Correction of over-focus in digital images using centered discrete imaginary-power fractional fourier transformations with high-accuracy orthonormal eigenvectors
View Patent ↗A method for correcting overfocus of a digital image created from coherent imaging using centered fractional Fourier transforms or mathematical equivalents is described. A received image is presented to a numerical processor, and a first numerical value for an imaginary power variable is selected and used in an iterative algorithm, numerical procedure, system architecture, etc. A centered discrete fractional Fourier transform of an imaginary power and a phase restore operator associated are applied to the image file to produce a modified image. A change in mis-focused is determined and used in adjusting the specified imaginary power for a next iteration.
1. 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, the mis-focused condition comprising an over-focus condition for at least a portion of the image;
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 reflection operator composed with a two-dimensional centered discrete fractional Fourier transform operator of imaginary power i χ to the image file and applying a phase correction operator associated with the two-dimensional centered discrete fractional Fourier transform matrix of power 2−i χ to the image file, where is a real number and i is the square root of −1;
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 χ ; and
beginning a next iteration of the numerical procedure on the numerical processor;
wherein a specific value of χ is chosen providing the best correction of the mis-focused condition.
2. The method of claim 1 , wherein the two-dimensional centered discrete fractional Fourier transform comprises two one-dimensional centered discrete fractional Fourier transform matrices, each such that the power of α=1 results in a centered discrete Fourier transform matrix and the power of α=2 results in a reflection matrix.
3. The method of claim 1 , wherein the reflection matrix reverses the sequence order of the elements of a vector on which the reflection matrix operates.
4. The method of claim 1 , wherein the two-dimensional centered discrete fractional Fourier transform operator of power iχ is applied after applying the phase correction operator associated with a two-dimensional centered discrete fractional Fourier transform matrix of power 2−iχ.
5. The method of claim 1 , wherein the two-dimensional centered discrete fractional Fourier transform operator of power iχ is applied before applying the phase correction operator associated with a two-dimensional centered discrete fractional Fourier transform matrix of power 2−iχ.
6. The method of claim 1 , wherein number χ is a positive number.
7. The method of claim 1 , wherein number χ is a negative number.
8. The method of claim 1 , wherein the image is an image created by coherent imaging.
9. The method of claim 1 , wherein the image is an image created by electron imaging.
10. The method of claim 1 , wherein the image is an image created by light imaging.
11. The method of claim 1 , wherein the two-dimensional centered discrete fractional Fourier transform operator is represented as a 4-dimensional array.
12. The method of claim 1 , wherein the two-dimensional centered discrete fractional Fourier transform operator is represented as a 4-dimensional tensor.
13. The method of claim 1 , wherein the two-dimensional centered discrete fractional Fourier transform operator is applied as a first one-dimensional centered discrete fractional Fourier transform matrix operating on columns of the image file and a second one-dimensional centered discrete fractional Fourier transform matrix operating on rows of the image file.
14. The method of claim 1 , wherein the phase correction operator is a 4-dimensional tensor.
15. The method of claim 1 , wherein the phase correction operator is a 4-dimensional array.
16. The method of claim 1 , wherein the phase correction is applied as a first one-dimensional phase correction operator matrix on columns of the at least portions of the image file and a second one-dimensional phase correction operator matrix operating on rows of the at least portions of the image file.
17. The method of claim 1 wherein the modified portion of the image file comprising the best correction of the mis-focused condition is provided as an output.