We investigate a Tikhonov method that embeds a graph Laplacian operator in the penalty term (graphLa+). The novelty lies in building the graph Laplacian based on a first approximation of the solution derived by any other reconstruction method. Consequently, the penalty term becomes dynamic, depending on and adapting to the observed data and noise. We demonstrate that graphLa+ is a regularization method and we rigorously establish both its convergence and stability properties. Moreover, we present some selected numerical experiments in 2D computerized tomography, where we combine the graphLa+ method with several reconstructors: Filter Back Projection (graphLa+FBP), standard Tikhonov (graphLa+Tik), Total Variation (graphLa+TV) and a trained deep neural network (graphLa+Net). The quality increase of the approximated solutions granted by the graphLa+ approach is outstanding for each given method. In particular, graphLa+Net outperforms any other method, presenting a robust and stable implementation of deep neural networks for applications involving inverse problems.

In the past decades, the use of Computerized Tomography (CT) has increased dramatically owing to its excellent diagnostic performance, easy accessibility, short scanning time, and cost-effectiveness. Enabling CT technologies with a reduced/null radiation dose while preserving/enhancing the diagnostic quality is a key challenge in modern medical imaging. Increased noise levels are, however, an expected downfall of all these new technologies.

In this series of two successive talks we will refer about our research focused on Diffuse Optical Tomography (DOT), a CT technology based on NIR light as investigating signal [1]. Strong light scattering in biological tissues makes the DOT reconstruction problem severely ill-conditioned, so that denoising is a crucial step. In the present talk, after a brief description of the DOT modality, first we will present our results in exploring variational approaches based on partial differential equation models endowed with different regularizers to compute a stable DOT-CT reconstruction [2,3]. Then, we will discuss our recent research on the use of DL-based generative models to produce more effective soft priors which, used in combination with standard forward problem solvers or DL-based forward problem solvers, allow to improve spatial resolution in high contrast zones and reduce noise in low-contrast zones, typical of medical imaging.

[1] S.R. Arridge. Optical tomography in medical imaging, Inverse problems 15(2): R41, 1999.

[2] P. Causin, M.G. Lupieri, G. Naldi, R.M. Weishaeupl. Mathematical and numerical challenges in optical screening of female breast, Int. J. Num. Meth. Biomed. Eng. 36(2): e3286, 2020.

[3] A. Benfenati, P. Causin, M.G. Lupieri, G. Naldi. Regularization techniques for inverse problem in DOT applications. In Journal of Physics: Conference Series (IOP Publishing) 1476(1): 012007, 2020.

Diffuse Optical Tomography is a medical imaging technique for functional monitoring of body tissues. Unlike other CT technologies (i.e. X-ray CT), DOT employs a non-ionizing light signal and thus can be used for multiple screenings [1]. DOT reconstruction in CW modality leads to an inverse problem for the unknown distribution of the optical absorption coefficient inside the tissue, which has diagnostic relevance.

The classic approach consists in solving an optimization problem, involving a fit-to-data functional (usually, the Least Square functional) coupled with a regularization (e.g., $l^1$, Tikhonov, Elastic Net [2]). In this talk, we refer about our research in adopting a deep learning approach, which exploits both data-driven and hybrid-physics driven techniques. In the first case, we employ neural networks to construct a Learned Singular Value Decomposition [3], whilst in the second case the network architecture is built upon \emph{a priori} knowledge on the physical phenomena. We will present numerical results obtained from synthetic datasets which show robustness even on noisy data.

[1] S. R. Arridge, J. C. Schotland. Optical tomography: forward and inverse problems, Inverse problems 25(12): 123010, 2009.

[2] A. Benfenati, P. Causin, M. G. Lupieri, G. Naldi. Regularization techniques for inverse problem in DOT applications, Journal of Physics: Conference Series (IOP Publishing) 1476(1): 012007, 2020.

[3] A. Benfenati, G. Bisazza, P. Causin. A Learned SVD approach for Inverse Problem Regularization in Diffuse Optical Tomography, 2021. [arXiv preprint arXiv:2111.13401]

Deep learning algorithms have recently emerged as the state-of-the-art in solving Inverse Problems, overcoming classical variational methods in terms of both accuracy and efficiency. However, most deep learning algorithms require supervised training, which necessitates a collection of matched low-quality and ground-truth data. This poses a significant challenge in medical imaging, as obtaining such a dataset would require subjecting the patient to approximately double the amount of radiation. As a result, it is common to mathematically simulate the degradation process, which can introduce biases that degrade model performance when tested on real data. To address this issue, we propose a general self-supervised procedure for training neural networks in a setting where the ground truth is missing, but the mathematical model is approximately known. We demonstrate that our proposed method produces results of comparable quality to supervised techniques while being more robust to perturbations. We will provide formal proof of the effectiveness of our proposed method.