M.Sc Student | Elad Asi |
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Subject | On Surface Flattening via MDS and FMMTD |

Department | Department of Computer Science |

Supervisor | Professor Ron Kimmel |

This research deals with two different problems, which at first glance seem to have little in common. The first is a basic problem in pattern classification. It deals with matching isometric surfaces, where `isometry' transforms one surface to another without stretching the surface. Most of existing algorithms for a partial or full matching between surfaces handle rigid surfaces, while we also deal with non-rigid (isometric) cases. The second problem is in medical imaging analysis. It deals with finding the best spherical coordinates that represent the curved convoluted outer surface of the brain, also known as the cortex. The purpose is to simplify the work and effort of medical doctors and deal with the analysis of the activity on the cortical surface and the morphological structure of the cortex.

The introduction of the Fast Marching Method on Triangulated Domains (FMMTD), as an efficient numerical technique to compute geodesic distances on triangulated surfaces, gave the opportunity to develop and implement new sophisticated algorithms in surface analysis. The algorithms we propose and explore

in this research use the geodesic distance as a basic characteristic measure of the surface. It is combined with Multi-Dimensional Scaling (MDS) techniques to solve these two problems. The algorithmic solutions use similar ingredients to construct a mapping that, in the first case, transform isometric surfaces into an invariant canonical form, and in the second case find the best spherical map. The mapping procedure is constructed by first measuring the inter geodesic distances between points on the given surfaces.

Next, a multi-dimensional scaling (MDS) technique is applied to perform the actual flattening. For the first problem, the actual flattening is provided by applying the MDS to compute new coordinates in a finite dimensional flat space, in which geodesic distances are represented as Euclidean ones. For the second spherical mapping problem, a constrained MDS technique is used to map the whole or a section of the cortex onto a sphere in a way that local and global distances on the surface are preserved.

**Keywords:**

** **

**FMMTD** (Fast Marching Method on
Triangulated Domains) - A recent technique for calculating

geodesic distances over triangulated domains.

**MDS** (Multi Dimensional Scaling) - A
family of methods that maps measurements of similarity or

dissimilarity among pairs of feature items, into distances between feature points with given

coordinates in a small-dimensional Euclidean space.