Optical Fourier Transform-Based Preconditioning for Large-Scale Inverse Problems

Abstract

Fast Fourier Transform (FFT)-based preconditioners provide an efficient approach for approximating inverse operators in large-scale linear inverse problems when the underlying operator structure can be exploited in the Fourier domain. Optical Fourier transforms (OFTs), the analog of digital Fourier transforms, exploit light propagation through optical systems and offer the potential for greater throughput and energy efficiency than conventional digital FFT implementations. However, practical optical systems are inherently imperfect and introduce a variety of errors into the computed output. Thus, OFTs can only provide approximations to their digital counterpart, and the implications of these inexact transforms for iterative solvers remains largely unexplored. This work investigates the feasibility of OFT-based preconditioning of iterative Krylov subspace methods for the solution of discrete inverse problems. We digitally simulate the optical Fourier transform using the FFT and incorporate sources of error expected in practical optical systems into the simulation. Numerical experiments suggest that given sufficient optical system quality, iterative methods utilizing OFT-based preconditioners can maintain reconstruction quality and convergence comparable to those with fully digital FFT-based preconditioners.