Computational aspects of Digital Plane and Hyperplane Recognition
In dimension 2, the arithmetical structure of discrete straight lines leads to efficient algorithms, as both their asymptotic computational cost and the efficiency of their implementations is studied. In higher dimension, arithmetical structures still exist in discrete planes and hyperplanes. However, to solve the recognition problem, we usually adapt algorithms from linear programming (LP) or computational geometry (CG). In this case, there is a gap between the time complexity bounds obtained for LP or CG problems, and the efficiency of these algorithms when applied discrete objects. Indeed, we can observe that complexity bounds are not tight at all when we perform experimental analysis. In this presentation, we will investigate computational and efficiency aspects of discrete plane and hyperplane recognition problems using results and tools from CG and Integer LP.
Polygonal Approximation of Point Sets
The presented approach is described in the statistical framework of Expectation Maximization (EM) and in cognitively motivated geometric framework. We use local support estimation motivated by human visual perception to evaluate support in data points of EM components after each EM step. Consequently, we are able to recognize a locally optimal solution, and modify the number of model components and their parameters. We will show that the proposed approach converges to a globally optimal solution independent of the initial number of model components and their initial parameters.
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