Abstract

Rules are elaborated by which curvature extrema can be used to infer deforming processes that have acted on a 2-D shape or a 3-D shape parametrized as a generalized cylinder. It is claimed that the inference of processes from extrema is mediated by a local symmetry analysis, but we find that the Blum symmetric axis transform and the Brady smooth local symmetry give incorrect results in 50 % of the cases. Thus we develop an alternative local symmetry analysis which overcomes this problem and yields a diagram of processes that have acted on the shape. We then develop a process-grammar of only six types of operation to express the deformational relationship between any two shapes such that one shape is described as the extrapolation of processes inferred in the other. Formalistically, the grammar expresses any deformation as a transformation of process-diagrams produced by the symmetry analysis. Functionally, the grammar can be regarded as explaining the curvature extrema in terms of a sequence of psychologically meaningful deformations. It is also found that the grammar organizes shape space into several intersecting strata-systems, each of which represents a history of successive modification by process extrapolation.

© 1986 Optical Society of America