6,813 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
An L1 Penalty Method for General Obstacle Problems
We construct an efficient numerical scheme for solving obstacle problems in
divergence form. The numerical method is based on a reformulation of the
obstacle in terms of an L1-like penalty on the variational problem. The
reformulation is an exact regularizer in the sense that for large (but finite)
penalty parameter, we recover the exact solution. Our formulation is applied to
classical elliptic obstacle problems as well as some related free boundary
problems, for example the two-phase membrane problem and the Hele-Shaw model.
One advantage of the proposed method is that the free boundary inherent in the
obstacle problem arises naturally in our energy minimization without any need
for problem specific or complicated discretization. In addition, our scheme
also works for nonlinear variational inequalities arising from convex
minimization problems.Comment: 20 pages, 18 figure
On the Rate of Convergence for the Pseudospectral Optimal Control of Feedback Linearizable Systems
In this paper, we prove a theorem on the rate of convergence for the optimal
cost computed using PS methods. It is a first proved convergence rate in the
literature of PS optimal control. In addition to the high-order convergence
rate, two theorems are proved for the existence and convergence of the
approximate solutions. This paper contains several essential differences from
existing papers on PS optimal control as well as some other direct
computational methods. The proofs do not use necessary conditions of optimal
control. Furthermore, we do not make coercivity type of assumptions. As a
result, the theory does not require the local uniqueness of optimal solutions.
In addition, a restrictive assumption on the cluster points of discrete
solutions made in existing convergence theorems are removed.Comment: 28 pages, 3 figures, 1 tabl
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