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

Analytical and Iterative Regularization Methods for Nonlinear Ill-posed Inverse Problems: Applications to Diffuse Optical and Electrical Impedance Tomography

Electrical impedance tomography (EIT) and Diffuse Optical Tomography (DOT) are imaging methods that have been gaining more popularity due to their ease of use and non-ivasiveness. EIT and DOT can potentially be used as alternatives to traditional imaging techniques, such as computed tomography (CT) scans, to reduce the damaging effects of radiation on tissue. The process of imaging using either EIT or DOT involves measuring the ability for tissue to impede electrical flow or absorb light, respectively. For EIT, the inner distribution of resistivity, which corresponds to different resistivity properties of different tissues, is estimated from the voltage potentials measured on the boundary of the object being imaged. In DOT, the optical properties of the tissue, mainly scattering and absorption, are estimated by measuring the light on the boundary of the tissue illuminated by a near-infrared source at the tissue\u27s surface. In this dissertation, we investigate a direct method for solving the EIT inverse problem using mollifier regularization, which is then modified and extended to solve the inverse problem in DOT. First, the mollifier method is formulated and then its efficacy is verified by developing an appropriate algorithm. For EIT and DOT, a comprehensive numerical and computational comparison, using several types of regularization techniques ranging from analytical to iterative to statistical method, is performed. Based on the comparative results using the aforementioned regularization methods, a novel hybrid method combining the deterministic (mollifier and iterative) and statistical (iterative and statistical) is proposed. The efficacy of the proposed method is then further investigated via simulations and using experimental data for damage detection in concrete

Advanced regularization and discretization methods in diffuse optical tomography

Diffuse optical tomography (DOT) is an emerging technique that utilizes light in the near infrared spectral region (650−900nm) to measure the optical properties of physiological tissue. Comparing with other imaging modalities, DOT modality is non-invasive and non-ionising. Because of the relatively lower absorption of haemoglobin, water and lipid at the near infrared spectral region, the light is able to propagate several centimeters inside of the tissue without being absolutely absorbed. The transmitted near infrared light is then combined with the image reconstruction algorithm to recover the clinical relevant information inside of the tissue. Image reconstruction in DOT is a critical problem. The accuracy and precision of diffuse optical imaging rely on the accuracy of image reconstruction. Therefore, it is of great importance to design efficient and effective algorithms for image reconstruction. Image reconstruction has two processes. The process of modelling light propagation in tissues is called the forward problem. A large number of models can be used to predict light propagation within tissues, including stochastic, analytical and numerical models. The process of recovering optical parameters inside of the tissue using the transmitted measurements is called the inverse problem. In this thesis, a number of advanced regularization and discretization methods in diffuse optical tomography are proposed and evaluated on simulated and real experimental data in reconstruction accuracy and efficiency. In DOT, the number of measurements is significantly fewer than the number of optical parameters to be recovered. Therefore the inverse problem is an ill-posed problem which would suffer from the local minimum trap. Regularization methods are necessary to alleviate the ill-posedness and help to constrain the inverse problem to achieve a plausible solution. In order to alleviate the over-smoothing effect of the popular used Tikhonov regularization, L1-norm regularization based nonlinear DOT reconstruction for spectrally constrained diffuse optical tomography is proposed. This proposed regularization can reduce crosstalk between chromophores and scatter parameters and maintain image contrast by inducing sparsity. This work investigates multiple algorithms to find the most computational efficient one for solving the proposed regularization methods. In order to recover non-sparse images where multiple activations or complex injuries happen in the brain, a more general total variation regularization is introduced. The proposed total variation is shown to be able to alleviate the over-smoothing effect of Tikhonov regularization and localize the anomaly by inducing sparsity of the gradient of the solution. A new numerical method called graph-based numerical method is introduced to model unstructured geometries of DOT objects. The new numerical method (discretization method) is compared with the widely used finite element-based (FEM) numerical method and it turns out that the graph-based numerical method is more stable and robust to changes in mesh resolution. With the advantages discovered on the graph-based numerical method, graph-based numerical method is further applied to model the light propagation inside of the tissue. In this work, two measurement systems are considered: continuous wave (CW) and frequency domain (FD). New formulations of the forward model for CW/FD DOT are proposed and the concepts of differential operators are defined under the nonlocal vector calculus. Extensive numerical experiments on simulated and realistic experimental data validated that the proposed forward models are able to accurately model the light propagation in the medium and are quantitatively comparable with both analytical and FEM forward models. In addition, it is more computational efficient and allows identical implementation for geometries in any dimension

Parametric Level-sets Enhanced To Improve Reconstruction (PaLEnTIR)

In this paper, we consider the restoration and reconstruction of piecewise constant objects in two and three dimensions using PaLEnTIR, a significantly enhanced Parametric level set (PaLS) model relative to the current state-of-the-art. The primary contribution of this paper is a new PaLS formulation which requires only a single level set function to recover a scene with piecewise constant objects possessing multiple unknown contrasts. Our model offers distinct advantages over current approaches to the multi-contrast, multi-object problem, all of which require multiple level sets and explicit estimation of the contrast magnitudes. Given upper and lower bounds on the contrast, our approach is able to recover objects with any distribution of contrasts and eliminates the need to know either the number of contrasts in a given scene or their values. We provide an iterative process for finding these space-varying contrast limits. Relative to most PaLS methods which employ radial basis functions (RBFs), our model makes use of non-isotropic basis functions, thereby expanding the class of shapes that a PaLS model of a given complexity can approximate. Finally, PaLEnTIR improves the conditioning of the Jacobian matrix required as part of the parameter identification process and consequently accelerates the optimization methods by controlling the magnitude of the PaLS expansion coefficients, fixing the centers of the basis functions, and the uniqueness of parametric to image mappings provided by the new parameterization. We demonstrate the performance of the new approach using both 2D and 3D variants of X-ray computed tomography, diffuse optical tomography (DOT), denoising, deconvolution problems. Application to experimental sparse CT data and simulated data with different types of noise are performed to further validate the proposed method.Comment: 31 pages, 56 figure

Computational methods for large-scale inverse problems:a survey on hybrid projection methods

This paper surveys animportant class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes abroad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems

Computational methods for large-scale inverse problems:a survey on hybrid projection methods

This paper surveys animportant class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes abroad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems

Computational Inverse Problems for Partial Differential Equations

The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges

Inverse Problems in Wave Scattering

The workshop treated inverse problems for partial differential equations, especially inverse scattering problems, and their applications in technology. While special attention was paid to sampling methods, decomposition methods, Newton methods and questions of unique determination were also investigated