The use of spherical harmonics for rigid and nonrigid shape representation is well known. This paper extends the method to surface harmonics defined on domains other than the sphere and to four-dimensional spherical harmonics. These harmonics enable us to represent shapes which cannot be represented as a global function in spherical coordinates, but can be in other coordinate systems. Prolate and oblate spheroidal harmonics and cylindrical harmonics are examples of surface harmonics which we find useful. Nonrigid shapes are represented as functions of space and time either by including the time-dependence as a separate factor or by using four-dimensional spherical harmonics. This paper compares the errors of fitting various surface harmonics to an assortment of synthetic and real data samples, both rigid and nonrigid. In all cases we use a linear least-squares approach to find the best fit to given range data. It is found that for some shapes there is a variation among geometries in the number of harmonics functions needed to achieve a desired accuracy. In particular, it was found that four-dimensional spherical harmonics provide an improved model of the motion of the left ventricle of the heart.
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