3 research outputs found
Deep synthesis regularization of inverse problems
Recently, a large number of efficient deep learning methods for solving
inverse problems have been developed and show outstanding numerical
performance. For these deep learning methods, however, a solid theoretical
foundation in the form of reconstruction guarantees is missing. In contrast,
for classical reconstruction methods, such as convex variational and
frame-based regularization, theoretical convergence and convergence rate
results are well established. In this paper, we introduce deep synthesis
regularization (DESYRE) using neural networks as nonlinear synthesis operator
bridging the gap between these two worlds. The proposed method allows to
exploit the deep learning benefits of being well adjustable to available
training data and on the other hand comes with a solid mathematical foundation.
We present a complete convergence analysis with convergence rates for the
proposed deep synthesis regularization. We present a strategy for constructing
a synthesis network as part of an analysis-synthesis sequence together with an
appropriate training strategy. Numerical results show the plausibility of our
approach.Comment: Submitted to IEEE Trans. Image Processin
Sparse aNETT for Solving Inverse Problems with Deep Learning
We propose a sparse reconstruction framework (aNETT) for solving inverse
problems. Opposed to existing sparse reconstruction techniques that are based
on linear sparsifying transforms, we train an autoencoder network
with acting as a nonlinear sparsifying transform and minimize a Tikhonov
functional with learned regularizer formed by the -norm of the encoder
coefficients and a penalty for the distance to the data manifold. We propose a
strategy for training an autoencoder based on a sample set of the underlying
image class such that the autoencoder is independent of the forward operator
and is subsequently adapted to the specific forward model. Numerical results
are presented for sparse view CT, which clearly demonstrate the feasibility,
robustness and the improved generalization capability and stability of aNETT
over post-processing networks.Comment: The original proceeding is part of the ISBI 2020 and only contains 4
pages due to page restriction
Regularization of Inverse Problems by Neural Networks
Inverse problems arise in a variety of imaging applications including
computed tomography, non-destructive testing, and remote sensing. The
characteristic features of inverse problems are the non-uniqueness and
instability of their solutions. Therefore, any reasonable solution method
requires the use of regularization tools that select specific solutions and at
the same time stabilize the inversion process. Recently, data-driven methods
using deep learning techniques and neural networks demonstrated to
significantly outperform classical solution methods for inverse problems. In
this chapter, we give an overview of inverse problems and demonstrate the
necessity of regularization concepts for their solution. We show that neural
networks can be used for the data-driven solution of inverse problems and
review existing deep learning methods for inverse problems. In particular, we
view these deep learning methods from the perspective of regularization theory,
the mathematical foundation of stable solution methods for inverse problems.
This chapter is more than just a review as many of the presented theoretical
results extend existing ones