Near-Optimal Spectral Filtering and Error Estimation for Solving Ill-Posed Problems

Abstract

We consider regularization methods for numerical solution of linear ill-posed problems, in particular image deblurring, when the singular value decomposition (SVD) of the operator is available. We assume that the noise-free problem satisfies the discrete Picard condition and define the Picard parameter, the index beyond which the data, expressed in the coordinate system of the SVD, are dominated by noise. We propose estimating the Picard parameter graphically or using standard statistical tests. Having this parameter available allows us to estimate the mean and standard deviation of the noise and drop noisy components, thus making filtered solutions much more reliable. We show how to compute a near-optimal choice of filter parameters for any filter. This includes the truncated SVD (TSVD) filter, the truncated singular component method (TSCM) filter, and several new filters which we define, including a truncated Tikhonov filter, a Tikhonov-TSVD filter, a Heaviside filter, and a spline filter. We show how to estimate the error in any spectral filter, regardless of how the filter parameters are chosen. We demonstrate the usefulness of our new filters, our near-optimal choice of parameters, and our error estimates for restoring blurred images.