Convergence Analysis of MAP Based Blur Kernel Estimation

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.

1. Introduction

Image blur due to camera shakes is an annoying artifact that severely degrades image quality. Image blur is often modeled as:

$$\begin{equation*} b=k*l+n, \tag{1} \end{equation*}$$ where is an observed blurry image, is a blur kernel, is a latent sharp image, is noise, and * is the convolution operator. Blind deconvolution is a problem to estimate and from a given blurry image b, which is severely ill-posed because the number of unknowns l and k exceeds the number of observed data b.