Using fractional derivatives as “degree of symmetry” to characterize natural shapes

Abstract

We propose to model shapes and features in order to recognize the unusual, or novel, in the presence of the ordinary especially when we lack a basis set of known patterns with which to conduct a search for the unknown. Our method is based on a novel abstraction called degree of symmetry and suggests that any feature or shape can be decomposed into two orthogonal components: one associated with a fractional degree of symmetry, and the other with a mirror, fractional degree of anti-symmetry. In this framework, we associate the measure of symmetry with fractional-order derivatives, and propose that every feature can be represented as f ( x ) = A G σ 1 α ( x ) + B G σ 2 n ∓ α ( ± x ) where n∈ N odd and G σ α denotes the αth order fractional derivative of a symmetric Schwartz function G with width σ. A/ B, α, and σ 1/ σ 2 are our parameters. These parameters induce a 3-D representation space to detect, classify and characterize features. We show that the footprint of wavelet transform coefficients particularly their decay characteristics help us determine the parameters of our model. We have processed 21 cm interstellar neutral hydrogen spectra collected synoptically by the 150-foot Stanford Radio Telescope to search for unusual structures, and examples obtained using our fractional symmetry transformations illustrate the utility of our approach.