Alignment of a pair of 3-D surfaces is an important problem especially in brain imaging, where scientists are interested in identifying disease specific effects on measures such as cortical thickness (which can be thought of a function defined on the surface).
Vertex to vertex correspondences can be matched by minimizing the L2 norm of wavelet coefficients from the vertex coordinates. (The proof is given in .)
Using sampled wavelet coefficients from a brain surface by the correspondences to a template, and reconstruct using the bases from the template yields a transformed brain surface with the same topology of the template.
1. D. Hammond et al, Wavelets on graphs via spectral graph theory, Applied and Computational Harmonic Analysis, 2011. 2. W. Kim et al, Multi-resolution Shape Analysis via Non-EuclideanWavelets: Applications to Mesh Segmentation and Surface Alignment Problems, CVPR, 2013.