High-accuracy centered fractional fourier transform matrix for optical imaging and other applications
AssigneeLester F. Ludwig
InventorLester F. Ludwig
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Abstract
Methods for numerically generating a centered discrete fractional Fourier transform matrix on a computer, the centered discrete fractional Fourier transform matrix of size N by N where N is an odd integer. Centering is obtained by simple barrel roll operations on eigenvectors. High-accuracy is obtained by numerically calculating the eigenvectors of the discrete fractional Fourier transform matrix from a closed-form mathematical formula and then iteratively performing a Gram-Schmidt orthogonalization procedure until a resulting set of improved-orthogonal eigenvectors is produced that is sufficiently orthogonal.
Claims20
A method for numerically generating a centered discrete fractional Fourier transform matrix on a computer for use in processing an image, the centered discrete fractional Fourier transform matrix of size N by N where N is an odd integer, the method comprising:
- numerically calculating the N eigenvectors of an N by N discrete fractional Fourier transform matrix from a closed-form mathematical formula, the calculation performed on a computer;
- performing a barrel shift operation on each of the N eigenvectors to produce N shifted eigenvectors;
- performing a Gram-Schmidt orthogonalization procedure on the N shifted eigenvectors to produce a first set of improved-orthogonal shifted eigenvectors, the Gram-Schmidt orthogonalization procedure;
- testing the resulting first set of improved-orthogonal shifted eigenvectors for mutually orthogonality performed on the computer;
- if the first set of improved-orthogonal shifted eigenvectors does not possess enough mutually orthogonality, applying another Gram-Schmidt orthogonalization procedure on the first set of improved-orthogonal shifted eigenvectors to produce a second set of improved-orthogonal shifted eigenvectors, and
- testing the resulting second set of improved-orthogonal shifted eigenvectors for mutually orthogonality;
- wherein if the first set of improved-orthogonal shifted eigenvectors does not possess enough mutually orthogonality, applying another Gram-Schmidt orthogonalization procedure, and testing the resulting improved-orthogonal shifted eigenvectors for mutually orthogonality, continuing until a resulting set of improved-orthogonal shifted eigenvectors is sufficiently orthogonal, and
- wherein the resulting set of improved-orthogonal shifted eigenvectors that is sufficiently orthogonal is used to create a centered discrete fractional Fourier transform matrix for use in processing the image.
A method for numerically generating a centered discrete fractional Fourier transform matrix on a computer for use in processing an image, the centered discrete fractional Fourier transform matrix of size N by N where N is an odd integer, the method comprising:
- numerically calculating the N eigenvectors of an N by N discrete fractional Fourier transform matrix from a closed-form mathematical formula, the calculation performed on a computer;
- performing a Gram-Schmidt orthogonalization procedure on the N shifted eigenvectors to produce a first set of improved-orthogonal shifted eigenvectors, the Gram-Schmidt orthogonalization procedure performed on the computer;
- testing the resulting first set of improved-orthogonal shifted eigenvectors for mutually orthogonality;
- if the first set of improved-orthogonal shifted eigenvectors does not possess enough mutually orthogonality, applying another Gram-Schmidt orthogonalization procedure on the first set of improved-orthogonal shifted eigenvectors to produce a second set of improved-orthogonal shifted eigenvectors, and
- testing the resulting second set of improved-orthogonal shifted eigenvectors for mutually orthogonality;
- wherein if the first set of improved-orthogonal shifted eigenvectors does not possess enough mutually orthogonality, applying another Gram-Schmidt orthogonalization procedure, and testing the resulting improved-orthogonal shifted eigenvectors for mutually orthogonality, continuing until a resulting set of improved-orthogonal shifted eigenvectors is sufficiently orthogonal;
- wherein the resulting set of improved-orthogonal shifted eigenvectors that is sufficiently orthogonal is subjected to a barrel shift operation on each of the N eigenvectors, resulting in N shifted eigenvectors, and
- wherein the N shifted eigenvectors are used to create a centered discrete fractional Fourier transform matrix for use in processing the image.
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