Ninth Annual Graduate Student Miniconference in Computational Mathematics
Joseph Daws, Jr.
University of Tennessee, Knoxville 

Abstract 
Compressed sensing for image reconstruction using hierarchical wavelets on lower sets
In this work we propose a compressed sensing approach for the reconstruction of functions or images represented by their expansion in a wavelet basis, from only a small number of samples. The success of applying wavelet representations for image reconstruction and compression has inspired many sparse recovery techniques. However, these approaches can be improved by exploiting the connections between different levels of wavelet coefficients in the hierarchical wavelet basis to find significant coefficients. In particular, we show that the important wavelet coefficients for a certain class of functions and images are concentrated on a lower set (i.e., a downward closed tree), and present a weighted $\ell _ 1$ minimization technique which takes advantage of this fact in order to reduce the overall complexity of our proposed compressed sensing recovery. Following some of the results in our previous effort we also present theoretical estimates related to the sampling complexity of our scheme as compared to unweighted $\ell _ 1$ minimization. Several numerical examples are provided to show the effectiveness of this weighted $\ell _ 1$ minimization scheme for solving the image inpainting problem as compared to several other established solution techniques. 