- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 10:55:50
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=IFB&vol=18&iss=4&update_since=2024-03-29
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
4
On the Gamma-limit of joint image segmentation and registration functionals based on phase fields
Benedikt
Wirth
Universität Münster, MÜNSTER, GERMANY
Image segmentation, image registration, phase fields, Mumford–Shah
A classical task in image processing is the following: Given two images, identify the structures inside (for instance detect all image edges or all homogeneous regions; this is called segmentation) and find a deformation which maps the structures in one image onto the corresponding ones in the other image (called registration). In medical imaging, for instance, one might segment the organs in two patient images and then identify corresponding organs in both images for automated comparison purposes. The image segmentation is classically performed variationally using the Mumford–Shah model, and the obtained structures are then mapped onto each other by minimizing a registration energy in which the deformation is regularized via elasticity. Experimentally it often seems more robust to perform segmentation and registration simultaneously so that both can benefit from each other. The question to be examined here is how phase field approximations of the Mumford–Shah model behave if used for the joint segmentation and registration problem.We mathematically analyze corresponding generic phase field models and reveal interesting phenomena that rule out some of the models. These phenomena are characteristic of coupling phase fields with deformations and thus are interesting in their own right. In essence, region-based segmentation and registration problems can well be approximated using phase fields, while edge-based approaches typically suffer from different types of vanishing or newly appearing edges. We conjecture how the introduction of a different scaling could remedy those shortcomings.
Calculus of variations and optimal control; optimization
441
477
10.4171/IFB/370
http://www.ems-ph.org/doi/10.4171/IFB/370
Convergence of a threshold-type algorithm using the signed distance function
Katsuyuki
Ishii
Kobe University, KOBE, JAPAN
Masato
Kimura
Kanazawa University, KANAZAWA, JAPAN
Threshold-type algorithm, curvature-dependent motions, signed distance function
We consider a threshold-type algorithm for curvature-dependent motions of hypersurfaces. This algorithm was numerically studied by [27], [9] and [35], where they used the signed distance function. It is also regarded as a variant of the Bence–Merriman–Osher algorithm for the mean curvature flow ( [4]). In this paper we prove the convergence of our algorithm under the nonfattening condition, applying the method of [30] which is based on the notion of the generalized flow due to [3]. Then we derive the rate of convergence of our algorithm to the smooth and compact curvature-dependent motions and show its optimality to the special case of a circle evolving by its curvature. We also give a local estimate on the convergence to a regular portion of the generalized curvature-dependent motion.
Partial differential equations
Numerical analysis
479
522
10.4171/IFB/371
http://www.ems-ph.org/doi/10.4171/IFB/371
A structure theorem for shape functions defined on submanifolds
Kevin
Sturm
Universität Duisburg-Essen, ESSEN, GERMANY
Shape optimisation, submanifolds, structure theorem
In this paper, we study shape functions depending on closed submanifolds. We prove a new structure theorem that establishes the general structure of the shape derivative for this type of shape function. As a special case we obtain the classical Hadamard–Zolésio structure theorem, but also the structure theorem for cracked sets can be recast into our framework. As an application we investigate several unconstrained shape functions arising from differential geometry and fracture mechanics.
Calculus of variations and optimal control; optimization
Operations research, mathematical programming
523
543
10.4171/IFB/372
http://www.ems-ph.org/doi/10.4171/IFB/372
Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination
Jean-François
Babadjian
Sorbonne Universités, UPMC Univ. Paris 6, PARIS CEDEX 05, FRANCE
Duvan
Henao
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Free discontinuity problems, functions of bounded deformation, $\Gamma$-convergence, fracture mechanics, thin films
In this work, we study the emergence of different crack modes in linearly elastic thin films by means of a $\Gamma$-convergence analysis as the thickness tends to zero. We first consider a purely elastic body made of a film deposited on an infinitely stiff substrate through a bonding layer. The displacement mismatch between the film and the substrate generates a cohesive type energy depending on the displacement jump. Then, we consider a single linearly elastic brittle thin film. We show that the limit admissible displacements are of Kirchhoff–Love type outside the cracks, which are themselves transverse. Finally, we study the interplay between transverse cracks and debonding. We come back to the first system made of a film, a bonding layer and a substrate, but now allow it to crack. In the simplified anti-plane setting, in addition to transverse cracks, a threshold criterion acting on the displacement activates either a cohesive or a delamination energy. Some partial results in the general vectorial case are discussed.
Calculus of variations and optimal control; optimization
Real functions
Partial differential equations
Mechanics of deformable solids
545
578
10.4171/IFB/373
http://www.ems-ph.org/doi/10.4171/IFB/373
A rigorous setting for the reinitialization of first order level set equations
Nao
Hamamuki
Hokkaido University, SAPPORO, JAPAN
Eleftherios
Ntovoris
Cité Descartes - Champs sur Marne, MARNE LA VALLÉE CEDEX 2, FRANCE
Viscosity solutions, level set equations, distance function, reinitialization, homogenization
In this paper we set up a rigorous justification for the reinitialization algorithm. Using the theory of viscosity solutions, we propose a well-posed Hamilton-Jacobi equation with a parameter, which is derived from homogenization for a Hamiltonian discontinuous in time which appears in the reinitialization. We prove that, as the parameter tends to infinity, the solution of the initial value problem converges to a signed distance function to the evolving interfaces. A locally uniform convergence is shown when the distance function is continuous, whereas a weaker notion of convergence is introduced to establish a convergence result to a possibly discontinuous distance function. In terms of the geometry of the interfaces, we give a necessary and sufficient condition for the continuity of the distance function.We also propose another simpler equation whose solution has a gradient bound away from zero.
Partial differential equations
579
621
10.4171/IFB/374
http://www.ems-ph.org/doi/10.4171/IFB/374