Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography: Part II

Abstract

Abstract-For imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems, discretization can be a major factor that determines the accuracy of imaging. In this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography. We model the forward problem by a pair of diffusion equations at the excitation and emission wavelengths and consider a finite element discretization method for the numerical solution of the forward problem. For the inverse problem, we use an optimization framework which allows incorporation of a priori information in the form of zeroth-and first-order Tikhonov regularization terms. Next, we convert the inverse problem into a variational problem and use Galerkin projection to discretize the inverse problem. Following the discretization, we analyze the error in reconstructed images due to the discretization of the forward and inverse problems and present two theorems which point out the factors that may lead to high error such as the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their positions with respect to fluorophore concentration, and the formulation of the inverse problem. Finally, we demonstrate the results and implications of our error analysis by numerical experiments. In the second part of the paper, we apply our results to design novel adaptive discretization algorithms.Index Terms-Adaptive meshing algorithms, error analysis, fluorescence diffuse optical tomography.

Reconstruction of an optical inhomogeneity map improves fluorescence diffuse optical tomography

“…In this section, we discuss the calculation of the Jacobian matrix J , forward estimate G , and weight matrix B required in (12), (13), and (16), respectively. All the three quantities require an appropriate model for the propagation of light within the object Ω.…”

“…The diffusion approximation is generally adopted to solve the forward problem. This partial differential equation, which may be solved by means of the finite element method [11,12] or the boundary element method [13,14], requires the knowledge of several optical parameters, principally the absorption coefficient m a and reduced scattering coefficient m ¢ s at each optical wavelength being measured, as well as the refractive index of the medium, insofar as it affects the determination of correct boundary conditions. One of the major difficulties of FDOT is to estimate the distribution of these optical parameters, which is even more critical when highly heterogeneous objects such as biological samples (e.g.…”

Performance evaluation of adaptive meshing algorithms for fluorescence diffuse optical tomography using experimental data

“…To address this trade-off, adaptive discretization techniques have been developed for FDOT imaging to improve the reconstruction accuracy while reducing the computational requirements [2][3][4][5][6]. In a series of papers, we first analyzed the effect of discretization on the accuracy of FDOT imaging and proposed novel adaptive meshing algorithms under the assumption that the measurements are noise free [2,5]. In [6], we extended our work to the case of noisy measurement and took into account noise statistics in designing adaptive meshes.…”