted butterfly scheme, showing that the proposed formulation of the weights is particularly useful in case of -channel lifting schemes. Experimental results demonstrate that the efficiency of a bit allocation process in a geometry coding of triangular meshes is increased thanks to the use of the weighted mean square error as distortion criterion: the PSNR gain reaches up to more than at some bitrates. Index Terms -channel biorthogonal filter bank, weighted mean square error (MSE), polyphase component, lifting scheme, butterfly scheme, quincunx filter, bit allocation, geometry coding, triangular meshes. Transactions on Image Processing EDICS Category: 2WAVP, 1STERIn this paper we propose a simple and efficient method to compute the weighted mean square error for a biorthogonal -channel wavelet coder for multidimensional signals. Indeed, biorthogonal filters weight the amount of quantization error which appears on the reconstructed output. We show that the mean square error of a reconstructed signal, resulting from the quantization errors of the cosets provided by an -channel wavelet coder, can be expressed as only a function of the polyphase components of the synthesis filters. Hence, the weights can be computed easily, in any dimension, for any lattice, and any downsampling. As examples, we deal with the computation of the weights for the two-dimensional non separable quincunx filters and for the lifted and unlifted butterfly scheme, showing that the proposed formulation of the weights is particularly useful in case of -channel lifting schemes. Experimental results demonstrate that the efficiency of a bit allocation process in a geometry coding of triangular meshes is increased thanks to the use of the weighted mean square error as distortion criterion: the PSNR gain reaches up to more than at some bitrates. Index Terms -channel biorthogonal filter bank, weighted mean square error (MSE), polyphase component, lifting scheme, butterfly scheme, quincunx filter, bit allocation, geometry coding, triangular meshes. Transactions on Image Processing EDICS Category: 2WAVP, 1STER October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 2 Mean Square Error for Biorthogonal -Channel Wavelet Coder I. INTRODUCTION Wavelet transforms are often exploited to perform efficient compression methods. Based on multiresolution analysis, wavelet coders achieve better compression than signal quantization methods: for instance, JPEG2000 in image coding [1], Lounsbery’s compression technique [2] or the zerotree-based coders for triangular meshes [3], [4]. Wavelet coders include a bit allocation dispatching the bits across the subbands. This bit allocation often minimizes a mean square error estimation of the reconstructed signal, according to the quantization error of each subband [5]. Relations between mean square error of the output signal and wavelet coefficients have been already studied in previous works [6], [7], [8], [9], [10]. These works have shown that using biorthogonal filters weights the amount of quantization error which appears on the reconstructed output. The mean square error of a reconstructed signal can be indeed formulated as a weighted sum of subband mean square errors. Among the previous works, Usevitch derived the weighting of the quantization error in the specific case of dyadic filtering of images [10]. More generally, Park and Haddad [8] have defined these weights for multidimensional signals across an -channel wavelet coder. All these works have formulated the weights in function of the coefficients of the synthesis filters. In this paper, we propose a novel approach to generalize the notion of weighted distortion related to biorthogonal -channel wavelet coders for multidimensional signals. We follow a deterministic approach, based on the additive noise model of quantizers [11], [12], unlike Park and Haddad [8] who propose a statistical approach based on the gain-plus-additive noise model advanced by Jayant [13]. Finally, we obtain an original formulation for the weights, that depends on only the polyphase components of the synthesis filter bank. Consequently, they can be computed in any dimension, for any lattice, and any downsampling. Moreover, we will see through two examples that the proposed formulation is very useful in case of wavelet transforms based on a lifting scheme [14]. The remainder of this paper is organized as follows. Section II explains the principle of an -channel wavelet coder and the notations used. Section III develops the notion of weighted mean square error of a reconstructed signal for -channel wavelet coders. To show the interest of the proposed formulation, section IV provides numerical values of weights for the two-dimensional non separable quincunx filters and for the lifted butterfly scheme. Finally, we experiment the effects of these weights in a bit allocation October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 3 for triangular mesh coders in section V, and conclude in section VI. II. -CHANNEL WAVELET CODER A. Principle of an -channel wavelet coder Fig. 1(a) shows the principle of an -channel wavelet coder. A signal is transformed into cosets on account of an -channel wavelet transform and a downsampling. The cosets are then quantized and the quantization error between the coset and its quantized value  is given by: (1) This formulation corresponds to the additive noise model of quantizers given by [11]. An upsampling followed by a synthesis wavelet transform provides the reconstructed signal . B. Notations Let us define a sampled signal as a sequence of real-valued numbers indexed by a finite set : (2) where with an inversible matrix permitting to obtain datas sampled on other lattices than the canonical lattice . For instance, the triangular edge lattice used in section IV. However, in the remainder of the paper, we assume is the identity, its only influence being in the choice of the neighborhoods for the filters [15]. A sublattice of can be obtained by where is a dilation matrix . The determinant of is an integer . Then, the lattice can be written as a sum of sublattices (3) with the shift related to the coset. Hence, we can define a coset as the set of elements of the signal corresponding to the sublattice , and given by (4) Note that is a sequence of real-valued numbers indexed by and not by [15]. According to the definition of a sublattice, an -channel filter bank on a lattice can be formulated according to the polyphase notation as: for (5) October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 4 with the polyphase component of the synthesis filters, defined by (6) and the shift related to the coset given by (7) and (8) The vector is the column vector of the matrix , and is given by: (9) III. MEAN SQUARE ERROR OF A RECONSTRUCTED SIGNAL This section develops the formulation of the mean square error of the reconstructed signal across an -channel wavelet coder. A. Case of a one-level -channel decomposition 1) Problem: In order to simplify the derivation, let us consider the source signal as a realization (or sample function) of a stationary and ergodic random process [11]. Hence, the quantization error can be considered as a deterministic quantity, and is defined by . Consequently, the mean square error between the input signal and the reconstructed signal can be written as: (10) where is the autocorrelation function of the reconstruction error , is the null vector of dimension , and is the number of samples of the input signal. is called the energy of the signal . The challenge is to obtain the mean square error according to the quantization error of each coset and the knowledge of the synthesis filter bank . For this purpose, we develop the expression of the autocorrelation function . The -transform of this function is given by (11) October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 5 with the -transform of the reconstruction error . According to Fig. 1(b), can be formulated in function of the error of each coset [16]: (12) where corresponds to the -transform of the quantization error , related to the coset . By assuming there is no cross-correlation between errors and (for all ) [11], [12], we can write Æ with the -transform of the autocorrelation function of the recontruction error , and Æ the Krönecker symbol defined by Æ si , si . Hence, Eq. (11) and (12) provide: (13) Applying the inverse -transform on Eq. (13) yields the formulation of the autocorrelation function of the reconstruction error: (14) The energy of the signal is then given by: (15) By assuming that the quantization error samplings are uncorrelated [11], if , and consequently, (16) Now, the problem is to deal with and . October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 6 2) Energy of the synthesis filter: We first deal with the energy of the synthesis filter , given by: (17) According to Eq. (5), Eq. (17) can be developed in (18) By using Eq. (6), and becomes: (19) (20) Hence, (18) can be developed in: (21) Using Cauchy theorem, that is, if else, the integral operator of Eq. (21) is equal to if it satisfies (22) The dilation matrix being invertible, the condition (22) becomes (23) From [15], we know that is restricted to the unit hypercube, that is, . On the other hand, . These two remarks yield that (24) We can observe these two definition domains involve that . Therefore, to satisfy the condition (22), we have to solve separately (25) and (26) October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 7 Consequently, the set of solutions of (25) is , and the set of solutions of (26) is . Finally, the energy of the synthesis filter is given by (27) with the coefficient of the polyphase component of the synthesis filter . 3) Energy of the quantization error: Now we have to deal with the energy of the quantization error . By assuming that the quantization error samplings are uncorrelated [11], the energy is: (28) where stands for the mean square error of the coset , and the number of samples of . 4) Solution: Merging (27), (28) and (16) in (10), we obtain the expression of the reconstructed mean square error: (29) Finally, the mean square error of the reconstructed signal is given by with (30) where represents the coefficient of the polyphase component of the synthesis filter , defined by . This formulation permits to compute the reconstruction mean square error of any multidimensional signal across any -channel wavelet coder, according to the quantization mean square error of each coset and the weights . Moreover, the weights can be deduced from only the coefficients of the polyphase matrix components of the synthesis filters, and thus can be designed in any dimension, for any lattice and any downsampling. This original formulation is useful, because in case of lifting schemes, the weights can be obtained directly from the polyphase components, without computing the corresponding synthesis filter bank. October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 8 B. Case of a multilevel decomposition Wavelet coders generally exploit several levels of decomposition by applying several times the wavelet transform on the coset of lowest frequency. For example, the -transform of the reconstruction error according to a two-level decomposition (see Fig. 2) can be written as: (31) where stands for the -transform of the quantization error related to the coset , with the level of decomposition and the channel index. By the same way as for the one-level decomposition, the mean square error across a two-level wavelet coder can be simplified in: (32) where is the number of samples of the coset. Thus, it is easy to generalize Eq. (32) to an -level decomposition: (33) with the weights due to the biorthogonal filters: (34) As in the case of a one-level decomposition (section III-A), we also observe in the case of a multilevel decomposition that the weights depend on only the coefficients of the polyphase components, and thus can be computed in any dimension, for any lattice and any downsampling. In the next section, we show that the proposed formulation for the weights is very useful in case of lifting schemes [14]. IV. COMPUTATION OF WEIGHTS FOR DIFFERENT -CHANNEL LIFTING SCHEMES In this section, we particularly deal with the computation of the weights for a two-channel lifting scheme, and a four-channel lifting scheme. October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 9 A. Polyphase matrix for an -channel lifting scheme As we said in the previous section, only the polyphase components are needed to compute the weights. In the case of an -channel lifting scheme [14], the polyphase matrix is [15]: .. .. .. . . . .. (35) with and the prediction and update operators associated to the coset. Hence, identifying this matrix with the operators and related to any -channel lifting scheme, and using the formulation (30) allows to compute directly the corresponding weights , without designing the synthesis filter bank. Moreover, the lifting scheme introduces gains on the cosets after the analysis step to satisfy the “normalization condition” [17]. To take into account these gains, the weights from Eq. (30) are rewritten as: (36) B. Example of a two-channel lifting scheme: the quincunx lattice The quincunx lattice applied to images is a two-dimensional non separable lattice ( ) with . Fig. 3 shows the neighborhood used to compute the wavelet coefficients. The values of the prediction and update operators for the and quincunx filters are given in table (I) [18]. Moreover, this scheme introduces gains and to satisfy the normalization condition. Each component of the polyphase matrix given by (35) (of dimension ) is computed according to the -transform of prediction and the update operators which depend on the neighborhood defined in Fig. 3, and the values given in table I. Then, the weights can be computed thanks to Eq.(36). The weights related to filters are: (37) The weights related to filters are: (38) October 15, 2004 DRAFT JOURNAL OF LATEX CLASS FILES, VOL. 1, NO. 11, OCTOBER 2004 10 TABLE I PREDICTION AND UPDATE OPERATORS FOR THE TWO-DIMENSIONAL QUINCUNX FILTERS AND filter QF(4,2) QF(6,2)