Murat Guven

Diffuse optical tomography with a priori anatomical information.

Diffuse optical tomography (DOT) poses a typical ill-posed inverse problem with a limited number of measurements and inherently low spatial resolution. In this paper, we propose a hierarchical Bayesian approach to improve spatial resolution and quantitative accuracy by using a priori information provided by a secondary high resolution anatomical imaging modality, such as magnetic resonance (MR) or x-ray. In such a dual imaging approach, while the correlation between optical and anatomical images may be high, it is not perfect. For example, a tumour may be present in the optical image, but may not be discernable in the anatomical image. The proposed hierarchical Bayesian approach allows incorporation of partial a priori knowledge about the noise and unknown optical image models, thereby capturing the function-anatomy correlation effectively. We present a computationally efficient iterative algorithm to simultaneously estimate the optical image and the unknown a priori model parameters. Extensive numerical simulations demonstrate that the proposed method avoids undesirable bias towards anatomical prior information and leads to significantly improved spatial resolution and quantitative accuracy.

Diffuse optical tomography with physiological and spatial a priori constraints.

Diffuse optical tomography is a typical inverse problem plagued by ill-condition. To overcome this drawback, regularization or constraining techniques are incorporated in the inverse formulation. In this work, we investigate the enhancement in recovering functional parameters by using physiological and spatial a priori constraints. More accurate recovery of the two main functional parameters that are the blood volume and the relative saturation is demonstrated through simulations by using our method compared to actual techniques.

Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging: II

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which influence the extent to which the discretization impacts on the accuracy of the reconstructed images. For example, the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their orientation with respect to optical absorptive heterogeneities, and the formulation of the inverse problem. As a result, our error analysis shows that the discretization of one problem cannot be considered independent of the other problem. While our analysis focuses specifically on the discretization error in DOT, the approach can be extended to quantify other error sources in DOT and other inverse parameter estimation problems.

An adaptive multigrid algorithm for region of interest diffuse optical tomography

Due to diffuse nature of light photons, diffuse optical tomography (DOT) image reconstruction is a challenging 3D problem with a relatively large number of unknowns and limited measurements. As a result, the computational complexity of the existing DOT image reconstruction algorithms remains prohibitive. In this work, we investigate an adaptive multigrid approach to improve the computational efficiency and the quantitative accuracy of DOT image reconstruction. The key idea is based on locally refined grid structure for region of interest (ROI). The ROI may be defined as diagnostically significant regions, strong background heterogeneities and/or deep optical edges, A 2-level mesh is generated to provide high resolution for ROI and sufficiently high resolution for the rest of the image. A least squares (LS) solution is formulated for the inverse problem. Fast adaptive composite (FAC) 2-grid algorithm is employed to solve the inverse problem. Conjugate gradient (CG) is used at the relaxation stage of FAC 2-grid. Same problem is also solved using direct CG and standard 2-grid method for globally fine grid structure. Our numerical studies demonstrate that the proposed FAC based adaptive 2-grid approach provides up to 90% reduction in computational requirements as compared to the direct iterative and standard 2-grid methods while providing better image quality. The fundamental ideas introduced in this study are directly applicable to other linear and nonlinear inverse problems with Newton type global linearization.

Two-level domain decomposition methods for diffuse optical tomography

Diffuse optical tomography (DOT) in the near infrared involves the reconstruction of spatially varying optical properties of a turbid medium from boundary measurements based on a forward model of photon propagation. Due to the nonlinear nature of DOT, high quality image reconstruction is a computationally demanding problem which requires repeated use of forward and inverse solvers. Therefore, it is desirable to develop methods and algorithms that are computationally efficient. In this paper, we develop two-level overlapping multiplicative Schwarz-type domain decomposition (DD) algorithms to address the computational complexity of the forward and inverse DOT problems. We use a frequency domain diffusion equation to model photon propagation and consider a nonlinear least-squares formulation with a general Tikhonov-type regularization for simultaneous reconstruction of absorption and scattering coefficients. In the forward solver, a two-grid method is used as a preconditioner to DD to enhance convergence. In the inverse solver, DD is initialized with a coarse grid solution to achieve local convergence. We show the strong local convexity of the nonlinear objective functional resulting from the inverse problem formulation and prove the local convergence of the DD algorithm for the inverse problem. We provide a computational cost analysis of the forward and inverse solvers and demonstrate their performance in numerical simulations.

Fluorescence optical tomography with a priori information

In this work, we discuss the incorporation of a priori information into the inverse problem formulation for fluorescence optical tomography. In this respect, we first formulate the inverse problem in the optimization framework which allows the incorporation of a priori information about the solution and its gradient. Then, we consider the variational problem, which is equivalent to the optimization problem and prove the existence and uniqueness of the solution. Finally, we discuss the design of the functions that incorporate the a priori information into the inverse problem formulation and present a model problem to illustrate the design procedure.

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

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.

Recursive least squares algorithm for optical diffusion tomography

Algebraic reconstruction techniques (ART) is a family of practical algorithms which sets algebraic equations for the unknowns in terms of the measured data and solves these equations iteratively. It is typical that the system of linear equations obtained in Diffuse Optical Tomography (DOT) is underdetermined and/or ill-conditioned. ART is one of the most popular image reconstruction techniques used in DOT to solve this kind of system of linear equations. There is, however, no natural way of including a priori information about the image in ART algorithm. Moreover ART requires a large number of iterations to reconstruct the image and hence convergence to the solution is slow. In this paper, for the inverse problem in DOT, we apply a Recursive Least Squares Algorithm (IUS) that converges in only one iteration and enables the use of a priori information such as image smoothness.We present comparison between the images reconstructed by ART and IUS.

Hierarchical Bayesian algorithm for diffuse optical tomography

Diffuse optical tomography (DOT) poses a typical ill-posed inverse problem with limited number of measurements and inherently low spatial resolution. In this paper, we propose a hierarchical Bayesian approach to improve spatial resolution and quantitative accuracy by using a priori information provided by a secondary high resolution anatomical imaging modality, such as magnetic resonance (MR) or X-ray. The proposed hierarchical Bayesian approach allows incorporation of partial a priori knowledge about the noise and unknown optical image models, thereby capturing the function-anatomy correlation effectively. Numerical simulations demonstrate that the proposed method avoids undesirable bias towards anatomical prior information and leads to significantly improved spatial resolution and quantitative accuracy

An adaptive V-grid algorithm for diffuse optical tomography

We investigate an adaptive multigrid approach to improve the computational efficiency and the quantitative accuracy of DOT image reconstruction. The key idea is based on a locally refined grid structure for region of interest (ROI). A least squares (LS) solution is formulated for the inverse problem. A fast adaptive composite (FAC) V-grid algorithm is employed to solve the inverse problem. The same problem is also solved using a fixed fine grid and FAC 2-grid scheme for a 2-level locally refined grid. Our numerical studies demonstrate that the proposed FAC based adaptive V-grid approach provides better image quality and up to 90% reduction in computational requirements as compared to the fixed grid and at least 10% reduction as compared to FAC 2-grid algorithms.