08.01.17 - 13.01.17, Seminar 17021

Functoriality in Geometric Data

The following text appeared on our web pages prior to the seminar, and was included as part of the invitation.

Motivation

This Dagstuhl Seminar aims to bring together researchers interested in the questions of similarity and correspondence across geometric data sets, including collections of images, 3D shapes, GPS traces and other types of data with geometric flavor. A recent trend, emerging independently in multiple theoretical and applied communities, is to understand networks of geometric data sets through their relations and interconnections, a point of view that can be broadly described as exploiting the functoriality of data, which has a long tradition associated with it in mathematics. Functoriality, in its broadest form, is the notion that in dealing with any kind of mathematical object, it is at least as important to understand the transformations or relations exhibited by the object or the family of objects to which it belongs, as it is to study the object itself.

Perhaps the clearest example of functoriality in analyzing and processing geometric data sets is the use of symmetries or self-similarities and self-maps for both gaining insight into the structure of the object and into its relation with other objects. Remarkably, many applied fields dealing with geometric data, including Computer Vision, Computer Graphics, and Computational Geometry have dedicated a significant amount of effort to extract and exploit self-similarity in data analysis. However, so far there has been very little interaction between the different communities and their use of these ideas, which is both surprising and unfortunate because the techniques that are developed have many points in common.

In this context, the organizers of the seminar hope to help bridge the large gaps that currently exist between various (especially applied and theoretic) fields that employ various notions of functoriality in geometric data analysis. The overall objective of this Dagstuhl Seminar, therefore, is to assemble experts in different fields that have considered similar ideas in the past and exchange their expertise, as well as provide a fertile ground for spawning new ideas and exciting future research directions.

Topics of interest in this seminar will include:

  • Defining and quantifying similarity measures for pairs and collections of images, 3D shapes, and other geometric data sets
  • Extracting shared structure from large geometric data collections
  • Representing and computing maps and correspondences between related data sets
  • Defining and computing various geometric and topological invariants for geometric data comparison
  • Automatically discovering relevant features for geometric data comparison.
  • Abstracting, summarizing, and correcting errors in geometric data collections

License
Creative Commons BY 3.0 Unported license
Mirela Ben-Chen and Frédéric Chazal and Leonidas J. Guibas and Maks Ovsjanikov