Distance Transform Gradient Density Estimation Using the Stationary Phase Approximation

Abstract

The complex wave representation (CWR) converts unsigned two-dimensional (2D) distance transforms into their corresponding wave functions. Here, the distance transform $S(X)$ appears as the phase of the wave function $\phi(X)$---specifically, $\phi(X)=\exp(\frac{iS(X)}{\tau})$, where $\tau$ is a free parameter. In this work, we prove a novel result using the higher-order stationary phase approximation: we show convergence of the normalized power spectrum (squared magnitude of the Fourier transform) of the wave function to the density function of the distance transform gradients as the free parameter $\tau\rightarrow0$. In colloquial terms, spatial frequencies are gradient histogram bins. Since distance transform gradients carry only orientation information (as their magnitudes are identically equal to one almost everywhere), the 2D Fourier transform values mainly lie on the unit circle in the spatial frequency domain as $\tau\rightarrow0$. The proof of the result involves standard integration techniques and requires proper ordering of limits. Our mathematical relation indicates that the CWR of distance transforms is an intriguing, new representation.