45794206
Rational Quadratic Parameterizations of Quadrics
Every irreducible quadric in E3 has infinitely many different rational quadratic parameterizations. These parameterizations and the relationships between them are investigated. It is shown that every faithful rational quadratic parameterization of a quadric can be generated by a stereographic projection from a point on the quadric, called the center of projection (COP). Two such parameterizations for the same quadric are related by a rational linear reparameterization if they have the same COP; otherwise they are related by a rational quadratic reparameterization. We also consider unfaithful parameterizations for which, in general, a one-to-one correspondence between points on the surface and parameters in the plane does not exist. It is shown that all unfaithful rational quadratic parameterizations of a properly degenerate quadric can be characterized by a simple canonical form, and there exist no unfaithful rational quadratic parameterizations for a nondegenerate quadric. In addition, given a faithful rational quadratic parameterization of a quadric, a new technique is presented to compute its base points and inversion formula. These results are applied to solve the problems of parameterizing the intersection of two quadrics and reparameterizing a given quadric parameterization with respect to a different COP without implicitization.
14471402
Computing quadric surface intersections based on an analysis of plane cubic curves
Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves. 2003 Elsevier Science (USA). All rights reserved.
2992545
Constrained 3D shape reconstruction using a combination of surface fitting and registration
We investigate 3D shape reconstruction from measurement data in the presence of constraints. The constraints may fix the surface type or set geometric relations between parts of an object's surface, such as orthogonality, parallelity and others. It is proposed to use a combination of surface fitting and registration within the geometric optimization framework of squared distance minimization (SDM). In this way, we obtain a quasi-Newton like optimization algorithm, which in each iteration simultaneously registers the data set with a rigid motion to the fitting surface and adapts the shape of the fitting surface. We present examples to show the applicability of our method to constrained 3D shape fitting for reverse engineering of CAD models and to high accuracy fitting with kinematic surfaces, which include surfaces of revolution (reconstructed from fragments of archeological pottery) and spiral surfaces, which are fitted to 3D measurement data of shells. Our optimization algorithm can combine registration of multiple scans of an object and model fitting into a single optimization process which is shown to be superior to the traditional procedure, which first registers the data and then fits a model to it.
16675497
Revisiting the [mu]-basis of a rational ruled surface
The µ-basis of a rational ruled surface P(s,t) = P0(s) + tP1(s) is defined in Chen et al. (Comput. Aided Geom. Design 18 (2001) 61) to consist of two polynomials p(x, y, z, s) and q(x, y, z, s) that are linear in x, y, z. It is shown there that the resultant of p and q with respect to s gives the implicit equation of the rational ruled surface; however, the parametric equation P(s, t) of the rational ruled surface cannot be recovered from p and q. Furthermore, the µ-basis thus defined for a rational ruled surface does not possess many nice properties that hold for the µ-basis of a rational planar curve (Comput. Aided Geom. Design 18 (1998) 803). In this paper, we introduce another polynomial r(x, y, z, s, t) that is linear in x, y, z and t such that p, q, r can be used to recover the parametric equation P(s, t) of the rational ruled surface; hence, we redefine the µ-basis to consist of the three polynomials p, q, r. We present an efficient algorithm for computing the newly-defined µ-basis, and derive some of its properties. In particular, we show that the new µ-basis serves as a basis for both the moving plane module and the moving plane ideal corresponding to the rational ruled surface.
205873026
Control point adjustment for B-spline curve approximation
Pottmann et al. propose an iterative optimization scheme for approximating a target curve with a B-spline curve based on square distance minimization, or SDM. The main advantage of SDM is that it does not need a parameterization of data points on the target curve. Starting with an initial B-spline curve, this scheme makes an active B-spline curve converge faster towards the target curve and produces a better approximating B-spline curve than existing methods relying on data point parameterization. However, SDM is sensitive to the initial B-spline curve due to its local nature of optimization. To address this, we integrate SDM with procedures for automatically adjusting both the number and locations of the control points of the active spline curve. This leads to a method that is more robust and applicable than SDM used alone. Furthermore, it is observed that the most time consuming part of SDM is the repeated computation of the foot-point on the target curve of a sample point on the active B-spline curve. In our implementation, we speed up the foot-point computation by pre-computing the distance field of the target curve using the Fast Marching Method. Experimental examples are presented to demonstrate the effectiveness of our method. Problems for further research are discussed. q 2003 Elsevier Ltd. All rights reserved.
17601595
Geocube – GPU accelerated real-time rendering of transparency and translucency
We present a new method based on GPU acceleration for real-time transparency and translucency rendering. Our method computes refraction at both the front and back sides of a transparent object, as well as internal reflection, thus delivering interactive realistic transparency effects on a commodity PC. The real-time performance is made possible by a new acceleration data structure, called geocube, that enables the use of GPU for fast ray-surface intersection testing. In addition, within the same framework, we introduce the novel use of the mip-map for a hierarchical representation of a sequence of key prefiltered environment maps to simulate translucency. By taking ray depth into account and using GPU to interpolate the key filtered maps to produce the desired blurring effects, we achieve real-time realistic translucency rendering of slightly scattering media that allows show-through of background details.
11186519
Edge-partitions of graphs of nonnegative characteristic and their game coloring numbers
Let G be a graph of nonnegative characteristic and let g(G) and @D(G) be its girth and maximum degree, respectively. We show that G has an edge-partition into a forest and a subgraph H so that (1) @D(H)= =11; (2) @D(H)= =7; (3) @D(H)= =5 or G does not contain 4-cycles and 5-cycles; (4) @D(H)= =11; (2) col"g(G)= =7; (3) col"g(G)= =5 or G contains no 4-cycles and 5-cycles; (4) col"g(G)=<10 if G does not contain 4-cycles.
11973783
Answering Approximate Range Aggregate Queries on OLAP Data Cubes with Probabilistic Guarantees
Approximate range aggregate queries are one of the most frequent and useful kinds of queries for Decision Support Systems (DSS). Traditionally, sampling- based techniques have been proposed to tackle this problem. However, its effectiveness will degrade when the underlying data distribution is skewed. Another approach based on the outlier management can limit the effect of data skew but fails to address other requirements of approximate range aggregate queries, such as error guarantees and query processing efficiency. In this paper, we present a technique that provide approximate answers to range aggregate queries on OLAP data cubes efficiently with theoretical error guarantees. Our basic idea is to build different data structures for outliers and the rest of the data. Experimental results verified the effectiveness of our proposed methods.
3193955
An Improved Biclustering Method for Analyzing Gene Expression Profiles
Microarrays are one of the latest breakthroughs in experimental molecular biology, which provide a powerful tool by which the expression patterns of thousands of genes can be monitored simultaneously and are already producing huge amount of valuable data. The concept of bicluster was introduced by Cheng and Church1 to capture the coherence of a subset of genes and a subset of conditions. A set of heuristic algorithms were also designed to either find one bicluster or a set of biclusters, which consist of iterations of masking null values and discovered biclusters, coarse and fine node deletion, node addition, and the inclusion of inverted data. These heuristics inevitably suffer from some serious drawback. The masking of null values and discovered biclusters with random numbers may result in the phenomenon of random interference which in turn impacts the discovery of high quality biclusters. To address this issue and to further accelerate the biclustering process, we generalize the model of bicluster to incorporate null values and propose a probabilistic algorithm (FLOC) that can discover a set of k possibly overlapping biclusters simultaneously. Furthermore, this algorithm can easily be extended to support additional features that suit different requirements at virtually little cost. Experimental study on the yeast gene expression data2 shows that the FLOC algorithm can offer substantial improvements over the previously proposed algorithm.
3416923
Efficient Computation of the Skyline Cube
Skyline has been proposed as an important operator for multi-criteria decision making, data mining and visualization, and user-preference queries. In this paper. we consider the problem of efficiently computing a SKYCUBE, which consists of skylines of all possible non-empty subsets of a given set of dimensions. While existing skyline computation algorithms can be immediately extended to computing each skyline query independently, such "shared-nothing" algorithms are inefficient. We develop several computation sharing strategies based on effectively identifying the computation dependencies among multiple related skyline queries. Based on these sharing strategies, two novel algorithms, Bottom-Up and Top-Down algorithms, are proposed to compute SKYCUBE efficiently. Finally, our extensive performance evaluations confirm the effectiveness of the sharing strategies. It is shown that new algorithms significantly outperform the naive ones.
11841640
Accelerating Profile Queries in Elevation Maps
Elevation maps are a widely used spatial data representation in geographical information systems (GIS). Paths on elevation maps can be characterized by profiles, which describe relative elevation as a function of distance. In this research, we address the inverse of this mapping - given a profile, how to efficiently find paths that could have generated it. This is called the profile query problem. Profiles have a wide variety of uses that include registering tracking information, or even other maps, to a given map. We describe a probabilistic model to characterize the maximal likelihood that a point lying on a path matches the query profile. Propagation of such probabilities to neighboring points can effectively prune the search space. This model enables us to efficiently answer queries of arbitrary profiles with user-specified error tolerances. When compared to existing spatial index methods, our approach supports more flexible queries with orders of magnitude speedup.