Lebesgue Anisotropic Image Denoising

摘要

The importance of image denoising can never be overemphasized due to the crucial role it plays in image processing and understanding. As the procedure to weed out noises from real visual signals, it stands as the actual foundation for other analysis schemes. A great many techniques have been developed by researchers from a wide array of disciplines such as signal processing, information theory, numerical analysis, computational physics, and computer vision. Extremely impressive progress has been made in the last several decades such as Gaussian and median filtering, Markov random field-based filtering. To avoid the oversmoothing artifact for filtering schemes, Perona and Malik developed the now-classic anisotropic denoising method. Thanks to Koenderink's insightful observation and other researchers' work, it was later discovered that just as Gaussian filtering is the solution to the diffusion-type partial differential equation (PDE), the anisotropic denoising can also be put as the solution to another PDE, which aligns change rate with spatial derivative. The PDE-based image denoising has thus received great attentions from mathematicians and computer scientists alike. In this paper, in the light of the problems of PDE-based scheme, we first retrace the mathematics underlying the Gaussian and median filtering, then instead of working on the original image macroblocks whereof the computation of derivatives is ill-posed, we take the Lebesgue's perspective by grouping intensity values that are neighbors to site p in value and location into a set. The median of this set then assumes the new value at p. Empirical studies suggested extremely encouraging performances by use of this simple Lebesgue anisotropic denoising method.