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 τ → 0. In colloquial terms, spatial frequencies are gradient histogram bins. Since distance transform gradients carry...

The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. In this paper, we will introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the c...

Very broadly speaking, the Fourier transform is a systematic way to decompose “generic” functions into a superposition of “symmetric” functions. These symmetric functions are usually quite explicit (such as a trigonometric function sin(nx) or cos(nx)), and are often associated with physical concepts such as frequency or energy. What “symmetric” means here will be left vague, but it will usually...