Inverse Scale Space Decomposition

With Marie Foged Schmidt, Denmark Technical University (DTU)

Inverse Scale Space Decomposition

In this talk we study the behaviour of the inverse scale space flow for computing an approximate solution to an inverse problem. The flow is a time-continuous version of Bregman iteration and has shown superior properties to standard regularization methods. The flow starts in the null space of the regularization functional and then incorporates finer and finer scales depending on the regularization. We want to study the inverse scale space flow for a specific structure of the measured data for linear inverse problems. To define the considered structure of the data we introduce what is called generalized singular vectors.

Generalized singular vectors arise from a generalization of the concept of singular vectors of linear operators to variational frameworks. The generalized singular vectors define a new concept of scale depending on the regularization functional. We show that the inverse scale space flow gives a decomposition into generalized singular vectors under certain conditions when the regularization functional is absolutely one-homogeneous and the data for the inverse problem is given as the forward operator applied to a linear combination of the singular vectors.

Finally, we address and discuss the question about when the first non-trivial solution of the inverse scale space flow is a generalized singular vector. At this point we define what we will call dual singular vectors which may actually be a better starting point for the structure of the data than generalized singular vectors.

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