Regularization Techniques for Inverse Problem in DOT Applications

Diffuse optical tomography (DOT) is an emerging diagnostic technique which uses near-infra-red light to investigate the optical coefficients distribution in biological tissues. The surface of the tissue is illuminated by light sources, then the outgoing light is measured by detectors placed at various locations on the surface itself. In order to reconstruct the optical coefficients, a mathematical model of light propagation is employed: such model leads to the minimization of the discrepancy between the detected data and the corresponding theoretical field. Due to severe ill-conditioning, regularization techniques are required: common procedures consider mainly \u2113 1-norm (LASSO) and \u2113 2-norm (Tikhonov) regularization. In the present work we investigate two original approaches in this context: The elastic-net regularization, previously used in machine learning problems, and the Bregman procedure. Numerical experiments are performed on synthetic 2D geometries and data, to evaluate the performance of these approaches. The results show that these techniques are indeed suitable choices for practical applications, where DOT is used as a cheap, first-level and almost real-Time screening technique for breast cancer detection

Convergence of distributed approximate subgradient method for minimizing convex function with convex functional constraints

In this paper, we investigate the distributed approximate subgradient-type method for minimizing a sum of differentiable and non-differentiable convex functions subject to nondifferentiable convex functional constraints in a Euclidean space. We establish the convergence of the sequence generated by our method to an optimal solution of the problem under consideration. Moreover, we derive a convergence rate of order $\mathcal{O}(N^{1-a})$ for the objective function values, where $a\in (0.5, 1)$. Finally, we provide a numerical example illustrating the effectiveness of the proposed method