Sparse recovery using variable band for compressive sensing algorithms

Inverse problems involve determining the causality of an effect from measurements obtained by observing the system. There are various topics in inverse problems with real-world applications, and wave imaging is one of them and finds applications in various fields of science, such as medical imaging, seismic imaging, sonar imaging, and more. Sparse reconstruction methods are commonly employed to solve such imaging and inverse problems. The objective of these methods is to recover the unknown from an under-determined system of linear measurements where the unknown and the measurement matrix need to satisfy certain conditions for successful reconstruction. In particular, the unknown should be sparse enough, and the measurement matrix should exhibit incoherence. In the real world, these conditions are rarely fully met, resulting in reduced reconstruction quality by these algorithms. Different approaches are discussed in the literature to address scenarios where CS methods fail under such realistic settings, and band exclusion can be considered as one of them. In the context of the array imaging problems, this band corresponds to a value that is closely related to the minimum separation between objects in the scene. This minimum separation information can then be utilized to improve the incoherence properties of the system, thus enhancing reconstruction quality. In this thesis, we propose using a variable band for each object in the scene instead of a single global band. This approach, embedded into different algorithms, can enhance their reconstruction quality, particularly when the scene contains relatively less sparse regions. We also discuss how to estimate these bands for each object in the scene and give theoretical justifications for their use. We then discuss implementations of the variable band approach for various known methods.