The Schrödinger distance transform (SDT) for point-sets and curves

Despite the ubiquitous use of distance transforms in the shape analysis literature and the popularity of fast marching and fast sweeping methods—essentially HamiltonJacobi solvers, there is very little recent work leveraging the Hamilton-Jacobi to Schrödinger connection for representational and computational purposes. In this work, we exploit the linearity of the Schrödinger equation to (i) design fast discrete convolution methods using the FFT to compute the distance transform, (ii) derive the histogram of oriented gradients (HOG) via the squared magnitude of the Fourier transform of the wave function, (iii) extend the Schrödinger formalism to cover the case of curves parametrized as line segments as opposed to point-sets, (iv) demonstrate that the Schrödinger formalism permits the addition of wave functions—an operation that is not allowed for distance transforms, and finally (v) construct a fundamentally new Schrödinger equation and show that it can represent both the distance transform and its gradient density—not possible in earlier efforts.