Online learning in optical tomography: a stochastic approach

TL;DR

This paper explores the use of stochastic gradient descent for efficiently solving the inverse problem in optical tomography, enabling faster and more memory-efficient reconstructions of optical parameters from light intensity data.

Contribution

It formulates the optical tomography inverse problem as a PDE-constrained optimization and applies stochastic gradient descent, providing convergence analysis for both nonlinear and linearized cases.

Findings

SGD effectively minimizes the mismatch in optical tomography.

The method reduces memory and computational costs.

Convergence performance of SGD is theoretically analyzed.

Abstract

We study the inverse problem of radiative transfer equation (RTE) using stochastic gradient descent method (SGD) in this paper. Mathematically, optical tomography amounts to recovering the optical parameters in RTE using the incoming-outgoing pair of light intensity. We formulate it as a PDE-constraint optimization problem, where the mismatch of computed and measured outgoing data is minimized with same initial data and RTE constraint. The memory and computation cost it requires, however, is typically prohibitive, especially in high dimensional space. Smart iterative solvers that only use partial information in each step is called for thereafter. Stochastic gradient descent method is an online learning algorithm that randomly selects data for minimizing the mismatch. It requires minimum memory and computation, and advances fast, therefore perfectly serves the purpose. In this paper we formulate the problem, in both nonlinear and its linearized setting, apply SGD algorithm and analyze the convergence performance.