Computing arbitrary fractional powers of a transform operator from selected precomputed fractional powers of the operator

1. A computer-implemented method for correcting misfocus in original particle beam image data, the method comprising:

assigning a correction parameter value to a fractional power parameter associated with a fractional Fourier transform operation (“FFTO”) to produce a resulting approximated F FTO;

operating on the original particle beam image data with the approximated FFTO to produce transformed image data;

determining a degree of improvement in the focus of the transformed image data over that of the original particle beam image data;

storing the original particle beam image data as a digital image; and

restoring phase information associated with the original particle beam image data;

wherein the particle beam is an electron beam associated with an electron microscope.

2. The method of claim 1 , further comprising adjusting the correction parameter value, and operating on the original particle beam image data with another resulting approximated FFTO having determined by the adjusted correction parameter value.

3. The method of claim 2 , further comprising: adjusting the correction parameter in response to a user input.

4. The method of claim 2 , further comprising: adjusting the correction parameter in response to a control system.

5. The method of claim 1 , wherein the FFTO is a numerical algorithm.

6. The method of claim 1 , wherein the fractional Fourier transform operation is approximated by any from the group of a Taylor series expansion, a Hermite function expansion, a perturbation approximation, a singular integral approximation, or an infinitesimal generator.

7. The method of claim 1 , wherein the original image data represents data from a first portion of an image, the method further comprising:

correcting misfocus for other image data representing data from a second portion of the image.

8. A system for correcting misfocus in original particle beam image data, said system comprising:

a fractional Fourier transform component for applying at least one fractional Fourier transform operation (“FFTO”) on the original particle beam image data, wherein at least a portion of the original particle beam image data is misfocused in at least one region of interest;

a parameter adjuster configured with said fractional Fourier transform component, said parameter adjuster providing a range of variation of at least one parameter of the at least one fractional Fourier transform operation to improve a corresponding range of misfocus in said original particle beam image data;

wherein the original particle beam image data is introduced to said fractional Fourier transform component and the at least one fractional Fourier transform operation is applied to the original particle beam image data,

resulting in improved particle beam image data that improves the focus of at least a portion of said misfocus in the original particle beam image data, and

wherein said fractional Fourier transform component is realized by a numerical algorithm,

wherein the numerical algorithm realizing the fractional Fourier transform component comprises an approximation.

9. The system of claim 8 , further comprising:

a phase reconstructing element to reconstruct phase information pertaining to the original particle beam image data as part of the correction of the misfocus.

10. The system of claim 8 , further comprising:

memory to store the original particle beam image data as a digital image.

11. The system of claim 8 , wherein the parameter adjuster is controlled by user input.

12. The system of claim 8 , wherein the parameter adjuster is controlled by an optimization algorithm.

13. The system of claim 8 , wherein the approximation employs any from the group of a Taylor series expansion, a Hermite function expansion, a perturbation approximation, a singular integral approximation, or an infinitesimal generator.

14. The system of claim 8 , wherein the original image data represents data from a first portion of an image, the method further comprising:

correcting misfocus for other image data representing data from a second portion of the image.

15. The method of claim 8 , wherein the particle beam is an electron beam associated with an electron microscope.