Phase correlation

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Phase correlation is a simple, fast way to find how much one image has shifted relative to another. It uses the Fourier transform to compare the two images in frequency space, which makes it robust to some noise and lighting changes.

How it works in brief:
- Optionally apply a window to reduce edge effects.
- Compute the 2D Fourier transform of both images.
- Form the cross-power spectrum by multiplying the first transform with the complex conjugate of the second and then normalizing to keep only the phase information.
- Apply the inverse Fourier transform to get a cross-correlation map.
- Locate the peak in this map—the peak position gives the translation between the images. Subpixel accuracy can be obtained by interpolating around the peak.

Why it’s useful:
- The shift corresponds to a simple phase change in frequency space, so the peak is a clear indicator of offset.
- Using FFTs makes it fast for large images.
- It works well in the presence of noise and other common image imperfections.

Notes and extensions:
- If the shift isn’t circular, windowing or zero-padding helps reduce border effects.
- You can handle rotation and scaling by transforming images to log-polar coordinates; then rotation and scale become translations.
- For periodic images, multiple peaks can appear, which may complicate peak detection.

Phase correlation is widely used for registering images and is valued for its speed and robustness in real-world imaging tasks.