NETT: solving inverse problems with deep neural networks

Abstract

Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (Network Tikhonov) approach to inverse problems. NETT considers data consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case.(1.4)Additionally, we derive convergence rates (quantitative error estimates) between R(V, · )minimizing solutions x + and regularized solutions x α,δ . As a consequence, (1.3) provides a stable solution scheme for (1.1) using data consistency and encoding a-priori knowledge via neural networks. For proving norm convergence and convergence rates, we introduce the absolute Bregman distance as a new generalization of the standard Bregman distance for non-convex regularization.

Computational cannula microscopy of neurons using neural networks

“…The inverse problem that we are attempting to solve in CCM is an ill-conditioned linear system of equations that can be represented approximately as y = b*x + c, where y is the recorded sensor image, b is the system transfer function, x is the unknown object and c is the noise. ANNs have been shown to be good candidates to solve poorly conditioned inverse problems such as these previously [8]. Specifically, the universal approximation theorem guarantees that an ANN is able to closely approximate any continuous function similar to the one outlined above [9].…”

Learned backprojection for sparse and limited view photoacoustic tomography

“…Deep learning and in particular deep convolutional neural networks (CNNs) have been very successfully applied to a great variety of pattern recognition and image processing tasks. Recently a lot of research has been done in solving inverse problems, incorporating deep learning techniques, including efficient and accurate image reconstruction methods in tomographic problems [2,5,12,13,14,28,17].…”

A Partially-Learned Algorithm for Joint Photo-acoustic Reconstruction and Segmentation

“…Many other learned iterative methods are based on classical iterative methods, but execute far fewer steps in their reconstruction procedure: variational networks [25]- [27] can be seen as a learned variant of proximal gradient methods, where kernel-function pairs of the regularisation term are learned. Learning the regularisation term is also the idea behind [28] and [29]. In [30], a learned variant of gradient-descent is applied, while learned proximal operators are investigated in [31].…”

“…This makes them precarious for implementation in many applications, especially in clinical practice. A convergence proof for a method in which the regulariser was learned is given in [29]. Very recently, Banert et al [36] presented a convergence proof for learned algorithms that make use of proximal operators.…”