1,109 research outputs found
Necessary conditions for variational regularization schemes
We study variational regularization methods in a general framework, more
precisely those methods that use a discrepancy and a regularization functional.
While several sets of sufficient conditions are known to obtain a
regularization method, we start with an investigation of the converse question:
How could necessary conditions for a variational method to provide a
regularization method look like? To this end, we formalize the notion of a
variational scheme and start with comparison of three different instances of
variational methods. Then we focus on the data space model and investigate the
role and interplay of the topological structure, the convergence notion and the
discrepancy functional. Especially, we deduce necessary conditions for the
discrepancy functional to fulfill usual continuity assumptions. The results are
applied to discrepancy functionals given by Bregman distances and especially to
the Kullback-Leibler divergence.Comment: To appear in Inverse Problem
Indirect Image Registration with Large Diffeomorphic Deformations
The paper adapts the large deformation diffeomorphic metric mapping framework
for image registration to the indirect setting where a template is registered
against a target that is given through indirect noisy observations. The
registration uses diffeomorphisms that transform the template through a (group)
action. These diffeomorphisms are generated by solving a flow equation that is
defined by a velocity field with certain regularity. The theoretical analysis
includes a proof that indirect image registration has solutions (existence)
that are stable and that converge as the data error tends so zero, so it
becomes a well-defined regularization method. The paper concludes with examples
of indirect image registration in 2D tomography with very sparse and/or highly
noisy data.Comment: 43 pages, 4 figures, 1 table; revise
Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing
For the solution of operator equations, Stevenson introduced a definition of
frames, where a Hilbert space and its dual are {\em not} identified. This means
that the Riesz isomorphism is not used as an identification, which, for
example, does not make sense for the Sobolev spaces and
. In this article, we are going to revisit the concept of
Stevenson frames and introduce it for Banach spaces. This is equivalent to
-Banach frames. It is known that, if such a system exists, by defining
a new inner product and using the Riesz isomorphism, the Banach space is
isomorphic to a Hilbert space. In this article, we deal with the contrasting
setting, where and are not identified, and
equivalent norms are distinguished, and show that in this setting the
investigation of -Banach frames make sense.Comment: 23 pages; accepted for publication in 'Numerical Functional Analysis
and Optimization
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