Estimation and Statistical Bounds for Three-Dimensional Polar Shapes in Diffuse Optical Tomography

Abstract

Voxel-based reconstructions in Diffuse Optical Tomography (DOT) tend to produce very smooth images due to the attenuation of information of high spatial frequency. This then causes difficulty in estimating the spatial extent and contrast of anomalous regions such as tumors. Given an assumption that the target image is piecewise constant, we can employ a parametric model to estimate the boundaries and contrast of an inhomogeneity directly. In this paper, we describe a method to directly reconstruct such a shape boundary from diffuse optical measurements. We parameterized the object boundary using a spherical harmonic basis, and derived a new method to compute sensitivities of measurements with respect to shape parameters, based on an integral equation formulation of the forward problem. We introduced a centroid constraint to ensure uniqueness of the combined shape/center parameter estimate, and a projected Newton method was utilized to optimize the object center position and shape parameters simultaneously. Using the shape Jacobian, we also computed the Cramér-Rao lower bound on the theoretical estimator accuracy given a particular measurement configuration, object shape, and level of measurement noise. Knowledge of the shape sensitivity matrix and of the measurement noise variance allows us to visualize the shape uncertainty region in three dimensions, giving a confidence region for our shape estimate. We have implemented our shape reconstruction method, using a finite-differencebased forward model to compute the forward and adjoint fields. Reconstruction results are shown for a number of simulated target shapes, and we investigate the problem of model order selection using realistic levels of measurement noise.

On the use of the Cramér–Rao lower bound for diffuse optical imaging system design

“…In Ref. 26, regularization was achieved by reparameterizing the problem using a spherical harmonic basis, which reduced the number of unknowns to be estimated from the data to an object center position and various shape parameters. The CRLB for the shape parameters was subsequently computed and used to provide confidence limits for the shape estimates.…”

Precision analysis based on Cramer–Rao bound for 2D acoustics and electromagnetic inverse scattering

“…Therefore, the CRB can be used to describe and predict the asymptotic behavior of the ML estimator. Finally, the derivation and the computation of the CRB are often easier than the ones associated to other performance bounds, which makes this bound attractive in various fields [8][9][10][11][12][13]. In the particular context of inverse scattering, the CRB has gained a significant interest recently.…”

A coupled finite element-boundary element method for modeling Diffusion equation in 3D multi-modality optical imaging

“…Using a level-set technique for image reconstruction, Schweiger et al [ 8 ] showed that detection and localization of small objects could be improved in 3D. Boverman et al [ 9 ] used a parametric approach to reconstruct shape and contrast of piece-wise constant regions in 3D with spherical harmonics for modeling sharp boundaries in tissue and demonstrated quantitative results in a domain with a single inclusion. Zacharopoulos et al [ 10 ] used a similar strategy and showed that they could accurately recover location and contrast of an anomaly in experiments on a domain with single inclusion.…”