Sergio Conti, Benedict Geihe, Martin Lenz and Martin Rumpf A posteriori modeling error estimates in the optimization of two-scale elastic composite materials ESAIM: Mathematical Modelling and Numerical Analysis, 52: 1457-1476 2018 10.1051/m2an/2017004
Alexander Effland, Martin Rumpf and Florian Schäfer Image extrapolation for the time discrete metamorphosis model -- existence and applications SIAM J. Imaging Sci. 2018 https://arxiv.org/abs/1705.04490
2017
Benjamin Berkels, Michael Buchner, Alexander Effland, Martin Rumpf and Steffen Schmitz-Valckenberg GPU Based Image Geodesics for Optical Coherence Tomography In Bildverarbeitung für die Medizin, Informatik aktuell
page 68--73.
Publisher: Springer
2017 http://dx.doi.org/10.1007/978-3-662-54345-0_21
Alexander Effland, Martin Rumpf and Florian Schäfer Time discrete extrapolation in a Riemannian space of images In Proc. of International Conference on Scale Space and Variational Methods in Computer Vision, Volume 10302
page 473--485.
Publisher: Springer, Cham
2017 https://dx.doi.org/10.1007/978-3-319-58771-4_38
Matthias Erbar, Martin Rumpf, Bernhard Schmitzer and Stefan Simon Computation of Optimal Transport on Discrete Metric Measure Spaces Unknown https://arxiv.org/abs/1707.06859
Behrend Heeren, Martin Rumpf and Benedikt Wirth Variational time discretization of Riemannian splines IMA J. Numer. Anal. 2017 https://arxiv.org/abs/1711.06069
Nora Lüthen, Martin Rumpf, Sascha Tölkes and Orestis Vantzos Branching Structures in Elastic Shape Optimization 2017 https://arxiv.org/abs/1711.03850
Abstract: Fine scale elastic structures are widespread in nature, for instances in plants or bones, whenever stiffness and low weight are required. These patterns frequently refine towards a Dirichlet boundary to ensure an effective load transfer. The paper discusses the optimization of such supporting structures in a specific class of domain patterns in 2D, which composes of periodic and branching period transitions on subdomain facets. These investigations can be considered as a case study to display examples of optimal branching domain patterns. In explicit, a rectangular domain is decomposed into rectangular subdomains, which share facets with neighbouring subdomains or with facets which split on one side into equally sized facets of two different subdomains. On each subdomain one considers an elastic material phase with stiff elasticity coefficients and an approximate void phase with orders of magnitude softer material. For given load on the outer domain boundary, which is distributed on a prescribed fine scale pattern representing the contact area of the shape, the interior elastic phase is optimized with respect to the compliance cost. The elastic stress is supposed to be continuous on the domain and a stress based finite volume discretization is used for the optimization. If in one direction equally sized subdomains with equal adjacent subdomain topology line up, these subdomains are consider as equal copies including the enforced boundary conditions for the stress and form a locally periodic substructure. An alternating descent algorithm is employed for a discrete characteristic function describing the stiff elastic subset on the subdomains and the solution of the elastic state equation. Numerical experiments are shown for compression and shear load on the boundary of a quadratic domain.
Jan Maas, Martin Rumpf and Stefan Simon Transport based image morphing with intensity modulation In Proc. of International Conference on Scale Space and Variational Methods in Computer Vision
Publisher: Springer, Cham
2017 http://dx.doi.org/10.1007/978-3-319-58771-4_45
2016
Sergio Conti, Martin Lenz and Martin Rumpf Hysteresis in Magnetic Shape Memory Composites: Modeling and Simulation 2016 10.1016/j.jmps.2015.12.010
Abstract: Magnetic shape memory alloys are characterized by the coupling between a structural phase transition and magnetic one. This permits to control the shape change via an external magnetic field, at least in single crystals. Composite materials with single-crystalline particles embedded in a softer matrix have been proposed as a way to overcome the blocking of the transformation at grain boundaries. We investigate hysteresis phenomena for small NiMnGa single crystals embedded in a polymer matrix for slowly varying magnetic fields. The evolution of the microstructure is studied within the rate-independent variational framework proposed by Mielke and Theil (1999). The underlying variational model incorporates linearized elasticity, micromagnetism, stray field and a dissipation term proportional to the volume swept by the phase boundary. The time discretization is based on an incremental minimization of the sum of energy and dissipation. A backtracking approach is employed to approximately ensure the global minimality condition. We illustrate and discuss the influence of the particle geometry (volume fraction, shape, arrangement) and the polymer elastic parameters on the observed hysteresis and compare with recent experimental results.
Patrick W. Dondl, Behrend Heeren and Martin Rumpf Optimization of the branching pattern in coherent phase transitions C. R. Math. Acad. Sci. Paris, 354(6): 639--644 2016 https://arxiv.org/abs/1512.06620
Abstract: Branching can be observed at the austenite-martensite interface of martensitic phase transformations. For a model problem, Kohn and Müller studied a branching pattern with optimal scaling of the energy with respect to its parameters. Here, we present finite element simulations that suggest a topologically different class of branching patterns and derive a novel, low dimensional family of patterns. After a geometric optimization within this family, the resulting pattern bears a striking resemblance to our simulation. The novel microstructure admits the same scaling exponents but results in a significantly lowered upper energy bound.
Peter Hornung, Martin Rumpf and Stefan Simon Material Optimization for Nonlinearly Elastic Planar Beams 2016 http://arxiv.org/abs/1604.02267
Jan Maas, Martin Rumpf and Stefan Simon Generalized optimal transport with singular sources 2016 http://arxiv.org/abs/1607.01186
2015
Benjamin Berkels, Alexander Effland and Martin Rumpf A Posteriori Error Control for the Binary Mumford-Shah Model ArXiv Preprint 2015 http://arxiv.org/abs/1505.05284
Abstract: The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non properly segmented region. To this end, a suitable strictly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin's functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the $L^2$-norm. In combination with a suitable cut out argument, a fully practical estimate for the area mismatch is derived. This estimate is incorporated in an adaptive meshing strategy. Two different adaptive primal-dual finite element schemes, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.
Benjamin Berkels, Alexander Effland and Martin Rumpf Time Discrete Geodesic Paths in the Space of Images SIAM J. Imaging Sci., 8(3): 1457-1488 2015 http://dx.doi.org/10.1137/140970719
Abstract: In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach, where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, Γ-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.
Sergio Conti, Janusz Ginster and Martin Rumpf A $BV$ Functional and its Relaxation for Joint Motion Estimation and Image Sequence Recovery ESAIM: Mathematical Modelling and Numerical Analysis, 49(5): 1463-1487 2015 http://dx.doi.org/10.1051/m2an/2015036
Abstract: The estimation of motion in an image sequence is a fundamental task in image processing. Frequently, the image sequence is corrupted by noise and one simultaneously asks for the underlying motion field and a restored sequence. In smoothly shaded regions of the restored image sequence the brightness constancy assumption along motion paths leads to a pointwise differential condition on the motion field. At object boundaries which are edge discontinuities both for the image intensity and for the motion field this condition is no longer well defined. In this paper a total-variation type functional is discussed for joint image restoration and motion estimation. This functional turns out not to be lower semicontinuous, and in particular fine-scale oscillations may appear around edges. By the general theory of vector valued $BV$ functionals its relaxation leads to the appearance of a singular part of the energy density, which can be determined by the solution of a local minimization problem at edges. Based on bounds for the singular part of the energy and under appropriate assumptions on the local intensity variation one can exclude the existence of microstructures and obtain a model well-suited for simultaneous image restoration and motion estimation. Indeed, the relaxed model incorporates a generalized variational formulation of the brightness constancy assumption. The analytical findings are related to ambiguity problems in motion estimation such as the proper distinction between foreground and background motion at object edges.
Alexander Effland, Martin Rumpf, Stefan Simon, Kirsten Stahn and Benedikt Wirth Bézier curves in the space of images In Proceedings Scale Space and Variational Methods in Computer Vision, Volume 9087 of Lecture Notes in Computer Science
page 372-384.
Publisher: Springer International
2015 http://dx.doi.org/10.1007/978-3-319-18461-6_30
Abstract: Bézier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bézier curves to the infinite-dimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model by Miller and Younes. Bézier curves are then computed via the Riemannian version of de Casteljau's algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bézier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bézier approach for the modulation of face interpolation and shape animation via image sketches is presented.
Benedict Geihe and Martin Rumpf A posteriori error estimates for sequential laminates in shape optimization In DCDS-S Special issue on Homogenization-Based Numerical Methods 2015 http://arxiv.org/abs/1501.07461
Abstract: A posteriori error estimates are derived in the context of two-dimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density.
Martin Rumpf and Benedikt Wirth Variational time discretization of geodesic calculus IMA J. Numer. Anal., 35(3): 1011-1046 2015 http://dx.doi.org/10.1093/imanum/dru027
Abstract: We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete logarithm, discrete exponential maps, and discrete parallel transport, and we prove convergence to their continuous counterparts. The presented analysis is based on the direct methods in the calculus of variation, on -convergence, and on weighted finite element error estimation. The convergence results of the discrete geodesic calculus are experimentally confirmed for a basic model on a two-dimensional Riemannian manifold. This provides a theoretical basis for the application to shape spaces in computer vision, for which we present one specific example.
2014
Martin Rumpf and Benedikt Wirth Discrete geodesic calculus in the space of viscous fluidic objects SIAM J. Imaging Sci., 6(4): 2581-2602 2014 http://www.arxiv.org/abs/1210.0822
Abstract: Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is developed in the context of shape spaces with shapes that are described via deformations of a given reference shape, and it is applied to a particular shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features.
2013
Benjamin Berkels, Tom Fletcher, Behrend Heeren, Martin Rumpf and Benedikt Wirth Discrete geodesic regression in shape space In Anders Heyden, Fredrik Kahl, Carl Olsson, Magnus Oskarsson, Xue-Cheng Tai, editor, Energy Minimization Methods in Computer Vision and Pattern Recognition, Volume 8081 of Lecture Notes in Computer Science
page 108-122.
Publisher: Springer International
2013 http://dx.doi.org/10.1007/978-3-642-40395-8_9
Abstract: A new approach for the effective computation of geodesic re- gression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity mea- sures between given two- or three-dimensional input shapes and corre- sponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is de- duced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.