Optimization of partition-based Weighted Sum filters and their application to image denoising

Partition-based Weighted Sum (P-WS) filtering is an effective method for processing nonstationary signals, especially those with regularly occurring structures, such as images. P-WS filters were originally formulated as Hard-partition Weighted Sum (HP-WS) filters and were successfully applied to image denoising. This formulation relied on intuitive arguments to generate the filter class. Here we present a statistical analysis that justifies the use of weighted sum filters after observation space partitioning. Unfortunately, the HP-WS filters are nondifferentiable and an analytical solution for their global optimization is therefore difficult to obtain. A two-stage suboptimal training procedure has been reported in the literature, but prior to this research no evaluation on the optimality of this approach has been reported. Here, a Genetic Algorithm (GA) HP-WS optimization procedure is developed that, in simulations, shows that the simpler two-stage training procedure yields near optimal results. Also developed in this paper are Soft-partition Weighted Sum (SP-WS) filters. The SP-WS filters utilize soft, or fuzzy, partitions that yield a differentiable filtering operation, enabling the development of gradient-based optimization procedures. Image denoising simulation results are presented comparing HP-WS and SP-WS filters, their optimization procedures, and wavelet-based image denoising. These results show that P-WS filters, in general, outperform traditional and wavelet-based image filters, and SP-WS filters utilizing soft partitioning not only allow for simple optimization, but also yields improved performance.