IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 7, JULY 2007 433 Row-Column Soft-Decision Feedback Algorithm for Two-Dimensional Intersymbol Interference Taikun Cheng, Benjamin J. Belzer, and Krishnamoorthy Sivakumar Abstract—We present a novel iterative row-column soft decision feedback algorithm (IRCSDFA) for detection of binary images corrupted by 2-D intersymbol interference and additive white Gaussian noise. The algorithm exchanges weighted soft infor- mation between row and column maximum a posteriori (MAP) detectors. Each MAP detector exploits soft-decision feedback from previously processed rows or columns. The new algorithm gains about 0.3 dB over the previously best published results for the 2 2 averaging mask. For a non-separable 3 3 mask, the IRCSDFA gains 0.8 dB over a previous soft-input/soft-output iterative algorithm which decomposes the 2-D convolution into 1-D row and column operations. Index Terms—Iterative algorithm, soft decision feedback, 2-D in- tersymbol interference. I. INTRODUCTION C ONSIDER the detection of an binary-equiprob- able 2-D independent and identically distributed (i.i.d.) image (with elements ) from re- ceived image (1) where is a finite-impulse-response 2-D blurring mask, the are zero mean i.i.d. Gaussian random variables (r.v.s) with variance , and the double sum is computed over the mask support region . It is assumed that a boundary of elements surrounds the image . This system model applies, e.g., to 2-D image storage systems, which suffer from 2-D ISI at high storage densities. Direct maximum likelihood (ML) detection of from requires comparison of with candidate images, and is therefore impractical for typical image dimen- sions. Standard Wiener filtering is significantly inferior to ML detection, especially at high SNR [1]. Hence, it is desirable to develop a low-complexity 2-D ISI detection algorithm that ap- proximates the performance of 2-D ML detection. For 1-D sig- nals, the Viterbi algorithm (VA) provides efficient ML detection Manuscript received July 6, 2006; revised October 18, 2006. This work was supported in part by the National Science Foundation under Grant CCR-0098357. This work was previously presented at the 39th Conference on Information Sciences and Systems (CISS’05), Johns-Hopkins University, Baltimore, MD, March 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Jitendra K. Tugnait. The authors are with the School of Electrical Engineering and Computer Sci- ence, Washington State University, Pullman, WA 99164-2752 USA (e-mail: tcheng@eecs.wsu.edu; belzer@eecs.wsu.edu; siva@eecs.wsu.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2006.891329 of ISI-corrupted data [2], but the VA does not generalize to two or higher dimensions. Union bounds on the performance of 2-D ML detection are described in [3]; these bounds are tight at high SNR, and are useful in assessing the performance of 2-D detec- tion algorithms. To our knowledge, [1] and [4] employed the first iterative algorithm for 2-D ISI reduction; a 2-D decision-feedback VA (DFVA) was run on rows and columns, and bits which agreed in both directions were fixed for subsequent iterations. Subse- quent work has employed the turbo principle (after [5]). In [6], the 2-D convolution operation is decomposed into two 1-D oper- ations, and an iterative decoding algorithm exchanges soft infor- mation between 1-D soft-input/soft-output (SISO) detectors. In [7], mask separability is exploited to construct an iterative row- column detector for low-density-parity-check (LDPC) coded bi- nary images, in which extrinsic information is exchanged be- tween a non-binary column SISO decoder, a binary row SISO decoder, and a LDPC decoder. In [8] and [9], soft information is exchanged between MAP detectors operating on multiple rows and multiple columns; this scheme handles nonseparable masks. The primary contribution of this paper is a new iterative soft- decision feedback (SDF) MAP detection algorithm for reduc- tion (or elimination) of 2-D ISI. Our scheme, while similar to that of [8], was developed independently, and has several key differences. First, we make decisions one row at a time and use SDF, rather than making decisions two or more rows at a time and using “feed-forward” [8]. Second, we weigh the extrinsic in- formation passed between SISOs, and increase the weights with each iteration; the weight schedule significantly improves the algorithm’s performance. Third, we achieve additional gains by adding rows (respectively, columns) to the state and input pixel blocks of the row (column) SISOs. And fourth, we demonstrate performance with both 2 2 and 3 3 masks on 128 128 and 64 64 images, whereas the maximum source image size con- sidered in [8], [9] is 5 5. The IRCSDFA achieves about 1.5 dB of SNR improvement over the hard-decision iterative algorithm of [1], for the 3 3 averaging mask. For a more rapidly decaying 3 3 mask, the IRCSDFA achieves about 0.8 dB gain over [6]. For the 2 2 averaging mask, the IRCSDFA gains about 0.3 dB over the separable algorithm of [7] (without coding), the previ- ously best published result for that mask. II. TRELLIS DEFINITION Two-dimensional convolution can be viewed as the inner product of image with inverted mask , with mask coefficient at pixel position . The inverted mask raster-scans through the image row-by-row or column-by-column. For the row-by-row case, we use Miller et al.’s method [1] to define the IRCSDFA trellis states and inputs as in Fig. 1. Trellis 1070-9908/$25.00 © 2007 IEEE