The examples here show snapshots over deformation iterations over minimization via scale-cascaded alignment, for a template (left column of example (a) or (b)), and the target (column marked final). SCA seeks to produce the most parsimonious deformation solution, whilst other approaches in the literature embed specific constraints.  To see how, read further... 

Field Alignment solutions depend strongly on the nature of constraints imposed. One common constraint for example is “smoothness” and optionally “divergence” -- for example Laplace-Beltrami flow.  One consequence is that complex explanations are produced for simple deformations. For example,  this video shows  how smoothness constraints produce a complex explanation for what should be just a rotation and translation.

Scale-cascaded alignment parameterizes deformation as a turbulent spectrum expanded as a Gabor basis. It then finds solutions from the smallest wave-number wavelet to the largest one, cascading across scale and thus returning the simplest explanation first. The result is to automatically produce the most dominant modes of motion, without explicit parameterization. The solution is perfect here for our cross:

Scale-cascaded alignment enables better tracking and deformation based object recognition. It enables better data assimilation by field alignment. You can read the papers here and watch examples: