Reconstruction of subdomain boundaries of piecewise constant coefficients of the radiative transfer equation from optical tomography data

Abstract

In this study we consider the reconstruction of the smooth subdomain boundaries of piecewise constant coefficients of the radiative transfer equation (RTE) from optical tomography data. The assumption is that the values of the absorption and scattering coefficients (μa, μs) of the different subdomains are known a priori but the smooth subdomain boundaries where μa and μs are discontinuous are unknown. For the reconstruction of (μa, μs) it is then sufficient to find the subdomain boundaries separating different values of the coefficients. This results in a nonlinear ill-posed inverse problem. In this study we propose a numerical algorithm for this inverse problem. The approach is based on the finite element discretization of the RTE. We formulate the forward problem as a mapping from a set of shape coefficients representing the shapes of the subdomain boundaries to optical tomography data, and derive the Jacobian of this forward mapping. Then an iterative Newton-type algorithm which seeks a boundary configuration minimizing the residual norm between measured and predicted data is implemented. The performance of the method is tested with simulated frequency domain optical tomography data from diffusive domains containing low-scattering (void) subdomains.