Two-dimensional block adaptive filtering algorithms with optimum convergence factors

In this paper, two new fast gradient algorithms which perform 2-D block adaptive filtering are presented. The 2-D adaptive filter coefficients are updated once for every block of input data and error measurement. The two algorithms employ variable convergence factors which are optimized in the least-squares (LS) sense to track the variations in an image's local statistics. These 2-D optimal algorithms are obtained from the one-dimensional optimal adaptive algorithms, which have been recently reported. In the first algorithm, the 2-D optimum block adaptive algorithm with individual adaptation of parameters (TDOBAI), the convergence factors are obtained, that are individually tailored for each 2-D filter weight and are updated once per block iteration. The second algorithm uses a convergence factor that is the same for all the 2-D coefficients at a particular block iteration, and is determined at each block iteration. This algorithm is called the 2-D optimum block adaptive algorithm (TDOBA). In both algorithms, the convergence factors are easily computed from readily available signals. The excellent performance characteristics of the optimal 1-D algorithms are shown to be retained in the proposed 2-D optimal algorithms. The convergence properties of the TDOBAI and the TDOBA algorithms are investigated and compared with the 2-D block least-mean-square (TDBLMS) algorithm which uses a convergence factor that is constant for each 2-D coefficient at each block iteration, using computer simulations. It is also shown that for the TDOBAI and TDOBA algorithms, the convergence, speed and accuracy of adaptation are greatly improved at the expense of a modest increase in computational complexity, as compared to the TDBLMS algorithm. The effectiveness of the algorithms is demonstrated in 2-D system modeling, restoration (2-D additive noise cancellation), and enhancement of artificially degraded images. Also, it is shown that the TDOBAI algorithm is a more general formulation, from which several other recently proposed 2-D sequential and block algorithms can be obtained as special cases, by trading performance with computational complexity.<
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