Sparse signal, image recovery in compressive sensing technique through l1 norm minimization
The classical Shannon Nyquist theorem tells us that, the number of samples required for a signal to reconstruct must be at least twice the bandwidth of the highest frequency for the signal of interest. In fact, this principle is used in all signal processing applications. Unfortunately, in most of the practical cases we end up with far too many samples. In such cases a new sampling method has been developed called Compressive Sensing (CS) or Compressive Sampling, where one can reconstruct certain signals and images from far fewer samples or measurements when compared to that of samples in classical theorem. CS theory primarily relies on sparsity principle and it exploits the fact that many natural signals or images are sparse in the sense that they have concise representations when expressed in the proper basis. Since CS theory relies on sparsity, we focused on reconstructing a sparse signal or sparse approximated image from its corresponding few measurements. In this document we focused on l1 norm minimization problem (convex optimization problem) and its importance in recovering a sparse signal or sparse approximated image in CS. To sparse approximate the image we have transformed the image form standard pixel domain to wavelet domain, because of its concise representation. The algorithms we used to solve the l1 norm minimization problem are primal-dual interior point method and barrier method. We came up with certain examples in Matlab to explain the differences between barrier method and primal-dual interior point method in solving a l1 norm minimization problem i.e. recovering a sparse signal or image from very few measurements. While recovering the images the approach we used is block wise approach and treating each block as vector.
Koppisetti, Phanendra, "Sparse signal, image recovery in compressive sensing technique through l1 norm minimization" (2012). ETD Collection for University of Texas, El Paso. AAI1512585.