Chapter 9: Shape Analysis of Left Ventricle Using Invariant 3-D Spherical Harmonics Shape Descriptors

This paper presents a new technique to generate triangular mesh surface parameterization and characterize 3-D surfaces by invariant spherical harmonic shape descriptors of objects with spherical topology. First, the surface is initially parameterized by defining a continuous one-to-one mapping from the surface of the object to the surface of a unit sphere. Then, the initial parameterization is optimized in a constrained optimization procedure. The obtained parameterized surfaces are expanded into spherical harmonics. The series coefficients are estimated in a least squares sense. Based on harmonic analysis and using results from representation theory, we compute the spherical Fourier transform on the unit sphere S² with the group of rotations SO(3) as the acting group. The shift theorem allows us to extract invariant 3-D rotation spherical harmonic shape descriptors. The new procedure is illustrated with modelling the left ventricle using the spherical harmonics model and myocardial scintigraphic data. The invariant shape descriptors are used to quantify the heart pathology level.