# 3-D Brain Surface Registration and Alignment

## Finding brain region correspndence

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).

## Registration by Wavelets in Non-Euclidean Space

Vertex to vertex correspondences can be matched by minimizing the L2 norm of wavelet coefficients from the vertex coordinates. (The proof is given in [2].)

##### Three different brain surfaces, and the eigenfunction of the first surface (template) is projected
onto the second and the third surface by the wavelet registration.

## Surface Alignment by Wavelets in Non-Euclidean Space

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.

##### The second and the third surface is aligned to the first surface (template) by wavelet coefficients
from the registration and the bases from the template surface.
As a result, all three surfaces now have the same topology.
The 20th eigenfunctions from each surface are plotted, which are now the same.

## Acknowledgment

This research is supported by NIH R01AG040396, NIH R01AG021155, NSF RI1116584, NSF RI1252725,
the Wisconsin Partnership Proposal, UW ADRC, and UW ICTR (1UL1RR025011).

## Reference

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.