Showing papers on "Polygon published in 1988"

The triangle processor and normal vector shader: a VLSI system for high performance graphics

Abstract: Current affordable architectures for high-speed display of shaded 3D objects operate orders of magnitude too slowly. Recent advances in floating point chip technology have outpaced polygon fill time, making the memory access bottleneck between the drawing processor and the frame buffer the most significant factor to be accelerated. Massively parallel VLSI system have the potential to bypass this bottleneck, but to date only at very high cost. We describe a new more affordable VLSI solution. A pipeline of triangle processors rasterizes the geometry, then a further pipeline of shading processors applies Phong shading with multiple light sources. The triangle processor pipeline performs 100 billion additions per second, and the shading pipeline performs two billion multiplies per second. This allows 3D graphics systems to be built capable of displaying more than one million triangles per second. We show the results of an anti-aliasing technique, and discuss extensions to texture mapping, shadows, and environment maps.

Hidden surface removal using polygon area sorting

Abstract: A polygon hidden surface and hidden line removal algorithm is presented. The algorithm recursively subdivides the image into polygon shaped windows until the depth order within the window is found. Accuracy of the input data is preserved.The approach is based on a two-dimensional polygon clipper which is sufficiently general to clip a concave polygon with holes to the borders of a concave polygon with holes.A major advantage of the algorithm is that the polygon form of the output is the same as the polygon form of the input. This allows entering previously calculated images to the system for further processing. Shadow casting may then be performed by first producing a hidden surface removed view from the vantage point of the light source and then resubmitting these tagged polygons for hidden surface removal from the position of the observer. Planar surface detail also becomes easy to represent without increasing the complexity of the hidden surface problem. Translucency is also possible.Calculation times are primarily related to the visible complexity of the final image, but can range from a linear to an exponential relationship with the number of input polygons depending on the particular environment portrayed. To avoid excessive computation time, the implementation uses a screen area subdivision preprocessor to create several windows, each containing a specified number of polygons. The hidden surface algorithm is applied to each of these windows separately. This technique avoids the difficulties of subdividing by screen area down to the screen resolution level while maintaining the advantages of the polygon area sort method.

A parallel algorithm for polygon rasterization

Abstract: A parallel algorithm for the rasterization of polygons is presented that is particularly well suited for 3D Z-buffered graphics implementations. The algorithm represents each edge of a polygon by a linear edge function that has a value greater than zero on one side of the edge and less than zero on the opposite side. The value of the function can be interpolated with hardware similar to hardware required to interpolate color and Z pixel values. In addition, the edge function of adjacent pixels may be easily computed in parallel. The coefficients of the "Edge function" can be computed from floating point endpoints in such a way that sub-pixel precision of the endpoints can be retained in an elegant way.

High-performance polygon rendering

Abstract: This paper describes a system architecture for realtime display of shaded polygons. Performance of 100,000 lighted, 4-sided polygons per second is achieved. Vectors and points draw at the rate of 400,000 per second. High-speed pan and zoom, alpha blending, realtime video input, and antialiased lines are supported. The architecture heavily leverages parallelism in several forms: pipeline, vector, and array processing. It is unique in providing efficient and balanced graphics that support interactive design and manipulation of solid models. After an overview of algorithms and computational requirements, we describe the details of the implementation. Finally, the unique features enabled by the architecture are highlighted.

Nonobtuse triangulation of polygons

Abstract: We show how to triangulate a polygon without using any obtuse triangles. Such triangulations can be used to discretize partial differential equations in a way that guarantees that the resulting matrix is Stieltjes, a desirable property both for computation and for theoretical analysis. A simple divide-and-conquer approach would fail because adjacent subproblems cannot be solved independently, but this can be overcome by careful subdivision. Overlay a square grid on the polygon, preferably with the polygon vertices at grid points. Choose boundary cells so they can be triangulated without propagating irregular points to adjacent cells. The remaining interior is rectangular and easily triangulated. Small angles can also be avoided in these constructions.

Synthetic texturing using digital filters

Abstract: Aliasing artifacts are eliminated from computer generated images of textured polygons by equivalently filtering both the texture and the edges of the polygons. Different filters can be easily compared because the weighting functions that define the shape of the filters are pre-computed and stored in lookup tables. A polygon subdivision algorithm removes the hidden surfaces so that the polygons are rendered sequentially to minimize accessing the texture definition files. An implementation of the texture rendering procedure is described.

Covering polygons is hard

Abstract: It is shown that the following minimum cover problems are NP-hard, even for polygons without holes: (1) covering an arbitrary polygon with convex polygons; (2) covering the boundary of an arbitrary polygon with convex polygons; (3) covering an orthogonal polygon with rectangles; and (4) covering the boundary of an orthogonal polygon with rectangles. It is noted that these results hold even if the polygons are required to be in general position. >

Polygon edge clipping

Abstract: A source polygon (40) is clipped to a view window (30). The valid (i.e. potentially visible) edges (40a,40d,33a,35a) or portions of edges of the source polygon (40) and of the view window (30) boundary edges are permitted to be sequentially output for further processing, while determining the valid parts in a predetermined direction around the perimeter of the source polygon, without having to store (other than for a first entry point) the value of exit points from or entry points to the view window. Only an end point (e.g. 42b) of an edge (40a) of source polygon (40) is considered at a time so that a maximum of two intersection point determinations between the source polygon edge and view window boundary planes (11,13,15,17) outside of which the source polygon edge lies are required to ascertain if a portion of the source polygon edge is valid.

Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines

Abstract: Catmull-Rom splines have local control, can be either approximating or interpolating, and are efficiently computable. Experience with Beta-splines has shown that it is useful to endow a spline with shape parameters, used to modify the shape of the curve or surface independently of the defining control vertices. Thus it is desirable to construct a subclass of the Catmull-Rom splines that has shape parameters.We present such a class, some members of which are interpolating and others approximating. As was done for the Beta-spline, shape parameters are introduced by requiring geometric rather than parametric continuity. Splines in this class are defined by a set of control vertices and a set of shape parameter values. The shape parameters may be applied globally, affecting the entire curve, or they may be modified locally, affecting only a portion of the curve near the corresponding joint. We show that this class results from combining geometrically continuous (Beta-spline) blending functions with a new set of geometrically continuous interpolating functions related to the classical Lagrange curves.We demonstrate the practicality of several members of the class by developing efficient computational algorithms. These algorithms are based on geometric constructions that take as input a control polygon and a set of shape parameter values and produce as output a sequence of Bezier control polygons that exactly describes the original curve. A specific example of shape design using a low-degree member of the class is given.

Method and apparatus for tracking, mapping and recognition of spatial patterns

Abstract: A method and apparatus for the identification of spatial patterns that occur in two or more scenes or maps. Each pattern comprises a set of points in a spatial coordinate system collectively represented by the geometrical figure formed by connecting all point pairs by straight lines. The pattern recognition process is one of recognizing congruent geometrical figures. Two geometrical figures are congruent if all the lines in one geometrical figure are of the same length as the corresponding lines in the other. This concept is valid in a spatial coordinate system of any number of dimensions. In two- or three-dimensional space, a geometrical figure may be considered as a polygon or polyhedron, respectively. Using the coordinates of the points in a pair of congruent geometrical figures, one in a scene and the other in a map, a least squares error transformation matrix may be found to map points in the scene into the map. Using the transformation matrix, the map may be updated and extended with points from the scene. If the scene is produced by the sensor system of a vehicle moving through an environment containing features at rest, the position and orientation of the vehicle may be charted, and, over a series of scenes, the course of the vehicle may be tracked. If the scenes are produced by a sensor system at rest, then moving objects and patterns in the field of view may be tracked.

Polygon Placement Under Translation and Rotation

Abstract: We present a gênerai algorithm which computes an exact description of the set of ail placements for a polygon I (with m edges) which is free to translate and/or to rotate but not to intersect another polygon E (with n edges). The time complexity of our algorithm is O(mn log mn) which is close to optimal in the worst-case. Moreover, in some practical situations, the time complexity is only O (n log n). This algorithm is rather simple and has been implemented. It can be used as an efficient tool in several applications such as cutting stock, inspection and motion planning for a two dimensional robot admidst polygonal obstacles. Résumé. Cet article présente un algorithme général qui calcule une description analytique exacte de Vensemble des placements d'un polygone I (ayant m arêtes) libre de se déplacer en translation et rotation dans le plan sans intersecter un polygone E (ayant n arêtes). La complexité de l'algorithme est O (m n log mn) ce qui est proche de l'optimal dans le cas le pire. On montre de plus que la complexité réelle de Valgorithme, dans certaines situations pratiques est O (n log n). Valgorithme présenté est assez simple et a été implanté. Il recouvre un champ d'applications varié incluant les problèmes de découpe automatique, de conformité de pièces industrielles ou les problèmes de planifications de trajectoires pour un robot mobile plan évoluant au milieu d'obstacles polygonaux.

Method of converting continuous three-dimensional geometrical representations of polygonal objects into discrete three-dimensional voxel-based representations thereof within a three-dimensional voxel-based system

Abstract: A method of converting continuous 3-D (three dimension) geometrical representations of polygonal objects into a discrete set of voxels in discrete 3-D voxel space. In one embodiment, a method is provided for converting a continuous 3-D polygon into a discrete set of voxels connected together in discrete 3-D voxel space. In this embodiment, voxel-based polygons having a wide variety of connectivities are generated. In another embodiment, a method is provided for converting a continuous 3-D polyhedron into a discrete set of voxels connected together in discrete 3-D voxel space. The method is incremental in nature and uses all integer arithmetic. The method is also characterized by decisional process loops, rather than brute-force type computational loops characteristic of prior art methodologies.

The size and number of rings on the square lattice

Abstract: The authors have calculated the number of self-avoiding polygons on the square lattice to 56 steps, and the caliper size to 54 steps. Analysis of the generating function permits estimates of the connective constant, mu =2.638 1585+or-10-6 and the critical exponents alpha =0.500 06+or-0.000 06 and v=0.753+or-0.007. The singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent Delta =1.5, as predicted by Nienhuis (1982, 1984). The confluent part, however, maps into the additive analytic term due to the value of the exponent alpha .

An efficient algorithm for finding the CSG representation of a simple polygon

Abstract: We consider the problem of converting boundary representations of polyhedral objects into constructive-solid-geometry (CSG) representations. The CSG representations for a polyhedron P are based on the half-spaces supporting the faces of P. For certain kinds of polyhedra this problem is equivalent to the corresponding problem for simple polygons in the plane. We give a new proof that the interior of each simple polygon can be represented by a monotone boolean formula based on the half-planes supporting the sides of the polygon and using each such half-plane only once. Our main contribution is an efficient and practical O(n log n) algorithm for doing this boundary-to-CSG conversion for a simple polygon of n sides. We also prove that such nice formulae do not always exist for general polyhedra in three dimensions.

Method of detecting and reviewing pattern defects

Abstract: A laser pattern inspection and/or writing system which writes or inspects a pattern on a target on a stage, by raster scanning the target pixels. Inspection can also be done by substage illumination with non-laser light. A database, organized into frames and strips, represents an ideal pattern as one or more polygons. Each polygon's data description is contained within a single data frame. The database is transformed into a turnpoint polygon representation, then a left and right vector representation, then an addressed pixel representation, then a bit-mapped representastion of the entire target. Most of the transformations are carried out in parallel pipelines. Guardbands around polygon sides are used for error filtering during inspection. Guardbands are polygons, and frames containing only guardband information are sent down dedicated pipelines. Error filtering also is done at the time of pixel comparisons of ideal with real patterns, and subsequently during defect area consolidation. Defect areas are viewed as color overlays of ideal and actual target areas, from data generated during real time. Defect areas can be de-zoomed to allow larger target areas to be viewed. An autofocus keeps the scanning laser beam in focus on the target. The inspection system is used to find fiducial marks to orient the target prior to raster scanning. IC bars are provided with alignment marks for locating each IC bar. Interferometers or glass scale encoders allow the stage position to be known.

Optimal parallel algorithms for polygon and point-set problems

Abstract: In this paper we give parallel algorithms for a number of problems defined on polygons and point sets. All of our algorithms have optimal T(n) *P(n) products, where T(n) is the time complexity and P(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. In addition, our algorithms provide parallel analogues to well known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently that point-set problems, and that one can solve nearest-neighbor problems without explicitly constructing a Voronoi diagram.

Pattern comparator with substage illumination and polygonal data representation

Abstract: A laser pattern inspection and/or writing system which writes or inspects a pattern on a target on a stage, by raster scanning the target pixels. Inspection can also be done by substage illumination with non-laser light. A database, organized into frames and strips, represents an ideal pattern as one or more polygons. Each polygon's data description is contained within a single data frame. The database is transformed into a turnpoint polygon representation, then a left and right vector representation, then an addressed pixel representation, then a bit-mapped representation of the entire target. Most of the transformations are carried out in parallel pipelines. Guardbands around polygon sides are used for error filtering during inspection. Guardbands are polygons, and frames containing only guardband information are sent down dedicated pipelines. Error filtering also is done at the time of pixel comparisons of ideal with real patterns, and subsequently during defect area consolidation. Defect areas are viewed as color overlays of ideal and actual target areas, from data generated during real time. Defect areas can be de-zoomed to allow larger target areas to be viewed. An autofocus keeps the scanning laser beam in focus on the target. The inspection system is used to find fiducial marks to orient the target prior to raster scanning. IC bars are provided with alignment marks for locating each IC bar. Interferometers or glass scale encoders allow the stage position to be known.

Path planning in 0/1/ weighted regions with applications

Abstract: We consider the terrain navigation problem in a two-dimensional polygonal subdivision consisting of obstacles, “free” regions (in which one can travel at no cost), and regions in which cost is proportional to distance traveled. This problem is a special case of the weighted region problem and is a generalization of the well-known planar shortest path problem in the presence of obstacles. We present an O(n2) exact algorithm for this problem and faster algorithms for the cases of convex free regions and/or obstacles. We generalize our algorithm to allow arbitrary weights on the edges of the subdivision. In addition, we present algorithms to solve a variety of important applications: (1) an O(n2W) algorithm for finding lexicographically shortest paths in weighted regions (with W different weights); (2) an O(k2n2) algorithm for planning least-risk paths in a simple polygon that contains k line-of-sight threats (this becomes O(k4n4) in polygons with holes); and (3) an O(k2n3) algorithm for finding least-risk watchman routes in simple rectilinear polygons (a watchman route is such that each point in the polygon is visible from at least one point along the route).

Guardbands for pattern inspector

Abstract: A laser pattern inspection and/or writing system which writes or inspects a pattern on a target on a stage, by raster scanning the target pixels. Inspection can also be done by substage illumination with non-laser light. A database, organized into frames and strips, represents an ideal pattern as one or more polygons. Each polygon's data description is contained within a single data frame. The database is transformed into a turnpoint polygon representation, then a left and right vector representation, then an addressed pixel representation, then a bit-mapped representation of the entire target. Most of the transformations are carried out in parallel pipelines. Guardbands around polygon sides are used for error filtering during inspection. Guardbands are polygons, and frames containing only guardband information are sent down dedicated pipelines. Error filtering also is done at the time of pixel comparisons of ideal with real patterns, and subsequently during defect area consolidation. Defect areas are viewed as color overlays of ideal and actual target areas, from data generated during real time. Defect areas can be de-zoomed to allow larger target areas to be viewed. An autofocus keeps the scanning laser beam in focus on the target. The inspection system is used to find fiducial marks to orient the target prior to raster scanning. IC bars are provided with alignment marks for locating each IC bar. Interferometers or glass scale encoders allow the stage position to be known.

Probing convex polygons with X-rays

Abstract: An X-ray probe through a polygon measures the length of intersection between a line and the polygon. This paper considers the properties of various classes of X-ray probes, and shows how they interact to give finite strategies for completely describing convex n-gons. It is shown that $({{3n} / 2}) + 6$ probes are sufficient to verify a specified n-gon, while for determining convex polygons ${{(3n - 1)} / 2}$ X-ray probes are necessary and $5n + O(1)$ sufficient, with $3n + O(1)$ sufficient given that a lower bound on the size of the smallest edge of P is known.

Partitioning polyhedral objects into nonintersecting parts

Abstract: An algorithm is described for partitioning intersecting polyhedrons into disjoint pieces and, more generally, removing intersections from sets of planar polygons embedded in three space. Polygons, or faces, need not be convex and may contain multiple holes. Intersections are removed by considering pairs of faces and slicing the faces apart along their regions of intersection. To reduce the number of face pairs examined, bounding boxes around groups of faces are checked for overlap. The intersection algorithm also computes set-theoretic operations on polyhedrons. Information gathered during face cutting is used to determine which portions of the original boundaries may be present in the result of an intersection, a union, or a difference of solids. The method includes provisions to detect and in some cases overcome, the effects of numerical inaccuracy on the topological decisions that the algorithm must make. The regions in which ambiguous results are possible are flagged so that the user can take appropriate action. >

A fast Las Vegas algorithm for triangulating a simple polygon

Abstract: We present an algorithm that triangulates a simple polygon on n vertices in O(n log*n) expected time. The algorithm uses random sampling on the input, and its running time does not depend on any assumptions about a probability distribution from which the polygon is drawn.

Decomposition and intersection of simple splinegons

Abstract: A splinegon is a polygon whose edges have been replaced by “well-behaved” curves. We show how to decompose a simple splinegon into a union of monotone pieces and into a union of differences of unions of convex pieces. We also show how to use a fast triangulation algorithm to test whether two given simple splinegons intersect. We conclude with examples of splinegons that make the extension of algorithms from polygons to splinegons difficult.

Fast algorithm for polygon decomposition

Abstract: An O(klog(k)+n) algorithm is developed, where n is the number of versions, to decompose rectilinear polygons into rectangles. This algorithm uses horizontal cuts only and reports nonoverlapping rectangles the union of which is the original rectilinear polygon. This algorithm has been programmed in Pascal on an Apollo DN320 workstation. Experimentation with rectilinear polygons from VLSI artwork indicate that the present algorithm is significantly faster than the plane sweep algorithm and the algorithm proposed by K.D. Gourley and D.M. Green (1983). >

Laser scanner using focusing acousto-optic device

Abstract: A laser pattern inspection and/or writing system which writes or inspects a pattern on a target on a stage, by raster scanning the target pixels. Inspection can also be done by substage illumination with non-laser light. A database, organized into frames and strips, represents an ideal pattern as one or more polygons. Each polygon's data description is contained within a single data frame. The database is transformed into a turnpoint polygon representation, then a left and right vector representation, then an addressed pixel representation, then a bit-mapped representation of the entire target. Most of the transformations are carried out in parallel pipelines. Guardbands around polygon sides are used for error filtering during inspection. Guardbands are polygons, and frames containing only guardband information are sent down dedicated pipelines. Error filtering also is done at the time of pixel comparisons of ideal with real patterns, and subsequently during defect area consolidation. Defect areas are viewed as color overlays of ideal and actual target areas, from data generated during real time. Defect areas can be de-zoomed to allow larger target areas to be viewed. An autofocus keeps the scanning laser beam in focus on the target. The inspection system is used to find fiducial marks to orient the target prior to raster scanning. IC bars are provided with alignment marks for locating each IC bar. Interferometers or glass scale encoders allow the stage position to be known.

Recognition of kidney glomerulus by dynamic programming matching method

Abstract: Dynamic programming was applied to locate the glomeruli in microscopic images of kidney tissue section. The glomeruli were modeled by a polygon whose sides could be varied within a given range of lengths. The objects were located by determining the best match of the model according to a so-called optimum criterion in which all possible shapes were evaluated at all possible positions in the input image. The best model was selected according to the maximum average gray level. To increase the probability of obtaining a closed contour, a distance criterion was added and the maximum gray-level requirement was relaxed somewhat. The optimum criterion was modified to include a directionality constraint in which the difference in angle between model segments and the edge values in the image was minimized, thereby increasing the performance of the method. A hierarchical multiresolution strategy was used to reduce calculation time. The cyclical property of a contour is also taken into account. >

Covering orthogonal polygons with star polygons: the perfect graph approach

Abstract: We consider the problem of covering simple orthogonal polygons with star polygons. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q.In general, orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques. We thus obtain a O (n10) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship.In the case where the polygon has at most three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a triangulated (chordal) graph with a minimum number of cliques. This gives us an O (n3) algorithm.

Extremum properties of hexagonal partitioning and the uniform distribution in Euclidean location

Abstract: A zero-sum game with a maximizer that selects a point x in given polygon R in the plane and a minimizer that selects K points $c_1 ,c_2 , \cdots ,c_K $ in the plane is considered; the payoff is the Euclidean distance from x to the closest of the points $c_j $, or any monotonically nondecreasing function of this quantity. Lower and upper bounds on the value of the game are derived by considering, respectively, the maximizer’s strategy of selecting a uniformly distributed random point in R and the minimizer’s strategy of selecting K members of a (uniformly) randomly positioned grid of centers that induce a covering of R by K congruent regular hexagons. The analysis shows that these strategies are asymptotically optimal (for $K \to \infty $). For Euclidean location problems with uniformly distributed customers, the results imply that hexagonal partitioning of the region is asymptotically optimal and that the uniform distribution is asymptotically the worst possible.

On maximum flows in polyhedral domains

Abstract: We introduce a new class of problems concerned with the computation of maximum flows through two-dimensional polyhedral domains. Given a polyhedral space (e.g., a simple polygon with holes), we want to find the maximum “flow” from a source edge to a sink edge. Flow is defined to be a divergence-free vector field on the interior of the domain, and capacity constraints are specified by giving the maximum magnitude of the flow vector at any point. Strang proved that max flow equals min cut; we address the problem of constructing min cuts and max flows. We give polynomial-time algorithms for maximum flow from a source edge to a sink edge through a simple polygon with uniform capacity constraint (with or without holes), maximum flow through a simple polygon from many sources to many sinks, and maximum flow through weighted polygonal regions. Central to our methodology is the intimate connection between the max-flow problem and its dual, the min-cut problem. We show how the continuous Dijkstra paradigm of solving shortest paths problems corresponds to a continuous version of Berge's algorithm for computation of maximum flow in a planar network.