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The pixels in each column are translated by an amount proportional to the x-index of that column.

By using the Fourier shift theorem (e.g., [1], p. 104) the shearing translation can be implemented in the Fourier domain using phase-shifts. In [18], which studies various methods for implementing shearing, it is shown the best-quality results can be obtained implementing the shears using a one-dimensional FFT.

Thus, a shear parallel to the x-axis can be implemented as

I (SXtt x) = !F—X{!FX {I (x)}ei2ntk'y }, where x is the Fourier transformation in the x-coordinate, 1 is its inverse. A shear parallel to the y-axis is performed similarly:

A rotation matrix is equal to the product of three shearing matrices, where c —s s c