Regularization by Architecture: A Deep Prior Approach for Inverse Problems

Abstract

The present paper studies the so called deep image prior (DIP) technique in the context of inverse problems. DIP networks have been introduced recently for applications in image processing, [50], also first experimental results for applying DIP to inverse problems have been reported [51]. This paper aims at discussing different interpretations of DIP and to obtain analytic results for specific network designs and linear operators. The main contribution is to introduce the idea of viewing these approaches as the optimization of Tiknonov functionals rather than optimizing networks. Besides theoretical results, we present numerical verifications for an academic example (integration operator) as well as for the inverse problem of magnetic particle imaging (MPI). The reconstructions obtained by deep prior networks are compared with state of the art methods.

Solving inverse problems using data-driven models

“…The images started to deteriorate slowly for more iterations. For implementation details, as well as further numerical examples also showing the limitation of the DIP approach, see Dittmer et al (2018).…”

“…We now briefly summarize the known theoretical foundations of DIP for inverse problems based on the recent paper by Dittmer, Kluth, Maass and Baguer (2018), who analyse and prove that certain network architectures in combination with suitable stopping rules do indeed lead to regularization schemes, which lead to the notion of ‘regularization by architecture’. We also include numerical results for the integration operator; more complex results for MPI are presented in Section 7.5.…”

“…The proximal mapping for the functional above is given by A rather lengthy calculation (see Dittmer, Kluth, Maass and Baguer 2018) yields an explicit formula for the derivative of with respect to in the iteration …”

Sub-Dip: Optimization On A Subspace With Deep Image Prior Regularization And Application To Superresolution

“…Several interpretations of its training objective in Eq. (3) have been discussed and theoretically analyzed in [6]. Of interest is the question if DIP can be further improved by adjusting the training objective.…”

“…As an alternative, we propose an approach based on an inverted perspective of Eq. (6). Specifically, we aim to search for a point in V that minimizes some distance to T , i.e., for an appropriate choice of some full rank matrix B ∈ R m×n , m ≤ n, we formulate the objective…”