Ph.D Student | Mashiah Ayelet |
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Subject | Surface and Volume Mapping via Mass Transport |

Department | Department of Electrical Engineering |

Supervisor | Mr. Allen Robert Tannenbaum |

Full Thesis text |

In this thesis, we present novel approaches for mapping surfaces and volumes to corresponding canonical domains with the same topology. In general we wish to find a bijection of the sampled surface or volume lying in 3D space with a simpler model, that respectively preserves the area or volume distribution and minimally distort the local geometry .

Many of the operations we wish to perform on the discrete surface or volume, for processing, analysis or visualization, may be greatly simplified if we perform them in a corresponding canonical domain with the same topology. For example, the surface of the brain is a topological sphere, but is highly convoluted. If one can find a good bijection of the brain surface onto the sphere, this would simplify many types of visualization and analysis tasks .

Our surface mapping (parameterization) method aims at being optimal for many applications by creating a map to the sphere that minimizes the geometrical distortion while being area preserving. Our approach is based on the technique of optimal mass transport (OMT), also known as the ``earth mover's problem.'' In our work we derived a novel formulation of the gradient descent minimizing flow for OMT problem. As well as being very useful for the efficient computation of surface parameterization, we believe that this flow is also quite interesting from a theoretical point of view, because it enables us to solve the optimal mass transport problem over any Riemannian manifold .

Our volume mapping method aims at being volume preserving. Using this method we can map any given solid to the cube. This mapping may be practical for simplifying calculations such as constrain analysis. In addition, the mapping process we use not only maps any given solid to a cube, but also generates a regular hexahedral mesh of the given solid, with very small variance of the volume of the hexahedral elements .