Resolution-First Scanning of Multidimensional Spaces

Abstract

Three methods are introduced for generating complete scans of multidimensional spaces. The traditional method is to use a raster (typically generated by nested iteration) which generates points at the maximum resolution and fills the space slowly. New methods are desirable, because in many applications it is desirable for the scanned points to be distributed throughout the space and for the resolution to increase with the number of points scanned. Three simple methods are introduced in this paper. Two of the methods are members of a class of methods in which the reverse-bit-order operator maps points from "R(esolution)-space" to the desired space. In "R-space" the distance from the origin determines the resolution level of the scanned point. The two scans occupy points in such a way that a distance measure such as the L¹ norm or the L^∞ norm increases with the progress of the scan. The third method uses iteration of primitive polynomials modulo 2 to generate a nonrepeating sequence of binary numbers which eventually fills the space. This method is most computationally efficient, but the L^∞ norm method generates partial scans which completely sample the space at intermediate levels of resolution. Applications are expected in scientific visualization, graphics rendering, multicriterion optimization, and progressive image transmission.