Variable Projection Methods for Solving Regularized Separable Inverse Problems with Applications to Semi-Blind Image Deblurring

TL;DR

This paper extends variable projection methods to regularized inverse problems, particularly semi-blind image deblurring, introducing new algorithms with convergence guarantees and demonstrating improved reconstruction accuracy.

Contribution

The work develops a novel extension of VarPro for regularized problems, including a quasi-Newton approach and inexact LSQR variant with convergence analysis.

Findings

Regularization prevents degenerate solutions in semi-blind deblurring.

Proposed methods achieve accurate image reconstructions.

Inexact LSQR variant offers a good accuracy-cost balance.

Abstract

Separable nonlinear least squares problems appear in many inverse problems, including semi-blind image deblurring. The variable projection (VarPro) method provides an efficient approach for solving such problems by eliminating linear variables and reducing the problem to a smaller, nonlinear one. In this work, we extend VarPro to solve minimization problems containing a differentiable regularization term on the nonlinear parameters, along with a general-form Tikhonov regularization term on the linear variables. Furthermore, we develop a quasi-Newton method for solving the resulting reduced problem, and provide a local convergence analysis under standard smoothness assumptions, establishing conditions for superlinear or quadratic convergence. For large-scale settings, we introduce an inexact LSQR-based variant and prove its local convergence despite inner-solve and Hessian approximations. Numerical experiments on semi-blind deblurring show that parameter regularization prevents degenerate no-blur solutions and that the proposed methods achieve accurate reconstructions, with the inexact variant offering a favorable accuracy-cost tradeoff consistent with the theory.