The visible-volume function of a set of cameras is continuous, piecewise rational, locally Lipschitz, and semi-algebraic in all dimensions

Titelangaben

Rambau, Jörg:
The visible-volume function of a set of cameras is continuous, piecewise rational, locally Lipschitz, and semi-algebraic in all dimensions.
In: Discrete & Computational Geometry. Bd. 70 (12 August 2023) . - S. 1038-1058.
ISSN 1432-0444
DOI der Verlagsversion: https://doi.org/10.1007/s00454-023-00541-w

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Abstract

The visible-volume function assigns to a configuration of cameras and a flexible environment in a convex room the volume that can be supervised by the cameras. It is of interest to configure the cameras and the environment in such a way that the visible volume is maximized. Some methods of global optimization can take profit from desirable analytic properties of the visible-volume function. Earlier work has only considered this function in dimensions two and three or for static environments implicitly defined by level-set functions. In this paper it is shown, that the visible-volume function for a flexible environment modeled explicitly by a parametrized simplicial complex is continuous, piecewise rational, locally Lipschitz, and semi-algebraic in all dimensions.