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Re-meshing of a given triangle mesh surface to a quad mesh using physically motivated methods based on the Ginzburg--Landau potential and solved efficiently solved via numerical splitting scheme

Periodic Reporting for period 1 - GLnQuadRemeshing (Re-meshing of a given triangle mesh surface to a quad mesh using physically motivated methods based on the Ginzburg--Landau potential and solved efficiently solved via numerical splitting scheme)

Reporting period: 2018-10-01 to 2020-09-30

Three dimensional (3D) curved objects are ubiquitous in computational science and engineering problems such as geometry acquisition, fabrication of shapes, virtual/augmented reality and computer aided design, to name a few. Often, only the surface, namely the surrounding shell of the object, is required in practice. To represent these curved surfaces in a computer, many algorithms approximate the object using a triangle mesh. That is, the given surface is tiled using small triangles glued over their edges. While triangle meshes are useful in many scenarios, in some cases such as computer modeling and numerical simulation, quad meshes, i.e. meshes with quadrangular faces are preferred. To this end, several conversion techniques were devised, allowing to transform an input triangle mesh to a quad mesh. To date, existing conversion methods work well on certain examples, but in other cases, the results could be greatly improved, motivating the following research. The main goal of this action is to explore new methods for generating a quadrangular surface that approximates a given triangle mesh.

Our investigation shows that the Ginzubrg—Landau potential is somewhat too sensitive to work with in practice. The main reason for the increased sensitivity is non-convexity of the potential which leads to less robust numerical schemes. Instead, we discovered that functionals based on the elastic and membrane energies are much more robust and easier to work with. While both energy functionals can be used for similar applications, the elastic energy we developed is especially useful to handle non-smooth surfaces which contain sharp corners or edges. In addition, we demonstrated that our core method can be utilized on data that originate from complex dynamical systems such as fluid flows and weather systems. Thus, we developed a simple method for analyzing the behavior of such complex systems in terms of their dominating main modes and their decay and growth rates
During the reporting period, I investigated in the outgoing phase at UCLA the Ginzburg—Landau potential and other related energy functionals for the purpose of computing correspondences between shell shapes and generating their quad tessellations. We report are scientific results in seven research manuscripts. While at UCLA, we published two conference papers and one technical report dealing with novel elastic energy functionals and consistency constraints for mapping problems. In addition, we published one journal paper, one conference paper (ICML) and two technical reports that exploit some of the results on maps toward a robust framework for predicting and analyzing dynamical systems. The technical reports are in various stages of a peer-review process. Our results include seven different novel algorithms which can be used in a standalone mode or incorporated to other systems. We published open-source code for four of these algorithms. I established new collaborations with the group Leo Guibas at Stanford, the group of Michael Mahoney at University of California Berkeley and Guy Gilboa at the Technion – Israel Institute of Technology. We communicated our results two international conferences and four international workshops. I received two competitive travel grants. I mentored 10 undergraduate students and two graduate students during their participation at the REU at UCLA. I taught undergraduate level programming courses at UCLA and prepared a new advanced course on dynamical systems. I accepted and started at Oct. 2020 a tenure-track position at the Computer Science Department at the Ben-Gurion University of the Negev, Israel.
I strengthened and enriched my technical knowledge on partial differential equations, optimization, Riemannian geometry and Koopman theory while conducting research in a new environment with new methodologies. Our research has yielded four accepted papers in top-tier venues such as SGP and ICML and three additional research reports which are still undergoing review which is well beyond our expected outcome. I have written most of those manuscripts and thus deepened my writing skills. I have participated in international conferences and professional workshops and meetings to present our research results and to widen my network connections. In addition, I taught several undergraduate-level courses and prepared a new graduate-level course during my outgoing phase. Finally, I also acquired project management skills as most of my works involved two or more research groups from various universities.

The project terminates before term by the end of the outgoing phase.
Quad remeshing of 3D objects
Quad remeshing of 3D objects