300 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
Anisotropic Total Variation Regularized L^1-Approximation and Denoising/Deblurring of 2D Bar Codes
We consider variations of the Rudin-Osher-Fatemi functional which are
particularly well-suited to denoising and deblurring of 2D bar codes. These
functionals consist of an anisotropic total variation favoring rectangles and a
fidelity term which measure the L^1 distance to the signal, both with and
without the presence of a deconvolution operator. Based upon the existence of a
certain associated vector field, we find necessary and sufficient conditions
for a function to be a minimizer. We apply these results to 2D bar codes to
find explicit regimes ---in terms of the fidelity parameter and smallest length
scale of the bar codes--- for which a perfect bar code is recoverable via
minimization of the functionals. Via a discretization reformulated as a linear
program, we perform numerical experiments for all functionals demonstrating
their denoising and deblurring capabilities.Comment: 34 pages, 6 figures (with a total of 30 subfigures); errors corrected
in Version 3, see Errata 1.1, 4.4, and 6.6 (v3 numbering) for more
informatio
Mumford-Shah and Potts Regularization for Manifold-Valued Data with Applications to DTI and Q-Ball Imaging
Mumford-Shah and Potts functionals are powerful variational models for
regularization which are widely used in signal and image processing; typical
applications are edge-preserving denoising and segmentation. Being both
non-smooth and non-convex, they are computationally challenging even for scalar
data. For manifold-valued data, the problem becomes even more involved since
typical features of vector spaces are not available. In this paper, we propose
algorithms for Mumford-Shah and for Potts regularization of manifold-valued
signals and images. For the univariate problems, we derive solvers based on
dynamic programming combined with (convex) optimization techniques for
manifold-valued data. For the class of Cartan-Hadamard manifolds (which
includes the data space in diffusion tensor imaging), we show that our
algorithms compute global minimizers for any starting point. For the
multivariate Mumford-Shah and Potts problems (for image regularization) we
propose a splitting into suitable subproblems which we can solve exactly using
the techniques developed for the corresponding univariate problems. Our method
does not require any a priori restrictions on the edge set and we do not have
to discretize the data space. We apply our method to diffusion tensor imaging
(DTI) as well as Q-ball imaging. Using the DTI model, we obtain a segmentation
of the corpus callosum
Small Volume Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional
We present the second of two articles on the small volume fraction limit of a
nonlocal Cahn-Hilliard functional introduced to model microphase separation of
diblock copolymers. After having established the results for the
sharp-interface version of the functional (arXiv:0907.2224), we consider here
the full diffuse-interface functional and address the limit in which epsilon
and the volume fraction tend to zero but the number of minority phases (called
particles) remains O(1). Using the language of Gamma-convergence, we focus on
two levels of this convergence, and derive first- and second-order effective
energies, whose energy landscapes are simpler and more transparent. These
limiting energies are only finite on weighted sums of delta functions,
corresponding to the concentration of mass into `point particles'. At the
highest level, the effective energy is entirely local and contains information
about the size of each particle but no information about their spatial
distribution. At the next level we encounter a Coulomb-like interaction between
the particles, which is responsible for the pattern formation. We present the
results in three dimensions and comment on their two-dimensional analogues
Variational methods and its applications to computer vision
Many computer vision applications such as image segmentation can be formulated in a ''variational'' way as energy minimization problems. Unfortunately, the computational task of minimizing these energies is usually difficult as it generally involves non convex functions in a space with thousands of dimensions and often the associated combinatorial problems are NP-hard to solve. Furthermore, they are ill-posed inverse problems and therefore are extremely sensitive to perturbations (e.g. noise). For this reason in order to compute a physically reliable approximation from given noisy data, it is necessary to incorporate into the mathematical model appropriate regularizations that require complex computations.
The main aim of this work is to describe variational segmentation methods that are particularly effective for curvilinear structures. Due to their complex geometry, classical regularization techniques cannot be adopted because they lead to the loss of most of low contrasted details. In contrast, the proposed method not only better preserves curvilinear structures, but also reconnects some parts that may have been disconnected by noise. Moreover, it can be easily extensible to graphs and successfully applied to different types of data such as medical imagery (i.e. vessels, hearth coronaries etc), material samples (i.e. concrete) and satellite signals (i.e. streets, rivers etc.). In particular, we will show results and performances about an implementation targeting new generation of High Performance Computing (HPC) architectures where different types of coprocessors cooperate. The involved dataset consists of approximately 200 images of cracks, captured in three different tunnels by a robotic machine designed for the European ROBO-SPECT project.Open Acces
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