Optical tomography on graphs

Abstract

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

Graph reconstruction from path correlation data

“…The form of our model and the assumptions we make originate from a body of work developed under the term Network Tomography [19] that seeks to infer link metrics and even the underlying network topology from the measured metric values on paths traversing the network between a set of routers at the network boundary, represented by V B . This setting is similar to other graph reconstruction problems, such as tomography of electrical resistance networks (see, e.g., [8,6]), optical networks [10], and graph reconstruction from Schrödinger-type spectral data (see, e.g., [3,4]). However, in a communication network model there is a single path between given origin and destination, in contrast to the electrical current flowing between two points a resistive medium via all possible paths.…”

Analysis of the inverse Born series: an approach

“…In previous work, we have applied the IBS to inverse problems on graphs. The motivation was to isolate the combinatorial structure in a simpler discrete setting [3]. To this end, we have shown that the IBS converges under qualitatively weaker conditions than those that hold in the continuum.…”

Norm-dependent convergence and stability of the inverse scattering series for diffuse and scalar waves

“…We are interested in studying the forward scattering Born series and the inverse scattering Born series of both the Helmholtz equation and the diffuse wave equation. This work is motivated by the paper of Moskow and Schotland [29], where such studies were conducted under the L p function spaces for both parameter and measurement in which the radius of convergence and the Lipschitz stability constant of the inverse problem change with respect to the choice of p. There have been many works extending this analysis to various inverse problems, including the optical diffusion tomography [28], diffuse waves [30], scalar waves [22], the electromagnetic scattering [23], the Calderón problem [2], the Schrödinger problem [6], the radiative transport equation [25] and optimal tomography on graphs [11]. In particular, studies of the radius of convergence and stability have been extended to general Banach spaces, and the parameter and data spaces do not have to be the same [31].…”