Title: Inverse Problems and Deep Learning for Mathematical Imaging
Date: 2022-01-24 11:57
Authors: Victorita Dolean
Job_Duration: 6 mois
Job_Employer: Université Côte d'Azur
Inverse problems appear naturally in different applications such as medical or geophysical imaging where one needs to reconstruct the unknown physical properties which can be human tissues or the subsurface. From the mathematical point of view, an inverse problem generally aims to identify model parameters from the knowledge of observed data and for this reason it appears in the form of a parameter estimation problem; it can hence be formulated as an optimization problem, and then solved using different optimization algorithms and techniques. In general, an inverse problem is ill- posed (the solution is not unique) and several local minima might be present. In this case reconstructing a unique solution that fits the observations is difficult or impossible without some prior knowledge about the data. Inverse problems can be made well-posed by adding a regularization term, using prior knowledge on desirable properties like smoothness.
Deep Learning (DL) is a subfield of Artificial Intelligence (AI) and type of machine learning technique aiming at building systems capable of operating in complex environments. DL relies on the use of deep neural networks (DNN) in order to approximate different quantities of interest. Supervised training of a DNN consists in the definition of a loss function that is minimized by means of gradient- based optimization techniques such as the stochastic gradient descent (SGD). DL techniques are often based on black-box approaches and they count on the availability of large amount of data.
DL and inverse problems are based on the same type of objects (definition of loss functions to be minimized) and techniques (optimization algorithms) and recent works have demonstrated how to integrate DL techniques and inverse problems where the available data is scarce. Expected results in this project are twofold: on one hand, DL algorithms can leverage large collections of training data to directly compute regularized image reconstructions and it has also been shown how they can be used to regularize the reconstructed image. On the other hand, DL algorithms can benefit from the vast inverse problem literature and well-known techniques in order to improve and better understand from practical and theoretical point of view the training process.