Xin Qi, Thesis Preliminary annoucement, 'Computational Conformal Geometry, Optimal Transport Theory and their Applications''

Friday, October 16, 2020 - 10:00am to 11:00am
Event Description: 

Computational conformal geometry covers topics in differential geometry, algebraic topology, PDE, and computer science, and it has been applied in many fields including visualization, computer vision, medical imaging and mechanical design and optimization. In this thesis proposal, we present a surface parameterization for non-trivial topology, optimal transport theory and applications, and further generalize the optimal transport theory into spherical situations.

In the first part, we will present an automatic skull registration technique based on discrete uniformization theory. Skull registration plays a fundamental role in forensic science and is crucial for craniofacial reconstruction. The complicated topology, lack of anatomical features, and low quality reconstructed mesh make skull registration challenging. We apply dynamic Yamabe flow to realize discrete uniformization, which modifies the mesh combinatorial structure during the flow and conformally maps the multiply connected skull surface onto a planar disk with circular holes, thus can handle complicated topologies and the technique is robust to low quality meshes. The 3D surfaces can be registered by matching their planar images using harmonic maps. This method is rigorous with theoretic guarantee, automatic without user intervention, and robust to low mesh quality.

In the second part, we will present optimal transport theory and its applications. Optimal transportation (OT) problem aims at finding the most economic way to transform one probability measure to the other, which plays a fundamental role in computer graphics, computer vision, machine learning, geometry processing and medical imaging. In this presentation, we will be talking about a high performance mesh decimation technique based on optimal transport theory. In addition to the euclidean situation, we will introduce a novel theoretic framework and computational algorithm to compute the optimal transport map on the sphere. Constructing with a variational principle approach, our spherical OT map is carried out by solving a convex energy minimization problem and building a spherical power diagram. Moreover, this theory will be generalized with various cost functions and is used in multiple applications. Our experimental results demonstrate the efficiency and efficacy of the methods.

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Hosted By: 
David Gu, CS Faculty
Computed Event Type: 
Event Title: 
Xin Qi, Thesis Preliminary annoucement, 'Computational Conformal Geometry, Optimal Transport Theory and their Applications''