Inverse problems in signal processing : Functional optimization, parameter estimation and machine learning

Sammanfattning: Inverse problems arise in any scientific endeavor. Indeed, it is seldom the case that our senses or basic instruments, i.e., the data, provide the answer we seek. It is only by using our understanding of how the world has generated the data, i.e., a model, that we can hope to infer what the data imply. Solving an inverse problem is, simply put, using a model to retrieve the information we seek from the data.In signal processing, systems are engineered to generate, process, or transmit signals, i.e., indexed data, in order to achieve some goal. The goal of a specific system could be to use an observed signal and its model to solve an inverse problem. However, the goal could also be to generate a signal so that it reveals a parameter to investigation by inverse problems. Inverse problems and signal processing overlap substantially, and rely on the same set of concepts and tools. This thesis lies at the intersection between them, and presents results in modeling, optimization, statistics, machine learning, biomedical imaging and automatic control.The novel scientific content of this thesis is contained in its seven composing publications, which are reproduced in Part II. In five of these, which are mostly motivated by a biomedical imaging application, a set of related optimization and machine learning approaches to source localization under diffusion and convolutional coding models are presented. These are included in Publications A, B, E, F and G, which also include contributions to the modeling and simulation of a specific family of image-based immunoassays. Publication C presents the analysis of a system for clock synchronization between two nodes connected by a channel, which is a problem of utmost relevance in automatic control. The system exploits a specific node design to generate a signal that enables the estimation of the synchronization parameters. In the analysis, substantial contributions to the identifiability of sawtooth signal models under different conditions are made. Finally, Publication D brings to light and proves results that have been largely overlooked by the signal processing community and characterize the information that quantized linear models contain about their location and scale parameters.