Signal-representation techniques as inspiration for designing optimization algorithms.

This research proposal is concerned with the study of techniques in two fields, signal processing and optimization, and in the formulation of signal processing problems as optimization ones. This formulation is extremely popular due to recent advances in the representation of signals (e.g., wavelet representations, sparse-coding techniques), in the study of the solutions obtained by such methods (e.g., compressed-sensing techniques), and in the optimization algorithms themselves (e.g., splitting methods). However, we argue that these two fields are connected in more ways than the formulation just described. In fact, the Fourier transform, the most fundamental tool of signal processing, can be shown to be equivalent to the Fenchel conjugate by formulating it in a min-plus algebra. We further argue that certain optimization problems themselves coincide with signal processing ones, and we propose to study this equivalence in depth in order to leverage some of the past advances in signal representation in the development of better optimization techniques, as well as past advances in optimization in the design of new representation techniques. Additionally, we plan to test the new developments in reconstruction problems from the emerging field of non–line-of-sight imaging. This field is concerned with techniques that allow one to see through obstacles and around corners, and it is starting to see its first applications in remote sensing, autonomous driving, and medical imaging.

Keywords:Mathematical optimization, signal representation, non–line-of-sight

Disciplines:Operations research and mathematical programming, Remote sensing, Signal processing, Image and language processing