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.
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.
Alberto Enciso, Daniel Peralta-Salas and Stefan Steinerberger Prescribing the nodal set of the first eigenfunction in each conformal class 2015 http://arxiv.org/abs/1503.05105
Matthias Erbar, Jan Maas and Prasad Tetali Ricci curvature bounds for Bernoulli-Laplace and random transposition models Ann. Fac. Sci. Toulouse Math., ArXiv e-prints, 24(4): 781-800 2015 http://arxiv.org/abs/1409.8605
Abstract: We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the $n$-dimensional discrete cube (the so-called Bernoulli-Laplace model) and the random transposition shuffle of the symmetric group of permutations on $n$ letters.
Matthias Erbar, Kazumasa Kuwada and Karl-Theodor Sturm On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces Invent. Math., 201(3): 993-1071 2015 http://dx.doi.org/10.1007/s00222-014-0563-7
Matthias Erbar and Martin Huesmann Curvature bounds for configuration spaces Calculus of Variations and Partial Differential Equations, 54(1): 397-430 2015 http://dx.doi.org//10.1007/s00526-014-0790-1
Juan J. L. Velázquez and Miguel Escobedo On the theory of Weak Turbulence for the Nonlinear Schrödinger Equation Memoirs of the AMS, 238 2015 http://dx.doi.org/10.1090/memo/1124
Abstract: We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior.
2014
Miguel Escobedo and Juan J. L. Velázquez Finite time blow-up and condensation for the bosonic Nordheim equation Inventiones mathematicae, 200(3): 761-847 2014 http://dx.doi.org/10.1007/s00222-014-0539-7
Abstract: The homogeneous bosonic Nordheim equation is a kinetic equation describing the dynamics of the distribution of particles in the space of moments for a homogeneous, weakly interacting, quantum gas of bosons. We show the existence of classical solutions of the homogeneous bosonic Nordheim equation that blow up in finite time. We also prove finite time condensation for a class of weak solutions of the kinetic equation.
Miguel Escobedo and Juan J. L. Velázquez On the Blow Up and Condensation of Supercritical Solutions of the Nordheim Equation for Bosons Communications in Mathematical Physics, 330(1): 331-365 2014 http://dx.doi.org/10.1007/s00220-014-2034-9
Abstract: In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons with bounded initial data blow up in finite time in the L ∞ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contain a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.