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

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 ...

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