Blind image deblurring using spectral properties of convolution operators

Abstract

Blind deconvolution is to recover a sharp version of a given blurry image or signal when the blur kernel is unknown. Because this problem is ill-conditioned in nature, effectual criteria pertaining to both the sharp image and blur kernel are required to constrain the space of candidate solutions. While the problem has been extensively studied for long, it is still unclear how to regularize the blur kernel in an elegant, effective fashion. In this paper, we show that the blurry image itself actually encodes rich information about the blur kernel, and such information can indeed be found by exploring and utilizing a well-known phenomenon, that is, sharp images are often high pass, whereas blurry images are usually low pass. More precisely, we shall show that the blur kernel can be retrieved through analyzing and comparing how the spectrum of an image as a convolution operator changes before and after blurring. Subsequently, we establish a convex kernel regularizer, which depends only on the given blurry image. Interestingly, the minimizer of this regularizer guarantees to give a good estimate to the desired blur kernel if the original image is sharp enough. By combining this powerful regularizer with the prevalent nonblind devonvolution techniques, we show how we could significantly improve the deblurring results through simulations on synthetic images and experiments on realistic images.