Conference abstracts

Session A1 - Approximation Theory

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Higher Order Total Variation, Multiscale Generalizations, and Applications to Inverse Problems

Toby Sanders

Arizona State University, United States   -   tlsander@asu.edu

In the realm of signal and image denoising and reconstruction, L1 regularization techniques have generated a great deal of attention with a multitude of variants. In this work, we are interested in higher order total variation (HOTV) approaches, which are motivated by the idea of encouraging low order polynomial behavior in the reconstruction. A key component for their success is that under certain assumptions, the solution of minimum L1 norm is a good approximation to the solution of minimum L0 norm. In this work, we demonstrate that this approximation can result in artifacts that are inconsistent with desired sparsity promoting L0 properties, resulting in subpar results in some instances. Therefore we have developed a multiscale higher order total variation (MHOTV) approach, which we show is closely related to the use of multiscale Daubechies wavelets. In the development of MHOTV, we confront a number of computational issues, and show how they can be circumvented in a mathematically elegant way, via operator decomposition and alternatively converting the problem into Fourier space. The relationship with wavelets, which we believe has generally gone unrecognized, is shown to hold in several numerical results, although subtle improvements in the results can be seen due to the properties of MHOTV. The results are shown to be useful in a number of computationally challenging practical applications, including image inpainting, synthetic aperture radar, and 3D tomography.

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