FoCM 2014 conference

Workshop B2 - Computational Topology and Geometry - Semi-plenary talk

December 15, 14:35 ~ 15:25 - Room A22

## Measuring the geometric similarities of genus-zero surfaces

### Patrice Koehl

### University of California, Davis, USA - pakoehl@ucdavis.edu

Finding efficient algorithms to describe, measure and compare shapes is a central problem in numerous disciplines that generate extensive quantitative and visual information. Among these, biology occupies a central place. Registration of brain anatomy for example is essential to many studies in neurobiology; at a molecular level, comparison of protein shapes is a key step in understanding the relationships between their functions. In this talk I will introduce the idea of a globally optimal conformal mapping between two (discrete) surfaces of genus zero as one method to solve this problem. In this approach, the whole mesh representing the source surface is warped onto the target surface, using the mapping defined through the composition of discrete conformal mappings of the surfaces onto the sphere and the M\"{o}bius transformation between these mappings. The M\"{o}bius transformation is then optimized to lead to minimal distortion between the source mesh and its image, where distortion is measured as difference from isometry. I will show that this approach leads to the definition of a metric in the space of genus-zero surfaces. I will describe the implementation of this approach and its applications on biological examples, from brain surface matching to testing how round proteins are.

Joint work with Joel Hass (University of California, Davis, USA).