$L_1$-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 5 convolution. Many blind deconvolution problems have a short-and-sparse (SaS) structure, i.e . the 6 signal (or its gradient) is sparse and the kernel size is much smaller than the signal size. While ℓ 1 -norm 7 relating regularizations have been widely used for solving SaS blind deconvolution problems, the so- 8 called region/edge selection technique brings great empirical improvement to such ℓ 1 -norm relating 9 regularizations in image deblurring. The essence of region/edge selection is during an alternative 10 iterative scheme of SaS blind deconvolution, one estimates the kernel on an estimate of the latent 11 image with well-separated image edges instead of the one with the least fitting error. In this paper, 12 we first examines the validity and soundness of ℓ 1 -norm relating regularization in the setting of 1D 13 SaS blind deconvolution. The analysis reveals the importance of the separation of non-zero signal 14 entries toward the soundness of such a regularization. The studies laid out the foundation of region 15 selection technique, i.e ., during the iteration, an estimate of the latent image with well-separated 16 edges is a better candidate for estimating the kernel than the one with least fitting error. Based 17 on the studies conducted in this paper, an alternating iterative scheme with region selection model 18 is developed for SaS blind deconvolution, which is then applied on blind motion deblurring. The 19 experiments showed its effectiveness over many existing ℓ 1 -norm relating approaches. 20