Correction of mis-focus in recorded images using centered discrete fractional fourier transformations with high-accuracy orthonormal eigenvectors

A method for correcting mis-focus of an image, the method comprising:

• receiving an image file at a numerical processor, the image file comprising a portion of the image in an original mis-focused condition;
• selecting a numerical value for a variable α;
• beginning a first iteration of a numerical procedure executing on the numerical processor;
• in the numerical procedure, applying a two-dimensional centered discrete fractional Fourier transform operator of power α to the image file and applying a phase correction operator associated with the two-dimensional centered discrete fractional Fourier transform matrix of power β−α to the image file;
• obtaining a modified portion of the image file from the above operations;
• inspecting the modified portion of the image file and determining a change in the mis-focused condition;
• based on the determined change, adjusting numerical value of α;
• and
• beginning a next iteration of the numerical procedure on the numerical processor;
• wherein if numerical values for the variable α between the real values of 0 and β do not result in a desired outcome, the numerical value for the variable α is adjusted to an imaginary value of the form iχ where χ is the imaginary part of α, wherein the two-dimensional centered discrete fractional Fourier transform operator is represented as a 4-dimensional array.

A method for correcting a mis-focus of image file created from coherent imaging using centered fractional Fourier transforms or mathematical equivalents, the method comprising:

• receiving the mis-focused image at a numerical processor;
• selecting a first numerical value for a variable α
• iterating a numerical procedure using the variable α;
• in the numerical procedure, applying a centered discrete fractional Fourier transform of power α and a phase restore operator associated with a centered discrete fractional Fourier transform of power β−α to the image file to produce a modified image;
• and
• determining a change in the mis-focused image and using the change in the mis-focused image in adjusting α for a next iteration,
• wherein if values for α between 0 and real number β do not result in a desired outcome, α is adjusted to an imaginary value of the form iχ where χ is the imaginary part of α.