Chapter 5 - Nonisometric motion analysis by variation of shape spectrum

Abstract

This chapter presents a novel approach based on spectral geometry to quantify and recognize non-isometric deformations of 3D surfaces by mapping two manifolds. The method can determine multi-scale, non-isometric deformations through the variation of Laplace–Beltrami spectrum of two shapes. Given two triangle meshes, the spectra can be varied from one to another with a scale function defined on each vertex. The variation is expressed as a linear interpolation of eigenvalues of the two shapes. In each iteration step, a quadratic programming problem is constructed, based on our derived spectrum variation theorem and smoothness energy constraint, to compute the spectrum variation. The derivation of the scale function is the solution of such a problem. Therefore, the final scale function can be solved by integral of the derivation from each step, which, in turn, quantitatively describes non-isometric deformations between two shapes. The extensive experiments on synthetic and real data show the accuracy and effectiveness of the method. The comparison to spatial registration-based methods, e.g., non-rigid Iterative Closest Point (ICP) and voxel-based method, also demonstrate the advantages of the method.