44 research outputs found
Interactive Hausdorff distance computation for general polygonal models
Figure 1: Interactive Hausdorff Distance Computation. Our algorithm can compute Hausdorff distance between complicated models at interactive rates (the first three figures). Here, the green line denotes the Hausdorff distance. This algorithm can also be used to find penetration depth (PD) for physically-based animation (the last two figures). It takes only a few milli-seconds to run on average. We present a simple algorithm to compute the Hausdorff distance between complicated, polygonal models at interactive rates. The algorithm requires no assumptions about the underlying topology and geometry. To avoid the high computational and implementa-tion complexity of exact Hausdorff distance calculation, we approx-imate the Hausdorff distance within a user-specified error bound. The main ingredient of our approximation algorithm is a novel polygon subdivision scheme, called Voronoi subdivision, combined with culling between the models based on bounding volume hier-archy (BVH). This cross-culling method relies on tight yet simple computation of bounds on the Hausdorff distance, and it discards unnecessary polygon pairs from each of the input models alterna-tively based on the distance bounds. This algorithm can approxi-mate the Hausdorff distance between polygonal models consisting of tens of thousands triangles with a small error bound in real-time, and outperforms the existing algorithm by more than an order of magnitude. We apply our Hausdorff distance algorithm to the mea-surement of shape similarity, and the computation of penetration depth for physically-based animation. In particular, the penetration depth computation using Hausdorff distance runs at highly interac-tive rates for complicated dynamics scene
Computation of the Hausdorff distance between sets of line segments in parallel
We show that the Hausdorff distance for two sets of non-intersecting line
segments can be computed in parallel in time using O(n)
processors in a CREW-PRAM computation model. We discuss how some parts of the
sequential algorithm can be performed in parallel using previously known
parallel algorithms; and identify the so-far unsolved part of the problem for
the parallel computation, which is the following: Given two sets of
-monotone curve segments, red and blue, for each red segment find its
extremal intersection points with the blue set, i.e. points with the minimal
and maximal -coordinate. Each segment set is assumed to be intersection
free. For this intersection problem we describe a parallel algorithm which
completes the Hausdorff distance computation within the stated time and
processor bounds
Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves
In this paper we introduce the notion of rational Hausdorff divisor, we
analyze the dimension and irreducibility of its associated linear system of
curves, and we prove that all irreducible real curves belonging to the linear
system are rational and are at finite Hausdorff distance among them. As a
consequence, we provide a projective linear subspace where all (irreducible)
elements are solutions to the approximate parametrization problem for a given
algebraic plane curve. Furthermore, we identify the linear system with a plane
curve that is shown to be rational and we develop algorithms to parametrize it
analyzing its fields of parametrization. Therefore, we present a generic answer
to the approximate parametrization problem. In addition, we introduce the
notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve
can always be parametrized with a generic rational parametrization having
coefficients depending on as many parameters as the degree of the input curve
Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement
The typical goal of surface remeshing consists in finding a mesh that is (1)
geometrically faithful to the original geometry, (2) as coarse as possible to
obtain a low-complexity representation and (3) free of bad elements that would
hamper the desired application. In this paper, we design an algorithm to
address all three optimization goals simultaneously. The user specifies desired
bounds on approximation error {\delta}, minimal interior angle {\theta} and
maximum mesh complexity N (number of vertices). Since such a desired mesh might
not even exist, our optimization framework treats only the approximation error
bound {\delta} as a hard constraint and the other two criteria as optimization
goals. More specifically, we iteratively perform carefully prioritized local
operators, whenever they do not violate the approximation error bound and
improve the mesh otherwise. In this way our optimization framework greedily
searches for the coarsest mesh with minimal interior angle above {\theta} and
approximation error bounded by {\delta}. Fast runtime is enabled by a local
approximation error estimation, while implicit feature preservation is obtained
by specifically designed vertex relocation operators. Experiments show that our
approach delivers high-quality meshes with implicitly preserved features and
better balances between geometric fidelity, mesh complexity and element quality
than the state-of-the-art.Comment: 14 pages, 20 figures. Submitted to IEEE Transactions on Visualization
and Computer Graphic
Efficient error control in 3D mesh coding
Our recently proposed wavelet-based L-infinite-constrained coding approach for meshes ensures that the maximum error between the vertex positions in the original and decoded meshes is guaranteed to be lower than a given upper bound. Instantiations of both L-2 and L-infinite coding approaches are demonstrated for MESHGRID, which is a scalable 3D object encoding system, part of MPEG-4 AFX. In this survey paper, we compare the novel L-infinite distortion estimator against the L-2 distortion estimator which is typically employed in 3D mesh coding systems. In addition, we show that, under certain conditions, the L-infinite estimator can be exploited to approximate the Hausdorff distance in real-time implementation