Marginalization-Based Regularization for Separable Nonlinear Least Squares Problems with Applications to Semi-Blind Deblurring
Separable nonlinear least squares problems arise in inverse problems such as semi-blind image deblurring, where the image enters the forward model linearly, and the blur parameters enter nonlinearly. Even when the image is regularized, joint estimation of the image and blur parameters can favor a degenerate no-blur solution, in which the blur operator tends to the identity and the reconstruction merely denoises the data without removing the blur. We address this by regularizing the blur parameters. Motivated by the determinant term arising from Gaussian marginalization, we construct a family of blur-parameter priors whose weight and auxiliary operators can be chosen independently of the image prior. The resulting log-determinant penalty is incorporated into regularized variable projection, which eliminates the image variables by minimization. Under periodic boundary conditions and assuming the matrices involved admit a joint spectral decomposition, we show that a sufficiently large determinant weight rules out the no-blur solution as a local minimizer. We also characterize the regularizer's behavior as the auxiliary regularization scale tends to zero or infinity. We solve the reduced problem with a quasi-Newton method that evaluates the objective and its derivatives in the Fourier domain. In experiments with Gaussian, defocus, and motion blur, the proposed prior improves image and blur recovery over Gaussian marginalization across several settings, with determinant weights computed from the true image. A single-sample experiment illustrates how to automatically choose parameters from the data using a conditional generalized cross-validation criterion.