Dual-grid parameter choice method with application to image deblurring

TL;DR

This work presents a dual-grid parameter choice method for variational image deblurring that eliminates the need for explicit noise level estimates by comparing reconstructions from two slightly shifted forward models using SSIM as the guiding criterion. By solving parallel regularized problems on a camera grid and a shifted grid with the same data $m$, the approach selects the smallest $\alpha$ that yields SSIM$\big(\mathbf{g}^{(\alpha)},\mathbf{f}^{(\alpha)}\big) \ge T$, where $T$ is a user-defined threshold, and demonstrates robustness across Tikhonov and TV regularization on simulated and real data. The paper compares this dual-grid method with the discrepancy principle and bilevel optimization, showing competitive parameter choices without noise estimates and highlighting practical considerations such as threshold selection and dataset-dependent behavior. The proposed framework is flexible, applicable to other forward models and regularizers, and offers a practical tool for automatic parameter tuning in ill-posed inverse problems with imaging applications.

Abstract

Variational regularization of ill-posed inverse problems is based on minimizing the sum of a data fidelity term and a regularization term. The balance between them is tuned using a positive regularization parameter, whose automatic choice remains an open question in general. A novel approach for parameter choice is introduced, based on the use of two slightly different computational models for the same inverse problem. Small parameter values should give two very different reconstructions due to amplification of noise. Large parameter values lead to two identical but trivial reconstructions. Optimal parameter is chosen between the extremes by matching image similarity of the two reconstructions with a pre-defined value. Efficacy of the new method is demonstrated with image deblurring using measured data and two different regularizers.

Figures (20)

• Figure 1: Left: Simulated ground truth with no noise or blur. Middle: Blurred with radius 4 kernel and 4% added noise. Right: Blurred with radius 4 kernel and added 8% noise.
• Figure 2: Left: "Ground truth", almost noiseless (ISO100). Middle: Slightly blurred, some noise (ISO1600). Right: Slightly blurred, more noise (ISO6400).
• Figure 3: "Ground truth" (ISO100) on the left. Slightly blurred and small noise (ISO6400) in the middle. Slightly blurred and more noise (ISO25600) on the right.
• Figure 4: 10 of the 230 natural images used for the numerical test of the similarity hypothesis.
• Figure 5: SSIM between images on the two grids. The blue graph indicates that for a regular photograph, a sub-pixel shift causes a significant change in the image as measured by SSIM. However, for images smoothed by Tikhonov or total variation denoising, the change caused by the shift is much weaker: their SSIM values are closer to 1.