Invariant surface alignment in the presence of affine and some nonlinear transformations

In this paper, we introduce a non-iterative geometric-based method to align 3D brain surfaces into standard coordinate system, which is based on a novel set of surface landmarks (e.g., inflection and/or zero torsion points residing on parabolic contours), which are intrinsic and are computed from the differential geometry of the surface. This is in contrast to existing methods that depend on anatomical landmarks that require expert intervention to locate--a very hard task. The landmarks are local and are preserved under affine transformations. To reduce the sensitivity of the landmarks to noise, we use a B-Spline surface representation that smoothes out the surface prior to the computation of the landmarks. The alignment is achieved by establishing correspondences between the landmarks after a sorting of the landmarks based on derived absolute invariants (volumes confined between parallelepipeds spanned by sets of the landmark point quadruplets). The method is tested for intra- and inter-brain alignments while entertaining affine, cubic and fourth-order polynomial nonlinear transformations using distance mapping as well as comparison with an expert alignment, and promising results are obtained. When comparing our automatic alignment with that of an expert we arrived at complete agreement for the more difficult case of partial alignment of sectional slab materials of five rats with an atlas (a whole brain of rat). This perfect alignment was only based on the surface structure for our procedure, whereas it was based on the staining and the external and internal structures for the expert.