| Filtering, often referred as any convolution type of operation, is widely used in signal processing, especially in denoising, restoration, interpolation and sharpening applications. Traditional linear filters, as well as many nonlinear filters, employ a common kernel to operate on observation signals of all structures and fail to produce high filtering performance when processing nonstationary signals.;We propose a class of Partition-based Weighted Sum filters that successfully process nonstationary signals and images. In this methodology, local observation signals are clustered into a finite set of regions that capture local signal structures. Next, a common weighted sum filtering operation is applied to observation signals in each partition. By exploiting the local signal structure, the Partition-based Weighted Sum filtering methodology processes signal nonstationarity properly, and produces excellent performance on signals comprised of various underlying structures. In this dissertation, we introduce the Partition-based Weighted Sum filtering framework, its optimization, and its specific formulations customized to address the needs of particular applications. Partition-based Weighted Sum filtering can be divided into two steps: (1) partitioning and (2) filtering. In the partitioning step, both hard and soft partitioning can be utilized. Hard partitioning divides the observation space into mutually exclusive regions, while soft partitioning assigns a real valued relation between an observation and each partition. In the filtering step, weighted sum operations are employed. The non-differentiable Hard-partition Weighted Sum filters are difficult to globally optimize. A two-stage decoupled procedure is developed as a suboptimal but convenient optimization method for this type of filters. Also, a global Genetic Algorithm is employed as an optimization benchmark. The Soft-partition Weighted Sum filters are differentiable, which allows gradient-based optimization methods to be applied to this type of filters.;This nonlinear filtering framework is successfully applied to three nonstationary signal processing applications: image denoising, image interpolation, and fetal electroencephalogram (EEG) extraction. In each application, Partition-based Weighted Sum filters yield results superior to traditional and advanced algorithms used in practice and reported in the literature. Therefore, Partition-based Weighted Sum filtering is an effective method to process nonstationary signals and images. |
