Complex sentence: At it's heart is the comparison of shape by recognizing in one shape the other; the Grenander abstraction within deformable templates is group action as the comparison mechanism; in CA the group action is diffeomorphic action.
Split into simpler sentences: At it's heart is the comparison of shape by recognizing in one shape the other. The Grenander abstraction within deformable templates is group action as the comparison mechanism; in CA the group action is diffeomorphic action.