Showing papers on "Polygon published in 1998"

Computational Geometry in C

Abstract: The first edition of this book is recognised as one of the definitive sources on the subject of Computational Geometry. In fact, O'Rourke has a long history in the field, has published many papers on the subject and is responsible for the computer graphics algorithms newsgroup which is where all computer geometers meet to discuss their ideas and problems. Typical problems discussed include how a polygon can be represented, how to calculate its area, how to detect if two polygons intersect and how to calculate the convex hull of a polygon. This leads onto more complex issues such as motion planning and seeing if a robot is able navigate from point x to point y without bumping into objects. The algorithms for these (and other) problems are discussed and many are implemented. In addition, many of the ideas are also discussed from the point of view of three and more dimensions. The only disappointment is that many problems are posed as questions at the end of the chapters and, as far as I could see, you cannot get the answers in the forms of a lecturer's supplement. This is fine in academia but not a lot of use for the commercial world. Due to the range of problems that incorporate computational geometry this book cannot be expected to answer every problem you might have. You will undoubtedly need access to other textbooks but I have been using the first edition of this book for many years and the second edition is a welcome addition to my bookshelf. If I was only allowed one computational geometry book then it would undoubtedly be this one.

Efficient View-Dependent Image-Based Rendering with Projective Texture-Mapping

Abstract: This paper presents how the image-based rendering technique of view-dependent texture-mapping (VDTM) can be efficiently implemented using projective texture mapping, a feature commonly available in polygon graphics hardware. VDTM is a technique for generating novel views of a scene with approximately known geometry making maximal use of a sparse set of original views. The original presentation of VDTM in by Debevec, Taylor, and Malik required significant per-pixel computation and did not scale well with the number of original images. In our technique, we precompute for each polygon the set of original images in which it is visible and create a "view map" data structure that encodes the best texture map to use for a regularly sampled set of possible viewing directions. To generate a novel view, the view map for each polygon is queried to determine a set of no more than three original images to blended together in order to render the polygon with projective texture-mapping. Invisible triangles are shaded using an object-space hole-filling method. We show how the rendering process can be streamlined for implementation on standard polygon graphics hardware. We present results of using the method to render a large-scale model of the Berkeley bell tower and its surrounding campus enironment.

Continuous shading of curved surfaces

Abstract: A procedure for computing shaded pictures of curved surfaces is presented. The surface is approximated by small polygons in order to solve easily the hidden-parts problem, but the shading of each polygon is computed so that discontinuities of shade are eliminated across the surface and a smooth appearance is obtained. In order to achieve speed efficiency, the technique developed by Watkins is used which makes possible a hardware implementation of this algorithm.

Watermarking three-dimensional polygonal models through geometric and topological modifications

Abstract: This paper discusses techniques for embedding data into three-dimensional (3-D) polygonal models of geometry. Given objects consisting of points, lines, (connected) polygons, or curved surfaces, the algorithms described in produce polygonal models with data embedded into either their vertex coordinates, their vertex topology (connectivity), or both. Such data embedding can be used, for example, for copyright notification, copyright protection, theft deterrence, and inventory of 3-D polygonal models. A description of the background and requirements is followed by a discussion of where, and by what fundamental methods, data can be embedded into 3-D polygonal models. The paper then presents several data embedding algorithms, with examples, based on these fundamental methods. By means of these algorithms and examples, we show that the embedding of data into 3-D polygonal models is a practicable technique.

Method and apparatus for displaying 3D ultrasound data using three modes of operation

Abstract: A method and an apparatus for allowing the operator of an ultrasound imaging system to switch between two-dimensional slices and three-dimensional projections in such a way that it is easy for the operator to visualize the relationship of the two-dimensional slice to the three-dimensional anatomy. In a "volume rotate" mode, the display screen displays an orientation box along with a three-dimensional projected image generated from a defined data volume. The orientation box provides a visual indication of the shape and orientation of that defined data volume. In a "cut plane" mode, a movable polygon representing a selected two-dimensional slice is displayed inside a stationary orientation box. The polygon provides a visual indication of the orientation and position of the slice relative to the defined data volume. In a "cut plane rotate" mode, a stationary polygon representing a selected two-dimensional slice is displayed inside a rotatable orientation box.

A marching method for the triangulation of surfaces

Abstract: Surface triangulations are necessary in applying finite element methods for solving mechanical problems and for displaying surfaces by ray tracing or other hidden line algorithms. A parametric surface can be a triangulated by triangulating its (plane) area of definition. However, the images of these triangles in object space may vary unacceptably for the application. Thus we need suitable methods of triangulation even for parametric surfaces. Triangulation algorithms for implicit surfaces are available in the literature. [ALGN'91; BL'88; LO'87; SC'93; WY'86]. All these methods divide the space into suitable polyhedrons (cubes, tetrahedrons) and determine the section of the given implicit surface with the edges of these polyhedrons. The intention of this paper is to introduce a marching method to build a mesh of triangles successively by starting with a point or a prescribed polygon. The triangulation is terminated by several bounding polygons (on the given surface) or a global bounding box. (A similar idea is used in the recently published paper [BAXU'97] on algebraic surfaces.) The method will be established for implicit surfaces. With the idea of numerical implicitization introduced in [HA'97], the triangulation is applicable to any surface for which foot points (i.e., points of minimal distance to the surface) can be determined.The main advantages of the triangulation presented in this paper are:

Efficient clipping of arbitrary polygons

Abstract: Clipping 2D polygons is one of the basic routines in computer graphics. In rendering complex 3D images it has to be done several thousand times. Efficient algorithms are therefore very important. We present such an efficient algorithm for clipping arbitrary 2D-polygons. The algorithm can handle arbitrary closed polygons, specifically where the clip and subject polygons may self-intersect. The algoirthm is simple and faster that Vatti's (1992) algorithm, which was designed for the general case as well. Simple modifications allow determination of union and set-theoretic differences of two arbitrary polygons.

An accurate method for voxelizing polygon meshes

Abstract: The process of generating discrete surfaces in a volumetric representation, termed voxelization, is confronted with topological considerations as well as accuracy and efficiency requirements. The authors introduce a new method for voxelizing planar objects which, unlike existing methods, provides topological conformity through geometric measures. They extend their approach to provide, for the first time, an accurate and coherent method for voxelizing polygon meshes. This method eliminates common voxelization artifacts at edges and vertices. They prove the method's topological attributes and report performance of their implementation. Finally, they demonstrate that this approach forms a basis for a new set of voxelization algorithms by voxelizing an example cubic object.

Aiming aid for optical data reading

Abstract: A data reading systems including an aiming aid system which creates a highly visible target or image in the scan volume at a preferred location for placement of the article to be scanned. In a preferred application, an overhead bar code scanner employs a rotating polygon mirror which scans one or more laser beams off pattern mirrors creating a complex pattern of scan lines down into the scan volume whereby the aiming aid is created by directing a laser beam onto a scanning mirror positioned on top of the polygon mirror generally along the rotational axis of the polygon mirror and then directing that beam out into the scan volume in the desired pattern, such as a circular aiming aid. Alternately, the aiming aid may be a multi-dimensional graphical image formed by holographic or diffractive optics.

Autosegmentation/autocontouring system and method for use with three-dimensional radiation therapy treatment planning

Abstract: An autosegmentation/autocontouring method that may be used for quickly and accurately contouring the regions and boundaries around regions for the development of cross sections that may be linearly disposed for the three-dimensional reconstruction of an image is described and claimed. The autosegmentation/autocontouring method includes at least four steps. The first step is to digitize a CT, MRI, or other suitable image and display it on a display screen. The two-dimensional image on the display screen will include gray-scale representations of the internal organs and tissue masses of the anatomical site through which the cross section was made. The second step is to select the interior of a ROI and draw a polygon within the boundaries of the cross-sectional view of this ROI. This polygon could also be drawn in the interior of a cancerous mass or other diseased tissue. The third step of the method of the present invention is to expand the polygon in a novel manner by iteratively testing pixels of the image on the display outside of, but adjacent to, the pixels that the polygon currently subtends. Pixels will be added to the polygon if the value of a decision rule function has a predetermined value. The expansion of the polygon is continued until none of the pixels at the perimeter of the polygon can satisfy the decision rule. Once it is found that none of the perimeter pixels satisfy the decision rule, the perimeter of the polygon is considered the boundary of the ROI. And the fourth step is that the boundary of the ROI is computed and a contour is developed based on this boundary. This same process is repeated for other ROIs that the user may select.

Interactive Volumetric Textures

Abstract: This paper presents a method for interactively rendering complex repetitive scenes such as landscapes, fur, organic tissues, etc. It is an adaptation to Z-buffer of volumetric textures, a ray-traced method, in order to use the power of existing graphics hardware. Our approach consists in slicing a piece of 3D geometry (one repetitive detail of the complex data) into a series of thin layers. A layer is a rectangle containing the shaded geometry that falls in that slice. These layers are used as transparent textures, that are mapped onto the underlying surface (e.g. a hill or an animal skin) with an extrusion offset. We show some results obtained with our first implementation, such as a scene of 13 millions of virtual polygons animated at 2.5 frames per second on a SGI O2.

Parameterized surface fitting technique having independent control of fitting and parameterization

Abstract: A method for creating a smooth parameterization and fitting it to an input surface in a 3-D computer graphics system comprises specifying a plurality of boundary curves on the surface that define a patch of the surface. The boundary curves are typically specified using a user-interactive curve editing procedure, but may also be specified automatically. The method then automatically generates a parameterization of the patch such that a discretized higher order energy functional defined on the surface is minimized. The method further comprises the step of fitting a smooth surface to the input surface, where the smooth surface has a parameterization as generated in the previous step. The fitting and parameterization steps are independently controlled to give the user a high degree of flexibility and control over both steps. The method is useful for converting dense irregular polygon meshes into surface models suitable for interactive modification and animation.

An optimal polygonal boundary encoding scheme in the rate distortion sense

Abstract: In this paper, we present fast and efficient methods for the lossy encoding of object boundaries that are given as eight-connect chain codes. We approximate the boundary by a polygon, and consider the problem of finding the polygon which leads to the smallest distortion for a given number of bits. We also address the dual problem of finding the polygon which leads to the smallest bit rate for a given distortion. We consider two different classes of distortion measures. The first class is based on the maximum operator and the second class is based on the summation operator. For the first class, we derive a fast and optimal scheme that is based on a shortest path algorithm for a weighted directed acyclic graph. For the second class we propose a solution approach that is based on the Lagrange multiplier method, which uses the above-mentioned shortest path algorithm. Since the Lagrange multiplier method can only find solutions on the convex hull of the operational rate distortion function, we also propose a tree-pruning-based algorithm that can find all the optimal solutions. Finally, we present results of the proposed schemes using objects from the Miss America sequence.

Converting sets of polygons to manifold surfaces by cutting and stitching

Abstract: Many real-world polygonal surfaces contain topological singularies that represent a challenge for processes such as simplification, compression, smoothing, etc. We present an algorithm for removing such singularities, thus converting non-manifold sets of polygons to manifold polygonal surfaces (orientable if necessary). We identify singular vertices and edges, multiply singular vertices, and cut through singular edges. In an optional stitching phase, we join surface boundary edges that were cut, or whose endpoints are sufficiently close, while guaranteeing that the surface is a manifold. We study two different stitching strategies called ”edge pinching” and ”edge snapping”; when snapping, special care is required to avoid re-creating singularities. The algorithm manipulates the polygon vertex indices (surface topology) and essentially ignores vertex coordinates (surface geometry). Except for the optional stitching, the algorithm has a linear complexity in the number of vertices edges and faces, and require no floating point operation. Key-words : Polygonal Surface, Manifold, Cutting, Stitching.

Voronoi diagrams and offset curves of curvilinear polygons

Abstract: This paper studies the practical generation of Voronoi diagrams and offset curves of simply-connected planar areas bounded by straight lines and circular arcs. We present and analyze a wavefront-propagation algorithm for the generation of Voronoi diagrams and compare it experimentally to a tuned version of Lee's divide-and-conquer algorithm. Key performance parameters of these two algorithms are compared based on machine-generated test data. We conclude that the wavefront-propagation algorithm can be expected to outperform the divide-and-conquer algorithm for all, but pathological test data. In particular, its practical running time seems to grow only linearly. We also used our implementation in order to gather statistics on the CPU-consumption of offsetting based on Voronoi diagrams. All tests clearly showed the practical suitability of using Voronoi diagrams for the offsetting of curvilinear polygons. The CPU-time consumptions recorded also compare very favorably with other published codes for computing Voronoi diagrams.

Numerical Conformal Mapping Using Cross-Ratios and Delaunay Triangulation

Abstract: We propose a new algorithm for computing the Riemann mapping of the unit disk to a polygon, also known as the Schwarz--Christoffel transformation. The new algorithm, CRDT (for cross-ratios of the Delaunay triangulation), is based on cross-ratios of the prevertices, and also on cross-ratios of quadrilaterals in a Delaunay triangulation of the polygon. The CRDT algorithm produces an accurate representation of the Riemann mapping even in the presence of arbitrary long, thin regions in the polygon, unlike any previous conformal mapping algorithm. We believe that CRDT solves all difficulties with crowding and global convergence, although these facts depend on conjectures that we have so far not been able to prove. We demonstrate convergence with computational experiments. The Riemann mapping has applications in two-dimensional potential theory and mesh generation. We demonstrate CRDT on problems in long, thin regions in which no other known algorithm can perform comparably.

Polygon representation in an integrated circuit layout

Abstract: An approach for representing polygons in an integrated circuit (IC) layout is provided. Polygons are represented by one or more wires, which in turn are each represented by one or more wire segments. Each wire segment is represented by a pair of directed line segments. A data structure hierarchy includes polygon data, wire data, wire segment data and branch data. The polygon data represents a set of IC devices to be represented in the IC layout. The wire data represents the wires that represent the polygons and specifies the associated wire segments and associated polygons. The wire segment data represents the wire segments and specifies the associated directed line segments for each wire segment that represent the wires and references the wire data. The branch data specifies connections between wires by specifying the connecting wire segments in the wires. A spacing check between a first polygon and a second polygon involves determining the canonical direction from the first polygon to the second polygon and testing the two closest faces between the polygons. To satisfy a spacing violation, an exclusion zone is constructed around the first polygon and the second polygon is moved a distance outside the exclusion zone which causes the minimum spacing required by a set of predetermined spacing criteria to be satisfied.

Conamara Chaos Region, Europa: Reconstruction of mobile polygonal ice blocks

Abstract: New Galileo images of Europa reveal regions of chaotic terrain in otherwise highly-lineated background plains. Examination of Conamara Chaos shows that 59% of the region is composed of fine-textured matrix material lying at low elevations and formed by destruction of lineated plains, while the remainder consists of 139 fragmented polygons of linear-textured background plains. Using through-trending linear features, we reconstruct the original positions of chaos polygons and find that significant lateral translation and rotation have occurred: 78% of the polygons have undergone horizontal translations with most moving between 1 and 5 km, and 81% have rotated (average rotation of ∼11°). Movement of polygons appears to be inward from the chaos margins and clockwise in the center, while polygon rotation is evenly clockwise and counterclockwise. Chaos formation in this region thus involves destruction of over half of the pre-existing terrain, and mobilization, translation, and rotation of the remaining polygons, implying elevated near-surface temperatures and a highly mobile substrate over lateral scales of ∼100 km.

Analytic motion blur coverage in the generation of computer graphics imagery

Abstract: A system and method for simulating motion blur. Intersections of pixel sampling points with leading and trailing edges of moving polygons are identified. The intersection points define segments of coverage of the pixel sampling point by a polygon. A segment list is sorted to identify segments that are associated with visible polygons. The segments of visible polygons are passed to shading and texturing functions which individually determine the temporal sampling points for a single polygon or a group of polygons.

A locational error model for spatial features

Abstract: A locational error model for spatial features in vector-based geographical information systems (GIS) is proposed in this paper. Using error in points as the fundamental building block, a stochastic model is constructed to analyse point, line, and polygon errors within a unified framework, a departure from current practices which treat errors in point and line separately. The proposed model gives, as a special case, the epsilon band model a true probabilistic meaning. Moreover, the model can also be employed to derive accuracy standards and cartographic estimates in GIS.

Folding and Unfolding in Computational Geometry

Abstract: Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which it folds. (3) Can every planar polygonal chain be straightened?

System and method for adjusting pixel parameters by subpixel positioning

Abstract: A method and system for simulating motion of a polygon on a display screen. The polygon may be included in a set of polygons used to model a three-dimensional object. The position of the polygon is defined by vertices tracked in a subpixel coordinate system existing in a computer-readable medium. The subpixel coordinates of the vertices are used to identify the pixels on the display screen having coordinates that correspond to subpixel coordinates lying within or, optionally, at the boundary of the polygon. The identified pixels are those that are to be lighted on the display screen to generate the image of the polygon. The display properties of the lighted pixels are selected by interpolation based on defined pixel display parameters assigned to the vertices of the triangle. As motion of the polygon is tracked in the subpixel coordinate system, the corresponding display on the display screen is repeatedly adjusted. The method of identifying and interpolating the display parameters of the pixels using the subpixel coordinate system provides the appearance of smooth polygon motion.

On fat partitioning, fat covering and the union size of polygons

Abstract: The complexity of the contour of the union of simple polygons can be O(n2) in general. In this paper, a necessary and sufficient condition is given for simple polygons which guarantees smaller union complexity. A δ-corridor in a polygon is a passage between two edges with width/length ratio δ. If a set of polygons with n vertices in total has no δ-corridors, then the union size is O((n log log n)/δ), which is close to optimal in the worst case. The result has many applications to basic problems in computational geometry, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a simple polygon P with n vertices into O(n) convex quadrilaterals, without introducing angles smaller than π/12 radians or narrow corridors. Furthermore, a convex quadrilateral can be covered (but not partitioned) with O(1/δ) triangles without introducing small angles. The maximum overlap of the triangles at any point is two. The algorithms take O(n log2n) and O(n log2n+n/δ time for partitioning and covering, respectively.

Method and apparatus for culling polygons

Abstract: A method and apparatus provide for preserving hardware resources in connection with a display of complex scenes. Polygons which make up portions of the display can be culled prior to use of the hardware resources. An occlusion parameter for use in the culling operation can be determined in accordance with a monitoring of a plurality of tiles which constitute a display. In particular, a maximum depth value associated with a given tile can be utilized to indicate whether a subsequently received polygon or primitive would otherwise be occluded and should therefore be discarded or ignored rather than rasterized.

Aperiodic tiling [computer graphics]

Abstract: One of the most interesting ways of assembling small units is along one of the lattices that make up crystals. In this column I live entirely in a 2D world, so the crystals are nothing but collections of polygons in the plane. It is well known that there are only three regular polygons that can tile the plane. Here the verb tile means to cover the infinite plane with a set of polygons so that no gaps or overlaps exist among the polygons. Each polygon is called a tile and the composite pattern is called a tiling.

Two Guard Walkability of Simple Polygons

Abstract: A pair of points s and g on the boundary of a simple polygon P admits a walk if two guards can simultaneously walk along the two boundary chains of P from s to g such that they are always visible to each other. The walk is a counter-walk if one guard moves from s to g while the other moves from g to s in the same direction along the boundary and they are always visible to each other. The (counter-)walk is straight if no backtracking is necessary during the (counter-)walk. In this paper, we show that, given a polygon with n vertices, to test if there exists (s,g) that admits a (straight) (counter-)walk can be solved in time O(n log n) and in linear space. Also we compute all (s,g)'s that admit a (straight) walk in O(n log n) time and all vertex pairs that admit a (straight) counter-walk in O(n log n + m), where m is O(n2).