2214669
Fast degree elevation and knot insertion for B-spline curves
We give a new, simple algorithm for simultaneous degree elevation and knot insertion for B-spline curves. The method is based on the simple approach of computing derivatives using the control points, resampling the knot vector, and then computing the new control points from the derivatives. We compare our approach with previous algorithms and illustrate it with examples.
10050571
Adding and Subtracting Eigenspaces
This paper provides two algorithms; one for adding eigenspaces, another for subtracting them, thus allowing for incremental updating and downdating of data models. Importantly, and unlike previous work, we keep an accurate track of the mean of the data, which allows our methods to be used in classification applications. The result of adding eigenspaces, each made from a set of data, is an approximation to that which would obtain were the sets of data taken together. Subtracting eigenspaces yields a result approximating that which would obtain were a subset of data used. Using our algorithms it is possible to perform “arithmetic” on eigenspaces without reference to the original data. We illustrate the use of our algorithms in three generic applications, including the dynamic construction of Gaussian mixture models.
750888
Detecting approximate incomplete symmetries in discrete point sets
Motivated by the need to detect design intent in approximate boundary representation models, we give an algorithm to detect incomplete symmetries of discrete points, giving the models' potential local symmetries at various automatically detected tolerances. Here, incomplete symmetry is defined as a set of incomplete cycles which are constructed by, e.g., a set of consecutive vertices of an approximately regular polygon, induced by a single isometry. All seven 3D elementary isometries are considered for symmetry detection. Incomplete cycles are first found using a tolerance-controlled point expansion approach. Subsequently, these cycles are clustered for incomplete symmetry detection. The resulting clusters have welldefined, unambiguous approximate symmetries suitable for design intent detection, as demonstrated experimentally.
14838227
Sweeping of three-dimensional objects
Evaluating the volume swept out by a three-dimensional (3D) object as it moves along an arbitrary path is of interest to many areas of CAD and CAM, such as mechanism design and robot path planning. This paper shows how envelope theory from differential geometry can be used to find the volumes swept out by the individual surfaces of a solid body, and how computer algebra methods may be of use to perform the computations involved. Finally, a new algorithm is presented which shows how the results of sweeping the individual surfaces of a solid body can be combined to form a new 3D model of the swept volume. This algorithm has strong resemblance to hidden line algorithms, but works in one dimension higher.
17590358
Constructing Regularity Feature Trees for Solid Models
Approximate geometric models, e.g. as created by reverse engineering, describe the approximate shape of an object, but do not record the underlying design intent. Automatically inferring geometric aspects of the design intent, represented by feature trees and geometric constraints, enhances the utility of such models for downstream tasks. One approach to design intent detection in such models is to decompose them into regularity features. Geometric regularities such as symmetries may then be sought in each regularity feature, and subsequently be combined into a global, consistent description of the model's geometric design intent. This paper describes a systematic approach for finding such regularity features based on recovering broken symmetries in the model. The output is a tree of regularity features for subsequent use in regularity detection and selection. Experimental results are given to demonstrate the operation and efficiency of the algorithm.
1359860
Skewed Mirror Symmetry in the 3D Reconstruction of Polyhedral Models
We aim to reconstruct three-dimensional polyhedra from axonometric line drawings. Existence of mirror symmetry in polyhedra can assist the reconstruction process. We present a new approach for determining planes of mirror symmetry of such polyhedral objects based on prior detection of their planar faces and any axes of symmetry of these faces. The axes are obtained from skewed facial symmetries, for which we also give a new method of determination.
34342205
Pfaffian Algorithms for Sampling Routings on Regions with Free Boundary Conditions
Sets of non-intersecting, monotonic lattice paths, or fixed routings, provide a common representation for several combinatorial problems and have been the key element for designing sampling algorithms. Markov chain algorithms based on routings have led to efficient samplers for tilings, Eulerian orientations [8] and triangulations [9], while an algorithm which successively calculates ratios of determinants has led to a very fast method for sampling fixed routings [12]. We extend Wilson’s determinant algorithm [12] to sample free routings where the number of paths, as well as the endpoints, are allowed to vary. The algorithm is based on a technique due to Stembridge for counting free routings by calculating the Pfaffian of a suitable matrix [11] and a method of Colbourn, Myrvold and Neufeld [1] for efficiently calculating ratios of determinants. As an application, we show how to sample tilings on planar lattice regions with free boundary conditions.
119154847
How many random edges make a dense graph hamiltonian
This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $\Theta(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.
17312370
A note on short cycles in a hypercube
How many edges can a quadrilateral-free subgraph of a hypercube have? This question was raised by Paul Erdos about 27 years ago. His conjecture that such a subgraph asymptotically has at most half the edges of a hypercube is still unresolved. Let f(n,C"l) be the largest number of edges in a subgraph of a hypercube Q"n containing no cycle of length l. It is known that f(n,C"l)=o(|E(Q"n)|), when l=4k, k>=2 and that f(n,C"6)>=13|E(Q"n)|. It is an open question to determine f(n,C"l) for l=4k+2, k>=2. Here, we give a general upper bound for f(n,C"l) when l=4k+2 and provide a coloring of E(Q"n) by four colors containing no induced monochromatic C"1"0.
5430944
Distributed selfish load balancing
Suppose that a set of m tasks are to be shared as equally as possible amongst a set of n resources. A game-theoretic mechanism to find a suitable allocation is to associate each task with a "selfish agent", and require each agent to select a resource, with the cost of a resource being the number of agents to select it. Agents would then be expected to migrate from overloaded to underloaded resources, until the allocation becomes balanced.Recent work has studied the question of how this can take place within a distributed setting in which agents migrate selfishly without any centralized control. In this paper we discuss a natural protocol for the agents which combines the following desirable features: It can be implemented in a strongly distributed setting, uses no central control, and has good convergence properties. For m ≫ n, the system becomes approximately balanced (an ε-Nash equilibrium) in expected time O(log log m). We show using a martingale technique that the process converges to a perfectly balanced allocation in expected time O(log log m + n4). We also give a lower bound of Ω (max{log log m, n}) for the convergence time.
16901231
A Noise Suppression System for the AMR Speech Codec
In this paper we describe a noise reduction preprocessing algorithm for the adaptive multirate (AMR) speech codec of the GSM system. The algorithm is based on spectral weighting and explicitly takes into account the properties of the human auditory system. The weighting rule results in the smallest possible speech distortion under the constraint that the background noise should exhibit no audible distortions. The algorithm was implemented in 16 Bit fixed-point arithmetic and submitted to the ETSI AMR noise reduction standardization contest. Compared to other algorithms, our noise reduction method gave very good results in CCR tests and good results in ACR tests.