Optimizing the Maximal Perturbation in Point Sets while Preserving the Order Type

• Main Authors: Luis Gerardo de la Fraga, Heriberto Cruz Hernández
• Format: Article
• Language: English
• Published: MDPI AG 2019-11-01
• Series: Mathematical and Computational Applications
• Subjects: order type; fiducial markers; optimization; computer vision; augmented reality
• Online Access: https://www.mdpi.com/2297-8747/24/4/97

Recently a new kind of fiducial marker based on order type (OT) has been proposed. Using OT one can unequivocally identify a set of points through its triples of point orientation, and therefore, there is no need to use metric information. These proposed order type tags (OTTs) are invariant under a projective transformation which allows identification of them directly from a photograph. The magnitude of noise in the point positions that a set of points can support without changing its OT, is named the maximal perturbation (MP) value. This value represents the maximal displacement that any point in the set can have in any direction without changing the triplet’s orientation in the set. A higher value of the MP makes an OTT instance more robust to perturbations in the points positions. In this paper, we address the problem of how to improve the MP value for sets of points. We optimize “by hand” the MP for all the 16 subsets of points in the set of OTs composed of six points, and we also propose a general algorithm to optimize all the sets of OTs composed of six, seven, and eight points. Finally, we show several OTTs with improved MP values, and their use in an augmented reality application.