Abstract:

Many applications using triangulated surface models benefit from a planar
parameterization, i.e. a mapping between the surface mesh and the plane.
Mapping of a texture onto a model is an essential part of model
representation in computer graphics. Providing a parameterization simplifies 
surface reconstruction from scanned data and 3D images. It improves the efficiency
of surface mesh generation and the quality of the resulting mesh. Computing
this mapping is a challenging problem, which has been receiving a lot of
attention in the last couple of years.

In the first part of the talk I will describe a new formulation of this problem
and introduce a parameterization method based on it. The Angle Based Flattening
(ABF) method, which I will describe, formulates the mapping problem in terms
of the flat mesh angles, and solves it in the angle space. The ABF algorithm
provides a quasi-conformal mapping and computes the optimal planar domain boundary 
in terms of minimal distortion. This method is guaranteed to compute a continuous
mapping and the numerical solution is proven to converge.

I will then discuss a new post-processing algorithm, aimed at reducing length
distortion of an existing parameterization and apply it to ABF results. The
post-processing is based on computing a mapping from the plane to itself
which has length distortion very similar to that of the ABF parameterization. 
By applying the inverse mapping to the result of the initial parameterization, 
we obtain a new parameterization with low length distortion.  We notice that the 
procedure for computing the inverse mapping can be applied to any other 
(convenient) mapping from the three-dimensional surface to the plane in order 
to improve it.

Time permitting, I will get to the "cutting" part of the talk, and describe
an algorithm for cutting the surfaces, which enables parameterization of
surfaces with high curvature as well as closed surfaces. The addition of seams 
reduces the surface curvature and hence moderates the metric distortion produced 
by the mapping.  The downside of cutting seams in the surface is the mapping 
discontinuity they generate.  The method minimizes the absolute length of the seams 
or alternatively minimizes their visual impact by placing them in less visible 
regions of the model.

(The talk is based on joint works with E. de Sturler and J. Hart.)