Fourier-space processing has been shown above to offer a powerful way to remove periodic noise from images. It is also used to perform convolutions, such as Gaussian smoothing or high-pass sharpening filters, more efficiently than can be done with large kernels applied directly to the pixel values. Deconvolution in which out-of-focus images are restored was also shown above.

Another very useful Fourier-space technique is cross-correlation. This requires two images, one of a scene containing objects of interest that may be difficult to locate or count because of their appearance or because of a complex surroundings, including camouflage. The second image contains just a single object of the type being sought. Fourier transforms of the two images are multiplied together with a phase shift and the result re-transformed to the pixel domain, to produce bright spots where the objects were located. The top hat filter, introduced above, may then be used to locate the spots.

In the Cross Correlation interactive tutorial, although human recognition of the various letters is easy, counting each one is difficult because of the irregular layout. Using each letter as a target, performing cross correlation, and applying a top hat filter produces a result that makes the counting process simple.

	Interactive Tutorial
Cross Correlation Discover how cross correlation can be used to locate simple targets.

A more realistic application appears in the Applying Cross Correlation tutorial, which shows an image of bubbles, with illumination from one side that produces bright and dark regions that make automatic thresholding to detect the features difficult. Cross correlation with an image of a single bubble locates the features, even though they vary somewhat in size and texture.

	Interactive Tutorial
Applying Cross Correlation Explore the use of cross correlation to locate a complex target.

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