Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

To apply the Generalized Cross-Validation (GCV) as a stopping rule for an iterative method, we must estimate the trace of the so-called in?uence matrix which appears in the denominator of the GCV function. In the case of conjugate gradient, unlike what happens with stationary iterative methods, the regularized solution has a nonlinear dependence on the noise which a?ects the data of the problem. This fact is often pointed out as a cause of poor performance of GCV. To overcome this drawback, in this paper we propose a new method which linearizes the dependence by computing the derivatives through iterative formulas along the lines of Perry and Reeves (1994) and Bardsley (2008). We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation both on 1D and on 2D test problems

Parameter selection in sparsity-driven SAR imaging

We consider a recently developed sparsity-driven synthetic aperture radar (SAR) imaging approach which can produce superresolution, feature-enhanced images. However, this regularization-based approach requires the selection of a hyper-parameter in order to generate such high-quality images. In this paper we present a number of techniques for automatically selecting the hyper-parameter involved in this problem. In particular, we propose and develop numerical procedures for the use of Stein’s unbiased risk estimation, generalized cross-validation, and L-curve techniques for automatic parameter choice. We demonstrate and compare the effectiveness of these procedures through experiments based on both simple synthetic scenes, as well as electromagnetically simulated realistic data. Our results suggest that sparsity-driven SAR imaging coupled with the proposed automatic parameter choice procedures offers significant improvements over conventional SAR imaging

A regularized GMRES method for inverse blackbody radiation problem

The inverse blackbody radiation problem is focused on determining temperature distribution of a blackbody from measured total radiated power spectrum. This problem consists of solving a first kind of Fredholm integral equation and many numerical methods have been proposed. In this paper, a regularized GMRES method is presented to solve the linear ill-posed problem caused by the discretization of such an integral equation. This method projects the orignal problem onto a lower dimensional subspaces by the Arnoldi process. Tikhonov regularization combined with GCV criterion is applied to stabilize the numerical iteration process. Three numerical examples indicate the effectiveness of the regularized GMRES method

REGULARIZATION PARAMETER SELECTION METHODS FOR ILL POSED POISSON IMAGING PROBLEMS

A common problem in imaging science is to estimate some underlying true image given noisy measurements of image intensity. When image intensity is measured by the counting of incident photons emitted by the object of interest, the data-noise is accurately modeled by a Poisson distribution, which motivates the use of Poisson maximum likelihood estimation. When the underlying model equation is ill-posed, regularization must be employed. I will present a computational framework for solving such problems, including statistically motivated methods for choosing the regularization parameter. Numerical examples will be included

Regularization by conjugate gradient of nonnegatively constrained least squares

In many image deconvolution applications the nonnegativity of the computed solution is required. Conjugate Gradient (CG), often used as a reliable regularization tool, may give solutions with negative entries, particularly evident when large nearly zero plateaus are present. <br>The active constrains set, detected by projection? onto the nonnegative quadrant, turns out to be largely incomplete and? poor effects on the accuracy of the reconstructed image may occur. In this paper an inner-outer method based on CG is proposed? to compute nonnegative reconstructed images with a strategy which enlarges subsequently the active constrains set. <br>This method appears to be especially suitable for the deconvolution of images having large nearly zero backgrounds. The numerical experimentation validates the effectiveness of the proposed method with respect to widely used classical algorithms for nonnegative reconstructio