Global image operations

A global operation on an image is a mapping of all input pixels f(m,n) into an output image g(i,j). Linear transformations (usually invertible) can be written:

The function O(i, j; m, n) is a function of the input and output coordinates, of the row coordinates i,m and the column coordinates j,n. Particularly interesting are the linear transformations with separable kernels:

in which case the two-dimensional transformation can be executed as the succession of two one-dimensional transforms, columns first and rows next:

or, in matrix notation, File:Hepa img424.gif .

All image transformations mentioned under orthogonal functions are of this type.

If F is an File:Hepa img425.gif image, this linear transformation represents O(N⁴) operations (multiplication and additions). For a Fourier transform the operations are complex. For a reasonably large N this becomes in practice a problem of computing time. If O is separable as above, the number of operations is reduced to 2N³. For further drastic reductions, Fast Transforms.

Another global image processing operation is the Hough transform.