Ancient Hindu Geometry: The Science of the Sulba by Bibhutibhushan Datta

By Bibhutibhushan Datta

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For instance, Laplace equations, derived from H 1 semi-norm, has been used for wavelet error concealment [24], TV based models are used for compression [5, 12], noise removal [19], post-processing to remove Gibbs’ oscillations [16], zooming [22], wavelet thresholding [11], wavelet inpainting [9], l1 norm optimization for sparse signal recovery [3, 4], anisotropic wavelet filters for denoising [14], variational image decomposition [27]. These studies have demonstrated promising results, which show clear advantages of the combinations of wavelet and variational PDE strategies over the traditional methods.

Weber’s law and Weberized TV restoration. Physica D: Nonlinear Phenomena, 175:241–251, 2003. 23. J. Shen. Bayesian video dejittering by BV image model. SIAM J. Appl. , 64(5):1691–1708, 2004. 24. L. A. Vese. A study in the BV space of a denoising-deblurring variational problem. Appl. Math. , 44(2):131–161, 2001. 25. L. A. Vese and S. J. Osher. Modeling textures with Total Variation minimization and oscillating patterns in image processing. J. Sci. , 19(1-3):553–572, 2003. 26. C. Vogel. Computational Methods for Inverse Problems.

Despite of the remarkable results obtained in combining variational PDE’s with wavelets, the theoretical understandings for those models remain limited, specially for the nonlinear TV based models. Most of the existing studies are focused on the existence and uniqueness (or non-uniqueness) of the solutions of the variational wavelet models. A few recent investigations have been conducted to address the recover properties, including the well-known results reported in [3], in which a probabilistic theory for the exact recovery conditions of sparse signals based on random frequency samples has been developed.

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