Dimensional Analysis and Variational Formulation of Diffuse Optical Tomography (DOT) Model

Diffuse Optical Tomography (DOT) is an emerging modality for soft tissue imaging with medical applications including breast cancer detection. DOT has many benefits, including its use of non ionizing radiation and its ability to produce high contrast images. However, it is well known that DOT image reconstruction is unstable and has low resolution. DOT uses near infra-red light waves to probe inside a body; for example, DOT can be used to measure the changes in the amount of oxygen in tissues, which can detect early stages of cancer in soft tissues such as the breast and brain. In this thesis, we perform dimensional analysis to obtain a dimensionless form of the ODE for the 1-d DOT model and the PDE for the 2-d DOT model. We later solve the 1-d cases using the finite element method (FEM) in MATLAB. We investigate whether the inverse problem using the dimensionless scaled forward DOT model will improve the ill-posedness of the image reconstruction problem in the 1-d case. We solve the inverse problem for DOT image reconstruction by reformulating the inverse problem as a variationally constrained non-linear optimization problem and compare solving the optimization problem for specific cases of the 1-d DOT model with Newton\u27s iteration versus the traditional Gauss-Newton method. We observe the effects of different regularization parameters and step lengths on the reconstructions for Newton\u27s iteration. We also observe the effect of moving the inclusion away from the boundary during image reconstruction. Using the optimally derived regularization parameter from the noise-free data, we reconstructed the parameter space by adding different levels of noise to the synthetic data. Based on our simulations in 1-d, we conclude that the scaled inverse problem is still ill-posed but that the variational approach provides a better reconstruction than the Gauss-Newton method

Truncated Total Least Squares Method with a Practical Truncation Parameter Choice Scheme for Bioluminescence Tomography Inverse Problem

In bioluminescence tomography (BLT), reconstruction of internal bioluminescent source distribution from the surface optical signals is an ill-posed inverse problem. In real BLT experiment, apart from the measurement noise, the system errors caused by geometry mismatch, numerical discretization, and optical modeling approximations are also inevitable, which may lead to large errors in the reconstruction results. Most regularization techniques such as Tikhonov method only consider measurement noise, whereas the influences of system errors have not been investigated. In this paper, the truncated total least squares method (TTLS) is introduced into BLT reconstruction, in which both system errors and measurement noise are taken into account. Based on the modified generalized cross validation (MGCV) criterion and residual error minimization, a practical parameter-choice scheme referred to as improved GCV (IGCV) is proposed for TTLS. Numerical simulations with different noise levels and physical experiments demonstrate the effectiveness and potential of TTLS combined with IGCV for solving the BLT inverse problem

Linearized Reconstruction for Diffuse Optical Spectroscopic Imaging

In this paper, we present a novel reconstruction method for diffuse optical spectroscopic imaging with a commonly used tissue model of optical absorption and scattering. It is based on linearization and group sparsity, which allows recovering the diffusion coefficient and absorption coefficient simultaneously, provided that their spectral profiles are incoherent and a sufficient number of wavelengths are judiciously taken for the measurements. We also discuss the reconstruction for imperfectly known boundary and show that with the multi-wavelength data, the method can reduce the influence of modelling errors and still recover the absorption coefficient. Extensive numerical experiments are presented to support our analysis.Comment: 18 pages, 7 figure

Enhanced diffuse optical tomographic reconstruction using concurrent ultrasound information

Multimodal imaging is an active branch of research as it has the potential to improve common medical imaging techniques. Diffuse optical tomography (DOT) is an example of a low resolution, functional imaging modality that typically has very low resolution due to the ill-posedness of its underlying inverse problem. Combining the functional information of DOT with a high resolution structural imaging modality has been studied widely. In particular, the combination of DOT with ultrasound (US) could serve as a useful tool for clinicians for the formulation of accurate diagnosis of breast lesions. In this paper, we propose a novel method for US-guided DOT reconstruction using a portable time-domain measurement system. B-mode US imaging is used to retrieve morphological information on the probed tissues by means of a semi-automatical segmentation procedure based on active contour fitting. A two-dimensional to three-dimensional extrapolation procedure, based on the concept of distance transform, is then applied to generate a three-dimensional edge-weighting prior for the regularization of DOT. The reconstruction procedure has been tested on experimental data obtained on specifically designed dual-modality silicon phantoms. Results show a substantial quantification improvement upon the application of the implemented technique. This article is part of the theme issue ‘Synergistic tomographic image reconstruction: part 2’

Sparsity promoting reconstructions via hierarchical prior models in diffuse optical tomography

Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating {\em a priori} information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $\ell^1$ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities. In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions

Computational Inverse Problems

Inverse problem typically deal with the identification of unknown quantities from indirect measurements and appear in many areas in technology, medicine, biology, finance, and econometrics. The computational solution of such problems is a very active, interdisciplinary field with close connections to optimization, control theory, differential equations, asymptotic analysis, statistics, and probability. The focus of this workshop was on hybrid methods, model reduction, regularization in Banach spaces, and statistical approaches

Hyperspectral Diffuse Optical Tomography Using the Reduced Basis Method and Sparsity Constraints

Diffuse Optical Tomography (DOT) has long been investigated as an effective imaging technique for soft tissue imaging, such as breast cancer detection. DOT has many benefits, including its use of non-ionizing light and its ability to produce high contrast images, but it also has low resolution. In recent years hyperspectral DOT (hyDOT) has been proposed, in an effort to improve that resolution by adding more information in the spectral domain. In this imaging modality, hundreds or even thousands of different wavelengths in the visible to near infrared range are used in the imaging process. Since tissue absorbs and scatters light differently at different wavelengths, it has been conjectured that this increase of information should provide images that give a better overall idea of the complete spatial reconstruction of the optical parameters. Although hyDOT has been investigated experimentally, a formal theoretical investigation into its mathematical foundations has not been thoroughly performed. This dissertation seeks to lay the groundwork for the mathematical formulation of this imaging modality. First, the forward problem for hyDOT is formulated and the spectral regularity of the solution investigated. We demonstrate that the solution to the governing PDE is very smooth with respect to wavelength. This spectral regularity allows for the application of a model reduction technique to the forward problem known as the Reduced Basis Method. Several proofs are given for the hyDOT forward solution and the spectral regularity term, including existence and uniqueness proofs and proofs showing the continuity of the solution with respect to the diffusion and absorption coefficients and the wavelength. The appropriate function spaces for the optical coefficients with respect to their dependence on the wavelength are explored and a new norm is proposed. Additionally, the hyDOT inverse problem is formulated. New cost functionals are proposed to solve the inverse problem that incorporate the spatial sparsity of the optical parameters and their spectral regularity. Finally, a gradient-based reconstruction algorithm that enforces the spatial sparsity with respect to wavelength, is shown to be very effective and robust in solving the hyDOT inverse problem when used on simulations with a simple geometry