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

TL;DR

This work extends Variable Projection (VarPro) to regularized separable nonlinear least squares (RSNLS) problems, enabling efficient recovery of both linear and nonlinear parameters in ill-posed inverse problems such as semi-blind image deblurring. It introduces a reduced nonlinear objective $\varphi({\bf y})$ via elimination of the linear variable and develops a robust quasi-Newton method (RGenVarPro) with convergence guarantees, plus an inexact LSQR-based variant (iRGenVarPro) suitable for large-scale cases. The RSNLS framework is instantiated with two regularization strategies on ${\bf y}$ (2-norm and logarithmic barriers) and demonstrated on a 512×512 Cameraman deblurring task, where parameter regularization prevents degenerate no-blur solutions and yields accurate reconstructions; the inexact variant achieves substantial cost savings with comparable accuracy under appropriate tolerances. Theoretical results are complemented by numerical experiments showing stable convergence, favorable accuracy-cost tradeoffs, and practical viability for large-scale semi-blind deblurring and related inverse problems.

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.

Figures (7)

• Figure 1: (Left) True signals: an edgy signal (top) and a smooth signal (bottom), and their corresponding blurred and noisy signals; (Middle) reduced functional $\varphi(\sigma)$ for selected values of $\lambda$; and (Right) convergence of $\sigma_k$ using the built-in MATLAB function fminsearch.
• Figure 1: Illustrative cameraman test problem: (a) true image, (b) PSF (cropped and upsampled by a factor of $4$ via nearest-neighbor interpolation), and (c) blurred and noisy image with $5\%$ Gaussian white noise.
• Figure 2: Plots of $\varphi({\bf y})$ without regularization (left), with $\mathcal{R}({\bf y})= \mu^2\|{\bf y} - {\bf y}_0\|^2$ (middle), and with $\mathcal{R}({\bf y})=-\sum_{j}\mu_j^2\log(y_j)$ (right). Regularization shifts the minimum from ${\bf y}=0$ toward the true value ${\bf y}\approx3$.
• Figure 3: Results for the 2-norm regularizer: (left) reconstructed image (SSIM = 0.66), (middle) RRE for ${\bf x}$, and (right) objective function value vs. iteration number.
• Figure 4: Results for the logarithmic regularizer: (left) reconstructed image (SSIM = 0.63), (middle) RRE for ${\bf x}$, and (right) objective function value vs. iteration number.

Theorems & Definitions (14)

• Theorem 2.1
• Proof 1
• Lemma 3.1
• Proof 2
• Lemma 3.2
• Proof 3
• Theorem 3.3
• Proof 4
• Corollary 3.4
• Proof 5