A limitation theorem for the differentiable prototypification of shape
In previous papers [the author, Biol. Cybern. 51, 141-153 (1984; Zbl 0572.92030), Comput. Vision Graphics Image Process. 31, 201-241 (1985; Zbl 0623.68080)] it was shown that subjects decompose the global linear prototypification of shape into separate stretch, shear, and rotation components, i.e., the Iwasawa decomposition of the full transformation. The purpose of the present paper is to extend these decompositional results to prototypification that varies differentiably across a region - a process that occurs in complex shape. As a first step we require that local symmetry axes become local eigenspaces in any assigned local transformations. This condition, called the Interaction Law, is substantially validated by previous experimental work. Clearly, the use of the Interaction Law requires, as input to the prototypification process, a local symmetry analysis. The analysis is the Smoothed Local Symmetry (SLS) of \textit{M. Brady} [Criteria for representations of shape. In A. Rosenfield and J. Beck (eds.), Human and machine vision (1983)]. Four alternative local frames are considered in order to characterize the SLS by a moving frame, but we find that only one of the alternatives satisfies the Interaction Law; i.e., the frame is a basis of eigenvectors that are local symmetry vectors. With this basis, the SLS is characterized as a trajectory through the General Linear Group. Prototypification can then take place by applying the Iwasawa decomposition at each point of the group trajectory, and factorizing the Iwasawa subgroups. The culmination of this analysis is the proof of a limitation theorem, which states that the factorization of one subgroup necessarily involves the factorization of one of the other subgroups. Thus one level of prototypification is impossible. Furthermore we find that two levels of prototypification can lead only to a global or flexed symmetry. Thus the theorem shows that the shapes that can result from the local use of the Iwasawa decomposition are much more restricted than the shapes that can result from the global use of the decomposition. Nevertheless, in the local case the resulting shapes are highly salient as prototypes, which strongly validates our method. The method is further validated by showing that if a moving frame were chosen in which the Interaction Law did not hold, the resulting shapes would not be psychological prototypes.
global linear prototypification of shape
trajectory through the General Linear Group
scientific article; zbMATH DE number 4027259