Camera placement in integer lattices

Camera Placement in Integer Lattices

The camera placement problem concerns the placement of a xed number of point-cameras on the integer lattice of d-tuples of integers in order to maximize their visibility. We give a caracterization of optimal conng-urations of size s less than 5 d and use it to compute in time O(s log s) an optimal abstract connguration under the assumption that the visibility of a connguration is computable in ...

Integer Optimization and Lattices

• Lattices. We will see basic properties of lattices, followed by Minkowski’s Theorem which guarantees that any symmetric convex set with volume bigger than 2 must have an non-zero integer point. We will show an application of Minkowski’s theorem to Diophantine approximations. Then we will introduce the powerful concept of Lattice basis reduction which finds an almost orthogonal basis for a lat...

BOOK OF ABSTRACTS Isoperimetry in integer lattices

The edge isoperimetric problem for a graph G is to determine, for each n, the minimum number of edges leaving any set of n vertices. Exact solutions are known only in special cases, for example when G is the usual integer lattice. The most natural open problem was to answer this question for the ‘strong lattice’, with edges between points at l∞ distance 1. Whilst studying this question we in fa...

On the Camera Placement Problem

We introduce a new probing problem: what is the minimum number of cameras at fixed positions necessary and sufficient to reconstruct any strictly convex polygon contained in a disk of radius 1 if cameras only see the silhouette of the polygon? The optimal number only depends on the largest angle α of the polygon. If no two camera tangents overlap, 3π π−α cameras are necessary and sufficient. Ot...

Completely Empty Pyramids on Integer Lattices