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



Alfred S. Carasso


Total variation (TV) image deblurring is a PDE-based technique that preserves edges, but often eliminates vital small-scale information or 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 (Besov) spaces Λ(α, p, q), wherein a wide class of non-smooth images can be accommodated, and develops a fast FFT-based deblurring method that can recover texture in cases where TV deblurring fails completely. Singular integral mollifiers, 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 belong to Λ(α, 1, {haze}{logical product}Λ(Β, 2, {haze}), with 0.2 2 error bounds that substantially improve on the Tikhonov-Miller method. This new so-called Poisson Singular Integral or PSI method is found to be well-behaved in both 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 significant improvements over both the total variation and Tikhonov-Miller methods.
NIST Interagency/Internal Report (NISTIR) - 7005
Report Number


Besov spaces, image deblurring, Lipschitz spaces, loss of texture, non-smooth images, Poisson kernel, PSI method, recovery of texture, semi-group approximation


Carasso, A. (2003), Singular Integrals, Image Smoothness, and the Recovery of Texture in Image Deblurring, NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online], (Accessed April 16, 2024)
Created June 1, 2003, Updated November 10, 2018