$L_1$-Norm Regularization for Short-and-Sparse Blind Deconvolution: Point Source Separability and Region Selection

Abstract

Blind deconvolution is about estimating both the convolution kernel and the latent signal from their convolution. Many blind deconvolution problems have a short-and-sparse (SaS) structure; i.e., the signal (or its gradient) is sparse and the kernel size is much smaller than the signal size. While $\ell_1$-norm relating regularizations have been widely used for solving SaS blind deconvolution problems, the so-called region/edge selection technique brings great empirical improvement to such $\ell_1$-norm relating regularizations in image deblurring. The essence of region/edge selection is during an alternative iterative scheme of SaS blind deconvolution: one estimates the kernel on an estimate of the latent image with well-separated image edges instead of the one with the least fitting error. In this paper, we first examine the validity and soundness of $\ell_1$-norm relating regularization in the setting of 1D SaS blind deconvolution. The analysis reveals the importance of the separation of nonzero signal entries toward the soundness of such a regularization. The studies laid out the foundation of region selection technique; i.e., during the iteration, an estimate of the latent image with well-separated edges is a better candidate for estimating the kernel than the one with the least fitting error. Based on the studies conducted in this paper, an alternating iterative scheme with region selection model is developed for SaS blind deconvolution, which is then applied to blind motion deblurring. The experiments show its effectiveness over many existing $\ell_1$-norm relating approaches.