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

A remarkable identity for lengths of curves

In this thesis we will prove the following new identity Σγ 1/(1 + exp |γ|) = 1/2, where the sum is over all closed simple geodesics γ on a punctured torus with a complete hyperbolic structure, and |γ| is the length of γ. Although it is well known that there are relations between the lengths of simple geodesics on a hyperbolic surface (for example the Fricke trace relations and the Selberg trace formula) this identity is of a wholly different character to anything in the literature. Our methods are purely geometric; that is, the techniques are based upon the work of Thurston and others on geodesic laminations rather than the analytic approach of Selberg. The first chapter is intended as an exposition of some relevant theory concerning laminations on a punctured surface. Most important of these results is that a leaf of a compact lamination cannot penetrate too deeply into a cusp region. Explicit bounds for the maximum depth are given; in the case of a torus a simple geodesic is disjoint from any cusp region whose bounding curve has length less than 4, and this bound is sharp. Another significant result is that a simple geodesic which enters a small cusp region is perpendicular to the horocyclic foliation of the cusp region. The second chapter is concerned with Gcusp, the set of ends of simple geodesics with at least one end up the cusp. A natural metric on Gcusp is introduced so that we can discuss approximation theory. We divide the geodesics of Gcusp into three classes according to the behaviour of their ends; each class also has a characterisation in terms of how well any member geodesic can be approximated. An example is given to demonstrate how this classification generalises some ideas in the classical theory of Diophantine approximation. The first class consists of geodesics with both ends up the cusp. Restricting to the punctured torus it is shown that for such a geodesic, γ, there is a portion of the cusp region surrounding each end which is disjoint from all other geodesics in Gcusp. We call such a portion a gap. The geometry of the gaps attached to γ is described and their area computed by elementary trigonometry. The area is a function of the length of the unique closed simple geodesic disjoint from γ. Next we consider the a generic geodesic in Gcusp, that is, a geodesic with a single end up the cusp and another end spiralling to a minimal compact lamination which is not a closed geodesic. We show that such a geodesic is the limit from both the right and left of other geodesics in Gcusp. Finally we give a technique for approximating a geodesic with a single end up the cusp and the other end spiralling to a closed geodesic. Essentially we repeatedly Dehn twist a suitable geodesic in Gcusp round this closed geodesic. The results of this chapter are then combined with a theorem of J. Birman and C. Series to yield the identity

The distribution of geodesic excursions into the neighborhood of a cone singularity on a hyperbolic 2-orbifold

A generic geodesic on a finite area, hyperbolic 2-orbifold exhibits an infinite sequence of penetrations into a neighborhood of a cone singularity, so that the sequence of depths of maximal penetration has a limiting distribution. The distribution function is the same for all such surfaces and is described by a fairly simple formula.Comment: 20 page

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

Analysis of Farthest Point Sampling for Approximating Geodesics in a Graph

A standard way to approximate the distance between any two vertices $p$ and $q$ on a mesh is to compute, in the associated graph, a shortest path from $p$ to $q$ that goes through one of $k$ sources, which are well-chosen vertices. Precomputing the distance between each of the $k$ sources to all vertices of the graph yields an efficient computation of approximate distances between any two vertices. One standard method for choosing $k$ sources, which has been used extensively and successfully for isometry-invariant surface processing, is the so-called Farthest Point Sampling (FPS), which starts with a random vertex as the first source, and iteratively selects the farthest vertex from the already selected sources. In this paper, we analyze the stretch factor $\mathcal{F}_{FPS}$ of approximate geodesics computed using FPS, which is the maximum, over all pairs of distinct vertices, of their approximated distance over their geodesic distance in the graph. We show that $\mathcal{F}_{FPS}$ can be bounded in terms of the minimal value $\mathcal{F}^*$ of the stretch factor obtained using an optimal placement of $k$ sources as $\mathcal{F}_{FPS}\leq 2 r_e^2 \mathcal{F}^*+ 2 r_e^2 + 8 r_e + 1$, where $r_e$ is the ratio of the lengths of the longest and the shortest edges of the graph. This provides some evidence explaining why farthest point sampling has been used successfully for isometry-invariant shape processing. Furthermore, we show that it is NP-complete to find $k$ sources that minimize the stretch factor.Comment: 13 pages, 4 figure