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Guest Editorial: Variational Models, Convex Analysis and Numerical Optimization in Mathematical Imaging
J Math Imaging Vis
Guest Editorial: Variational Models, Convex Analysis and Numerical Optimization in Mathematical Imaging
Antonin Chambolle 0 1 2 3
Michael Hintermller 0 1 2 3
Thomas Pock 0 1 2 3
Christoph Schnrr 0 1 2 3
Gabriele Steidl 0 1 2 3
0 T. Pock Graz University of Technology , Graz , Austria
1 M. Hintermuller Humboldt-University of Berlin , Berlin , Germany
2 A. Chambolle CMAP, Ecole Polytechnique , Palaiseau , France
3 G. Steidl Technical University of Kaiserslautern , Kaiserlautern , Germany
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Models (doi:10.1007/s10851-013-0419-6) rigorously
investigate from a continuous variational perspective various
probabilistic models of local data likelihood, beyond the
basic Gaussian assumption. These models are relevant to a
range a practical imaging scenarios. An alternating
numerical scheme for model evaluation and model parameter
estimation is devised as well.
A connection between convex lower level feasible sets
constraining a convex objective function and the
regularization parameter of the corresponding penalty formulation
is worked out in the paper Homogeneous Penalizers and
Constraints in Convex Image Restoration (doi:10.1007/
s10851-012-0392-5). The problem class considered
encompasses a broad range of variational approaches to image
processing of current interest and also makes a connection to the
problem of tracing the regularization path in machine
learning.
In the paper Prox-Regularity of Rank Constraint Sets
and Implications for Algorithms
(doi:10.1007/s10851-0120406-3), the prox-regularity of lower level sets of the rank
function is established. This result enables the direct
application of basic iterative, locally converging algorithms to
rank constrained problems. It covers a significant portion of
recovery problems of current research, without the need to
resort to convex relaxations or heuristics.
A priori performance bounds for the continuous
formulation of the variational image labeling (partitioning)
problem are established in the paper Optimality Bounds for a
Variational Relaxation of the Image Partitioning Problem
(doi:10.1007/s10851-012-0390-7), for a broad class of
regularizers. In particular, the probabilistic approach to convert
optima of convex relaxations to high-quality combinatorial
solutions is considered. Corresponding image partitions
better preserve geometric image structure due to the underlying
continuous formulation.
Regularization of signals with periodic values, related to
orientation and angles, phase, etc., is studied in the paper
Total Cyclic Variation and Generalizations (doi:10.1007/
s10851-012-0396-1). Specifically, a natural total variation
measure on the 1D torus is devised that is shift invariant and
thus avoids commonly encountered wrap-around artefacts.
Convex techniques of relaxation and optimization enable the
efficient application of the approach to problem class.
Authors of the paper Convex relaxation of a class of
vertex penalizing functionals
(doi:10.1007/s10851-0120347-x) explore the functional lifting approach in
connection with curvature based regularization, a topic with links to
earlier work in many fields, ranging from visual perception
to PDE-based and variational models. In particular, the
paper provides a striking example for the sophisticated
reformulation of an apparently intractable problem class within a
convex variational relaxation framework.
The editors of this issue hope that readers of JMIV will
enjoy these contributions to mathematical imaging and
vision.
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