Abstract
Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operators and fixed regularizers and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to reflect various practical aspects and take into account discretization of the data space, the solution space, the forward operator and the neural network defining the regularizer. We show the asymptotic convergence of the discretized NETT approach for decreasing noise levels and discretization errors. Additionally, we derive convergence rates and present numerical results for a limited data problem in photoacoustic tomography.
Highlights
In this paper, we are interested in neural network based solutions to inverse problems of the formFind x from data yδ = Ax + η . (1)Here A is a potentially non-linear operator between Banach spaces X and Y, yδ are the given noisy data, x is the unknown to be recovered, η is the unknown noise perturbation and δ ≥ 0 indicates the noise level
We expect that the NETT functional will yield better results due to data consistency, which is mainly helpful outside the masked center diagonal
We performed numerical experiments using a limited data problem for Photoacoustic Tomography (PAT) that is the combination of an inverse problem for the wave equation and an inpainting problem
Summary
We are interested in neural network based solutions to inverse problems of the form. A is a potentially non-linear operator between Banach spaces X and Y, yδ are the given noisy data, x is the unknown to be recovered, η is the unknown noise perturbation and δ ≥ 0 indicates the noise level. Special challenges in solving inverse problems are the non-uniqueness of the solutions and the instability of the solutions with respect to the given data. To overcome these issues, regularization methods are needed, which select specific solutions and at the same time stabilize the inversion process
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