IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006 1 Wiener Filter-based Error Resilient Time Domain Lapped Transform Jie Liang, Member, IEEE, Chengjie Tu, Member, IEEE, Lu Gan, Member, IEEE, Trac D. Tran, Member, IEEE, and Kai-Kuang Ma, Senior Member, IEEE

Abstract— In this paper, the design of the error resilient time- DCT and reducing the blocking artifact associated with DCT- domain lapped transform is formulated as a linear minimal based schemes. On the other hand, the postfilter can also mean squared error (LMMSE) problem. The optimal Wiener be designed to spread out the information of a block to its solution and several simplifications with different tradeoffs be- tween complexity and performance are developed. We also prove neighboring blocks. This is helpful when we need to recover the persymmetric structure of these Wiener filters. The existing a lost block during image transmission. mean reconstruction method is proven to be a special case of In [10], various techniques were proposed to estimate the the proposed framework. Our method also includes as a special lost data when the lapped orthogonal transform (LOT) was case the linear interpolation method used in DCT-based systems used. A mean reconstruction method and a nonlinear sharp- when there is no pre/postfiltering and when the quantization noise is ignored. The design criteria in our previous results are ening method were found to be quite effective, under the scrutinized and improved solutions are obtained. Various design assumption that DC coefficients were intact. In particular, in examples and multiple description image coding experiments the mean reconstruction method, each lost block was estimated are reported to demonstrate the performance of the proposed by averaging its available neighboring blocks. In [11], it was method. found that the extended lapped transform (ELT) has better robustness against transmission error than the conventional I.INTRODUCTION LOT. However, the complexity of the ELT is higher than the LOT and it does not have linear phase [9], thus its application With the rapid development of Internet, computer and wire- in image coding is limited. less communications technologies, there have been growing The methods in [10], [11] performed error concealment at demands for delivering compressed images over Internet and the decoder. The transforms used there were still optimized for wireless networks. This poses new challenges to conven- the best compression performance. In [12], the optimization tional algorithms, which are extremely of the error resilient LOT for a specific error concealment vulnerable to transmission errors. On the other hand, perfect technique was addressed. It was shown that the lost blocks can reception of all data is usually not necessary due to the intrinsic be better recovered this way. In addition, the transform can be structures present in most natural images. Special algorithms designed to achieve different trade-offs between compression known as error concealment can be employed to produce efficiency and error resilience. reasonable visual quality in the presence of transmission error. Since the main purpose of [12] was to verify the feasibility Among the error concealment techniques that have been of error resilient lapped transform, it still used the simple mean proposed [1], some methods, such as the reversible variable reconstruction method in [10]. To improve the smoothness length coding [2], introduce error resilience at the encoder. of the reconstructed images, the maximally smooth recovery Some of them focus on the error concealment at the de- (MSR) method proposed in [13] was used in [14], and a coder side by estimating the lost data with methods such as multiple description was developed using the trans- interpolation and projection onto convex sets [3]–[8]. Other forms designed in [12]. However, the transforms in [12] approaches tackle the problem by a joint design of the encoder were not optimal for the MSR method because they were and decoder, for which the lapped transform provides a useful designed for the mean reconstruction method. This problem framework [9]. was resolved in [15] by incorporating the maximally smooth In the original lapped transform [9], a postfilter is applied recovery constraint in the objective function of the transform at block boundaries after the DCT. The postfilter is usually optimization. Better transforms were obtained to achieve the designed to remove the remaining redundancy between neigh- same reconstruction quality with lower . boring blocks, thereby improving the coding efficiency of the Despite the improvements in [12], [14], [15], some limita- J. Liang is with the School of Engineering Science, Simon Fraser Univer- tions still exist. First, only orthogonal lapped transform was sity, Burnaby, BC V5A 1S6, Canada (Email: [email protected]). C. Tu is with Mi- considered. Therefore the MSE of the reconstructed image crosoft Corporation, Redmond, WA 98052 (Email: [email protected]). is always the same [12], which seriously confines the error L. Gan is with the School of EECS, the University of Newcastle, N.S.W. 2308, Australia (Email: [email protected]). T. D. Tran is with the ECE Department, concealment capability of the system. Second, the entire M- the Johns Hopkins University, Baltimore, MD 21218. (Email: [email protected]). band, 2M-tap (M 2M for short) lapped transform matrix is K.-K. Ma is with the School of EEE, Nanyang Technological University, optimized directly× (M is the block size) , which increases the Singapore 639798 (Email: [email protected]). This work was supported by SFU President’s Research Grant, the NSERC Discovery Grant and NSF complexity of both optimization and implementation. CAREER Grant CCR-0093262. Recently, a new family of lapped transforms, the time- 2 IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006

M/2 domain lapped transform (TDLT) [16], has been developed. -1 x(n-1) P q(n-1) P x(n-1) In the TDLT, a prefilter is applied at each block boundary D y(n-1) y(n-1) ID M/2 s(n-1) C + s(n-1) before the DCT. At the decoder side, a postfilter is applied T CT M M -1 x(n) P q(n) P x(n) at the same location after the inverse DCT. This framework D y(n) y(n) is more compatible to DCT-based infrastructures, since the s(n) C + ID s(n) T CT x(n+1) P q(n+1) -1 pre/postfilters can be easily incorporated into existing DCT P x(n+1) D y(n+1) y(n+1) software or hardware implementations. The TDLT also offers s(n+1) C + ID s(n+1) T CT competitive compression performance compared to JPEG 2000 x(n+2) P -1 P x(n+2) [17]. As a result, it has been used in Microsoft Windows Media 9 codec (WMV9) [18]. WMV9 has been accepted by Fig. 1. Forward and inverse time-domain lapped transform. the DVD Forum as one of the three mandatory formats for the next-generation HD DVD. It is also being standardized by the Society of Motion Picture and Television Engineers (SMPTE) Correspondingly, a postfilter T is applied by the decoder at as its VC-1 video coding standard. Therefore, the TDLT will each block boundary after inverse DCT. In this paper, we play an important role in future image and video coding choose the following structure of P and T so that they yield applications, and it is necessary to develop error resilient linear-phase perfect reconstruction filter bank [16]: TDLT so that it can be used in error prone environments. Error resilient TDLT is first considered in [19], [20]. Thanks to the structure of the TDLT, the design of error resilient P = WM diag I M , V M WM , lapped transform reduces to that of the pre/postfilters. The { 2 2 } −1 −1 (1) problem is therefore more tractable. Biorthogonal solutions T = P = WM diag I M , V M WM , { 2 2 } with lower MSE can be easily obtained by using biorthogonal pre/postfilters. In addition, the decoder can employ two post- filters - one for perfectly received blocks and another for lost where blocks. One remaining problem in [19], [20] is that the mean I M J M 1 2 2 reconstruction method is still used to estimate the lost blocks. WM = . (2) √2 J M I M In this paper, we present a general framework of error resilient " 2 − 2 # TDLT. We formulate the filter design as a linear minimal MSE (LMMSE) problem, and derive the corresponding Wiener filter M M solution, which unleashes the full potential of the error re- The I M and J M above are identity matrix and reversal 2 2 2 × 2 identity matrix, respectively. The matrix V M can be optimized silient lapped transform. Several simplifications of the general 2 structure and their optimal solutions are then developed. The to improve the compression performance of the system. mean reconstruction method is revealed to be a trivial special In this paper, we use x(i), s(i), y(i) and q(i) to denote case of the general solution. As a by-product, we also show the i-th block of prefilter input, DCT input, DCT output, and that the linear interpolation method used in DCT systems is a quantization noise, respectively. Notice that x(i) is aligned special case of the proposed scheme when the pre/postfilters with the prefilter, whereas the rest are aligned with the DCT. are disabled and when quantization error is ignored. In ad- Compared with the notations in [12], [14], [15], [19], [20], the dition, we prove that these Wiener filters have persymmetric definition here can simplify the problem formulation and the structure and can be implemented efficiently. We also revisit derivation of the optimal solution. some of the design criteria used in our preliminary results The structure of the time-domain lapped transform allows in [21], [22], such as the reconstruction gain and postfilter an effective strategy for error concealment. In this paper, we switching, and improved solutions are presented. assume that all coefficients of a block are either received Compared to the mean reconstruction method in [19], [20], perfectly or lost entirely. This scenario can happen in, for the reconstruction error can be reduced by as much as 80% by example, multiple description coding [23]. In [19], [20], a the Wiener filter method, and up to 7 8 dB improvement can mean reconstruction method as in [12] is used, where the be achieved in multiple description image∼ coding experiments. lost block is estimated by averaging the received neighboring Our method also shows noticeably improvements over the blocks. It is shown in [19], [20] that the pre/postfilters can be results in [14], [15]. jointly designed such that the reconstructed quality is improved in the presence of transmission error. Moreover, two postfilters II. GENERAL PRE/POSTFILTERING STRUCTURE FOR can be designed - one for perfectly received blocks and another ERROR CONCEALMENT one for lost blocks, as shown in Fig. 2 (a). Fig. 1 illustrates the time-domain lapped transform-based To further improve the reconstruction quality, notice that image compression system with block size of M (M is when yˆ(n) is lost, the error concealment problem can be even). An M M prefilter P is applied at the boundary viewed as the estimation of x(n) and x(n + 1) from the of two neighboring× blocks before the DCT. Therefore the observations yˆ(n 1) and yˆ(n + 1), or equivalently ˆs(n 1) basis functions of the forward transform cover two blocks. and ˆs(n+1). Therefore− the general filter should be a 2M −2M × LIANG et al.: WIENER FILTER-BASED ERROR RESILIENT TIME DOMAIN LAPPED TRANSFORM 3

M/2

-1 -1 -1 -1 P x(n-1) P x(n-1) P x(n-1) P x(n-1) y(n-1) ID y(n-1) ID s(n-1) y(n-1) ID s(n-1) y(n-1) ID CT CT CT CT 1/2 T x(n) x(n) T T + x(n) + x(n) H 1/2 0 H1 s(n) H2 s(n) + + T x(n+1) x(n+1) T x(n+1) T x(n+1) y(n+1) ID y(n+1) ID s(n+1) y(n+1) ID s(n+1) y(n+1) ID CT CT CT CT -1 -1 -1 -1 P x(n+2) P x(n+2) P x(n+2) P x(n+2) (a) (b) (c) (d)

Fig. 2. (a) Existing decoder side error concealment design; (b) General structure for error concealment; (c) A close approximation of (b); (d) Further simplification of (c). matrix H0, as shown in Fig. 2 (b). If we define also assume that the quantization noises of different subbands R T T T are uncorrelated, i.e., q3q3 is a diagonal matrix. At high x2 = x (n) x (n + 1) , rates, the noise variance of the k-th subband can be written as T T T T T x4 =  x (n 1) x (n) x (n + 1) x (n + 2) , [25] − − T 2 2 2Rk T T σqk = c σyk 2 , (11) ˆs2 =  ˆs (n 1) ˆs (n + 1) ,  − T T T T s3 =  s (n 1) s (n) s (n + 1) , where c is a constant that depends on the input statistics, − 2 T T T T σyk is the subband variance that can be obtained from the ˆs3 =  ˆs (n 1) ˆs (n) ˆs (n + 1)  , − input statistics and the forward transform, and Rk is the bit T T T T q3 =  q (n 1) q (n) q (n + 1) , rate allocated to the k-th subband. The bit allocation can be − (3) optimized during the filter design. We will demonstrate in Sec.   VI-C that the quantization noise can be safely ignored in the the auto-correlation of the reconstruction error can be written Wiener filter expression, since the final error is dominated by as T the transmission loss. Ree = E (H ˆs x )(H ˆs x ) . (4) { 0 2 − 2 0 2 − 2 } When applied to image error concealment, an important The linear MMSE solution of H0 is one that minimizes the requirement is that the Wiener filter should maintain the DC following MSE expression: component of the local region, i.e., 1 = trace Ree . (5) H0d = d, (12) E 2M { } The optimal solution is given by the Wiener filter [24] where d = [1, 1,..., 1]T . Therefore we need to minimize ∗ − the MSE subject to the constraint in (12). The solution can H = R R 1 . (6) 0 x2ˆs2 ˆs2ˆs2 be found by the Lagrangian method, which constructs the following objective function: Matrices Rx2ˆs2 and Rˆs2ˆs2 in (6) can be obtained as follows. We first partition P into 2M−1 ′ 1 T T T = trace Ree + λi(h0,id 1), (13) P = P0 P1 , (7) E 2M { } − i=0 X where P0 and P1 contain the top and the bottom M/2 rows where h0,i is the i-th row of H0. The corresponding LMMSE of the prefilter P, respectively. Let C3 = diag C, C, C solution is represents the DCT operations of three neighboring{ blocks (C} ¯ ∗ T −1 is the M-point DCT). We have H = (Rx2s2 Mλd )R , (14) 0 ˆ − ˆs2ˆs2 T T ˆs3 = s3 + C3 q3 = P34x4 + C3 q3, (8) where where T λ = [λ0, λ1,..., λ2M−1] P34 = diag P1, P, P, P0 . (9) { } 1 −1 (15) = Rx2s2 R I d. T −1 ˆ ˆs2ˆs2 − From (8) we get M(d Rˆs2ˆs2 d) T  Rx4ˆs3 = Rx4s3 = Rx4x4 P34, Another way to ensure the DC condition is to scale each T row of the Wiener filter such that the sum of every row is Rs3ˆs3 = Rs3s3 = P34Rx4x4 P34, (10) unity, i.e., R = P R PT + CT R C , ˆs3ˆs3 34 x4x4 34 3 q3q3 3 ˆ ∗ ∗ H0 = ΘH0, (16) where Rx4x4 and Rq3q3 are correlation matrices of x4 and q3, respectively. In deriving (10), we assume that the input is where Θ is a 2M 2M diagonal matrix whose i-th diagonal × 2M−1 ∗ uncorrelated with the quantization noise. Matrices Rx2ˆs2 and entry is Θ(i, i) = 1/ j=0 H0(i, j). Our experimental Rˆs2ˆs2 in (6) can thus be obtained from the appropriate sub- results show that the difference between the two solutions is P matrices of Rx4ˆs3 and Rˆs3ˆs3 . In this paper, Rx4x4 in (10) is negligible. Therefore the scaling method in (16) is chosen in obtained by assuming the input follows an AR(1) model. We the rest of this paper due to its simplicity. 4 IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006

III. SIMPLIFICATIONS AND FACTORIZATIONS Since ˆs1(n 1) and ˆs0(n + 1) are parts of ˆs2, we have −−1 −1 R − R I and R R I , thus the In this section, we show that the general solution in (6) ˆs1(n 1)ˆs2 ˆs2ˆs2 = b1 ˆs0(n+1)ˆs2 ˆs2ˆs2 = b2 can be approximately factorized into two stages, which can general Wiener filter reduces to the two-stage method given simplify the implementation. Further simplifications are also by (20) and (19). presented, leading to the conclusion that the mean reconstruc- In [20], it is found that applying two postfilters can improve tion method is a special case of our method. We also prove the performance of the mean reconstruction method. One that these Wiener filters are persymmetric, a property that can postfilter is for the correctly received blocks and another be used to further reduce the implementation cost. one is for the lost blocks. When the M 2M Wiener filter (19) is used, the analysis above shows that× there is no need A. A close approximation of the general solution to switch between two postfilters. The perfect reconstruction postfilter P−1 is sufficient. Therefore the implementation can In [19], [20], the lost blocks are first estimated by the simple be simplified. average of neighboring blocks before applying postfilter. This is clearly not optimal in the light of estimation theory. To find the optimal estimate of s(n) in this two-stage approach, we B. Further Simplifications define an M 2M matrix H1, i.e., The complexity of (20) and Fig. 2 (c) can be further reduced × by imposing the following constraint on H1: ¯s(n)= H1ˆs2. (17)

H = H H = H IM IM , (24) The optimal solution for H1 can be found by minimizing 1 2 2 2

1 T where the size of H2 is M M. This is equivalent to 1 = trace E (H1ˆs2 s(n))(H1ˆs2 s(n)) , (18) × E M { { − − }} estimating s(n) by and the solution is also a Wiener filter, which can be written ¯s(n)= H (ˆs(n 1)+ ˆs(n + 1)) , H ˆsa. (25) as 2 − 2 ∗ −1 H1 = Rs(n)ˆs2 Rˆs2ˆs2 , (19) The structure is shown in Fig. 2 (d). Again, Wiener solution exists in this case and is given by where Rs(n)ˆs2 is a sub-matrix of Rs3ˆs3 . Once ¯s(n) is obtained, the postfilter is applied as usual. A ∗ −1 H2 = Rs(n)ˆsa Rˆsaˆsa . (26) different postfilter can be used around lost blocks to further improve the visual quality. This scheme, as shown in Fig. 2 The matrices involved can be obtained from (10) by simple (c), can be viewed as imposing the following structure to the manipulations. general matrix H0 in Fig. 2 (b): In this case, our experimental results show that using two postfilters is still necessary, because the performance of the I T 0 b1 M M filter (26) is far below that of (19), and a special H = M M H , (20) 0 0 T 1 postfilter× is required to further reduce the error. M M  I    b2 It is clear from (24) that the mean reconstruction method   where Ib1 and Ib2 are defined by used in [19], [20] is simply a special case of the already sub- 1 optimal approach in (24) with H2 = IM . Therefore it can I = 0 M I M 0 M 0 M , , 2 b1 2 2 2 2 be expected that the error concealment performance can be (21) h i ∗ ∗ ∗ 0 M 0 M I M 0 M , improved considerably if H2, H1 or H0 are used. Ib2 = 2 2 2 2 . When applied to ˆsh, I simply extracts thei second half of 2 b1 C. Persymmetry of the Wiener Filters ˆs(n 1) and Ib2 extracts the first half of ˆs(n + 1). Due− to the structure of the lapped transform, the result given The Wiener filters HL×N derived in the last two sections, by the two-stage method in (20) approximates the general including the constrained solutions (14) and (16), satisfy the solution (6) very well. In fact, the two structures are equivalent following persymmetric condition (also known as centrosym- if T = P−1 and if we ignore the quantization noise. In this metric) [26]: case, xˆ(n) = x(n), ˆs(n) = s(n). It can be seen from Fig. 1 that HL×N = JL×LHL×N JN×N , (27) ˆs (n 1) T 0 1 − which means Hi, j = HL− −i, N− −j , for i = 0,...,L 1, x2 = xˆ2 = ˆs(n) , (22) 1 1 0 T   j = 0,...,N 1, i.e., the matrix is symmetric with respect−   ˆs0(n + 1) −   to its center. The proof is given in Appendix A and relies on where ˆs1(n 1) and ˆs0(n + 1) denote the second half of the linear-phase property of the lapped transform. The per- ˆs(n 1) and− the first half of ˆs(n + 1), respectively. Plugging − symmetric structure allows the Wiener filters to be factorized (22) into the definition of Rx2ˆs2 , the general Wiener filter in as [27] (6) becomes −1 HL×N = WL×Ldiag A L × N , B L × N WN×N . (28) − 2 2 2 2 Rˆs1(n 1)ˆs2 Rˆs2ˆs2 { } ∗ T 0 −1 H0 = Rˆs(n)ˆs2 Rˆs2ˆs2 . (23) This factorization can be exploited to reduce the complexity 0 T  −1    Rˆs0(n+1)ˆs2 Rˆs2ˆs2 of implementation by roughly 50%.   LIANG et al.: WIENER FILTER-BASED ERROR RESILIENT TIME DOMAIN LAPPED TRANSFORM 5

IV. ESTIMATION FROM ONE NEIGHBORING BLOCK Using the adjoint matrix method, the first column of the inverse matrix R−1 can be found to be Sometimes it is necessary to estimate a lost block from only xˆBxˆB one neighboring block. This can happen at the image boundary 1 ρ2 ρ6 + ρ8 or when contiguous blocks are lost. A Wiener solution can also 1 ρ(1− ρ2− ρ6 + ρ8)  − − −  . (35) be obtained for this scenario. When only the previous block det(RxˆB xˆB ) 0 ˆs(n 1) is available after the inverse DCT, the objective is to  0  − ∗   find an M M matrix H which minimizes   × 3 It can be easily verified that the first column of the product −1 T RxnxˆB R becomes all zero. Since the Wiener filter is still trace E (H3ˆs(n 1) s(n))(H3ˆs(n 1) s(n)) . (29) xˆBxˆB { { − − − − }} persymmetric in this case, the last column is also zero. When The solution is thus given by ρ =0.95, the filter becomes

∗ −1 0 0.67 0.33 0 H = R − R . (30) H = , (36) 3 s(n)ˆs(n 1) ˆs(n−1)ˆs(n−1) DCT 0 0.33 0.67 0   The matrices involved can be readily obtained from Rs3ˆs3 and which represents the linear interpolation between the two Rˆs3ˆs3 in (10). immediate neighboring pixels of the lost block. Similarly, if only the next block ˆs(n + 1) is available, the When M =8, the Wiener filter (32) only has the following ∗ corresponding Wiener solution H4 becomes non-zero entries in the 8-th and the 9-th columns: T H∗ R R−1 (31) 0.88 0.77 0.65 0.54 0.43 0.32 0.21 0.11 4 = s(n)ˆs(n+1) ˆs(n+1)ˆs(n+1). , 0.11 0.21 0.32 0.43 0.54 0.65 0.77 0.88   From that facts that Rˆs(n−1)ˆs(n−1) = Rˆs(n+1)ˆs(n+1), (37) Rˆs(n−1)ˆs(n−1) is persymmetric, and Rs(n)ˆs(n−1) = which is also a linear interpolation operator. JRs(n)ˆs(n+1)J, as shown in Appendix A, it is straightforward However, if quantization noise is included in RxˆB xˆB , its ∗ ∗ inverse R−1 would lose the nice analytical expression in to verify that H3 = JH4J, i.e., the two filters are xˆB xˆB persymmetric to each other. However, each of them is not (35), and the Wiener filter would in general be a full matrix, persymmetric; therefore they cannot be factorized as in (28). meaning that the linear interpolation is no longer optimal in the DCT systems.

V. RELATIONSHIP WITH LINEAR INTERPOLATION IN DCT VI. DESIGN EXAMPLES AND APPLICATIONS SYSTEMS A. Design Criteria Another special case of the proposed framework is when the In this section, we show various design examples and their pre/postfilters are turned off. In this case, the lapped transform applications in multiple description image coding. In [20] and reduces to the DCT. If we still estimate a lost block x(n) by T T T our preliminary results in [21], [22], a Matlab optimization its two neighboring blocks xˆB = xˆ (n 1), xˆ (n + 1) , program is used to find the optimal TDLT that maximizes the the Wiener filter (19) becomes −   following objective function −1 HDCT = Rx(n)xˆB RxˆB xˆB . (32) J = GTC α + β GR, (38) − E An interesting fact is that when the quantization noise in which is a weighted average of the coding gain GTC of the R is ignored, only the M-th and the (M +1)-th columns transform, the residual MSE in (5) after transmission error xˆBxˆB E of the Wiener filter are non-zero, and the result is therefore a and Wiener filter-based error concealment, and the reconstruc- linear interpolation of the missing block using the last pixel tion gain GR. Notice that the classical lapped transform is of xˆ(n 1) and the first pixel of xˆ(n + 1). This justifies the only optimized for coding gain. The coding gain measures wide adoptions− of linear interpolation method in, for example, the maximum distortion reduction of a transform over PCM [4], [5], [7]. scheme. With optimal bit allocation the coding gain reduces To illustrate this property and to gain more insights, we to [25] assume RxˆB xˆB = RxBxB and consider the expression of the 2 , σx Wiener filter when the block size is 2. Assuming an AR(1) GTC 10 log10 1 , (39) − M M 1 2 2 input with unit variance and correlation coefficient ρ, one can σ fi i=0 yi || || show that 2 Q 2  where σx is the variance of the input, σyi is the variance of the ρ2 ρ ρ2 ρ3 2 R = , (33) i-th subband, and fi is the norm of the i-th synthesis basis xnxˆB ρ3 ρ2 ρ ρ2 || ||   function. The input is assumed to follow an AR(1) model with correlation coefficient ρ =0.95. 1 ρ ρ4 ρ5 The reconstruction gain is first defined in [12] to control ρ 1 ρ3 ρ4 the error distribution across transform coefficients when a R = . (34) xˆBxˆB  ρ4 ρ3 1 ρ  block is lost. This is pivotal to the performance of orthogonal  ρ5 ρ4 ρ 1  transforms because the MSE is always the same in different     6 IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006

0.45 0.45 P4 P41 P3 P31 0.4 P2 0.4 P21 P1 P11

0.35 0.35

0.3 0.3

0.25 0.25

0.2 0.2 MSE at each pixel location 0.15 MSE at each pixel location 0.15

0.1 0.1

0.05 0.05

0 0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Pixel location Pixel location (a)(b)

Fig. 3. (a) Error distributions of the TDLT designs in [20]; (b) Error distributions of new designs.

512 x 512 Lena: 4 descriptions 512 x 512 Lena: 3 descriptions 45 38

P41 36 P4 P31 40 P3 34 P21 P2 P11 32 P1 35 P41 PSNR (dB) PSNR (dB) 30 P4 P31

P3 28 30 P21 P2 P11 26 P1

25 24 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Bit Rate (bpp) Bit Rate (bpp) (a)(b)

512 x 512 Lena: 2 descriptions 512 x 512 Lena: 1 descriptions 36 32

34 30 P41 P41 P4 P4 32 P31 P31 P3 28 P3 P21 30 P21 P2 P2 P11 26 P11 PSNR (dB) 28 P1 PSNR (dB) P1

24 26

22 24

22 20 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Bit Rate (bpp) Bit Rate (bpp) (c)(d)

Fig. 4. R-D curves of the 512 × 512 Lena image under different loss patterns and different TDLT configurations: (a) with four descriptions; (b) with three descriptions; (c) with two descriptions; (d) with one description. LIANG et al.: WIENER FILTER-BASED ERROR RESILIENT TIME DOMAIN LAPPED TRANSFORM 7

TABLE I DESIGN EXAMPLESOF DIFFERENT CONFIGURATIONS WITH M = 8, β = 0 AND DIFFERENT α IN (38)

Cfg. P1 P10 P11 P12 P2 P20 P21 P22 P3 P30 P31 P32 P4 P40 P41 P42 α - 92 92 111 - 26 26 37.3 - 12.5 12.5 18.5 - 1 1 1 GT C 6.96 6.96 6.96 6.96 8.41 8.42 8.42 8.42 9.17 9.17 9.17 9.17 9.61 9.61 9.61 9.61 MSE 0.14 0.03 0.03 0.08 0.15 0.06 0.06 0.10 0.16 0.10 0.10 0.13 0.21 0.17 0.17 0.18

TABLE II V SOME OPTIMIZED RESULTS FOR THE FREE MATRIX M/2 IN Eq. (1) WITH M = 8

Cfg. P11 P21

¿ ¾ ¿

¾ −0.9672 0.4692 0.3425 0.1418 0.9448 0.1452 −0.7385 −0.0213

7 6 7

6 −0.1651 0.6042 −0.7937 0.0394 −0.0809 1.0449 −0.2754 −0.2339

7 6 7

V M 6

5 4 5 2 4 0.4408 0.5179 0.0867 0.7004 0.7003 0.2656 0.6545 −0.1442 −0.2171 −0.5711 −0.3456 0.7543 0.1620 0.2895 −0.0939 0.9174

Cfg. P31 P41

¿ ¾ ¿

¾ 0.9588 0.6245 −0.0743 −0.1501 0.9490 0.7662 0.3260 0.2031

7 6 7

6 −0.4781 0.9028 0.3171 −0.1448 −0.5546 0.8881 0.5931 0.2007

7 6 7

V M 6

5 4 5 2 4 0.1968 −0.2192 0.9954 0.0638 0.1099 −0.3708 1.0738 0.3596 0.0278 0.1226 −0.0712 1.0049 −0.0307 0.0048 −0.1267 1.1664

(a)(b)(c)(d)

(e)(f)(g)(h)

(i)(j)(k)(l)

Fig. 5. Decoding results at 1bpp and 50% loss. (a) Loss pattern; (b) Original TDLT (23.87 / 39.94 dB); (c) P22 (27.54 / 38.38 dB); (d) P12 (28.68 / 35.59 dB); (e) P4 (24.01 / 39.92 dB); (f) P3 (25.28 / 39.61 dB); (g) P2 (26.00 / 38.11 dB); (h) P1 ( 26.19 / 34.47 dB); (i) P41 (24.59 / 39.96 dB); (j) P31 (27.45 / 39.84 dB); (k) P21 (30.32 / 39.03 dB); (l) P11 (33.20 / 35.91 dB). 8 IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006

(a) (b) (c) (d)

Fig. 6. Portions of reconstruction results with P11 design and the 512 × 512 Barbara image at 1 bpp: (a) four descriptions (32.72 dB); (b) three descriptions (29.09 dB); (c) two descriptions (27.12 dB); (d) one description (24.11 dB). methods. The reconstruction gain in [12] attains its maximal B. Comparison with Mean Reconstruction Method value when the error is uniformly distributed across all trans- In this section, we compare the performance of the Wiener form coefficients. filter-based TDLT design with that of the mean reconstruction- Since biorthogonal transform is used in [20], the estimation based results in [20]. The entries in matrix V of Eq. (1) are the error in the transform domain is no longer equal to the final optimization parameters. Different solutions can be obtained reconstruction error. Therefore the following spatial domain by varying α in (38) (β is fixed as 0). The quantization noise reconstruction gain is defined [20] is ignored in Wiener filter expressions, for example, (19). We

1 will show in Sec. VI-C that this is a reasonable choice. 2M 2M−1 2 Four families with M =8 are designed. Their coding gains i=0 σei GR = − , (40) and residual MSE after error concealment are summarized in  1 2M 1 2 2QM i=0 σei Table I. Some optimized matrices V in the prefilter are shown in Table II, and the error distribution of some designs across 2 P where σei is the i-th diagonal entry of Ree in (4), i.e., the the two affected blocks x(n) and x(n + 1) are plotted in Fig. final expected reconstruction error of the i-th pixel in the two 3. The Wiener filters H∗, H∗, and H∗ in (6), (19) and (26) are T 0 1 2 blocks xT (n) xT (n + 1) . used to obtain configurations Pi0, Pi1, and Pi2 (i =1, ..., 4), When Wiener filter is applied, our experimental results show respectively. Two postfilters are optimized for configurations   ∗ that the reconstruction error can be reduced so dramatically Pi2, since H2 is sub-optimal. The configurations in each that there is no need to control the error distribution explicitly. family are designed to have similar coding gains so that their Therefore we always fix β = 0 in this paper. In addition, performance can be compared fairly. The first group yields the our experiments indicate that the error distribution control lowest coding gain but also the lowest MSE in the presence imposes serious constraints on the filter design. Removing this of transmission error. On the contrary, the last group has the criterion in the optimization can improve the coding gain or highest coding gain (close to the best result in [16]) but is the reconstruction quality, as will be shown later. most vulnerable to transmission loss. In fact, higher spatial domain reconstruction gain as defined Also included in Table I are the configurations P1 to P4 1 above does not always lead to pleasant visual quality, since it in [20] with the mean reconstruction method (H2 = 2 I). tends to generate similar error at each pixel, thereby creating Among them, P3 and P4 use two postfilters. The first one a clear visual artifact around the lost block, especially if the is the inverse of the prefilter and is applied to healthy blocks, neighboring blocks have very small quantization error. This and the second one is applied around lost blocks to further suggests that visually pleasant error distribution should have reduce the reconstruction error. a smooth transition between healthy blocks and the error Compared with P1 to P4, we can see from these results that concealed blocks. In [14], [15], this is achieved by solving the final reconstruction error in the presence of transmission for each block a underdetermined equation with a smoothness error can be reduced substantially by the 2M 2M and × constraint. In our framework, if the control of error distribution M 2M Wiener filters. The improvement is more pronounced × is necessary, we can include a curve fitting term in the as coding gain decreases, since more correlations among objective function such that the optimized reconstruction error neighboring blocks are introduced. For example, compared across the two blocks follows a desired shape, for example, with P1, the MSE after error concealment is reduced by 80% a Gaussian curve. Moreover, the template can be defined in P10 and P11. It can still be reduced by 20% even when the to match the neighboring quantization noise at the block coding gain is at its highest value, as given by P40 and P41. boundary. Further discussion can be found in [22]. However, Notice also that even the MSE of P3i is less than that of P1, when the error distribution constraint is introduced, the coding although the coding gain of P3i is much higher than P1. ∗ gain or the MSE has to be sacrificed; therefore it becomes Table I also shows that the MSE given by H1 in (19) is more complicated to find a good trade-off. identical to that of the general 2M 2M solution H∗ in × 0 LIANG et al.: WIENER FILTER-BASED ERROR RESILIENT TIME DOMAIN LAPPED TRANSFORM 9

(6). This verifies their relationship proved in Section III. The due to the average operator in mean reconstruction method. ∗ performance of the M M Wiener filter H2 lies roughly Portions of the multiple description decoding results for the halfway between the mean× reconstruction and the M 2M 512 512 Barbara image are given in Fig. 6, when P11 is used. Wiener filter. × It can× be seen that the quality of the reconstruction decreases Some results in Table I are also better than our preliminary gradually when more descriptions are lost. This shows that results in [21], [22], due to the elimination of reconstruction the AR(1) model-based Wiener filter is quite robust even for gain in the objective function and due to the switch of two images of rich textures. postfilters for Pi2 only. The improvement can also be observed in the image coding experiments below. C. Robustness of the Wiener Filter to Quantization Noise To verify the error concealment performance of the Wiener As mentioned in Sec. VI-B, the Wiener filter examples in filters, we implement a multiple description codec following this paper are obtained by ignoring the quantization noise. In the approach in [14], where the transformed image is split this subsection, we investigate the robustness of the Wiener into four descriptions at block level. For example, all (even, filter to quantization noise. A prefilter and the corresponding even)-indexed blocks are grouped into the first description, Wiener filter are designed by including the quantization noise all (even, odd)-indexed blocks are grouped into the second (11) in (19). This requires us to choose a typical average description, and so on. The context-adaptive binary arithmetic bit rate, because in practice we prefer to have a prefilter coding in [17] is applied to each description independently. At and a Wiener filter that can be used at all bit rates. In our the decoder side, the inverse DCT is applied to all received optimization, the average bit rate is fixed as 1 bit/pixel. Notice blocks first. After that, the Wiener filter is used to estimate that the bit rate does not have to be specified in the coding each lost block before applying postfiltering. The Wiener gain optimization part, because the bit rate can be canceled filters derived in this paper are based on 1-D signal model. out under optimal bit allocation [25]. To apply them to 2-D images, we first estimate each row of The parameter α in (38) is chosen as 30 so that the coding a lost block from its horizontal neighboring blocks, and then gain of the optimized lapped transform is the same as P21 estimate each column from its vertical neighbors. The final in Table I. The multiple description coding performances of result is the weighted average of the two estimations, and the two configurations for the images Lena and Barbara are the weighting factors are decided by the number of available compared in Fig. 7, which shows that the result depends neighbors in each direction. on the nature of the image. In particular, P21 gives better The scenario of losing three out of the four descriptions concealment performance for the Lena image, whereas con- requires special treatments. In this case, a three-step method sidering the quantization noise improves the performance of is employed to estimate the lost data. We first estimate those the Barbara image. Therefore a better quantization noise model blocks that have available horizontal neighbors, followed by is necessary in order to get a consistent gain. However, the fact those blocks with available vertical neighbors. The rest are that the difference of the two methods is less than 0.4 dB in then estimated using previously estimated neighbors. We will all cases suggests that the Wiener filter is not sensitive to the show later that this approach achieves significant improvement quantization noise, since the error caused by transmission loss over the maximally smooth recovery method in [14], [15]. is usually much higher than the quantization noise. Fig. 4 plots the R-D curves of different TDLT configurations with different number of available descriptions when the 512 512 Lena image is used. Fig. 4 (a) shows that all D. Comparison with MSR Method new× designs yield better compression performance than their In this section, we compare the performance of our Wiener counterparts in [20] with the same coding gain. In addition, filter-based TDLT with the results in [14] and [15]. The they provide much better reconstruction results than the mean method in [14] uses the maximally smooth recovery in the reconstruction method. For example, at 2bpp, the PSNRs by estimation part and examples T6 to T9 in [12] in the transform P11 and P21 is about 7 8 dB and 4 dB higher than that of part. Since our codec uses arithmetic coding, whereas adaptive P1 and P2, respectively. We∼ can also see that the improvement is used in [14], we will only compare the of P1 over P4 is less than 3 dB in most cases, whereas P11 performance of the two methods in the absence of quantization can be 10 dB better than P41. This agrees with the theoretical error. This is to ensure fair comparison of the transform and MSE in Table I. Moreover, it can be seen that even P31 can the error concealment parts. Reconstructed PSNRs for the achieve better R-D performance than P1. Overall, P21 offers 256 256 Lena image with different number of received a good trade-off between the compression performance and descriptions× are reported in [14] and is duplicated in Table error resilience. III. For fair comparison, we design four M 2M Wiener ∗ × Fig. 5 shows the reconstruction results of different methods filters using H1 in (19) to match the coding gains of T6 when two descriptions of the 512 512 Lena image coded to T9, respectively. As discussed before, β is fixed as 0 in at 1bpp are lost. The PSNRs with× and without error are the objective function (38) and the postfilter is chosen as the reported for each case. Satisfactory results are produced by inverse of the prefilter. P11 and P21. Notice that the coding performance of P21 is As shown in Table III, the Wiener filter based TDLT only 0.9 dB below the compression-optimized TDLT, but the achieves an average of 2.98 dB improvement over [14]. reconstruction is 6.5 dB higher. The results of P1 to P4 are More improvements are achieved at higher quite blurred and exhibit strong ghost artifacts near the edges, gain. In addition, the most significant improvement, 3.94 dB 10 IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006

Lena: PSNRs of P21 and Wiener filter with quant. noise (α=30, rate=1bpp) Barb: PSNRs of P21 and Wiener filter with quant. noise (α=30, rate=1bpp) 55 55

50 50

4 desc: P21 45 45 4 desc: With Q 3 desc: P21 4 desc: P21 3 desc: With Q 4 desc: With Q 40 2 desc: P21 40 3 desc: P21 2 desc: With Q 3 desc: With Q 1 desc: P21 2 desc: P21 1 desc: With Q

PSNR (dB) PSNR (dB) 2 desc: With Q 35 35 1 desc: P21 1 desc: With Q

30 30

25 25

20 20 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Bit Rate (bpp) Bit Rate (bpp) (a) (b) Fig. 7. Performance comparison between P21 (with quantization noise ignored) and a configuration without ignoring quantization noise. The two configurations have the same coding gain. (a) Results with the image Lena; (b) Results with the image Barbara. on average, happens when only one description is received. also show that both the mean reconstruction method and the Therefore our method offers more graceful quality degradation linear interpolation method are special cases of the proposed when more descriptions are lost. framework. We next compare our method with the results in [15], The performance of the proposed method can be further where the maximally smooth recovery constraint is applied improved in several ways. First of all, the Wiener filters in in the transform design. The re-designed transforms in [15] this paper are designed based on 1-D signal model. Better can achieve the same error resilience with lower bit rate than performance can be achieved by designing 2-D Wiener filter the transforms used in [14]. directly. Some preliminary result have been reported in [28]. In Table IV, we use the 256 256 Lena image to compare Secondly, the closed-form Wiener solutions lend themselves the quality degradation behavior× of our method with that of the naturally to adaptive error concealment. How to implement the MSR method in [15] when the error-free reconstruction quality adaptive algorithm with reasonable complexity is our ongoing is 33 dB. Although different entropy coding methods are used, work. we believe the performance degradations of the two methods with respect to the same error-free quality can be compared VIII. ACKNOWLEDGEMENT fairly. The first part in Table IV is taken from [15]. The second The authors would like to thank the anonymous reviewers part is given by the M 2M Wiener filter H∗-based TDLT × 1 for their suggestions that have significantly enhanced the with the same coding gains as those in [15]. The average bit quality of this paper. rate of our method is 19% lower than that in [15] (up to 33% in the case of M2). Note that this fact alone should not be used APPENDIX A to judge against the MSR method since arithmetic coding is PROOFOFTHEPERSYMMETRICSTRUCTUREOFTHE used in our method and adaptive Huffman coding is used in [15] (In fact, the actual bit rate saving is even higher since the WIENER FILTER table overhead for the adaptive Huffman code is not counted In this section, we prove the persymmetric relationship in ∗ in [15]). In terms of error concealment quality, our average (27) for the 2M 2M Wiener filter H in (6) and the × 0 PSNR improvement over [15] is 0.74 dB, with 1.07 dB at the constrained solutions in (14) and (16). The proof for other highest coding gain and 1.46 dB when only one description is simplified Wiener filers can be obtained similarly. ∗ available. These behaviors are also consistent with Table III, The expression of H0 is reproduced below: confirming the more graceful performance degradation of our ∗ − H = R R 1 . (41) method when more data are lost. 0 x2ˆs2 ˆs2ˆs2 ∗ Since JJ = I, H0 satisfies (27) if the following are true: VII. CONCLUSIONS Rx2ˆs2 = JRx2ˆs2 J, (42) This paper analyzes the error concealment design of the − − R 1 = JR 1 J. (43) time-domain lapped transform from the perspective of es- ˆs2ˆs2 ˆs2ˆs2 timation theory. The general LMMSE solution and various Since Rx2ˆs2 is a submatrix of Rx4ˆs3 , (42) can be obtained T simplifications are proposed. Design examples and image if Rx4ˆs3 = Rx4x4 P34 = JRx4ˆs3 J, which in turn needs T T coding results show that the reconstruction error can be Rx4x4 = J4M Rx4x4 J4M and P34 = J4M P34J3M . The reduced dramatically, compared to the mean reconstruction former is obvious since Rx4x4 is a symmetric Toeplitz matrix. method and the maximally smooth recovery method. We The latter is a direct result of P = JPJ, which has already LIANG et al.: WIENER FILTER-BASED ERROR RESILIENT TIME DOMAIN LAPPED TRANSFORM 11

TABLE III PSNR(dB) UNDERDIFFERENTLOSSESFORTHE 256 × 256 LENAIMAGEWITHOUTQUANTIZATIONERRORS Available Results in [14] α in (38) (β = 0) Average Descriptions T6 T7 T8 T9 69.0 74.0 112.0 143.0 Improvement (dB) 3 29.33 31.90 32.95 33.45 32.15 34.72 35.31 35.56 2.48 2 26.22 28.90 30.02 30.57 29.16 31.71 32.32 32.58 2.52 1 21.08 23.57 24.75 25.34 25.50 27.75 28.46 28.78 3.94 Avg. Improvement (dB) - - - - 3.39 3.27 2.79 2.52 2.98 Coding Gain (dB) 7.83 7.18 6.76 6.51 7.83 7.18 6.76 6.51 -

TABLE IV PSNR(dB) UNDERDIFFERENTLOSSESFORTHE 256 × 256 LENAIMAGEWHENTHECODINGDISTORTIONIS 33 dB Available Results in [15] α in (38) (β = 0) Average Descriptions M2 M8 M49 M58 15.7 30.8 81.6 102.2 Improvement (dB) 4 33.05 33.07 33.04 33.04 33.05 33.07 33.04 33.04 - 3 26.99 28.99 30.96 31.11 27.62 29.37 31.00 31.20 0.29 2 24.37 26.77 29.47 29.67 25.22 27.35 29.65 29.93 0.47 1 20.20 22.34 25.57 26.14 21.94 24.26 26.74 27.13 1.46 Average Improvement (dB) - - - - 1.07 0.96 0.46 0.45 0.74 Bit Rate (bpp) 0.77 0.88 1.14 1.27 0.51 0.65 1.08 1.14 -19% Coding Gain (dB) 8.96 8.26 7.08 6.85 8.96 8.26 7.08 6.85 - been reinforced in the TDLT in order to obtain linear-phase REFERENCES filer bank [16]. [1] Y. Wang and Q. F. Zhu, “Error control and concealment for video Similarly, (43) can be established if Rˆs3ˆs3 = JRˆs3ˆs3 J. communication: a review,” Proceedings of IEEE, vol. 86(5), pp. 974– Recall that Rˆs3ˆs3 is given by 997, May 1998. T T [2] Y. Takishima, M. Wada, and H. Murakami, “Reversible variable length Rˆs3ˆs3 = P34Rx4x4 P34 + C3 Rq3q3 C3. (44) codes,” IEEE Trans. Commun., vol. 43(234), pp. 158–162, 1995. [3] H. Sun and W. Kwok, “Concealment of damaged block transform coded The first part is persymmetric since both P34 and Rx4x4 are images using projections onto convex sets,” IEEE Trans. Image Proc., persymmetric. To show the persymmetry of the second part, vol. 4(4), pp. 470–477, Apr. 1995. we assume the subband quantization noises of different blocks [4] P. Salama, N. B. Shroff, E. J. Coyle, and E. J. Delp, “Error concealment techniques for encoded video streams,” icip, vol. 1, pp. 9–12, Oct. 1995. are uncorrelated. Thus [5] J.-W. Suh and Y.-S. Ho, “Error concealment based on directional interpolation,” IEEE Trans. Consumer Electronics, vol. 43(3), pp. 295– Rq3q3 = diag Rqq, Rqq, Rqq . (45) { } 302, Aug. 1997. The block noise correlation after the inverse DCT can be [6] J. W. Park and S. U. Lee, “Recovery of corrupted image data based on the NURBS interpolation,” IEEE Trans. Image Proc., vol. 9(7), pp. rewritten as 1003–1008, Oct. 1999. M−1 [7] Z. Alkachouh and M. G. Bellanger, “Fast DCT-based spatial domain T 2 T interpolation of blocks in images,” IEEE Trans. Image Proc., vol. 9(4), Rvv , C RqqC = σ c ci, (46) qi i pp. 729–732, Apr. 2000. i=0 X [8] X. Li and M. Orchard, “Novel sequential error concealment using where ci is the i-th basis function of the DCT, which orientation adaptive interpolation,” IEEE Trans. Circ. Syst. for Video Tech., vol. 12(10), pp. 857–864, Oct. 2002. has linear-phase. Therefore ci is either symmetric or anti- [9] H. S. Malvar, Signal Processing with Lapped Transforms. Norwood, symmetric. Let MA: Artech House, 1992. , T Bi ci ci. (47) [10] P. Haskell and D. Messerschmitt, “Reconstructing lost video data in a lapped orthogonal transform based coder,” Proc. IEEE Int. Conf. Acoust., By the symmetry of ci, we have Speech, and Signal Proc., vol. 4, pp. 1985–1988, Apr. 1990. B c c [11] R. L. de Queiroz and K. R. Rao, “Extended lapped transform in image i(j, k)= i,j i,k coding,” IEEE Trans. Image Proc., vol. 4(6), pp. 828–832, Jun. 1995. = ci,M−1−j ci,M−1−k = Bi(M 1 j, M 1 k). [12] S. S. Hemami, “Reconstruction-optimized lapped transforms for robust − − − − (48) image transmission,” IEEE Trans. Circ. Syst. for Video Tech., vol. 6 (2), pp. 168–181, Apr. 1996. This shows that Bi is also persymmetric, which leads to the [13] Y. Wang, Q.-F. Zhu, and L. Shaw, “Maximally smooth image recovery T in transform coding,” IEEE Trans. Commun., vol. 41, pp. 1544–1551, persymmetry of Rvv and C3 Rq3q3 C3, and therefore the ∗ Oct. 1993. persymmetry of the Wiener filter H0. The proof for other [14] D. Chung and Y. Wang, “Multiple description image coding using simplified Wiener filters can be conducted in a similar fashion. signal decomposition and reconstruction based on lapped orthogonal Once the persymmetry of the Wiener filter is established, transforms,” IEEE Trans. Circ. Syst. for Video Tech., vol. 9 (6), pp. 895–908, Sept. 1999. the persymmetry of the constrained Wiener solution in (14) [15] ——, “Lapped orthogonal transform designed for error resilient image becomes evident after substituting (15) into (14) and noticing coding,” IEEE Trans. Circ. Syst. for Video Tech., vol. 12(9), pp. 752–764, that ddT is persymmetric. The scaled Wiener filter in (16) is Sept. 2002. [16] T. D. Tran, J. Liang, and C. Tu, “Lapped transform via time-domain also persymmetric since Θ(i, i) = Θ(L 1 i, L 1 i) − − − − pre- and post-processing,” IEEE Trans. Signal Proc., vol. 51 (6), pp. for an L N persymmetric Wiener filter. 1557–1571, Jun. 2003. × 12 IEEE TRANSACTIONS ON IMAGE PROCESSING, SUBMITTED JAN. 2006

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