Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise

Abstract

Abstract-In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration and the discretization of the forward and inverse problems. In this paper, based on these error estimates, we develop two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT, with the objective of error reduction in the reconstructed optical images due to discretization while keeping the size of the discretized forward and inverse problems within the allowable limits. We present three-dimensional numerical simulations to demonstrate the improvements in accuracy, resolution and detectability of small heterogeneities in reconstructed images provided by the use of the new adaptive mesh generation algorithms. Finally, we compare our algorithms both analytically and numerically with the existing conventional adaptive mesh generation algorithms.Index Terms-Adaptive meshing algorithms, error analysis, fluorescence diffuse optical tomography.

Basis mapping methods for forward and inverse problems

“…We assumed the forward model to operate on an unstructured piecewise polynomial finite element basis representation, while the solution of the inverse problem is represented by a regular grid of basis functions, be it a piecewise constant or polynomial pixel or voxel representation, or a blob basis representation with radially symmetric basis functions arranged in a regular grid. Adaptive approaches to the numerical solution of inverse problems, such as those described in the introduction, that could potentially benefit from the basis mapping algorithms derived in this paper include electromagnetic imaging, where the forward model is given by Maxwell's equations [23,24], diffuse optical tomography [4,25] and fluorescence tomography [26,27], electrical impedance tomography [28], as well as in adaptive inverse electrocardiography [29] and electroencephalography problems [30]. Applications can also be found in geophysics, using adaptive methods for seismic tomography [31], local earthquake tomography [32] and magnetic susceptibility mapping [33].…”

Preconditioning of the fluorescence diffuse optical tomography sensing matrix based on compressive sensing

“…Fluorescence diffuse optical tomography (FDOT) is an imaging modality that uses near infrared light to measure 3D fluorophore activity inside biological tissue [1]. The FDOT inverse problem involves recovering the unknown fluorophore yield μ in a domain Ω from the measurements obtained on the domain boundary ∂Ω based on the following integral equation [1]:…”

“…The FDOT inverse problem involves recovering the unknown fluorophore yield μ in a domain Ω from the measurements obtained on the domain boundary ∂Ω based on the following integral equation [1]:…”

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

“…The numerical solutions pose a trade-off between the accuracy and the computational efficiency of image reconstruction. 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 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. In this Letter, we evaluate the performance of the adaptive meshing algorithms developed in [6] using real data obtained from a phantom experiment and demonstrate the practical advantages of our algorithms in FDOT imaging:…”