| Wiener filters are commonly used for image restoration. They are mean square based optimization filters, and are effective when the signal and noise are jointly Gaussian and stationary. The previously proposed partition-based weighted sum (PWS) filters combine vector quantization (VQ) and linear finite impulse response (FIR) Wiener filtering concepts. By partitioning the observation space and applying a tuned Wiener filter to each partition, the PWS filter is spatially adaptive and has been shown to perform well in noise reduction applications.; In this dissertation, subspace PWS (SPWS) filters are proposed, and their efficacy in image deconvolution and noise reduction applications is evaluated. In the SPWS filter, the observation vectors are projected into a subspace using principal component analysis (PCA), or other methods, prior to partitioning. This subspace projection can dramatically reduce the computational burden associated with partitioning, especially for large window sizes. In some cases, performance is also enhanced due to improved partitioning.; The Soft-PWS filters use information from all of the partition filters to process each observation vector. Soft-PWS filters have been shown to be capable of outperforming PWS filters in noise reduction applications. To date, however, the optimization of the Soft-PWS filters has received only limited attention. Prior work has focused on a stochastic-gradient based method that requires a large number of iterations, making it computationally prohibitive in many applications. Furthermore, its convergence is highly sensitive to the step size and no reliable method of determining a suitable step size has been presented.; In this dissertation, a novel radial basis function interpretation of the Soft-PWS filters is described, and an efficient optimization procedure that we believe makes the Soft-PWS filters far more practical to implement is presented. To demonstrate the efficacy of the Soft-PWS filters with the new optimization procedure, we apply the filters to the problem of noise reduction. Simulation results show that, for the data used here, the Soft-PWS filter outperforms the standard PWS filter and Wiener filter under the mean squared error criterion. |
