MEAN SQUARE ERROR FOR BIORTHOGONAL M-CHANNEL WAVELET CODER

Transcription

1 LABORATOIRE INFORMATIQUE, SIGNAU ET SYSTÈMES E SOPHIA ANTIPOLIS UMR 6070 MEAN SQUARE ERROR FOR BIORTHOGONAL M-CHANNEL WAVELET COER Frédéric PAYAN, MarcANTONINI Projet CREATIVE Rapport de recherche ISRN I3S/RR FR Janvier 2005 LABORATOIRE I3S: Les Algorithmes / Euclide B 2000 route des Lucioles B.P Sophia-Antipolis Cedex, France Tél. (33) Télécopie : (33)

2 RÉSUMÉ : MOTS CLÉS : filtres d ondelettes biorthogonales M-canaux, Erreur Quadratique Moyenne (EQM) pondérée, composantes polyphases, schéma lifting, Transformée en ondelettes de butterfly, filtres quinconces 2, allocation binaire, codage géométrique, maillages triangulaires ABSTRACT: 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 WORS : M-channel biorthogonal filter bank, weighted mean square error (MSE), polyphase component, lifting scheme, butterfly scheme, quincunx filter, bit allocation, geometry coding, triangular meshes

3 Mean Square Error for Biorthogonal M-Channel Wavelet Coder Frédéric Payan, Marc Antonini Laboratoire I3S - UPRES-A 6070 CNRS Université de Nice - Sophia Antipolis Route des Lucioles - F Sophia-Antipolis FRANCE Phone: +33 (0) , Fax: +33 (0) ffpayan,amg@i3s.unice.fr This work was supported by a grant from the Région Provence Alpes Côte d Azur (France). RAFT

4 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER Mean Square Error for Biorthogonal M-Channel Wavelet Coder Abstract In this paper 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. Index Terms M-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 - EICS Category: 2- WAVP, 1- STER RAFT

5 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER Mean Square Error for Biorthogonal M-Channel Wavelet Coder I. INTROUCTION 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 M-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 M-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 M-channel wavelet coder and the notations used. Section III develops the notion of weighted mean square error of a reconstructed signal for M-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 RAFT

6 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER for triangular mesh coders in section V, and conclude in section VI. II. M -CHANNEL WAVELET COER A. Principle of an M-channel wavelet coder Fig. 1(a) shows the principle of an M-channel wavelet coder. A signal s is transformed into M cosets fs i ;i = 0; :::; M 1g on account of an M-channel wavelet transform fh i ;i = 0; :::; M 1g and a downsampling. The cosets are then quantized and the quantization error " i between the i th coset s i and its quantized value ^s i is given by: " i =(s i ^s i ): (1) This formulation corresponds to the additive noise model of quantizers given by [11]. An upsampling followed by a synthesis wavelet transform g i provides the reconstructed signal ^s. B. Notations Let us define a sampled signal s as a sequence of real-valued numbers indexed by a finite set K: s = fs(k) 2 R j k 2Kg; (2) where K = Z d with an inversible d d matrix permitting to obtain datas sampled on other lattices than the canonical lattice Z d. 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 K can be obtained by Z d where is a dilation matrix d d. The determinant of is an integer M 2 Z. Then, the lattice Z d can be written as a sum of sublattices Z d = [ j=0 Z d + tj ; (3) with tj 2 Z d the shift related to the j th coset. Hence, we can define a coset s i as the set of elements of the signal s corresponding to the sublattice L = Z d + ti, and given by s i (L) =fs(k + ti) j k 2 Z d g: (4) Note that s i (L) is a sequence of real-valued numbers indexed by Z d and not by L [15]. According to the definition of a sublattice, an M-channel filter bank fg i g on a lattice K can be formulated according to the polyphase notation as: G i (z) = j=0 z tj G i;j (z ) for i 2f0; :::; M 1g ; (5) RAFT

7 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER with G i;j (z) the i; j th polyphase component of the synthesis filters, defined by and z tj and G i;j (z) = the shift related to the j th coset given by k g i (k + tj) z ; (6) k2z d z tj = dy n=1 The vector dj is the j th column vector of the matrix, and z dj tj (n) z n ; (7) z = fz d1 ; z d2 ; :::; z dd g: (8) z dj = dy n=1 is given by: dj (n) zn : (9) III. MEAN SQUARE ERROR OF A RECONSTRUCTE SIGNAL This section develops the formulation of the mean square error of the reconstructed signal across an M-channel wavelet coder. A. Case of a one-level M-channel decomposition 1) Problem: In order to simplify the derivation, let us consider the source signal s 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 " = f" (k) =(s (k) ^s (k)) 2 R j k 2Kg. Consequently, the mean square error between the input signal and the reconstructed signal can be written as: ff 2 " = 1 [r " (0)] ; (10) N s where r " (t) is the autocorrelation function of the reconstruction error ", 0 is the null vector of dimension d, and N s is the number of samples of the input signal. r " (0) is called the energy of the signal ". The challenge is to obtain the mean square error according to the quantization error of each coset s i and the knowledge of the synthesis filter bank fg i g. For this purpose, we develop the expression of the autocorrelation function r " (t). The z-transform of this function is given by R E (z) =E(z) E(z 1 ); (11) RAFT

8 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER with E(z) the z-transform of the reconstruction error ". According to Fig. 1(b), E(z) can be formulated in function of the error of each coset s i [16]: E(z) = M 1 i=0 G i (z) E i (z ); (12) where E i (z) corresponds to the z-transform of the quantization error " i, related to the coset s i.by assuming there is no cross-correlation between errors " i (k) and " i (k 0 ) (for all k 6= k 0 ) [11], [12], we can write E i (z )E j (z )=ffi i;j R Ei (z ) with R Ei (z) the z-transform of the autocorrelation function of the recontruction error " i, and ffi i;j Krönecker symbol defined by ffi i;j = 8 < : 1 si i = j, 0 si i 6= j. the Hence, Eq. (11) and (12) provide: R E (z) = i=0 R Gi (z) R Ei (z ): (13) Applying the inverse z-transform on Eq. (13) yields the formulation of the autocorrelation function of the reconstruction error: r " (t) = i=0 The energy r " (0) of the signal " is then given by: r " (0) = fi i=0 r gi (fi ) r "i (t fi ): (14) fi r gi (fi ) r "i ( fi ): (15) By assuming that the quantization error samplings are uncorrelated [11], r "i ( fi )=0if fi 6= 0, and consequently, r " (0) = i=0 Now, the problem is to deal with r gi (0) and r "i (0). r gi (0)r "i (0): (16) RAFT

9 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER ) Energy of the synthesis filter: We first deal with the energy of the synthesis filter r gi (0), given by: I r gi (0) = 1 G i (z) G i (z 1 ) z 1 dz: (17) 2ßj According to Eq. (5), Eq. (17) can be developed in r gi (0) = 1 M 1 M 1I G i;u (z ) G i;v (z ) z ( tu+tv 1) dz: (18) 2ßj u=0 v=0 By using Eq. (6), G i;u (z ) and G i;u (z ) becomes: G i;u (z k ) = g i (k + tu) z ; (19) k2z d G i;v (z k ) = g i (k + tv) z : (20) k2z d Hence, (18) can be developed in: r gi (0) = 1 M 1 M 1 I g i (k + tu) g i (k + tv) z ( k+k0 tu+tv 1) dz: (21) 2ßj u=0 v=0 k2z d 0 k 2Z d Using Cauchy theorem, that is, I 8 < 1 1 if l = 0; z l 1 dz = 2ßj : 0 else, the integral operator of Eq. (21) is equal to 1 if it satisfies The dilation matrix being invertible, the condition (22) becomes k + k 0 tu + tv = 0: (22) ( k + k 0 ) ( 1 tu 1 tv) =0 (23) From [15], we know that 1 tj is restricted to the unit hypercube, that is, [0; 1) d. On the other hand, k 2 Z d. These two remarks yield that 8 < : ( k + k 0 ) 2 Z d ( 1 tu 1 tv) 2 ( 1; 1) d (24) We can observe these two definition domains involve that Z d condition (22), we have to solve separately ( 1; 1) d = 0. Therefore, to satisfy the ( k + k 0 )=0; (25) and ( 1 tu 1 tv) =0; (26) RAFT

10 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER Consequently, the set of solutions of (25) is k = k 0, and the set of solutions of (26) is u = v. Finally, the energy of the synthesis filter r gi (0) is given by r gi (0) = j=0 g i (k + tj) 2 ; (27) k2z d with g i (k + tj) =g i;j (k) the coefficient k of the j th polyphase component of the synthesis filter i. 3) Energy of the quantization error: Now we have to deal with the energy of the quantization error r "i (0). By assuming that the quantization error samplings are uncorrelated [11], the energy r "i (0) is: r "i (0) = 2 " i (k) = Nsi ff 2 " i ; (28) k2z d where ff 2 " i stands for the mean square error of the coset s i, and N si the number of samples of s i. 4) Solution: Merging (27), (28) and (16) in (10), we obtain the expression of the reconstructed mean square error: M 1 ff 2 " = 2 M 1 4 N s i ff 2 2 " N i g i;j (k) 5 : (29) s i=0 j=0 k2z d Finally, the mean square error of the reconstructed signal is given by N ff 2 " = si w i ff 2 " N i with w i = g i;j (k) 2 : (30) s i=0 j=0 k2z d where g i;j (k) represents the coefficient k of the j th polyphase component of the synthesis filter i, defined by g i;j (k) =g i (k + tj). This formulation permits to compute the reconstruction mean square error of any multidimensional signal across any M-channel wavelet coder, according to the quantization mean square error of each coset and the weights w i. Moreover, the weights w i 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. 3 RAFT

11 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER 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 z-transform of the reconstruction error " according to a two-level decomposition (see Fig. 2) can be written as: E(z) =G 0 (z) G l (z ) E i;j (z 2 )+ l=0 l=1 G l (z) E i;j (z 2 ); (31) where E i;j (z) stands for the z-transform of the quantization error " i;j related to the coset (i; j), with i the level of decomposition and j 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: ff N 2 " = s 0;0» Ns1;l w 0 w l ff 2 " N s N 1;l + s0;0 where N si;j N-level decomposition: l=0 l=1» Ns0;l N s w l ff 2 " 0;l ; (32) is the number of samples of the i; j th coset. Thus, it is easy to generalize Eq. (32) to an ff 2 " = N s N 1;0 N s W N 1;0 ff 2 " N 1;0 + with W i;l the weights due to the biorthogonal filters: N 1 i=0 M 1 l=1 N si;l N s W i;l ff 2 " i;l ; (33) W i;l =(w 0 ) i w l : (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 IFFERENT M -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. RAFT

12 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER A. Polyphase matrix for an M-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 M-channel lifting scheme [14], the polyphase matrix is [15]: G = 0 1 p 1 p 2 ::: p u 1 1 u 1 p 1 u 1 p 2 ::: u 1 p u 2 u 2 p 1 1 u 2 p 2 ::: u 2 p u u p 1 u p 2 ::: 1 u p 1 C A (35) with p i and u i the prediction and update operators associated to the i th coset. Hence, identifying this matrix with the operators p i and u i related to any M-channel lifting scheme, and using the formulation (30) allows to compute directly the corresponding weights w i, without designing the synthesis filter bank. Moreover, the lifting scheme introduces gains K i 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: w i = 1 M 1 K i j=0 B. Example of a two-channel lifting scheme: the quincunx lattice g i;j (k) 2 : (36) k2z d The quincunx lattice applied to images is a two-dimensional non separable lattice (d =2) with M =2. Fig. 3 shows the neighborhood used to compute the wavelet coefficients. The values of the prediction and update operators for the (4; 2) and (6; 2) quincunx filters are given in table (I) [18]. Moreover, this scheme introduces gains K 0 =1and K 1 =1=2 to satisfy the normalization condition. Each component of the polyphase matrix given by (35) (of dimension 2) is computed according to the z-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 (4; 2) are: The weights related to filters (6; 2) are: 8 < : w 0 = 1: w 1 = 3: : 8 < : w 0 = 1: w 1 = 3: : (37) (38) RAFT

13 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER TABLE I PREICTION AN UPATE OPERATORS FOR THE TWO-IMENSIONAL QUINCUN FILTERS (4; 2) AN (6; 2) filter QF(4,2) QF(6,2) a =39=2 7 a = 675=2 11 b = 3=2 7 b = 165=2 12 p c = 1=2 7 c = 85=2 13 d =15=2 12 e =15=2 13 f =3=2 13 u a =1=8 a =1=8 We observe the weights resulting from our method correspond to the weights computed in [19], but we do not have to design the entire filter bank to compute them, since only the polyphase components need to be known. C. A four-channel lifting scheme: the triangular edge lattice The lifting scheme for a triangular mesh is based on a triangular edge lattice [15]. In this case, d =2, and M =4, providing four cosets. In this section, we focus on the well-known butterfly scheme [20]. This lifting scheme is based on the interpolating subdivision scheme butterfly [21], and exists in two different versions: the lifted butterfly transform (a prediction step and an update step), and the unlifted butterfly transform (only a prediction step). The prediction and update operator filters of the lifted butterfly scheme are presented in Fig. 4 [15]. Moreover, the gains K 0 = 2 and K 1 = K 2 = K 3 = 1 are introduced to respect the normalization condition. The weights for the lifted butterfly transform, computed by substituting the z-transform of the prediction and update operators in each component of the polyphase matrix given by (35) (of dimension 4) and by using Eq.(36), are: 8 >< >: w 0 = w 1 = w 2 = w 3 = ' 0: ' 0: ' 0: ' 0: : (39) RAFT

14 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER Similarly, the weights for the unlifted butterfly transform are: 8 >< >: w 0 = w 1 = 1 w 2 = 1 w 3 = 1: ' 0: (40) V. EPERIMENTAL RESULTS In this section, experimental results are given to show the interest and the efficiency of the weighted distortion, presented in section III-B. We exploit this weighted distortion in a bit allocation process for a multiresolution geometry coder of semi-regular triangular meshes. For the two-dimensional non separable quincunx filters proposed in section IV-B, experimental results can be found in [19], where the authors propose another method to compute the same corresponding weights. A. A multiresolution geometry coder for triangular meshes The proposed geometry coder of meshes includes a model-based bit allocation [5] optimizing the quantization steps used to quantize the wavelet coefficients. The global scheme of this coder [22] presented in Fig. 5 includes a remeshing technique to obtain a semi-regular version of the original mesh, a wavelet transform, a bit allocation optimizing the quantization, and finally an entropy coder. The general purpose of the bit allocation process is to determine the best set of quantization steps that minimizes the total distortion at a given rate. The total distortion depends on the weighted mean square error of each subband, according to Eq. (33) and Eq. (34). In parallel, the topology information of the coarse mesh is encoded with the topology coder of [23]. We deal with two different versions of this coder. The first version, named the MAPS coder, exploits the remesher MAPS [24] and the lifted butterfly transform [20]. Consequently, the weighted distortion depends on the values of Eq. (39). The second version, named the NORMAL coder, exploits the Normal Remesher [25] and the unlifted butterfly transform [4]. In that case, the weighted distortion depends on the values of Eq. (40). To show the improvement in coding performances related to the use of the weighted distortion, we compare the quality of the reconstructed meshes according to two cases: (1) the total distortion is only the sum of the mean square errors across the subbands (w i = 1 8 i, asinthe orthogonal case); (2) the total distortion is the sum of the mean square errors weighted by the values computed in section IV-C. RAFT

15 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER The comparison criterion generally used in mesh compression [3], [4] is the specific PSNR depending on the RMSE between two surfaces based on the Hausdorff distance [26]. It is given by PSNR =20log 10 peak d S ; where peak is the bounding box diagonal of the original object, and d s is the RMSE based on the Hausdorff distance between the original mesh and the reconstructed one (computed with MESH [27]). Fig. 6 shows the PSNR measures for the model BUNNY encoded with the MAPS coder. Fig. 7 shows the PSNR measures for the model VENUS encoded with the NORMAL coder. Theoretically the quantization error assumptions used during derivation in section III are only valid for high bitrates [11], and we could expect that the coding gain is limited for low bitrates. However, we globally observe that using the weighted distortion improves the coding performances for any model, even at low bitrates. We particularly observe an improvement up to more than 3 db for BUNNY at low bitrates. To show the visual impact, Fig. 8 shows the visual differences between BUNNY when the weighted mean square error is used as distortion criterion during the bit allocation (see Fig. 8(b)), and BUNNY when the unweighted MSE is used (see Fig. 8(c)). We observe that for a specific bitrate, the mesh quantized with the weighted MSE is visually closer to the original one (see Fig. 8(a)), than the mesh quantized with the unweighted mean square error. In addition, to prove the interest of using the weighted mean square error in mesh coding, we have also drawn the PSNR curves of the state-of-the-art zerotree-based mesh coders, PGC [3], and NMC [4]. PGC includes the MAPS, while NMC includes the Normal Remesher. Therefore we compare the proposed MAPS coder with PGC, and the proposed NORMAL coder with NMC. We observe that each version of the proposed coder provides better results than the corresponding zerotree-based coder, while each version of the proposed coder provides worse results when the bit allocation process exploits an unweighted distortion. VI. CONCLUSIONS In this paper we provide a novel formulation of the weighted mean square error introduced by a biorthogonal M-channel wavelet coder on multidimensional signals. Consequently, the weights due to the non-orthogonal filters can be easily computed in any dimension, for any lattice and any downsampling. Also, the proposed formulation for the weights is particularly useful in case of M-channel lifting schemes, as shown in the two proposed examples. Indeed, the weights can be expressed as only a function of the RAFT

16 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER polyphase components of the synthesis filters. Finally, to prove the interest of using the weighted mean square error as distortion measure in a bit allocation criterion, some experimental results of two original geometry coders of triangular meshes, the MAPS coder and the NORMAL coder are given. Thanks to the weighted mean square error, these coders provide better results in term of PSNR than the corresponding state-of-the-art zerotree-based methods, respectively PGC and NMC. ACKNOWLEGMENTS The VENUS model is courtesy of Cyberware, the BUNNY model is courtesy of Stanford University. We are particularly grateful to Igor Guskov for providing us with his normal meshes, Aaron Lee for providing us with his MAPS meshes, and Andrei Khodakovsky for providing us with his executables. We also want to acknowledge the anonymous reviewers for their advises which permitted to improve the quality of the paper. REFERENCES [1] Information technology JPEG 2000 image coding system part 1: Core coding system, ISO/IEC :2000. [2] M. Lounsbery, T. erose, and J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, Trans. on Graphics 16,1, vol. 99, [3] A. Khodakovsky, P. Schröder, and W. Sweldens, Progressive geometry compression, Proceedings of SIGGRAPH, [4] A. Khodakovsky and I. Guskov, Normal mesh compression, Geometric Modeling for Scientific Visualization, Springer- Verlag, [5] C. Parisot, M. Antonini, and M. Barlaud, EBWIC: A low complexity and efficient rate constrained wavelet image coder, Proceedings of IEEE ICIP, september [6] J. Woods and T. Naveen, A filter based bit allocation scheme for subband compression of hdtv, IEEE Trans. on Image Processing, vol. 1, no. 3, [7] C. Cheong, K. Aizawa, T. Saito, and M. Hatori, Subband image coding with biorthogonal wavelets, IEICE Trans. Fundamentals, vol. E75-A, no. 7, july [8] R. Haddad and K. Park, Modeling and optimal compensation of quantization in multidimensional m-band filter bank, in Proceedings of the IEEE International Conference Acoustic, Speech, and Signal Processing, april [9] P. Moulin, A multiscale relaxation algorithm for snr maximization in nonorthogonal subband coding, IEEE Trans. on Image Processing, vol. 4, no. 9, [10] B. Usevitch, Optimal bit allocation for biorthogonal wavelet coding, IEEE ata Compression Conference, April [11] A. Gersho and R. Gray, Vector quantization and signal compression, Norwell, Massachusetts: Kluwer Academic Publishers, [12] R. Gray and T. Stockham, ithered quantizers, IEEE trans. on Information theory, vol. 39, no. 3, may 93. [13] N. S. Jayant and P. Noll, igital Coding of Waveforms, E. Cliffs, Ed. New Jersey: Prentice Hall, [14] W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM Journal on Mathematical Analysis, vol. 29, no. 2, pp , RAFT

17 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER [15] J. Kovacevic and W. Sweldens, Wavelet families of increasing order in arbitrary dimensions, IEEE Trans. on IP, [16] M. Vetterli and J. Kovacevic, Wavelets and Subband coding. Prentice Hall PTR, Engelwood Cliffs, New Jersey [17] W. Sweldens, The lifting scheme: A custom-design construction of biorthogonal wavelets, Appl. Comput. Harmon. Anal., vol. 3, no. 2, pp , [18] A. Gouze, M. Antonini, M. Barlaud, and B. Macq, esign of signal-adapted multidimensional lifting scheme for lossy coding, IEEE Trans. on Image Processing, ecember 2004, to appear. [19] A. Gouze, C. Parisot, M. Antonini, and M. Barlaud, Optimal weighted model-based bit allocation for quincunx sampled images, Proceedings of IEEE International Conference in Image Processing (ICIP), September [20] P. Schröder and W. Sweldens, Spherical wavelets: Efficiently representing functions on the sphere, Proceedings of SIGGRAPH 95, pp , [21] N. yn,. Levin, and J. Gregory, A butterfly subdivision scheme for surface interpolation with tension control, ACM Transaction. on Graphics, vol. 9,2, pp , [22] F. Payan and M. Antonini, Model-based bit allocation for normal mesh compression, in Proceedings of IEEE international workshop on MultiMedia Signal Processing, Siena, Italy, september [23] C. Touma and C. Gotsman, Triangle mesh compression, Graphics Interface 98, pp , [24] A. Lee, W. Sweldens, P. Schröder, P. Cowsar, and. obkin, MAPS: Multiresolution Adaptive Parametrization of Surfaces, SIGGRAPH 98, [25] I. Guskov, K. Vidimce, W. Sweldens, and P. Schröder, Normal meshes, in Siggraph 2000, Computer Graphics Proceedings, K. Akeley, Ed. ACM Press / ACM SIGGRAPH / Addison Wesley Longman, 2000, pp [26] P. Cignoni, C. Rocchini, and R. Scopigno, Metro: Measuring error on simplified surfaces, Computer graphics Forum, vol. 2, no. 17, pp , [27] N. Aspert,. Santa-Cruz, and T. Ebrahimi, MESH: Measuring Errors between Surfaces using the Hausdorff distance, in Proceedings of the IEEE International Conference on Multimedia and Expo, vol. I, 2002, pp Frédéric PAYAN is a Ph. student in signal processing since 2000 at the I3S laboratory, directed by Marc Antonini, in the CREATIVE research group, directed by Mr. Barlaud, at the University of Nice-Sophia Antipolis (France). His work was supported by a grant from the region Provence Alpes Côte d Azur and Opteway Corporation in Sophia Antipolis (France). His research interests include signal processing, multiresolution analysis, wavelet transform, medical image coding, computational geometry and mesh compression. Marc ANTONINI received the Ph. degree in electrical engineering in 1991 and the Habilitation à iriger des Recherches from the University of Nice-Sophia Antipolis (France) in He was a postdoctoral fellow at the Centre National d Etudes Spatiales (Toulouse, France), in 1991 and Since 1993, he is working with CNRS at the I3S laboratory both from CNRS and University of Nice- Sophia Antipolis. He is a regular reviewer for several journals (IEEE Transactions on Image Processing, Information Theory and Signal Processing, IEE Electronics Letters) and participated to the organization of the IEEE Workshop Multimedia and Signal Processing 2001 in Cannes (France). He also participates to several national research and development projects with French industries, and in several international academic collaborations. His research interests include multidimensional image processing, wavelet analysis, lattice vector quantization, information theory, still image and video coding, joint source/channel coding, inverse problem for decoding, multispectral image coding, multiresolution 3 mesh coding. RAFT

18 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER h 0 s 0 Q s^ 0 g 0 S h 1 s 1 Q s^ 1 g 1 + ^ S h M-1 s M-1 Q s^ M-1 g M-1 (a) Filter bank representation. z t0 s 0 Q s^ 0 z -t0 S z t1 H s 1 Q s^ 1 G z -t1 + S ^ z tm-1 s M-1 Q s^ M-1 z -tm-1 (b) Corresponding polyphase representation. Fig. 1. Principle of an M-channel wavelet coder. (a) The signal s is splitted in M cosets according to an analysis step h i and a downsampling #. Once the cosets quantized, an upsampling " and a synthesis step g i are processed to obtain the reconstructed signal. (b) This scheme can be also represented by a polyphase representation. h 0 Q g 0 h 1 Q g 1 h 0 + g 0 S h M-1 Q g M-1 + S ^ h 1 Q g 1 h M-1 Q g M-1 Fig. 2. Principle of a wavelet coder including a two-level M-channel wavelet transform. After one decomposition in M cosets, another decomposition is processed on the 0 th coset, i.e., the coset of lowest frequency. RAFT

19 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER f e e n 2 d c d d b b d e b a b e f c a C a c f e b a b e d b b d d c d e e f n 1 Fig. 3. Neighborhood used to compute the wavelet coefficient C on a two-dimensional quincunx lattice. The values of the prediction and update operators for the (4; 2) and (6; 2) quincunx filters are given in table (I). n 1 and n 2 represent the frame of the quincunx lattice, used to design the prediction and update operators. 1/4-1/16-1/16 n 2 n 2 1/2 1/2 n 1-1/8-1/8 n 1-1/16 1/4-1/16 Fig. 4. Neighborhood used to design the prediction operator p 1 (left) and the update operator u 1 (right) of the lifted butterfly scheme related to the coset s 1. The operators related to the cosets s 2 and s 3 are obtained by the same way, but rotated according to their orientation. RAFT

20 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER Semi-regular mesh remeshing Wavelet Transform Topology encoding (coarse mesh) Geometry encoding M U L T I P L E I N G Bitstream Irregular mesh Bit allocation Target bitrate Fig. 5. Global scheme of the proposed mesh coder. It includes a remeshing technique to obtain a semi-regular version of the original mesh, a wavelet transform, a bit allocation optimizing the quantization, and finally an entropy coder. The coarse mesh connectivity is encoded with [23]. Fig. 6. PSNR curve for the model BUNNY (remeshed with MAPS), when the lifted butterfly transform is used. peak for this object is equal to 0: RAFT

21 JOURNAL OF LATE CLASS FILES, VOL. 1, NO. 11, OCTOBER Fig. 7. PSNR curve for the model VENUS (remeshed with the Normal Remesher), when the unlifted butterfly transform is used. peak for this object is equal to 1:5648. (a) Original. (b) Weighted MSE. (c) Unweighted MSE. Fig. 8. Visual difference between the original BUNNY (zoom on the left ear), the quantized one with the weighted mean square error as distortion criterion during the bit allocation, and the quantized one with the unweighted mean square error as distortion criterion. The bitrate of the quantized meshes is 0.4 bits per irregular vertex. RAFT