Graph- and finite element-based total variation models for the inverse problem in diffuse optical tomography

Total variation (TV) is a powerful regularization method that has been widely applied in different imaging applications, but is difficult to apply to diffuse optical tomography (DOT) image reconstruction (inverse problem) due to complex and unstructured geometries, non-linearity of the data fitting and regularization terms, and non-differentiability of the regularization term. We develop several approaches to overcome these difficulties by: i) defining discrete differential operators for unstructured geometries using both finite element and graph representations; ii) developing an optimization algorithm based on the alternating direction method of multipliers (ADMM) for the non-differentiable and non-linear minimization problem; iii) investigating isotropic and anisotropic variants of TV regularization, and comparing their finite element- and graph-based implementations. These approaches are evaluated on experiments on simulated data and real data acquired from a tissue phantom. Our results show that both FEM and graph-based TV regularization is able to accurately reconstruct both sparse and non-sparse distributions without the over-smoothing effect of Tikhonov regularization and the over-sparsifying effect of L$_1$ regularization. The graph representation was found to out-perform the FEM method for low-resolution meshes, and the FEM method was found to be more accurate for high-resolution meshes.Comment: 24 pages, 11 figures. Reviced version includes revised figures and improved clarit

Instability of an inverse problem for the stationary radiative transport near the diffusion limit

In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean free path (the Knudsen number) 0 < \eps \ll 1. It is well-known that there is a transition of stability from H\"{o}lder type to logarithmic type with \eps\to 0, the theory of this transition of stability is still an open problem. In this study, we show the transition of stability by establishing the balance of two different regimes depending on the relative sizes of \eps and the perturbation in measurements. When \eps is sufficiently small, we obtain exponential instability, which stands for the diffusive regime, and otherwise we obtain H\"{o}lder instability instead, which stands for the transport regime.Comment: 20 page

Mixed Total Variation and L

Optical tomography is an emerging and important molecular imaging modality. The aim of optical tomography is to reconstruct optical properties of human tissues. In this paper, we focus on reconstructing the absorption coefficient based on the radiative transfer equation (RTE). It is an ill-posed parameter identification problem. Regularization methods have been broadly applied to reconstruct the optical coefficients, such as the total variation (TV) regularization and the L1 regularization. In order to better reconstruct the piecewise constant and sparse coefficient distributions, TV and L1 norms are combined as the regularization. The forward problem is discretized with the discontinuous Galerkin method on the spatial space and the finite element method on the angular space. The minimization problem is solved by a Jacobian-based Levenberg-Marquardt type method which is equipped with a split Bregman algorithms for the L1 regularization. We use the adjoint method to compute the Jacobian matrix which dramatically improves the computation efficiency. By comparing with the other imaging reconstruction methods based on TV and L1 regularizations, the simulation results show the validity and efficiency of the proposed method

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

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

Multiplexed fluorescence diffuse optical tomography

Fluorescence tomography (FT) is an emerging non-invasive in vivo molecular imaging modality that aims at quantification and three-dimensional (3D) localization of fluorescent tagged inclusions, such as cancer lesions and drug molecules, buried deep in human and animal subjects. Depth-resolved 3D reconstruction of fluorescent inclusions distributed over the volume of optically turbid biological tissue using the diffuse fluorescent photons detected on the skin poses a highly ill-conditioned problem, as depth information must be extracted from boundary data. Due to this ill-posed nature of FT reconstructions, noise and errors in the data can severely impair the accuracy of the 3D reconstructions. Consequently, improvements in the signal-to-noise ratio (SNR) of the data significantly enhance the quality of the FT reconstructions. Furthermore, enhancing the SNR of the FT data can greatly contribute to the speed of FT scans. The pivotal factor in the SNR of the FT data is the power of the radiation illuminating the subject and exciting the administered fluorescent agents. In existing single-point illumination FT systems, the illumination power level is limited by the skin maximum radiation exposure levels. In this research, a multiplexed architecture governed by the Hadamard transform was conceptualized, developed, and experimentally implemented for orders-of-magnitude enhancement of the SNR and the robustness of FT reconstructions. The multiplexed FT system allows for Hadamard-coded multi-point illumination of the subject while maintaining the maximal information content of the FT data. The significant improvements offered by the multiplexed FT system were validated by numerical and experimental studies carried out using a custom-built multiplexed FT system developed exclusively in this work. The studies indicate that Hadamard multiplexing offers significantly enhanced robustness in reconstructing deep fluorescent inclusions from low-SNR FT data.Ph.D