1,759 research outputs found
Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics
In this paper we present a spatially-adaptive method for image reconstruction
that is based on the concept of statistical multiresolution estimation as
introduced in [Frick K, Marnitz P, and Munk A. "Statistical multiresolution
Dantzig estimation in imaging: Fundamental concepts and algorithmic framework".
Electron. J. Stat., 6:231-268, 2012]. It constitutes a variational
regularization technique that uses an supremum-type distance measure as
data-fidelity combined with a convex cost functional. The resulting convex
optimization problem is approached by a combination of an inexact alternating
direction method of multipliers and Dykstra's projection algorithm. We describe
a novel method for balancing data-fit and regularity that is fully automatic
and allows for a sound statistical interpretation. The performance of our
estimation approach is studied for various problems in imaging. Among others,
this includes deconvolution problems that arise in Poisson nanoscale
fluorescence microscopy
Shape Constrained Regularisation by Statistical Multiresolution for Inverse Problems: Asymptotic Analysis
This paper is concerned with a novel regularisation technique for solving
linear ill-posed operator equations in Hilbert spaces from data that is
corrupted by white noise. We combine convex penalty functionals with
extreme-value statistics of projections of the residuals on a given set of
sub-spaces in the image-space of the operator. We prove general consistency and
convergence rate results in the framework of Bregman-divergences which allows
for a vast range of penalty functionals. Various examples that indicate the
applicability of our approach will be discussed. We will illustrate in the
context of signal and image processing that the presented method constitutes a
locally adaptive reconstruction method
Level Set Jet Schemes for Stiff Advection Equations: The SemiJet Method
Many interfacial phenomena in physical and biological systems are dominated
by high order geometric quantities such as curvature.
Here a semi-implicit method is combined with a level set jet scheme to handle
stiff nonlinear advection problems.
The new method offers an improvement over the semi-implicit gradient
augmented level set method previously introduced by requiring only one
smoothing step when updating the level set jet function while still preserving
the underlying methods higher accuracy. Sample results demonstrate that
accuracy is not sacrificed while strict time step restrictions can be avoided
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