Algorithmes basés sur les moments adaptatifs pour résoudre le problème inverse en tomographie optique diffuse - Nada Chakhim (Faculté des Sciences et Techniques, Settat - Doctorat ou Doctrat National, 2022)
The inverse problem in Diffusive Optical Tomography (DOT) is nonlinear and severely ill-posed, therefore, only low resolution reconstructions are feasible when noise is added to the data nowadays. The purpose of this thesis is to improve image reconstruction in DOT of the main optical properties of tissues with some novel mathematical methods. We hâve used the Adaptive moment (Adam) optimizer, the Nesterov-Adaptive moment (Nadam) optimizer and its improved AmsGrad optimizer for single image reconstructions of the absorption coefficient. In the first instance, we will compare the behavior of three gradient descent-based optimizers on solving DOT inverse problem by running randomized simulation and analyzing the generated data in order to shed light on any significant difference-if existing at ail- in performance among these optimizers in our spécifie context of DOT. The major practical problems when selecting or using an optimization algorithm in a production context for a DOT System is to be confident that the algorithm will hâve a high convergence rate to the true solution, reasonably fast speed and high quality of the reconstructed image in terms of good localization of the inclusions and good agreement with the true image. In this work we hamessed a carefully designed randomized simulations to tackle the practical problem of choosing the right optimizer with the right parameters in the context of practical DOT applications, and derived statistical results conceming rate of convergence, speed and quality of image reconstruction. The statistical analysis performed on the generated data and the main results for convergence rate, reconstruction speed and quality between three optimization algorithms are presented. Then, we will explore a different way to construct optimizer algorithms for solving the inverse problem of Diffuse Optical Tomography by using diversification of two stochastic gradient-based algorithms, namely NADAM and AMSGrad. We will study the speed of convergence of the proposed new breed of algorithms, also we will discuss the quality of reconstructed images in both cases of free of noise and noisy measurement data. For analysis and exploration of the potential of the proposed algorithm, we use statistical simulations and analysis approach. Whereas most of the approaches for solving the nonlinear problem of DOT make use of the diffusion approximation (DA) to the radiative transfer équation (RTE) to model the propagation of the light in tissue. Therefore, we hâve solved the inverse problem in DOT by the more accurate continuons wave Diffusion équation in two dimensions
Keywords: Problème inverse; Reconstruction d'image; Algorithmes de moments adaptatifs; Optimisation; Tomographie optique diffuse; Équation de transfert radiatif; Mathématiques et informatique; Applications mathématiques