300 research outputs found
Exact Geosedics and Shortest Paths on Polyhedral Surface
We present two algorithms for computing distances along a non-convex polyhedral surface. The first algorithm computes exact minimal-geodesic distances and the second algorithm combines these distances to compute exact shortest-path distances along the surface. Both algorithms have been extended to compute the exact minimalgeodesic paths and shortest paths. These algorithms have been implemented and validated on surfaces for which the correct solutions are known, in order to verify the accuracy and to measure the run-time performance, which is cubic or less for each algorithm. The exact-distance computations carried out by these algorithms are feasible for large-scale surfaces containing tens of thousands of vertices, and are a necessary component of near-isometric surface flattening methods that accurately transform curved manifolds into flat representations.National Institute for Biomedical Imaging and Bioengineering (R01 EB001550
Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework
This paper introduces a set of numerical methods for Riemannian shape
analysis of 3D surfaces within the setting of invariant (elastic) second-order
Sobolev metrics. More specifically, we address the computation of geodesics and
geodesic distances between parametrized or unparametrized immersed surfaces
represented as 3D meshes. Building on this, we develop tools for the
statistical shape analysis of sets of surfaces, including methods for
estimating Karcher means and performing tangent PCA on shape populations, and
for computing parallel transport along paths of surfaces. Our proposed approach
fundamentally relies on a relaxed variational formulation for the geodesic
matching problem via the use of varifold fidelity terms, which enable us to
enforce reparametrization independence when computing geodesics between
unparametrized surfaces, while also yielding versatile algorithms that allow us
to compare surfaces with varying sampling or mesh structures. Importantly, we
demonstrate how our relaxed variational framework can be extended to tackle
partially observed data. The different benefits of our numerical pipeline are
illustrated over various examples, synthetic and real.Comment: 25 pages, 16 figures, 1 tabl
Solid NURBS Conforming Scaffolding for Isogeometric Analysis
This work introduces a scaffolding framework to compactly parametrise solid structures with conforming NURBS elements for isogeometric analysis. A novel formulation introduces a topological, geometrical and parametric subdivision of the space in a minimal plurality of conforming vectorial elements. These determine a multi-compartmental scaffolding for arbitrary branching patterns. A solid smoothing paradigm is devised for the conforming scaffolding achieving higher than positional geometrical and parametric continuity. Results are shown for synthetic shapes of varying complexity, for modular CAD geometries, for branching structures from tessellated meshes and for organic biological structures from imaging data. Representative simulations demonstrate the validity of the introduced scaffolding framework with scalable performance and groundbreaking applications for isogeometric analysis
Fast exact and approximate geodesics on meshes
The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.Engineering and Applied Science
Adaptive Event Horizon Tracking and Critical Phenomena in Binary Black Hole Coalescence
This work establishes critical phenomena in the topological transition of
black hole coalescence. We describe and validate a computational front tracking
event horizon solver, developed for generic studies of the black hole
coalescence problem. We then apply this to the Kastor - Traschen axisymmetric
analytic solution of the extremal Maxwell - Einstein black hole merger with
cosmological constant. The surprising result of this computational analysis is
a power law scaling of the minimal throat proportional to time. The minimal
throat connecting the two holes obeys this power law during a short time
immediately at the beginning of merger. We also confirm the behavior
analytically. Thus, at least in one axisymmetric situation a critical
phenomenon exists. We give arguments for a broader universality class than the
restricted requirements of the Kastor - Traschen solution.Comment: 13 pages, 20 figures Corrected labels on figures 17 through 20.
Corrected typos in references. Added some comment
Gauge Invariant Framework for Shape Analysis of Surfaces
This paper describes a novel framework for computing geodesic paths in shape
spaces of spherical surfaces under an elastic Riemannian metric. The novelty
lies in defining this Riemannian metric directly on the quotient (shape) space,
rather than inheriting it from pre-shape space, and using it to formulate a
path energy that measures only the normal components of velocities along the
path. In other words, this paper defines and solves for geodesics directly on
the shape space and avoids complications resulting from the quotient operation.
This comprehensive framework is invariant to arbitrary parameterizations of
surfaces along paths, a phenomenon termed as gauge invariance. Additionally,
this paper makes a link between different elastic metrics used in the computer
science literature on one hand, and the mathematical literature on the other
hand, and provides a geometrical interpretation of the terms involved. Examples
using real and simulated 3D objects are provided to help illustrate the main
ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern
Analysis and Machine Intelligence in a better resolutio
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