טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentSaba Shadi
SubjectBarycentric Spherical Embeddings - Algorithms and Theory
DepartmentDepartment of Computer Science
Supervisors Professor Irad Yavneh
Ms. Alla Sheffer
Professor Chaim Craig Gotsman


Abstract

Parameterization of 3D meshes over a parameter domain is important in many applications in the computer graphics field, such as remeshing, compression, morphing and texture mapping.


In our research we examined the parameterization of genus-0 3D meshes over a sphere. The set of equations that represents the general spherical embedding

problem was formulated previously by Gotsman, Gu and Sheffer as a natural extension of planar barycentric embedding. We present a full overview of the theory behind these equations and explain the algebraic, physical and geometrical meanings of the formulas of Gotsman, Gu and Sheffer.


Then, we construct a robust and stable numerical scheme that approximates a valid solution for the problem. Our method uses several optimization methods, combined with an algebraic multigrid technique. Using these, we are able to spherically parameterize meshes containing up to a hundred thousand vertices in a matter of minutes. We demonstrate our results for many models.


We also investigate the conformal transformation space of the spherical embeddings and show that it can be used to minimize different distortion metrics, including the case of the centralized distortion metric. We then solve the reverse-engineering problem of finding the barycentric weights for a given spherical embedding, consequently obtaining two new recipes for reproducing the weights. We demonstrate the accuracy of our reproduction for various models. Finally we show that there is actually a whole family of barycentric weights reproducing each spherical embedding. This is a new theoretical result.