Convergence Analysis of MAP based Blur Kernel Estimation

TL;DR

This paper analyzes the convergence of MAP-based blind deconvolution methods with sparsity priors, revealing conditions under which they favor sharp images and demonstrating their practical applications.

Contribution

It introduces a modified energy function that clarifies the convergence behavior of MAP-based blind deconvolution and explains its success in certain scenarios.

Findings

The reformulated energy function can favor the correct sharp solution.

Conditions are identified under which MAP approaches converge to the right solution.

Applications include automatic blur kernel size selection and defocus estimation.

Abstract

One popular approach for blind deconvolution is to formulate a maximum a posteriori (MAP) problem with sparsity priors on the gradients of the latent image, and then alternatingly estimate the blur kernel and the latent image. While several successful MAP based methods have been proposed, there has been much controversy and confusion about their convergence, because sparsity priors have been shown to prefer blurry images to sharp natural images. In this paper, we revisit this problem and provide an analysis on the convergence of MAP based approaches. We first introduce a slight modification to a conventional joint energy function for blind deconvolution. The reformulated energy function yields the same alternating estimation process, but more clearly reveals how blind deconvolution works. We then show the energy function can actually favor the right solution instead of the no-blur solution under certain conditions, which explains the success of previous MAP based approaches. The reformulated energy function and our conditions for the convergence also provide a way to compare the qualities of different blur kernels, and we demonstrate its applicability to automatic blur kernel size selection, blur kernel estimation using light streaks, and defocus estimation.