  Construction of spline surfaces from given boundary curves is one of the
classical problems in computer aided geometric design, which regains much
attention in isogeometric analysis in recent years and is called domain
parameterization. However, for most of the state-of-the-art parameterization
methods, the rank of the spline parameterization is usually large, which
results in higher computational cost in solving numerical PDEs. In this paper,
we propose a low-rank representation for the spline parameterization of planar
domains using low-rank tensor approximation technique, and apply
quasi-conformal map as the framework of the spline parameterization. Under
given correspondence of boundary curves, a quasi-conformal map with low rank
and low distortion between a unit square and the computational domain can be
modeled as a non-linear optimization problem. We propose an efficient algorithm
to compute the quasi-conformal map by solving two convex optimization problems
alternatively. Experimental results show that our approach can produce a
bijective and low-rank parametric spline representation of planar domains,
which results in better performance than previous approaches in solving
numerical PDEs.
