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Wiener Filter-Based Error Resilient Time Domain Lapped Transform Jie Liang, Member, IEEE, Chengjie Tu, Member, IEEE, Lu Gan, Member, IEEE, Trac D

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Wiener Filter-Based Error Resilient Time Domain Lapped Transform Jie Liang, Member, IEEE, Chengjie Tu, Member, IEEE, Lu Gan, Member, IEEE, Trac D 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 image compression 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 codec 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 bit rate. 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 Video 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.
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