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I will talk about a new approach to linear ill-posed inverse problems with data-driven regularization. Instead of learning a stable inverse, unrolling standard algorithms into neural nets, or learning projectors for iterative schemes, we still compute the solution as a minimizer of a regularized cost functional, albeit non-convex. Our regularizer promotes "correct" conditional statistics in some feature space. As feature transform we choose the non-linear multiscale scattering transform---a complex convolutional network which discards the phase and thus exposes spectral correlations otherwise hidden beneath the phase fluctuations. We need scale separation in order to guarantee stability to deformations. For a given realization, the feature-space representation is linearly estimated from a reconstruction in a stable subspace and it represents the unstable part of the signal. We demonstrate that our approach stably recovers the missing spectrum in super-resolution and tomography.
This presentation is part of Minisymposium “MS42 - Low dimensional structures in imaging science (3 parts)”
organized by: Wenjing Liao (Georgia Institute of Technology) , Haizhao Yang (Duke University) , Zhizhen Zhao (University of Illinois Urbana-Champaign) .