A mathematical semantics of rendering I: ideal rendering

Abstract The diversity of ways in which scenes can be mapped into raster images necessitates an examination into precisely what is meant by the process of rendering . The development of 3D graphics standards and graphical programming languages requires the ability to specify, evaluate, and rigorously prove properties of renderings. This paper presents a mathematical semantics of rendering using the basic mathematical notion of measure. Well known, intuitive concepts such as fractional coverage, weight, intensity contribution, filtering, and all sampling techniques are concisely captured measure-theoretically. The uniformity of their specification endows the semantics with considerable explanatory power. The sequel to this paper considers approximations to ideal rendering techniques.

[1]  Rupert Gnatz.  Approaching a formal framework for graphics software standards , 1984, Comput. Graph..

[2]  Eugene Fiume.  Bit-mapped graphics: A semantics and theory , 1987, Comput. Graph..

[3]  J. Marsden,et al.  Elementary classical analysis , 1974 .

[4]  James D. Foley,et al.  Fundamentals of interactive computer graphics , 1982 .

[5]  William Roberts Mallgren.  Formal Specification of Interactive Graphics Programming Languages , 1983 .

[6]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[7]  Mark A. Z. Dippé,et al.  Antialiasing through stochastic sampling , 1985, SIGGRAPH.

[8]  D. S. Jones.  The theory of generalised functions: Table of Laplace transforms , 1982 .

[9]  Eugene Fiume,et al.  Constant-time filtering with space-variant kernels , 1988, SIGGRAPH.

[10]  Don P. Mitchell,et al.  Generating antialiased images at low sampling densities , 1987, SIGGRAPH.

[11]  Robert L. Cook,et al.  Stochastic sampling in computer graphics , 1988, TOGS.

[12]  George S. Carson.  An approach to the formal specification of computer graphics systems , 1984, Comput. Graph..

[13]  William R. Mallgren,et al.  Formal Specification of Graphic Data Types , 1982, TOPL.

[14]  Franklin C. Crow,et al.  The aliasing problem in computer-generated shaded images , 1977, Commun. ACM.

[15]  David A. Duce,et al.  Better Understanding through Formal Specification , 1985, Comput. Graph. Forum.

[16]  David A. Duce,et al.  Formal specification of a small example based on GKS , 1988, TOGS.

[17]  George Carson.  The Specification of Computer Graphics Systems , 1983, IEEE Computer Graphics and Applications.

[18]  Norman I. Badler,et al.  Temporal anti-aliasing in computer generated animation , 1983, SIGGRAPH.

[19]  Eugene Fiume,et al.  The Mathematical Structure of Raster Graphics , 1989 .