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, one or more selected precomputed fractional powers of the transform operator are stored in memory. Computation is simplified by associating precomputed powers of the numerical transform operator with the binary values of individual digits in a binary fraction representation of the fractional exponent. The numerical transform operator may be a discrete Fourier transform operator, discrete fractional Fourier transform operator, and the like. This numerical calculation is 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 particle beams (for example, in electron microscopy or ion lithography).