A pyramidal data structure for triangle-based surface ... - IEEE Xplore

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Surface Description A Pyramidal Data Structure for

Triangle-Based Surface Description This article describes a hierarchical model for approximating 2%-dimensionalsurfaces. This model,

called a Delaunay pyramid, is a method for compression of spatial data and representation of a surface at successively finer levels of detail. A Delaunay pyramid is based on a sequence of Delaunay triangulations of suitably defined subsets of the set of data points. We will present a triangleoriented encoding structure for a Delaunay pyramid and evaluate its storage complexity. This article also

urface representation plays an important role in a variety of disciplines, including computer graphics, computer-aided design, computer vision, and geographic data processing. The surface reconstruction problem in 2xD is considered here for application to digital surface modeling. This problem can be mathematically stated as that of interpolating a function of two variables when its values are given either at the vertices of a uniformly spaced grid or at a set of irregularly distributed points in the x-y plane. In many applications with large amounts of data available, like computer vision or geographic data processing, a surface model capable of compressing spatial data according to an accuracy-based criterion is required. Thus, hierarchical surface models have been developed in the last few years. Hierarchical structures are used in a wide range of applications for modeling point data, planar regions, surfaces, and 3D objects (see Samet' for a survey). Such models allow the manipulation of an entity at increasingly higher levels of resolution as well as the application of e f f i c i e n t algorithms based o n a divide-and-conquer approach. Hierarchical models of 2D or 3D objects can be classified into domaindependent representations, like the quadtree' or the octree,' which are based on the decomposition of the space occupied by the object, and object-dependent models, like the p r i ~ m t r e e the , ~ hierarchical triangulaMarch 1989

Leila De Floriani Institute of Applied Mathematics of the National Research Council, Italy

ti on^,^.^ or the Delaunay tree,'" which provide object descriptions in an object-centered coordinate frame. Hierarchical surface models belong to this latter group and provide representations of a surface at different levels of resolution, thus also allowing a reduction in the number of points needed to describe the shape of the surface and the application of ray-tracing algorithms for an efficient surface rendering.",'* Existing hierarchical surface models, however, are either well-suited for surface reconstruction only when the data points are regularly q amp led,^^^^^^'^ or they produce triangle-based surface approximations that are

0272-171618910300-0067$01.00 1989 I E E E

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numerically inaccurate because of the elongated shape can be classified into hierarchical triangulations, which of their constituent triangles.fiBeing based on a fixed are based on a triangular subdivision of an initial enclossurface-splitting rule, all these models introduce edges ing triangle, and quadtreelike models, which make use of on the resulting surface. These edges are strictly depen- domain partition techniques based on rectangles. dent on the subdivision and often do not disappear by Hierarchical triangulations try to combine the features increasing the resolution. On the other hand, the Delau- of triangular grids for domain discretization with the use nay triangulation has been extensively applied to build of the hierarchy. A hierarchical triangulation is defined surface approximations at fixed accuracy because of its by the recursive subdivision of an initial triangle (with local definition and because of the equiangularity vertices at three of the data points and containing all the pr~perty.'~.'~ other points inside] into a set of nested subtriangles with We will discuss the properties of several hierarchical vertices at data points. surface representations and define a hierarchical The most common forms of hierarchical triangulaDelaunay-based model. This model combines the useful tions are the ternary and the quaternary hierarchical trifeatures of the Delaunay triangulation with the advan- angulations. In a ternary a subdivision tages of using a hierarchical surface description. Given of a triangle t consists of joining an internal point P to a set S of points describing a surface, we define an the three vertices o f t , thus forming three subtriangles ordered sequence of Delaunay triangulations built on incident at point P. In a quaternary t r i a n g ~ l a t i o n ~ . ~ ~ " suitably chosen subsets of S, each containing an increas- each triangle is subdivided into four subtriangles formed ing number of points and providing a more accurate sur- by joining three points, each lying on a different trianface description. A set of links is defined between any gle side. two consecutive triangulations in the sequence, resultA hierarchical triangulation is described by a segmening in a pyramidal structure, which we call a Delaunay tation tree, in which the root corresponds to the initial pyramid. The benefits of such a model include all the enclosing triangle, whereas any other tree node common advantages of hierarchical representations represents a triangle resulting from the subdivision of without the problems caused by fixed splitting rules. its parent node. A ternary triangulation will be described We will study the properties of the Delaunay pyramid by a ternary tree, whereas a quaternary one will be repand compare it with the hierarchical searching structure resented by a tree in which every node has exactly four defined by Kirkpatrick for the point location problem on children, and thus is also referred to as a quadtreelike triarbitrary triangulations." We will propose a data strucangular tesselation. Figures 1 and 2 show examples of a ture for encoding the Delaunay pyramid. This combines ternary and a quaternary triangulation respectively. Note a triangle-oriented representation of the individual trithat each triangle in a ternary triangulation is adjacent angulations with a pointer structure describing the links to, at most, one triangle along each edge, whereas a tribetween consecutive levels in the pyramid. We will presangle in a quaternary triangulation may have several ent algorithms for the construction and traversal of a edge-adjacent triangles. Delaunay pyramid. The time complexity of the pyramid In each triangle t of a hierarchical triangulation the construction algorithm proves to be comparable to that surface is approximated by a planar triangular patch of any incremental Delaunay triangulation methodI4or any algorithm that builds a hierarchical triang~lation.~.' defined by the three data points whose projections are vertices oft. The approximation error associated with a triangle t is evaluated as the maximum value of the errors computed at the data points whose projections fall Hierarchical surface models: An overview inside t. The absolute value of the difference between the In the last few years hierarchical models for surface interpolated and the given z value at a point P is used as description have been d e v e l ~ p e d ~to - ~represent