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Encyclopedia > Phase correlation

Phase correlation is a frequency domain approach to determine the relative translative movement between two images. Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... In common usage, an image (from Latin imago) or picture is an artifact that reproduces the likeness of some subject—usually a physical object or a person. ...

Contents


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).

Window functions are applied to avoid discontinuities at the beginning and the end of a set of data. ... An edge effect is the effect of the juxtaposition of contrasting environment on an ecosystem. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

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}
(Δx,Δy) = argmaxΔx,Δy{PC}

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}) }
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). Image:Phase correlation.png Image File history File links Example Image of the Phase Correlation technique --Fredrik Orderud 21:24, 11 Mar 2005 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...


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

  Results from FactBites:
 
U.S. Patent: 5610940 - Method and apparatus for noncoherent reception and correlation of a continous phase modulated ... (15128 words)
The I correlation signal 408 and the Q correlation signal 410 are coupled to a summer 411, which combines its inputs and produces a unified correlation signal 412.
When the entire correlation sequence appears in the received signal s*(t), the sum of the I correlation signal 408 and the Q correlation signal 410 is at a maximum, and may be compared against a predetermined threshold to allow recognition of the chip sequence.
The outputs of all 32 CPM correlators may be summed and, when the sum is at a predetermined maximum level, the CPM correlator 402 with the highest magnitude output may be chosen by a best-of-M detector or similar means as indicative of the data stream d(t).
  More results at FactBites »

 

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