August 18, Thursday 11:00, Room 570, Education Building

Title: A Complex View of Barycentric Coordinates with Applications to Planar Shape Deformation

Lecturer: Ofir Weber

Lecturer homepage : http://www.cs.technion.ac.il/~weber/

Affiliation : Courant Institute of Mathematical Sciences, New York University

Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to satisfy a number of key properties, the first being that they are real. In this work we relax this requirement, allowing the barycentric coordinates to be complex numbers. This allows us to generate new families of barycentric coordinates, which have some powerful advantages over traditional ones and are especially useful for creating detail-preserving planar shape deformations.

We first use Cauchy.s theorem to construct complex barycentric coordinates that can be used to generate holomorphic functions. Such functions can be interpreted as conformal planar mappings as long as the derivative of the function doesn.t vanish.

We then construct another type of complex barycentric coordinates based on the Hilbert transform. Combined with a novel 2D shape deformation system, we show how to generate .foldovers free. pure conformal planar deformations. Beyond deforming a given shape into a new one at each key frame, our method also provides the ability to interpolate between shapes in a very natural way, such that also the intermediate deformations are conformal.

Finally, we extend and generalize these results by investigating the complex representation of real-valued barycentric coordinates, when applied to planar domains. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex-weighted combination of edge-to-edge similarity transformations, allowing the design of ``home-made'' barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.

Joint work with: Craig Gotsman, Kai Hormann and Mirela Ben-Chen`

Short Bio: Ofir Weber received his Ph.D. from the Department of Computer Science, Technion in 2010. He is currently a postdoc researcher at the Courant Institute of Mathematical Sciences, New York University. His main research interests are theoretical and computational methods in geometry and their applications to problems in computer graphics, shape/image deformation, parameterization and quadrilateral remeshing.