Raster Set Representations with Eecient Join Operator 2.1 Set Operators

Some 3D computer graphics algorithms are using front-to-back scene traversal (for example hidden-line removal algorithms or fast form-factor computation). In such methods each pixel has to be processed at most once. For this purpose various raster set representations can be used. This paper gives a survey of selected data structures for raster set representation. Some new non-symmetrical methods useful for scanline processing of 2D objects are described. EEciency of DIFFERENCE and UNION operators (and compound JOIN operator) in typical rendering environment is our key point of view. Quantitative results measured in diierent computer architectures are included. 1 Motivation There are several computer graphics algorithms where compact and eecient raster set representation is wanted. We can point out methods using front-to-back traversals of 3D scene in the case that multiple processing of a single pixel cannot be accepted or shape algebras used in window systems and graphics libraries. Generally all methods with high pixel-processing cost are relevant. Applications in hidden surface removal are described in Foley90] and Hrrz96], eecient form-factor computation based on UNION set operator can be found in Nechvvle97]. A little diierent situation is also mentioned in Steinhart91], where a coherency between adjacent scanlines plays an important role. An immediate stimulation for this study was lack of any practical information about data structure eeciency in a typical comput-ing/rendering environment. One can read about a number of smart data structures in classical literature (eg. Preparata85] or Samet89]) but often a theoretical complexity doesn't tell well about the real beneet of the method (example: almost every searching algorithm has O(log 2 N) complexity but real gains of individual methods can vary a lot). Using only theoretical results, there is little chance to nd the most suitable data representation for a given problem, especially in context of modern computer architectures with memory cache hierarchies and advanced CPU pipelines. 2 Basic concepts In 2D raster graphics generally, "set" stands for a subset of some regular 2D lattice. In this text we are dealing with integer numbers neglecting other geometric properties of a lattice: term "1D set" is used for a subset of Z, "2D set" denominates a subset of Z 2. Members of such sets are called "pixels", processing units (graphical primitives) are called "ob-jects". Objects use to be compact and can be successfully surrounded by "minimal bounding rect-angles". In most cases objects are planar polygons , sometimes they are limited to triangles. …