Figure: A mapping of data (coloured diamonds) on the surface of a cylinder to the surface of a ring.

A new mathematical approach to mapping surfaces

The motivation – Imagine trying to compare brain data across multiple subjects.  A fundamental problem is to find the correspondence (or map) between one part of a cerebral cortex to the same part of another cortex. This is a difficult task because brains are different sizes and shapes. The uniqueness of each person’s brain depends on characteristics such as age, health, and stature. More generally, finding a map from locations on any surface to locations on any other surface is a problem of interest in numerous fields.

MSc student Nathan King

The discovery – Nathan King (MSc from SFU in Applied Math) and supervisor Steve Ruuth solved this mapping challenge.  Their research generated an accurate and efficient computer algorithm for solving the mathematical equations that describe how to map locations from one surface to another.  The method is faster and gives better quality results than state-of-the-art level set methods in current use.  This new algorithm will potentially impact diverse subjects where finding maps between surfaces arise.

Its significance –Computing these maps between different surfaces is an essential task in medical applications where tissues or organs are compared. In image processing, a map between two surfaces can be used to repair a damaged photo. These maps can also be used in physics to describe the principles of liquid crystal display (LCD) TVs and superconductors in wind turbines.

Read the paper“Solving variational problems and partial differential equations that map between manifolds via the closest point method” by Nathan D. King and Steven J. RuuthJournal of Computational Physics 336:330–346 (2017).

Website article compiled by Jacqueline Watson with Theresa Kitos