Ultrasound-modulated optical tomography is modeled by a linear integral equation and an inverse problem involving a diffusion equation in n spatial dimensions, n=2, 3. Based on measured data, the optical absorption coefficient μ is reconstructed inside of a given domain. We make a two-step mathematical model. First, we solve a linear integral equation. Assuming the energy fluence rate has been recovered from the previous equation, the absorption coefficient μ is then reconstructed by solving an inverse problem. Numerical experiments are presented for the case n=2. Two methods are used for the numerical experiments, gradient descent and levelset. At the end, advantages and disadvantages of those two methods are mentioned.

M/EEG mechanisms allow determining changes in the brain activity, which is useful in diagnosing brain disorders such as epilepsy. They consist of measuring the electric potential at the scalp and the magnetic field around the head. The measurements are related to the underlying brain activity by a linear model that depends on the lead-field matrix. Localizing the sources, or dipoles, of M/EEG measurements consists of inverting this linear model. However, the non-uniqueness of the solution (due to the fundamental law of physics) and the low number of dipoles make the inverse problem ill-posed. Solving such problem requires some sort of regularization to reduce the search space. The literature abounds of methods and techniques to solve this problem, especially with variational approaches. This thesis develops Bayesian methods to solve ill-posed inverse problems, with application to M/EEG. The main idea underlying this work is to constrain sources to be sparse. This hypothesis is valid in many applications such as certain types of epilepsy. We develop different hierarchical models to account for the sparsity of the sources. Theoretically, enforcing sparsity is equivalent to minimizing a cost function penalized by an l0 pseudo norm of the solution. However, since the l0 regularization leads to NP-hard problems, the l1 approximation is usually preferred. Our first contribution consists of combining the two norms in a Bayesian framework, using a Bernoulli-Laplace prior. A Markov chain Monte Carlo (MCMC) algorithm is used to estimate the parameters of the model jointly with the source location and intensity. Comparing the results, in several scenarios, with those obtained with sLoreta and the weighted l1 norm regularization shows interesting performance, at the price of a higher computational complexity. Our Bernoulli-Laplace model solves the source localization problem at one instant of time. However, it is biophysically well-known that the brain activity follows spatiotemporal patterns. Exploiting the temporal dimension is therefore interesting to further constrain the problem. Our second contribution consists of formulating a structured sparsity model to exploit this biophysical phenomenon. Precisely, a multivariate Bernoulli-Laplacian distribution is proposed as an a priori distribution for the dipole locations. A latent variable is introduced to handle the resulting complex posterior and an original Metropolis-Hastings sampling algorithm is developed. The results show that the proposed sampling technique improves significantly the convergence. A comparative analysis of the results is performed between the proposed model, an l21 mixed norm regularization and the Multiple Sparse Priors (MSP) algorithm. Various experiments are conducted with synthetic and real data. Results show that our model has several advantages including a better recovery of the dipole locations. The previous two algorithms consider a fully known leadfield matrix. However, this is seldom the case in practical applications. Instead, this matrix is the result of approximation methods that lead to significant uncertainties. Our third contribution consists of handling the uncertainty of the lead-field matrix. The proposed method consists in expressing this matrix as a function of the skull conductivity using a polynomial matrix interpolation technique. The conductivity is considered as the main source of uncertainty of the lead-field matrix. Our multivariate Bernoulli-Laplacian model is then extended to estimate the skull conductivity jointly with the brain activity. The resulting model is compared to other methods including the techniques of Vallaghé et al and Guttierez et al. Our method provides results of better quality without requiring knowledge of the active dipole positions and is not limited to a single dipole activation.

Partial Differential Equations (PDEs) are commonly used to model complex systems that arise for example in biology, engineering, chemistry, and elsewhere. The parameters (or coefficients) and the source of PDE models are often unknown and are estimated from available measurements. Despite its importance, solving the estimation problem is mathematically and numerically challenging and especially when the measurements are corrupted by noise, which is often the case. Various methods have been proposed to solve estimation problems in PDEs which can be classified into optimization methods and recursive methods. The optimization methods are usually heavy computationally, especially when the number of unknowns is large. In addition, they are sensitive to the initial guess and stop condition, and they suffer from the lack of robustness to noise. Recursive methods, such as observer-based approaches, are limited by their dependence on some structural properties such as observability and identifiability which might be lost when approximating the PDE numerically. Moreover, most of these methods provide asymptotic estimates which might not be useful for control applications for example. An alternative non-asymptotic approach with less computational burden has been proposed in engineering fields based on the so-called modulating functions. In this dissertation, we propose to mathematically and numerically analyze the modulating functions based approaches. We also propose to extend these approaches to different situations. The contributions of this thesis are as follows. (i) Provide a mathematical analysis of the modulating function-based method (MFBM) which includes: its well-posedness, statistical properties, and estimation errors. (ii) Provide a numerical analysis of the MFBM through some estimation problems, and study the sensitivity of the method to the modulating functions' parameters. (iii) Propose an effective algorithm for selecting the method's design parameters. (iv) Develop a two-dimensional MFBM to estimate space-time dependent unknowns which is illustrated in estimating the source term in the damped wave equation describing the physiological characterization of brain activity. (v) Introduce a moving horizon strategy in the MFBM for on-line estimation and examine its effectiveness on estimating the source term of a first order hyperbolic equation which describes the heat transfer in distributed solar collector systems.

There has been great interest in recent times in the use of elastography for the characterization of human tissue. Digital Image Elasto-Tomography is a novel breast cancer pre-screening technique under development at the University of Canterbury, which aims to identify and locate stiff areas within the breast that require further investigation using images of the surface motion alone. A calibrated array of five digital cameras is used to capture surface motion of the breast under harmonic actuation. The forward problem, that is the resulting motion for a given mechanical property distribution, is calculated using the Finite Element Method. The inverse problem is to find the mechanical properties which reproduce the measured surface motion through numerical simulation. A reconstruction algorithm is developed using a shape based description to reduce the number of parameters in the inverse problem. A parallel Genetic Algorithm is developed for parameter optimization. A geometric method termed Fitness Function Analysis is shown to improve the inclusion location optimization problem. The ensemble of solutions generated using the Genetic Algorithm is used to produce an optimal and a credible region for inclusion location. Successful single frequency phantom reconstructions are presented. An effective way of combining information from multi-frequency phantom data by examining the characteristics of the measured surface motion using data quality metrics is developed and used to produce improved reconstructions. Results from numerical simulation datasets and a two inclusion phantom used to test the optimization of multiple and ellipsoidal inclusions indicate that although two inclusions can be successfully reconstructed, the single inclusions assumption may suffice even in irregular, heterogeneous cases. This assumption was used to successfully locate the stiffest inclusion in a phantom containing multiple inclusions of differing stiffness based on three multi-frequency datasets. The methods developed in phantoms are applied to three in vivo cases for both single and multi-frequency data with limited success. This thesis builds on previous work undertaken at the University of Canterbury. The original contributions in this work are as follows. A new reconstruction algorithm combining a genetic algorithm with fitness function analysis is developed. The most realistic tissue mimicking phantoms to date are used. An ellipsoidal shape-based description is presented, and applied to the first multi-inclusion reconstructions in DIET. This work presents the first reconstruction using meshes created directly from data using a meshing algorithm developed by Jonas Biehler. A multi-frequency cost function is developed to produce the first multi-frequency and in vivo reconstructions using DIET data.

Cardiac disease is one of the leading causes of death in the world, with an increase in cardiac arrhythmias in recent years. In addition, myocardial ischemia, which arises from the lack of blood in the cardiac tissue, can lead to cardiac arrhythmias and even sudden cardiac death. Cardiac arrhythmias, such as atrial fibrillation, are characterised by abnormal wave excitation and repolarization patterns in the myocardial tissue. These abnormal patterns are usually diagnosed through non-invasive electrical measurements on the surface of the body, i.e., the electrocardiogram (ECG). However, the most common lead configuration of the ECG, the 12-lead ECG, has its limitations in providing sufficient information to identify and locate the origin of cardiac arrhythmias. Therefore, there is an increasing need to develop novel methods to diagnose and find the origin of arrhythmic excitation, which will increase the efficacy of the treatment and diagnosis of cardiac arrhythmias. The objective of this research was to develop a family of multi-scale computational models of the human heart and thorax to simulate and investigate the effect of arrhythmic electrical activity in the heart on the electric and magnetic activities on the surface of the body. Based on these simulations, new theoretical algorithms were developed to non-invasively diagnose the origins of cardiac arrhythmias, such as the location of ectopic activities in the atria or ischemic regions within the ventricles, which are challenging to the clinician. These non-invasive diagnose methods were based on the implementation of multi-lead ECG systems, magnetocardiograms (MCGs) and electrocardiographic imaging.

Blood oxygen level dependent (BOLD) functional magnetic resonance imaging is a non-invasive technique used to trace changes in neural dynamics in reaction to mental activity caused by perceptual, motor or cognitive tasks. The BOLD response is a complex signal, a consequence of a series of physiological events regulated by increased neural activity. A method to infer from the BOLD signal onto underlying neuronal activity (hemodynamic inverse problem) is proposed in Chapter 2 under the assumption of a previously proposed mathematical model on the transduction of neural activity to the BOLD signal. Also, in this chapter we clarify the meaning of the neural activity function used as the input for an intrinsic dynamic system which can be viewed as an advanced substitute for the impulse response function. Chapter 3 describes an approach for recovering neural timing information (mental chronometry) in an object interaction decision task via solving the hemodynamic inverse problem. In contrast to the hemodynamic level, at the neural level, we were able to determine statistically significant latencies in activation between functional units in the model used. In Chapter 4, two approaches for regularization parameter tuning in a regularized-regression analysis are compared in an attempt to find the optimal amount of smoothing to be imposed on fMRI data in determining an empirical hemodynamic response function. We found that the noise autocorrelation structure can be improved by tuning the regularization parameter but the whitening-based criterion provides too much smoothing when compared to cross-validation. Chapter~5 illustrates that the smoothing techniques proposed in Chapter 4 can be useful in the issue of correlating behavioral and hemodynamic characteristics. Specifically, Chapter 5, based on the smoothing techniques from Chapter 4, seeks to correlate several parameters characterizing the hemodynamic response in Broca's area to behavioral measures in a naming task. In particular, a condition for independence between two routes of converting print to speech in a dual route cognitive model was verified in terms of hemodynamic parameters.

Blood oxygen level dependent (BOLD) functional magnetic resonance imaging is a non-invasive technique used to trace changes in neural dynamics in reaction to mental activity caused by perceptual, motor or cognitive tasks. The BOLD response is a complex signal, a consequence of a series of physiological events regulated by increased neural activity. A method to infer from the BOLD signal onto underlying neuronal activity (hemodynamic inverse problem) is proposed in Chapter 2 under the assumption of a previously proposed mathematical model on the transduction of neural activity to the BOLD signal. Also, in this chapter we clarify the meaning of the neural activity function used as the input for an intrinsic dynamic system which can be viewed as an advanced substitute for the impulse response function. Chapter 3 describes an approach for recovering neural timing information (mental chronometry) in an object interaction decision task via solving the hemodynamic inverse problem. In contrast to the hemodynamic level, at the neural level, we were able to determine statistically significant latencies in activation between functional units in the model used. In Chapter 4, two approaches for regularization parameter tuning in a regularized-regression analysis are compared in an attempt to find the optimal amount of smoothing to be imposed on fMRI data in determining an empirical hemodynamic response function. We found that the noise autocorrelation structure can be improved by tuning the regularization parameter but the whitening-based criterion provides too much smoothing when compared to cross-validation. Chapter~5 illustrates that the smoothing techniques proposed in Chapter 4 can be useful in the issue of correlating behavioral and hemodynamic characteristics. Specifically, Chapter 5, based on the smoothing techniques from Chapter 4, seeks to correlate several parameters characterizing the hemodynamic response in Broca's area to behavioral measures in a naming task. In particular, a condition for independence between two routes of converting print to speech in a dual route cognitive model was verified in terms of hemodynamic parameters.

Dans le problème inverse d’électrocardiographie, le cible est faire la reconstruction de l’activité électrophysiologique dans le cœur sans mesurer directement dans sa surface (sans interventions avec cathéter). Il est important remarque que en l’actualité la solution numérique du problème inverse est résolu avec le modèle quasi-statique. Ce modèle ne considère pas la dynamique du cœur et peut produire des erreurs dans la reconstruction de la solution sur la surface du cœur. Dans cette thèse, différents méthodologies était investigue pour résoudre le problème inverse d’électrocardiographie comme intelligence artificielle, et modèles dynamiques limites. Aussi, les effets de différents opérateurs en utilisant méthodes d’éléments de frontière , et méthodes d’élément finis était investigue. / In the inverse problem of electrocardiography, the target is to make the reconstruction of electrophysiological activity in the heart without measuring directly in its surface (without interventions with catheter). It is important to note that the current numerical solution of the inverse problem is solved with the quasi-static model. This model does not consider the dynamics of the heart and can cause errors in the reconstruction of the solution on the surface of the heart. This thesis investigates different methodologies was to solve the inverse problem of electrocardiography as artificial intelligence and dynamic models limits. Also, the effects of different operators using boundary element methods, finite element methods, and was investigates.

Observers are well known in the theory of dynamical systems. They are used to estimate the states of a system from some measurements. However, recently observers have also been developed to estimate some unknowns for systems governed by Partial differential equations. Our aim is to design an observer to solve inverse source problem for a one dimensional wave equation. Firstly, the problem is discretized in both space and time and then an adaptive observer based on partial field measurements (i.e measurements taken form the solution of the wave equation) is applied to estimate both the states and the source. We see the effectiveness of this observer in both noise-free and noisy cases. In each case, numerical simulations are provided to illustrate the effectiveness of this approach. Finally, we compare the performance of the observer approach with Tikhonov regularization approach.

In this thesis, we introduce a novel method of reconstructing fibre directions from diffusion images. By modelling the Principal Diffusion Direction PDD (the fibre direction) directly, we are able to apply regularization to the fibre direction explicitly, which was not possible before. Diffusion Tensor Imaging (DTI) is a technique which extracts information from multiple Magnetic Resonance Images about the amount and orientation of diffusion within the body. It is commonly used for brain connectivity studies, providing information about the white matter structure. Many methods have been represented in the literature for estimating diffusion tensors with and without regularization. Previous methods of regularization applied to the source images or diffusion tensors. The process of extracting PDDs therefore required two or three numerical procedures, in which regularization (including filtering) is applied in earlier steps before the PDD is extracted. Such methods require and/or impose smoothness on all components of the signal, which is inherently less efficient than using regularizing terms that penalize non-smoothness in the principal diffusion direction directly. Our model can be interpreted as a restriction of the diffusion tensor model, in which the principal eigenvalue of the diffusion tensor is a model variable and not a derived quantity. We test the model using a numerical phantom designed to test many fibre orientations in parallel, and process a set of thigh muscle diffusion-weighted images. / Thesis / Master of Science (MSc)