Scalar Scattering Theory and Physics-inspired Optimization for Computational Imaging

Abstract

This thesis explores the realm of computational imaging, focusing on the critical problems of phase retrieval and optical scattering—essential for accurately extracting physical information from photons. It aims to enhance the understanding and computational efficiency of existing models by addressing the fundamental challenges encountered due to diffraction effects, multiple scattering, and noise. Specifically, the thesis proposes improvements and comprehensive analyses of models related to phase retrieval, such as the Transport-of-Intensity Equation (TIE), and optical scattering approximations, including the Lippmann-Schwinger Equation (LSE).

For phase retrieval, this work introduces mathematical approaches to reduce the TIE's sensitivity to experimental conditions and provides a quantitative comparison with other methods to clarify its applicability. It also explores the adjoint method for solving the TIE, which significantly enhances numerical stability, and discusses the analytical relationship between non-paraxial formulations and conventional phase retrieval methods, deepening our understanding of the field.

In the domain of optical scattering, where information in photons is further encoded via complex light-matter interactions, this thesis examines several models derived from the scalar wave equation such as the LSE, the Born series, and the beam propagation method. It provides a direct and quantitative analysis of their relationships and numerical stability, highlighting the strengths and weaknesses of these models in various experimental contexts, which has not been discussed thoroughly in previous studies.

Additionally, the thesis tackles the computational challenges associated with the LSE by proposing numerical strategies and integrating neural networks as a learnable regularization. This approach aims to reduce computational demands while maintaining generalizability across different scattering objects.

Overall, this work contributes to the field of computational imaging by offering a deeper understanding of phase retrieval and optical scattering models, alongside presenting solutions to overcome their limitations. It sets the stage for further theoretical analysis and practical applications in physics, where accurate information retrieval from photons is crucial.