354 research outputs found
Point triangulation through polyhedron collapse using the l∞ norm
Multi-camera triangulation of feature points based on a minimisation of the overall l(2) reprojection error can get stuck in suboptimal local minima or require slow global optimisation. For this reason, researchers have proposed optimising the l(infinity) norm of the l(2) single view reprojection errors, which avoids the problem of local minima entirely. In this paper we present a novel method for l(infinity) triangulation that minimizes the l(infinity) norm of the l(infinity) reprojection errors: this apparently small difference leads to a much faster but equally accurate solution which is related to the MLE under the assumption of uniform noise. The proposed method adopts a new optimisation strategy based on solving simple quadratic equations. This stands in contrast with the fastest existing methods, which solve a sequence of more complex auxiliary Linear Programming or Second Order Cone Problems. The proposed algorithm performs well: for triangulation, it achieves the same accuracy as existing techniques while executing faster and being straightforward to implement
3-manifolds efficiently bound 4-manifolds
It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for
instance, every 3-manifold has a surgery diagram. There are several proofs of
this fact, including constructive proofs, but there has been little attention
to the complexity of the 4-manifold produced. Given a 3-manifold M of
complexity n, we show how to construct a 4-manifold bounded by M of complexity
O(n^2). Here we measure ``complexity'' of a piecewise-linear manifold by the
minimum number of n-simplices in a triangulation. It is an open question
whether this quadratic bound can be replaced by a linear bound.
The proof goes through the notion of "shadow complexity" of a 3-manifold M. A
shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M.
We prove that, for a manifold M satisfying the Geometrization Conjecture with
Gromov norm G and shadow complexity S, c_1 G <= S <= c_2 G^2 for suitable
constants c_1, c_2. In particular, the manifolds with shadow complexity 0 are
the graph manifolds.Comment: 39 pages, 21 figures; added proof for spin case as wel
Lower bounds on volumes of hyperbolic Haken 3-manifolds
In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds
with various topological conditions. Let V_3 = 1.01494 denote the volume of a
regular ideal simplex in hyperbolic 3-space. As a special case of the main
theorem, if a hyperbolic manifold M contains an acylindrical surface S, then
Vol(M)>= -2 V_3 chi(S). We also show that if beta_1(M)>= 2, then Vol(M)>= 4/5
V_3.Comment: 20 pages, 15 figure
Polyhedral realisation of hyperbolic metrics with conical singularities on compact surfaces
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant
under the action of a Fuchsian group of isometries (i.e. a group of isometries
leaving globally invariant a totally geodesic surface, on which it acts
cocompactly). The induced metric on a convex Fuchsian polyhedron is isometric
to a hyperbolic metric with conical singularities of positive singular
curvature on a compact surface of genus greater than one. We prove that these
metrics are actually realised by exactly one convex Fuchsian polyhedron (up to
global isometries). This extends a famous theorem of A.D. Alexandrov.Comment: Some little corrections from the preceding version. To appear in Les
Annales de l'Institut Fourie
On a new conformal functional for simplicial surfaces
We introduce a smooth quadratic conformal functional and its weighted version
where
is the extrinsic intersection angle of the circumcircles of the
triangles of the mesh sharing the edge and is the valence of
vertex . Besides minimizing the squared local conformal discrete Willmore
energy this functional also minimizes local differences of the angles
. We investigate the minimizers of this functionals for simplicial
spheres and simplicial surfaces of nontrivial topology. Several remarkable
facts are observed. In particular for most of randomly generated simplicial
polyhedra the minimizers of and are inscribed polyhedra. We
demonstrate also some applications in geometry processing, for example, a
conformal deformation of surfaces to the round sphere. A partial theoretical
explanation through quadratic optimization theory of some observed phenomena is
presented.Comment: 14 pages, 8 figures, to appear in the proceedings of "Curves and
Surfaces, 8th International Conference", June 201
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