The following image demonstrates the usage of phase correlation to determine relative translative movement between two images corrupted by independent Gaussian noise. The image was translated by (30,33) pixels. Accordingly, one can clearly see a peak in the phase-correlation representation at approximately (30,33).
Given two input images and :
Where is the Hadamard product (entry-wise product).
Obtain the normalized cross-correlation by applying the inverse Fourier transform.
Determine the location of the peak in .
Commonly, interpolation methods are used to estimate the peak location to non-integer values, despite the fact that the data are discrete and is often termed 'subpixel registration'. Because the Fourier representation of the data has already been computed, it is especially convenient to use the Fourier shift theorem with real-valued shifts for this purpose. It is also possible to infer the peak location from phase characteristics in Fourier space without the inverse transformation, as noted by Stone 
The method is based on the Fourier shift theorem. Let the two images and be circularly-shifted versions of each other:
(where the images are in size).
Then, the discrete Fourier transforms of the images will be shifted relatively in phase:
One can then calculate the normalized cross-power spectrum to factor out the phase difference:
since the magnitude of an imaginary exponential always is one, and the phase of always is zero.
The inverse Fourier transform of a complex exponential is a Kronecker delta, i.e. a single peak:
This result could have been obtained by calculating the cross correlation directly. The advantage of this method is that the discrete Fourier transform and its inverse can be performed using the fast Fourier transform, which is much faster than correlation for large images.
Unlike many spatial-domain algorithms, the phase correlation method is resilient to noise, occlusions, and other defects typical of medical or satellite images.
The method can be extended to determine rotation and scaling differences between two images by first converting the images to log-polar coordinates. Due to properties of the Fourier transform, the rotation and scaling parameters can be determined in a manner invariant to translation.
In practice, it is more likely that will be a simple linear shift of , rather than a circular shift as required by the explanation above. In such cases, will not be a simple delta function, which will reduce the performance of the method. In such cases, a window function (such as a Gaussian or Tukey window) should be employed during the Fourier transform to reduce edge effects, or the images should be zero padded so that the edge effects can be ignored. If the images consist of a flat background, with all detail situated away from the edges, then a linear shift will be equivalent to a circular shift, and the above derivation will hold exactly. The peak can be sharpened by using edge or vector correlation. 
For periodic images (such as a chessboard), phase correlation may yield ambiguous results with several peaks in the resulting output.
Phase correlation is the preferred method for television standards conversion, as it leaves the fewest artifacts.
- Harold S. Stone, "A Fast Direct Fourier-Based Algorithm for Subpixel Registration of Images", IEEE Transactions on Geoscience and Remote Sensing, V. 39, No. 10, Oct. 2001, pp.2235-2242.
- 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
- B. S Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration”, IEEE Transactions on Image Processing 5, no. 8 (1996): 1266–1271.