Segmentation and Restoration of Images on Surfaces by Parametric Active Contours with Topology Changes

In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are mathematically described using a parametric approach. For image restoration, a diffusion equation with Neumann boundary conditions is solved in a postprocessing step in the individual regions. Numerical schemes are presented which allow to efficiently compute segmentations and denoised versions of images on surfaces. Also topology changes of the evolving curves are detected and performed using a fast sub-routine. Finally, several experiments are presented where the developed methods are applied on different artificial and real images defined on different surfaces

Geodesics in Heat

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required

Quantum Gravity via Causal Dynamical Triangulations

"Causal Dynamical Triangulations" (CDT) represent a lattice regularization of the sum over spacetime histories, providing us with a non-perturbative formulation of quantum gravity. The ultraviolet fixed points of the lattice theory can be used to define a continuum quantum field theory, potentially making contact with quantum gravity defined via asymptotic safety. We describe the formalism of CDT, its phase diagram, and the quantum geometries emerging from it. We also argue that the formalism should be able to describe a more general class of quantum-gravitational models of Horava-Lifshitz type.Comment: To appear in "Handbook of Spacetime", Springer Verlag. 31 page

Discrete approaches to quantum gravity in four dimensions

The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of space-time and the Einstein action. I review here three major areas of research: gauge-theoretic approaches, both in a path-integral and a Hamiltonian formulation, quantum Regge calculus, and the method of dynamical triangulations, confining attention to work that is strictly four-dimensional, strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the author welcomes any comments and suggestion

Dynamical Optimal Transport on Discrete Surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows

Lattice Quantum Gravity: Review and Recent Developments

We review the status of different approaches to lattice quantum gravity indicating the successes and problems of each. Recent developments within the dynamical triangulation formulation are then described. Plenary talk at LATTICE 95 July 11-15, Melbourne, Australia.Comment: 12 pages, 8 figure

Variational Methods for Biomolecular Modeling

Structure, function and dynamics of many biomolecular systems can be characterized by the energetic variational principle and the corresponding systems of partial differential equations (PDEs). This principle allows us to focus on the identification of essential energetic components, the optimal parametrization of energies, and the efficient computational implementation of energy variation or minimization. Given the fact that complex biomolecular systems are structurally non-uniform and their interactions occur through contact interfaces, their free energies are associated with various interfaces as well, such as solute-solvent interface, molecular binding interface, lipid domain interface, and membrane surfaces. This fact motivates the inclusion of interface geometry, particular its curvatures, to the parametrization of free energies. Applications of such interface geometry based energetic variational principles are illustrated through three concrete topics: the multiscale modeling of biomolecular electrostatics and solvation that includes the curvature energy of the molecular surface, the formation of microdomains on lipid membrane due to the geometric and molecular mechanics at the lipid interface, and the mean curvature driven protein localization on membrane surfaces. By further implicitly representing the interface using a phase field function over the entire domain, one can simulate the dynamics of the interface and the corresponding energy variation by evolving the phase field function, achieving significant reduction of the number of degrees of freedom and computational complexity. Strategies for improving the efficiency of computational implementations and for extending applications to coarse-graining or multiscale molecular simulations are outlined.Comment: 36 page

A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface

Continuum or hybrid modeling of bilayer membrane morphological dynamics induced by embedded proteins necessitates the identification of protein-membrane interfaces and coupling of deformations of two surfaces. In this article we developed (i) a minimal total geodesic curvature model to describe these interfaces, and (ii) a numerical one-one mapping between two surface through a conformal mapping of each surface to the common middle annulus. Our work provides the first computational tractable approach for determining the interfaces between bilayer and embedded proteins. The one-one mapping allows a convenient coupling of the morphology of two surfaces. We integrated these two new developments into the energetic model of protein-membrane interactions, and developed the full set of numerical methods for the coupled system. Numerical examples are presented to demonstrate (1) the efficiency and robustness of our methods in locating the curves with minimal total geodesic curvature on highly complicated protein surfaces, (2) the usefulness of these interfaces as interior boundaries for membrane deformation, and (3) the rich morphology of bilayer surfaces for different protein-membrane interfaces