981 research outputs found
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
The Residual Method for Regularizing Ill-Posed Problems
Although the \emph{residual method}, or \emph{constrained regularization}, is
frequently used in applications, a detailed study of its properties is still
missing. This sharply contrasts the progress of the theory of Tikhonov
regularization, where a series of new results for regularization in Banach
spaces has been published in the recent years. The present paper intends to
bridge the gap between the existing theories as far as possible. We develop a
stability and convergence theory for the residual method in general topological
spaces. In addition, we prove convergence rates in terms of (generalized)
Bregman distances, which can also be applied to non-convex regularization
functionals. We provide three examples that show the applicability of our
theory. The first example is the regularized solution of linear operator
equations on -spaces, where we show that the results of Tikhonov
regularization generalize unchanged to the residual method. As a second
example, we consider the problem of density estimation from a finite number of
sampling points, using the Wasserstein distance as a fidelity term and an
entropy measure as regularization term. It is shown that the densities obtained
in this way depend continuously on the location of the sampled points and that
the underlying density can be recovered as the number of sampling points tends
to infinity. Finally, we apply our theory to compressed sensing. Here, we show
the well-posedness of the method and derive convergence rates both for convex
and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur
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