PhaseCorrelation

PhaseCorrelation is a Fourier-domain technique for estimating the relative translational displacement between two signals or images. It derives a translation estimate from the inverse Fourier transform of the normalized cross-power spectrum, producing a sharp impulse-like peak whose location corresponds to the offset between the inputs.

Given two images f and g with Fourier transforms F and G, the cross-power spectrum is computed as (F · conj(G)) / |F · conj(G)|. The inverse transform of this quantity yields the phase-correlation surface; the peak position indicates integer-pixel shift. Subpixel accuracy is commonly obtained by interpolating the peak (parabolic fit, centroid) or by upsampling via zero-padding in the frequency domain. The method is computationally efficient because it leverages fast Fourier transforms and is robust to uniform illumination changes and additive noise, since phase information, rather than magnitude, governs the estimate.

Limitations include sensitivity to large rotations, scale changes, occlusions, and periodic structures that produce ambiguous peaks. Extensions address these issues: log-polar transforms combined with phase correlation can estimate rotation and scale, while windowing, high-pass filtering, and phase-only correlation variants sharpen peaks and reduce artifacts. Applications span image registration, panorama stitching, motion estimation, video stabilization, remote sensing, and medical image alignment. The technique generalizes to one- and multi-dimensional signals and is widely used where shift-only registration and computational efficiency are required.