  The parameterization of open and closed anatomical surfaces is of fundamental
importance in many biomedical applications. Spherical harmonics, a set of basis
functions defined on the unit sphere, are widely used for anatomical shape
description. However, establishing a one-to-one correspondence between the
object surface and the entire unit sphere may induce a large geometric
distortion in case the shape of the surface is too different from a perfect
sphere. In this work, we propose adaptive area-preserving parameterization
methods for simply-connected open and closed surfaces with the target of the
parameterization being a spherical cap. Our methods optimize the shape of the
parameter domain along with the mapping from the object surface to the
parameter domain. The object surface will be globally mapped to an optimal
spherical cap region of the unit sphere in an area-preserving manner while also
exhibiting low conformal distortion. We further develop a set of spherical
harmonics-like basis functions defined over the adaptive spherical cap domain,
which we call the adaptive harmonics. Experimental results show that the
proposed parameterization methods outperform the existing methods for both open
and closed anatomical surfaces in terms of area and angle distortion. Surface
description of the object surfaces can be effectively achieved using a novel
combination of the adaptive parameterization and the adaptive harmonics. Our
work provides a novel way of mapping anatomical surfaces with improved accuracy
and greater flexibility. More broadly, the idea of using an adaptive parameter
domain allows easy handling of a wide range of biomedical shapes.
