The Complex Wave Representation of Distance Transforms

Abstract

The complex wave representation (CWR) converts unsigned 2D distance transforms into their corresponding wave functions. The underlying motivation for performing this maneuver is as follows: the normalized power spectrum of the wave function is an excellent approximation (at small values of Planck’s constant—here a free parameter τ) to the density function of the distance transform gradients. Or in colloquial terms, spatial frequencies are gradient histogram bins. Since the distance transform gradients have only orientation information, the Fourier transform values mainly lie on the unit circle in the spatial frequency domain. We use the higher-order stationary phase approximation to prove this result and then provide empirical confirmation at low values of τ. The result indicates that the CWR of distance transforms is an intriguing and novel shape representation.Keywordsdistance transformsVoronoiHamilton-Jacobi equationSchrödinger wave functioncomplex wave representation (CWR)stationary phase (method of)gradient densitypower spectrum