An invariance in the partial visibility due to mobile source in planar scenes.

Abstract

Four intrinsic properties on the illumination of a partially visible environment in 2D are studied in this thesis: the bounds on the number of critical directions, the bounds on the number of permanently invisible regions, or umbrae, the linear equations satisfied by critical directions, and the Lum-Penumbra graph. A direction of illumination is said to be critical if in that direction the morphology of the visible or the invisible changes. By characterizing the changes in the morphology of the visible and the invisible, the invariant relations in a 2D polygonal scene with no interior objects are shown to be G = M + 2 $- U\sb{PS}$ and T = S + 2 $- U\sb{\rm PS}$ where G, M, S and T are the numbers of the critical directions of types generate, merge, split and terminate respectively, and $U\sb{PS}$ is the number of umbrae in the scene. It is proved that these two relations are the only linearly independent relations among G, T, M, S and $U\sb{PS}.$ The invariant relations in a 2D curved scene with no interior closed curves are shown to be G = M + 2 $- U\sb{CS}$ and T = S + 2 $- U\sb{CS}$ where $U\sb{CS}$ is the number of permanently invisible regions in the scene. The bounds on the total number of critical directions in the scenes are established. In a single-window polygonal scene of $n\sb{PS}$ vertices with no interior polygons, the upper bound of the critical directions is $(2n\sb{PS} - 4)$ if $n\sb{PS}$ is even; $(2n\sb{PS} - 2)$ if $n\sb{PS}$ is odd. In a multi-window polygonal scene with no interior polygons, the number of the critical directions is shown to be $(2\omega n\sb{MPS} - 2\omega)$ where $\omega$ is the number of windows and $n\sb{MPS}$ is the number of vertices. In a single-window polygonal scene with interior polygons, the number of the critical directions is shown to be $(2m\sb{PS} n\sb{PS} + 4n\sb{PS})$ where $m\sb{PS}$ is the number of interior polygons. In a multi-window polygonal scene with interior polygons, the number of the critical directions is shown to be $(2\omega\ m\sb{MPS}\ n\sb{MPS} + 4\omega\ n\sb{MPS})$ where $m\sb{MPS}$ is the number of interior polygons. In a single-window curved scene with no closed curves, the total number of the critical directions is shown to be $(2\omega\ m\sb{MPS}\ n\sb{MPS} + 4\omega n\sb{MPS})$ where $m\sb{MPS}$ is the number of interior polygons. The morphological changes in the scene are represented as a Lum-Penumbra graph. It is shown that in a single-window scene with no interior objects, there exists an all dark path if and only if there is at least an umbra in the scene and that in a single-window scene with interior objects, there exists at least an all dark path. (Abstract shortened by UMI.)