Singular Integrals, Image Smoothness, and the Recovery of Texture in Image Deblurring

Abstract

Total variation (TV) image deblurring is a PDE-based technique that preserves edges, but often eliminates vital small-scale information, or {\em texture}. This phenomenon reflects the fact that most natural images are not of bounded variation. The present paper reconsiders the image deblurring problem in Lipschitz spaces $\Lambda(\alpha, p, q)$, wherein a wide class of nonsmooth images can be accommodated. A new and fast FFT-based deblurring method is developed that can recover texture in cases where TV deblurring fails completely. Singular integrals, such as the Poisson kernel, are used to create an effective new image analysis tool that can calibrate the lack of smoothness in an image. It is found that a rich class of images $\in \Lambda(\alpha, 1, \infty) \cap \Lambda(\beta, 2, \infty)$, with $0.2 < \alpha, \beta < 0.7$. The Poisson kernel is then used to regularize the deblurring problem by appropriately constraining its solutions in $\Lambda(\alpha, 2, \infty)$ spaces, leading to new L2 error bounds that substantially improve on the Tikhonov--Miller method. This so-called Poisson Singular Integral or PSI method is only one of an infinite variety of singular integral deblurring methods that can be constructed. The method is found to be well-behaved in both the L1 and L2 norms, producing results closely matching those obtained in the theoretically optimal, but practically unrealizable, case of true Wiener filtering. Deblurring experiments on synthetically defocused images illustrate the PSI method's very significant improvements over both the total variation and Tikhonov--Miller methods. In addition, successful reconstructions with inexact prior Lipschitz space information, highlight the robustness and practicality of the PSI method.