Invariant curve/surface alignment in the presence of affine and some nonlinear transformations with application in image registrations

This thesis focuses on the development of a new nonlinear registration technique for biomedical image. We introduce a non-iterative geometric-based method to align both 2D sectional contours and 3D surfaces into standard coordinate system (SCS), which is based on a novel set of curve/surface landmarks, which are intrinsic and are computed from the differential geometry of the curve/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 smoothed out the surface prior to the computation of the landmarks. The alignment is achieved by establishing correspondences between the landmarks after a conformal sorting based on derived absolute invariant (volumes/areas confined between parallelogram/parallelepipeds spanned by sets of the landmark point triplets/quadruplets). The method is tested for intra- and inter-subject alignments while entertaining cubic nonlinear transformations. The intra-subject 2D-to-2D registration was performed on MRI of human liver. The inter-subject (animal) 3D-to-3D alignment was tested on rat brain data. The experiments have shown that the purposed methods are robust and promising even in the presence of noise.