Sigurd Angenent

Bio: Sigurd Angenent is an academic researcher from University of Wisconsin-Madison. The author has contributed to research in topics: Ricci flow & Mean curvature flow. The author has an hindex of 38, co-authored 86 publications receiving 6224 citations. Previous affiliations of Sigurd Angenent include Leiden University & Australian National University.

The zero set of a solution of a parabolic equation

Abstract: On etudie l'ensemble nul d'une solution u(t,x) de l'equation u t =a(x,t)u xx +b(x,t)u x +C(x,t)u, sous des hypotheses tres generales sur les coefficients a, b, et c

Optimal Mass Transport for Registration and Warping

Abstract: Image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times. In this paper we present a method for computing elastic registration and warping maps based on the Monge–Kantorovich theory of optimal mass transport. This mass transport method has a number of important characteristics. First, it is parameter free. Moreover, it utilizes all of the grayscale data in both images, places the two images on equal footing and is symmetrical: the optimal mapping from image A to image B being the inverse of the optimal mapping from B to A. The method does not require that landmarks be specified, and the minimizer of the distance functional involved is uniques there are no other local minimizers. Finally, optimal transport naturally takes into account changes in density that result from changes in area or volume. Although the optimal transport method is certainly not appropriate for all registration and warping problems, this mass preservation property makes the Monge–Kantorovich approach quite useful for an interesting class of warping problems, as we show in this paper. Our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the L2 Kantorovich–Wasserstein or “Earth Mover's Distance” under a mass preservation constraint. We show how this approach leads to practical algorithms, and demonstrate our method with a number of examples, including those from the medical field. We also extend this method to take into account changes in intensity, and show that it is well suited for applications such as image morphing.

Conformal surface parameterization for texture mapping

Abstract: We give an explicit method for mapping any simply connected surface onto the sphere in a manner which preserves angles. This technique relies on certain conformal mappings from differential geometry. Our method provides a new way to automatically assign texture coordinates to complex undulating surfaces. We demonstrate a finite element method that can be used to apply our mapping technique to a triangulated geometric description of a surface.

Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface

Abstract: Paper 1 [1988 g][1] of this series began an investigation whose goal is a thermomechanics of two-phase continua based on Gibbs’s notion of a sharp phase-interface endowed with thermomechanical structure. In that paper a balance law, balance of capillary forces, was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics; the chief results are thermodynamic restrictions on constitutive equations, exact and approximate free-boundary conditions at the interface, and a hierarchy of free-boundary problems. The simplest versions of these problems (the Mullins-Sekerka problems) are essentially the classical Stefan problem with the free-boundary condition u = 0 for the temperature replaced by the condition u = h K, where K is the curvature of the free-boundary and h > 0 is a material constant. This dependence on curvature renders the problem difficult, and apart from numerical studies involving linearization stability, there are almost no supporting theoretical results.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Geodesic active contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

Optimal Transport: Old and New

Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Geodesic Active Contours

Abstract: A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.