109,351 research outputs found
`Optimal' triangulation of surfaces and bodies
A new criterion is given for constructing an optimal triangulation of surfaces and bodies. The triangulation, called the {\em tight} triangulation, is convexity preserving and accepts long, thin triangles whenever they are useful. Both properties are not shared by the maxmin triangulation, which in the plane is called the Delaunay triangulation
Optimal area triangulation
Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in time and space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area
Fast Optimal Three View Triangulation
We consider the problem of L2-optimal triangulation from three separate views. Triangulation is an important part of numerous computer vision systems. Under gaussian noise, minimizing the L2 norm of the reprojection error gives a statistically optimal estimate. This has been solved for two views. However, for three or more views, it is not clear how this should be done. A previously proposed, but computationally impractical, method draws on Gröbner basis techniques to solve for the complete set of stationary points of the cost function. We show how this method can be modified to become significantly more stable and hence given a fast implementation in standard IEEE double precision. We evaluate the precision and speed of the new method on both synthetic and real data. The algorithm has been implemented in a freely available software package which can be downloaded from the Internet
Dyck path triangulations and extendability
We introduce the Dyck path triangulation of the cartesian product of two
simplices . The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever , any triangulation of
extends to a unique triangulation of
. Moreover, with an explicit construction, we
prove that the bound is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.Comment: 15 pages, 14 figures. Comments very welcome
Bound and Conquer: Improving Triangulation by Enforcing Consistency
We study the accuracy of triangulation in multi-camera systems with respect
to the number of cameras. We show that, under certain conditions, the optimal
achievable reconstruction error decays quadratically as more cameras are added
to the system. Furthermore, we analyse the error decay-rate of major
state-of-the-art algorithms with respect to the number of cameras. To this end,
we introduce the notion of consistency for triangulation, and show that
consistent reconstruction algorithms achieve the optimal quadratic decay, which
is asymptotically faster than some other methods. Finally, we present
simulations results supporting our findings. Our simulations have been
implemented in MATLAB and the resulting code is available in the supplementary
material.Comment: 8 pages, 4 figures, Submitted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
- …