THE USE OF 3-DIMENSIONAL AND 4-DIMENSIONAL SURFACE HARMONICS FOR RIGID AND NONRIGID SHAPE RECOVERY AND REPRESENTATION

The use of spherical harmonics for rigid and nonrigid shape representa tion is well known, This paper extends the method to surface harmonics defined on domains other than the sphere and to four-dimensional sphe rical harmonics, These harmonics enable us to represent shapes which c annot be represented as a global function in spherical coordinates, bu t can be in other coordinate systems, Prolate and oblate spheroidal ha rmonics and cylindrical harmonics are examples of surface harmonics wh ich we find useful, Nonrigid shapes are represented as functions of sp ace and time either by including the time-dependence as a separate fac tor or by using four-dimensional spherical harmonics, This paper compa res the errors of fitting various surface harmonics to an assortment o f synthetic and real data samples, both rigid and nonrigid. In all cas es we use a linear least-squares approach to find the best fit to give n range data. It is found that for some shapes there is a variation am ong 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 lef t ventricle of the heart.