The Complex Wave Representation of Distance Transforms Karthik S. Gurumoorthy, Anand Rangarajan and Arunava Banerjee Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, USA {ksg,anand,arunava}@cise.ufl.edu 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. Key words: distance transforms, Voronoi, Hamilton-Jacobi equation, Schr¨ odinger wave function, complex wave representation (CWR), stationary phase (method of), gradient density, power spectrum 1 Introduction Over the past three decades, image analysis has borrowed numerous formalisms, methodologies and techniques from classical physics. These include variational and level-set methods for active contours and surface reconstruction [1, 12], Markov Chain Monte Carlo (MCMC) [17], mean-field methods in image segmentation and matching [9, 20, 8], fluidic flow formulations for image registration [6] etc. Curiously, there has been very little interest in adapting approaches from quantum mechanics. This is despite the fact that linear Schr¨ odinger equations are the quantum counterpart to nonlinear Hamilton-Jacobi equations [4] and the knowledge that the quantum approaches the classical as Planck’s constant ~ tends to zero [2]. The principal theme in this work is the introduction of complex wave representations (CWRs) of shapes. We begin by reconstructing the well known bridge between the Hamilton-Jacobi and Schr¨ odinger equations as adapted to the problem of Euclidean distance transform computation. As expected, the familiar nonlinear, static Hamilton-Jacobi equation emerges from a linear, static Schr¨ odinger equation in the limit as τ → 0. This paves the way for the complex wave representation (CWR) of distance transforms: here the wave function ψ(x, y) is equal to exp n i S(x,y) τ o where S (x, y) is the distance transform in 2D. Since distance transform gradients (when they exist) are unit vectors [13], their appropriate representation is the space of orientations. The centerpiece of this work is the following statement 1