Tree expectation propagation for ml decoding of LDPC codes over the BEC

We propose a decoding algorithm for LDPC codes that achieves the maximum likelihood (ML) solution over the bi- nary erasure channel (BEC). In this channel, the tree-structured expectation propagation (TEP) decoder improves the peeling decoder (PD) by processing check nodes of degree one and two. However, it does not achieve the ML solution, as the tree structure of the TEP allows only for approximate inference. In this paper, we provide the procedure to construct the structure needed for exact inference. This algorithm, denoted as generalized tree-structured expectation propagation (GTEP), modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph. The GTEP decoder upon completion either provides the unique ML solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ML solution. We also explain the algorithm as a Gaussian elimination method, relating the GTEP to other ML solutions. Compared to previous approaches, it presents an equivalent complexity, it exhibits a simpler graphical message-passing procedure and, most interesting, the algorithm can be generalized to other channels

Bayesian equalization for LDPC channel decoding

We describe the channel equalization problem, and its prior estimate of the channel state information (CSI), as a joint Bayesian estimation problem to improve each symbol posterior estimates at the input of the channel decoder. Our approach takes into consideration not only the uncertainty due to the noise in the channel, but also the uncertainty in the CSI estimate. However, this solution cannot be computed in linear time, because it depends on all the transmitted symbols. Hence, we also put forward an approximation for each symbol's posterior, using the expectation propagation algorithm, which is optimal from the Kullback-Leibler divergence viewpoint and yields an equalization with a complexity identical to the BCJR algorithm. We also use a graphical model representation of the full posterior, in which the proposed approximation can be readily understood. The proposed posterior estimates are more accurate than those computed using the ML estimate for the CSI. In order to illustrate this point, we measure the error rate at the output of a low-density parity-check decoder, which needs the exact posterior for each symbol to detect the incoming word and it is sensitive to a mismatch in those posterior estimates. For example, for QPSK modulation and a channel with three taps, we can expect gains over 0.5 dB with same computational complexity as the ML receiver

Approaching the DT Bound Using Linear Codes in the Short Blocklength Regime

The dependence-testing (DT) bound is one of the strongest achievability bounds for the binary erasure channel (BEC) in the finite block length regime. In this paper, we show that maximum likelihood decoded regular low-density parity-check (LDPC) codes with at least 5 ones per column almost achieve the DT bound. Specifically, using quasi-regular LDPC codes with block length of 256 bits, we achieve a rate that is less than 1% away from the rate predicted by the DT bound for a word error rate below 10^−3. The results also indicate that the maximum-likelihood solution is computationally feasible for decoding block codes over the BEC with several hundred bits

Transitively Consistent and Unbiased Multi-Image Registration Using Numerically Stable Transformation Synchronisation

Abstract. Transitive consistency of pairwise transformations is a desir- able property of groupwise image registration procedures. The transfor- mation synchronisation method [4] is able to retrieve transitively con- sistent pairwise transformations from pairwise transformations that are initially not transitively consistent. In the present paper, we present a numerically stable implementation of the transformation synchronisa- tion method for a ne transformations, which can deal with very large translations, such as those occurring in medical images where the coor- dinate origins may be far away from each other. By using this method in conjunction with any pairwise (a ne) image registration algorithm, a transitively consistent and unbiased groupwise image registration can be achieved. Experiments involving the average template generation from 3D brain images demonstrate that the method is more robust with re- spect to outliers and achieves higher registration accuracy compared to reference-based registration