The Ultimate Phase correlation - American History Information Guide and Reference

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

$\textbf{I}_a = \mathcal{F}\{i_a\}, \; \textbf{I}_b = \mathcal{F}\{i_b\}$

$NCS = \frac{ \textbf{I}_a \textbf{I}_b^*}{|\textbf{I}_a \textbf{I}_b^*|}$

$PC = \mathcal{F}^{-1}\{NCS\}$

(Δ x,Δ y) = a r g m a x Δ x,Δ y {P C}

Proof

The technique is based on the Fourier shift theorem.

$i_a(x,y), \; i_b(x,y) = i_a(x - \Delta x, y - \Delta y)$

$I_a(u,v), \; I_b(u,v) = I_a(u,v) e^{-2 \pi i (\frac{u \Delta x}{M} + \frac{v \Delta y}{N}) }$

$NCS = \frac{ \textbf{I}_a \textbf{I}_b^*}{|\textbf{I}_a \textbf{I}_b^*|} = e^{2 \pi i (\frac{u \Delta x}{M} + \frac{v \Delta y}{N}) }$

P C = δ(x - Δ x, y - Δ y)

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

Categories: Computer vision

Last updated: 10-16-2005 13:06:31

Your American History Reference Guide! - Phase correlation

Phase correlation

Method

Mathematical derivation

Proof

Example

References

Your American History Reference Guide!
- Phase correlation