Sigurd Angenent

Conformal surface parameterization for texture mapping

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.

Membrane Tension Maintains Cell Polarity by Confining Signals to the Leading Edge during Neutrophil Migration

Little is known about how neutrophils and other cells establish a single zone of actin assembly during migration. A widespread assumption is that the leading edge prevents formation of additional fronts by generating long-range diffusible inhibitors or by sequestering essential polarity components. We use morphological perturbations, cell-severing experiments, and computational simulations to show that diffusion-based mechanisms are not sufficient for long-range inhibition by the pseudopod. Instead, plasma membrane tension could serve as a long-range inhibitor in neutrophils. We find that membrane tension doubles during leading-edge protrusion, and increasing tension is sufficient for long-range inhibition of actin assembly and Rac activation. Furthermore, reducing membrane tension causes uniform actin assembly. We suggest that tension, rather than diffusible molecules generated or sequestered at the leading edge, is the dominant source of long-range inhibition that constrains the spread of the existing front and prevents the formation of secondary fronts.

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

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.

Optimal Mass Transport for Registration and Warping

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 unique; 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 Movers 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.

On the Laplace-Beltrami operator and brain surface flattening

In this paper, using certain conformal mappings from uniformization theory, the authors give an explicit method for flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, the authors indicate how the procedure may be implemented using finite elements. Further, they show how the geometry of the brain surface may be studied using this approach.

On the spontaneous emergence of cell polarity.

Diverse cell polarity networks require positive feedback for locally amplifying distributions of signalling molecules at the plasma membrane. Additional mechanisms, such as directed transport or coupled inhibitors, have been proposed to be required for reinforcing a unique axis of polarity. Here we analyse a simple model of positive feedback, with strong analogy to the ‘stepping stone’ model of population genetics, in which a single species of diffusible, membrane-bound signalling molecules can self-recruit from a cytoplasmic pool. We identify an intrinsic stochastic mechanism through which positive feedback alone is sufficient to account for the spontaneous establishment of a single site of polarity. We find that the polarization frequency has an inverse dependence on the number of signalling molecules: the frequency of polarization decreases as the number of molecules becomes large. Experimental observation of polarizing Cdc42 in budding yeast is consistent with this prediction. Our work suggests that positive feedback can work alone or with additional mechanisms to create robust cell polarity.

Nondistorting flattening maps and the 3-D visualization of colon CT images

Considers a novel three-dimensional (3-D) visualization technique based on surface flattening for virtual colonoscopy. Such visualization methods could be important in virtual colonoscopy because they have the potential for noninvasively determining the presence of polyps and other pathologies. Further, the authors demonstrate a method that presents a surface scan of the entire colon as a cine, and affords the viewer the opportunity to examine each point on the surface without distortion. The authors use certain angle-preserving mappings from differential geometry to derive an explicit method for flattening surfaces obtained from 3-D colon computed tomography (CT) imagery. Indeed, the authors describe a general method based on a discretization of the Laplace-Beltrami operator for flattening a surface into the plane in a conformal manner. From a triangulated surface representation of the colon, the authors indicate how the procedure may be implemented using a finite element technique, which takes into account special boundary conditions. They also provide simple formulas that may be used in a real-time cine to correct for distortion.

Nonlinear analytic semiflows

In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), and quasilinear initial value problems ( x ′( t )= A ( x ( t )) x ( t ) + f ( x ( t ))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.

Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions

We describe a theory for parabolic equations for immersed curves on surfaces, which generalizes the curve shortening or flow-by-mean-curvature problem, as well as several models in the theory of phase transitions in two dimensions. A class of equations is described for which the initial value problem is well-posed for rough initial data, for which one can give a description of the way a smooth solution becomes singular, and for which one can define generalized solutions, i.e., solutions which are smooth, except at a discrete set of times. The methods which are used in this paper are more geometrical than those of Part I. By comparing arbitrary solutions with certain special solutions, and by considering the way they intersect, we derive estimates for the curvature and the tangent, which allow one to study the initial value problem, and the way solutions become singular.

Finsler Active Contours

In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery.