2,648 research outputs found
Computing Contour Generators of Evolving Implicit Surfaces
The contour generator is an important visibility feature of a smooth object seen under parallel projection. It is the curve on the surface which seperates front-facing from back-facing regions. The apparent contour is the projection of the contour generator onto a plane perpendicular to the view direction. Both curves play an important role in computer graphics. Our goal is to obtain fast and robust algorithms that compute the contour generator with a guarantee of topological correctness. To this end, we first study the singularities of the contour generator and apparent contour for both generic views and generic time-dependent projections, for example, when the surface is rotated or deformed. The singularities indicate when components of the contour generator merge or split as time evolves. We present an algorithm to compute an initial contour generator by using a dynamic step size. An interval test guarantees the topological correctness. This initial contour generator can thus be maintained under a time-dependent projection by examining its singularities
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
Second-order Shape Optimization for Geometric Inverse Problems in Vision
We develop a method for optimization in shape spaces, i.e., sets of surfaces
modulo re-parametrization. Unlike previously proposed gradient flows, we
achieve superlinear convergence rates through a subtle approximation of the
shape Hessian, which is generally hard to compute and suffers from a series of
degeneracies. Our analysis highlights the role of mean curvature motion in
comparison with first-order schemes: instead of surface area, our approach
penalizes deformation, either by its Dirichlet energy or total variation.
Latter regularizer sparks the development of an alternating direction method of
multipliers on triangular meshes. Therein, a conjugate-gradients solver enables
us to bypass formation of the Gaussian normal equations appearing in the course
of the overall optimization. We combine all of the aforementioned ideas in a
versatile geometric variation-regularized Levenberg-Marquardt-type method
applicable to a variety of shape functionals, depending on intrinsic properties
of the surface such as normal field and curvature as well as its embedding into
space. Promising experimental results are reported
Gauge-invariant approach to quark dynamics
The main aspects of a gauge-invariant approach to the description of quark
dynamics in the nonperturbative regime of QCD are first reviewed. In
particular, the role of the parallel transport operation in constructing
gauge-invariant Green's functions is presented, and the relevance of Wilson
loops for the representation of the interaction is emphasized. Recent
developments, based on the use of polygonal lines for the parallel transport
operation, are then presented. An integro-differential equation is obtained for
the quark Green's function defined with a phase factor along a single, straight
line segment. It is solved exactly and analytically in the case of
two-dimensional QCD in the large limit. The solution displays the
dynamical mass generation phenomenon for quarks, with an infinite number of
branch-cut singularities that are stronger than simple poles.Comment: 21 pages, 5 figures. Based on the talk given at the Workshop
Dyson-Schwinger equations in modern mathematics and physics, ECT*, Trento,
22-26 September 2014. Review article contribution to the special issue of
Frontiers of Physics (Eds. M. Pitschmann and C. D. Roberts
Numerical relativity with the conformal field equations
I discuss the conformal approach to the numerical simulation of radiating
isolated systems in general relativity. The method is based on conformal
compactification and a reformulation of the Einstein equations in terms of
rescaled variables, the so-called ``conformal field equations'' developed by
Friedrich. These equations allow to include ``infinity'' on a finite grid,
solving regular equations, whose solutions give rise to solutions of the
Einstein equations of (vacuum) general relativity. The conformal approach
promises certain advantages, in particular with respect to the treatment of
radiation extraction and boundary conditions. I will discuss the essential
features of the analytical approach to the problem, previous work on the
problem - in particular a code for simulations in 3+1 dimensions, some new
results, open problems and strategies for future work.Comment: 34 pages, submitted to the Proceedings of the 2001 Spanish Relativity
meeting, eds. L. Fernandez and L. Gonzalez, to be published by Springer,
Lecture Notes in Physics serie
Analytical Solutions of Open String Field Theory
In this work we review Schnabl's construction of the tachyon vacuum solution
to bosonic covariant open string field theory and the results that followed.
We survey the state of the art of string field theory research preceding this
construction focusing on Sen's conjectures and the results obtained using level
truncation methods.
The tachyon vacuum solution can be described in various ways. We describe its
geometric representation using wedge states, its formal algebraic
representation as a pure-gauge solution and its oscillator representation. We
also describe the analytical proofs of some of Sen's conjectures for this
solution.
The tools used in the context of the vacuum solution can be adapted to the
construction of other solutions, namely various marginal deformations. We
present some of the approaches used in the construction of these solutions.
The generalization of these ideas to open superstring field theory is
explained in detail. We start from the exposition of the problems one faces in
the construction of superstring field theory. We then present the cubic and the
non-polynomial versions of superstring field theory and discuss a proposal
suggesting their classical equivalence. Finally, the bosonic solutions are
generalized to this case. In particular, we focus on the (somewhat surprising)
generalization of the tachyon solution to the case of a theory with no
tachyons.Comment: Invited review for Physics Reports. v1: 106 p., 8 fig. v2: 108 p.,
minor changes. v3: 117 p., 9 fig., presentation modified and expanded in
several places, typos corrected, ref. added and updated. v4: Published
version. 125 p., 10 fig., further modifications of the presentation, ref.
added and update
- …