Online learning in optical tomography: a stochastic approach

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.

Reconstructing the thermal phonon transmission coefficient at solid interfaces in the phonon transport equation

“…This is because most PDE-based inverse problems eventually are formulated as minimization problem with the loss function being in the format of a summation of multiple smaller loss functions. This is the exactly the format where SGD outperforms other optimization methods [14].…”

On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

“…, n}, and η k > 0 is the corresponding step size. This algorithm has demonstrated very encouraging numerical results in [2] for diffuse optical tomography (with radiative transfer equation). It is also worth noting that a variant of the algorithm, i.e., randomized Kaczmarz method (RKM) (see, e.g., [20,11] for the equivalence result between RKM and Algorithm 1), has been extremely successful in the computed tomography community [7,8] (see, e.g., [24] and [26] for interesting linear convergence results of the RKM for least-squares regression and phase retrieval with "well-conditioned" matrix and exact data).…”

“…Remark 4.4. The constants in Lemma 4.5 involve an unpleasant dependence on the number of equations n as n 1 2 . This is due to the variance inflation of the stochastic gradient estimate instead of the true gradient.…”

On diffusive scaling in acousto-optic imaging

“…In this situation, the forward model is the classical elliptic type equation, and mathematically, recovering the optical parameter amounts to reconstructing the diffusion coefficient in the elliptic equation using the Dirichlet-to-Neumann map [39], and is proven mathematically to be logarithmically unstable [1]. Multiple strategies are adopted to "stabilize" the problem [33,37], both by adjusting the experimental modalities upfront, or introducing image deblurring techniques as a post-processing [16,27,36,38].…”