Structured Total Least Squares And Image Processing
Abstract
Although Gauss-Markov linear least squares leads to a simple linear algebra formulation and solution, it is not always the best choice for deblurring problems arising in image processing. Its weaknesses include uncertainty and possible ill-conditioning in the deblurring matrix, and an overemphasis on outliers.
As alteratives to the Gauss-Markov approach, nonlinear least squares models are formulated for two image processing problems. For the first, the superresolution problem, structured total least squares (STLS) leads to an elegant nonlinear least squares formulation. For the second, color image restoration, a different structured nonlinear least squares problem is formulated and shown to be superior to the standard Gauss-Markov approach.
Both of these nonlinear least squares problems are solved using a Gauss-Newton iteration. Each Newton step involves the solution of a large scale least squares problem which is solved very efficiently by the preconditioned conjugate gradient method. For the superresolution problem, a preconditioner is constructed using the family of block Toeplitz preconditioners. For the color image restoration problem, we choose a Discrete Cosine Transform based preconditioner. In both cases, these preconditioners are shown to lead to efficient algorithms.