Advanced Techniques in Regularization for Partial Differential Equations and Imaging

Theresa Scarnati

Arizona State University

In recent years, l1 regularization has received considerable attention in designing image recon- struction algorithms from under-sampled and noisy data for images that have some sparsity proper- ties. In this study we use l1 regularization techniques, often seen in compressed sensing, along with polynomial annihilation to develop advanced techniques for both enhancing the numerical solutions of nonlinear hyperbolic partial differential equations (PDEs) and for the accurate reconstruction of synthetic aperture radar (SAR) imagery. Nonlinear hyperbolic PDEs have solutions that develop discontinuities regardless of smooth initial conditions. By exploiting the fact that these discontinues are sparse in the physical domain, we are able to increase the accuracy of current, modern numerical methods while also relaxing the restrictive time stepping conditions usually required to maintain stability in solutions. In SAR image formation, autofocusing is required to estimate the phase errors that result from the imperfect knowledge of the position of the imaging platform. Also inherent in the SAR imaging system is speckle. Speckle is a, granular, salt and pepper multiplicative noise that corrupts the contrast of images. We use alternating minimization to jointly estimate the phase errors and the SAR image. Within this minimization process, we exploit the edge information present in the data while simultaneously taking into account the fact that the phase errors to be estimated are correlated. With these techniques we are able to reduce speckle, estimate phase errors and enhance SAR imagery, for the purpose of improving autonomous classification algorithms.

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