Fully Nonlinear ${SP}_{3}$ Approximation Based Fluorescence Optical Tomography

In fluorescence optical tomography, many works in the literature focus on the linear reconstruction problem to obtain the fluorescent yield or the linearized reconstruction problem to obtain the absorption coefficient. The nonlinear reconstruction problem, to reconstruct the fluorophore absorption coefficient, is of interest in imaging studies as it presents the possibility of better reconstructions owing to a more appropriate model. Accurate and computationally efficient forward models are also critical in the reconstruction process. The SP
_N
approximation to the radiative transfer equation (RTE) is gaining importance for tomographic reconstructions owing to its computational advantages over the full RTE while being more accurate and applicable than the commonly used diffusion approximation. This paper presents Gauss-Newton-based fully nonlinear reconstruction for the SP
₃
approximated fluorescence optical tomography problem with respect to shape as well as the conventional finite-element method-based representations. The contribution of this paper is the Frechet derivative calculations for this problem and demonstration of reconstructions in both representations. For the shape reconstructions, radial-basis-function represented level-set-based shape representations are used. We present reconstructions for tumor-mimicking test objects in scattering and absorption dominant settings, respectively,for moderately noisy data sets in order to demonstrate the viability of the formulation. Comparisons are presented between the nonlinear and linearized reconstruction schemes in an element wise setting to illustrate the benefits of using the former especially for absorption dominant media.