Abstract

Fast wave-front reconstruction methods are becoming increasingly important, for example, in large astronomical adaptive optics systems and high spatial resolution shear interferometry, where pseudoinverse matrix methods scale poorly with problem size. Wave-front reconstruction from difference measurements can be achieved by use of fast implementations of the discrete Fourier transform (DFT), obtaining performance comparable with that of the pseudoinverse in terms of the noise propagation coefficient. Existing methods that are based on the use of the DFT give exact results (in the absence of noise) only for the particular case in which the shear is a divisor of the number of samples to be reconstructed. We present two alternate solutions for the more general case when the shear is any integer. In the first solution the dimensions of the problem are enlarged, and in the second the problem is subdivided into a set of smaller problems with shear amplitude equal to one. We also show that the retrieved solutions have minimum norm and calculate the noise propagation coefficient for both methods. The proposed algorithms are implemented and timed against pseudoinverse multiplication. The results show a speed increase by a factor of 50 over the pseudoinverse multiplication for a grid with N = 3 × 103 samples.

© 2004 Optical Society of America

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References

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2002

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1999

1991

1986

1980

1977

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Brase, J. M.

C. Ghiglia, D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998).

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Elster, C.

Fried, D. L.

Gavel, D. T.

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Hudgin, R. H.

Koliopoulos, C. L.

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Poyneer, L. A.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998).

R. Freischlad, K.

Roddier, C.

Roddier, F.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

Southwell, W. H.

V. Oppenheim, A.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Weingärtner, I.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, D. Sorensen, LAPACK Users’ Guide, 3rd ed. (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1999).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, New York, 1998).

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Figures (9)

Fig. 1
Fig. 1

Illustration of the method for extending the difference data outside the pupil by copying the difference values in the boundary of the pupil to the area outside the pupil. The black spots are the points over which the wave front ϕ is to be evaluated, the gray area indicates the pupil P, and the curved arrows indicate the two values of the grid that define the difference with coordinates at the origin of the arrow.

Fig. 2
Fig. 2

Example of the interlaced subgrids for a grid with dimensions N x = 6 by N y = 7 and shears s x = 3 and s y = 2. The relative phases between points in different subgrids are undetermined by the data.

Fig. 3
Fig. 3

Plot of the number of real multiplications needed to perform the wave-front reconstruction over a square grid with a total of N points and a shear s < N by means of the pseudoinverse (solid line) and the FFT algorithms (dashed line).

Fig. 4
Fig. 4

Plot of wave-front reconstruction time performance over square grids with unit shears by means of the pseudoinverse (solid line), the extension (dashed line), and the subdivision methods (dotted line) for different numbers of grid sample points. Note that for the pseudoinverse this does not include the time for the evaluation of the pseudoinverse itself, which is only carried out once for a given geometry.

Fig. 5
Fig. 5

Plot of the noise coefficient versus the number of grid samples for wave-front reconstruction in (a) square and (b) circular pupils for pseudoinverse reconstruction (solid line) and DFT reconstruction methods (dotted line).

Fig. 6
Fig. 6

Effect of extending the data by enlarging a circular pupil of 32 samples across and with unit shear against N ext, the number of columns and rows of padding for extension from two sides of the grid only (dots) and for symmetric extension from all four sides of the grid (circles). The noise coefficient of the pseudoinverse is shown as a solid line for reference.

Fig. 7
Fig. 7

Effect of extending the data by enlarging a square pupil of 32 samples across and with unit shear against N ext, the number of columns and rows of padding for extension from two sides of the grid only (dots) and for symmetric extension from all four sides of the grid (circles). The noise coefficient of the pseudoinverse is shown as a solid line for reference.

Fig. 8
Fig. 8

Spatial distribution of the noise variance for DFT methods for a square pupil of 32 samples across.

Fig. 9
Fig. 9

Spatial distribution of the noise variance for DFT methods for a centered square pupil of 32 samples across and 2 columns and rows of padding around each of the four sides of the pupil.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

Dxn, m=ϕn+sx, m-ϕn, m,
Dyn, m=ϕn, m+sy-ϕn, m,
ε2=n,m|Sxn, m-Dxn, m|2+|Syn, m-Dyn, m|2.
ϕ˜k, l=exp-2πi ksxNx-1S˜xk, l+exp-2πi lsyNy-1S˜yk, l4sin2π ksxNx+sin2π lsyNy,
SxNx-sx+kx, m=-j=1Nxsx-1 Sxsxj-1+ky, m,
Syn, Ny-sy+ky=-j=1Nysy-1 Syn, syj-1+ky,
Dxn, m+Dyn+sx, m-Dxn, m+sy-Dyn, m=0.
ϕ˜k, l=0whenksxNxandlsyNyare integersexp-2πi ksxNx-1D˜xk, l+exp-2πi lsyNy-1D˜yk, l4sin2π ksxNx+sin2π lsyNyelsewhere.
C=1Npupilω RωRω,

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