Phase correlation is a frequency domain approach to determine the relative translative movement between two images.

Contents

1 Method 2 Mathematical derivation 3 Proof 4 Example 5 References

Method

Given two input images a and b:

• Apply a window function (e.g the Hamming window) on both images to reduce edge effects
• Calculate the discrete 2D Fourier transform of both images
• Take the conjugate of the second image
• Multiply the Fourier transforms together elementwise
• Normalize this produce elementwise (yielding a normalized cross power spectrum )
• Inverse transform the normalized cross power spectrum
• Determine peak in inverse transform (possible using sub-pixel methods ).

Mathematical derivation

(Δx,Δy) = argmax_Δx,Δy{PC}

Proof

The technique is based on the Fourier shift theorem.

PC = δ(x - Δx,y - Δy)

Example

The following image demonstrates the usage of phase-correlation to determe relative translative movement between two images corrupted by independent gaussian noise. One can clearly see a peak in the phase-correlation spectrum approximately at (30,33).

References

• E. De Castro and C. Morandi "Registration of Translated and Rotated Images Using Finite Fourier Transforms", IEEE Transactions on pattern analysis and machine intelligence, Sept. 1987