Image processing utilizing numerical calculation of fractional exponential
powers of a diagonalizable numerical transform operator for use in an iterative
or other larger computational environments. In one implementation, a computation
involving a similarity transformation is partitioned so that one part remains fixed
and may be reused in subsequent iterations. The numerical transform operator may
be a discrete Fourier transform operator, discrete fractional Fourier transform
operator, centered discrete fractional Fourier transform operator, and other operators,
modeling propagation through physical media. Such iterative environments for these
types of numerical calculations are useful in correcting the focus of misfocused
images which may originate from optical processes involving light (for example,
with a lens or lens system) or from particle beams (for example, in electron microscopy
or ion lithography).