Abstract

We derive the analytic gradient of a phase retrieval error metric with respect to the sampling factor or the f-number that produced the measured point-spread function. This allows us to efficiently optimize over the sampling factor, thereby improving the accuracy of the phase estimate. Computer simulation results show its effectiveness.

© 2014 Optical Society of America

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References

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  1. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
    [CrossRef]
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  3. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef]
  4. L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).
  5. R. D. Fiete, “Image quality and λfn/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
    [CrossRef]
  6. T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.
  7. T. P. Zielinski, “Robust image-based wavefront sensing,” Ph.D. thesis (University of Rochester, 2011).
  8. A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed. (SIAM, 2008).
  9. A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014).
    [CrossRef]
  10. A. S. Jurling and J. Fienup, “F/# estimation using a chirp z-transform and analytic gradients for wavefront sensing,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.1.
  11. M. D. Bergkoetter and J. R. Fienup, “A computational model for phase retrieval of narrow-band chromatic aberrations,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.2.
  12. L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
    [CrossRef]
  13. L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
    [CrossRef]
  14. D. Bailey and P. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1991).
    [CrossRef]
  15. R. Tong and R. W. Cox, “Rotation of NMR images using the 2D chirp-z transform,” Magn. Reson. Med. 41, 253–256 (1999).
    [CrossRef]
  16. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15, 5631–5640 (2007).
    [CrossRef]
  17. S. T. Thurman and J. R. Fienup, “Phase retrieval with signal bias,” J. Opt. Soc. Am. A 26, 1008–1014 (2009).
    [CrossRef]
  18. S. J. Wright and J. Nocedal, “Quasi-Newton methods,” in Numerical Optimization (Springer, 1999), Chap. 8, p. 198.

2014

2009

2007

R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15, 5631–5640 (2007).
[CrossRef]

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

1999

R. D. Fiete, “Image quality and λfn/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[CrossRef]

R. Tong and R. W. Cox, “Rotation of NMR images using the 2D chirp-z transform,” Magn. Reson. Med. 41, 253–256 (1999).
[CrossRef]

1993

1991

D. Bailey and P. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1991).
[CrossRef]

1982

1970

L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

1969

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[CrossRef]

Acton, D. S.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Aronstein, D. L.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.

Bailey, D.

D. Bailey and P. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1991).
[CrossRef]

Bergkoetter, M. D.

M. D. Bergkoetter and J. R. Fienup, “A computational model for phase retrieval of narrow-band chromatic aberrations,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.2.

Bluestein, L.

L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

Bowers, C. W.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Carey, L.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Contos, A.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Cox, R. W.

R. Tong and R. W. Cox, “Rotation of NMR images using the 2D chirp-z transform,” Magn. Reson. Med. 41, 253–256 (1999).
[CrossRef]

Dean, B. H.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.

Feinberg, L. D.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Fienup, J.

A. S. Jurling and J. Fienup, “F/# estimation using a chirp z-transform and analytic gradients for wavefront sensing,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.1.

Fienup, J. R.

A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014).
[CrossRef]

S. T. Thurman and J. R. Fienup, “Phase retrieval with signal bias,” J. Opt. Soc. Am. A 26, 1008–1014 (2009).
[CrossRef]

J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
[CrossRef]

J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

M. D. Bergkoetter and J. R. Fienup, “A computational model for phase retrieval of narrow-band chromatic aberrations,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.2.

T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.

Fiete, R. D.

R. D. Fiete, “Image quality and λfn/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[CrossRef]

Griewank, A.

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed. (SIAM, 2008).

Hayden, W.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Jurling, A. S.

A. S. Jurling and J. R. Fienup, “Applications of algorithmic differentiation to phase retrieval algorithms,” J. Opt. Soc. Am. A 31, 1348–1359 (2014).
[CrossRef]

A. S. Jurling and J. Fienup, “F/# estimation using a chirp z-transform and analytic gradients for wavefront sensing,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.1.

Lyon, R. G.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Marron, J. C.

Meza, L.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Muffoletto, R. P.

Nocedal, J.

S. J. Wright and J. Nocedal, “Quasi-Newton methods,” in Numerical Optimization (Springer, 1999), Chap. 8, p. 198.

Rabiner, L.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[CrossRef]

Rader, C.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[CrossRef]

Sabatke, E.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Schafer, R.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[CrossRef]

Schulz, T. J.

Schwenker, J.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Seldin, J. H.

Shi, F.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Shields, D.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Shiri, R.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Smith, J. S.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.

Swarztrauber, P.

D. Bailey and P. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1991).
[CrossRef]

Thurman, S. T.

Tohline, J. E.

Tong, R.

R. Tong and R. W. Cox, “Rotation of NMR images using the 2D chirp-z transform,” Magn. Reson. Med. 41, 253–256 (1999).
[CrossRef]

Towell, T.

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

Tyler, J. M.

Walther, A.

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed. (SIAM, 2008).

Wright, S. J.

S. J. Wright and J. Nocedal, “Quasi-Newton methods,” in Numerical Optimization (Springer, 1999), Chap. 8, p. 198.

Zielinski, T. P.

T. P. Zielinski, “Robust image-based wavefront sensing,” Ph.D. thesis (University of Rochester, 2011).

T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.

Appl. Opt.

IEEE Trans. Audio Electroacoust.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17, 86–92 (1969).
[CrossRef]

L. Bluestein, “A linear filtering approach to the computation of discrete Fourier transform,” IEEE Trans. Audio Electroacoust. 18, 451–455 (1970).
[CrossRef]

J. Opt. Soc. Am. A

Magn. Reson. Med.

R. Tong and R. W. Cox, “Rotation of NMR images using the 2D chirp-z transform,” Magn. Reson. Med. 41, 253–256 (1999).
[CrossRef]

Opt. Eng.

R. D. Fiete, “Image quality and λfn/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[CrossRef]

Opt. Express

Proc. SPIE

L. D. Feinberg, B. H. Dean, D. L. Aronstein, C. W. Bowers, W. Hayden, R. G. Lyon, R. Shiri, J. S. Smith, D. S. Acton, L. Carey, A. Contos, E. Sabatke, J. Schwenker, D. Shields, T. Towell, F. Shi, and L. Meza, “TRL-6 for JWST wavefront sensing and control,” Proc. SPIE 6687, 668708 (2007).

SIAM Rev.

D. Bailey and P. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1991).
[CrossRef]

Other

A. S. Jurling and J. Fienup, “F/# estimation using a chirp z-transform and analytic gradients for wavefront sensing,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.1.

M. D. Bergkoetter and J. R. Fienup, “A computational model for phase retrieval of narrow-band chromatic aberrations,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper OW3A.2.

S. J. Wright and J. Nocedal, “Quasi-Newton methods,” in Numerical Optimization (Springer, 1999), Chap. 8, p. 198.

T. P. Zielinski, B. H. Dean, J. S. Smith, D. L. Aronstein, and J. R. Fienup, “Determination of the sampling factor in a phase-diverse phase retrieval algorithm,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2010), paper FWJ3.

T. P. Zielinski, “Robust image-based wavefront sensing,” Ph.D. thesis (University of Rochester, 2011).

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed. (SIAM, 2008).

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Figures (3)

Fig. 1.
Fig. 1.

Simulated (a) true PSFs, (b) true wavefront, (c) fit PSFs, and (d) fit wavefront. The color bar is in units of waves.

Fig. 2.
Fig. 2.

(a) Q, (b) RMS wavefront error from truth, and (c) phase-retrieval error metric as a function of iteration number.

Fig. 3.
Fig. 3.

Capture range for Q retrieval. The color bar indicates the fraction of cases that converged to a correct solution for Q. The horizontal axis is the true value of Q and the vertical axis is the initial guess for Q.

Equations (49)

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

W=nanZn,
g=Aexp(i2πλW),
G=FFT{g}.
I=|G|2=GG*.
E=m(ImDm)2,
a^=argmina[E(a)].
I¯=2(ID),G¯=2GI¯,g¯=IFFT{G¯},W¯=2πλI{g*g¯},a¯n=p(W¯Zn)p,
α=1QNc,
Xm=n=0N1xne2πi(mΔm)(nΔn)α.
nm=n2+m2(nm)22.
(nΔn)(mΔm)=12[(nΔn)2+(mΔm)2(mnΔm+Δn)2].
βn=exp[iπ(nΔn)2α],β=exp(iπn^2α),
γm=exp[iπ(mΔm)2α],γ=exp(iπm^2α),
zmn=exp[iπ(mnΔm+Δn)2α],zp=exp[iπ(pΔm+Δn)2α],
Xm=γmn=0N1xnβnzmn.
X=γ[(xβ)z].
LN+M1,
w^=extend(w,L)
w=truncate(w^,M)
z^p={exp{iπ[pΔm+Δn]2α},0pM1exp{iπ[pLΔm+Δn]2α},LN+1pL1arbitrary,otherwise.
z^p=exp{iπ[pΔm+Δn]2α},z^q=exp{iπ[qLΔm+Δn]2α}.
y=βx,y^=extend(y,L),Y=FFT(y^),Z=FFT(z^),H=YZ,h^=IFFT(H),h=truncate(h^,M),X=hγ.
h¯=X¯γ*,h¯^=extend(h¯,L),H¯=GIFFT(h¯^),Y¯=H¯Z^*,y¯^=GFFT(Y¯),y¯=truncate(y¯^,N),x¯=y¯β*.
FFT{x}m=an=0N1xnexp(i2πnmN),IFFT{x}m=bn=0N1xnexp(i2πnmN),
GFFT{x¯}=abIFFT{x¯},GIFFT{x¯}=baFFT{x¯}.
y=exp(ax),
x¯=n(a*y*y¯)n.
γ¯=X¯h*,α¯γ=m[(iπm^2)γ*γ¯]m,
β¯=y¯x*,α¯β=n[(iπn^2)β*β¯]n.
Z¯=H¯Y*,z¯^=GFFT(Z¯),α¯p=p=0M1{iπ[pΔm+Δn]2}z^p*z¯^p,α¯q=q=LNL1{iπ[qLΔm+Δn]2}z^q*z¯^q,
α¯=α¯β+α¯γ+α¯p+α¯q.
Xmr,mc=nr=0Nr1,nc=0Nc1,xnr,nce2πi[(mrΔmr)(nrΔnr)αr+(mcΔmc)(ncΔnc)αc],
Xmr,mc=γmrγmcnr=0Nr1,nc=0Nc1,xnr,ncβnrβnczmrnrzmcnc.
w^=extend(w,Lr,Lc)
w=truncate(w^,Mr,Mc),
d=a,b,cdnr,nc=anrbnr,nccnc,
b¯=a*,d¯,c*,a¯nr=nccnc*bnr,nc*d¯nr,nc,c¯nc=nranr*bnr,nc*d¯nr,nc.
y=βr,x,βc,y^=extend(y,Lr,Lc),Y=FFT(y^),H=Zr,Y,Zc,h^=IFFT(H),h=truncate(h^,Mr,Mc),X=γr,h,γc.
h¯=γr*,X¯,γc*,h¯^=extend(h¯,Lr,Lc),H¯=GIFFT(h¯^),Y¯=Z^r*,H¯,Z^c*,y¯^=GFFT(Y¯),y¯=truncate(y¯^,Nr,Nc),x¯=βr*,y¯,βc*.
γ¯r,nr=ncγc,nc*hnr,nc*X¯nr,nc,γ¯c,nc=nrγr,nr*hnr,nc*X¯nr,nc,
β¯r,nr=ncβc,nc*ynr,nc*y¯nr,nc,β¯c,nc=nrβr,nr*ynr,nc*y¯nr,nc.
Z¯r,nr=ncZc,nc*Ynr,nc*H¯nr,nc,Z¯c,nc=nrZr,nr*Ynr,nc*H¯nr,nc.
α¯r,γ=mr[(iπm^r2)γr*γ¯r]mr,α¯c,γ=mc[(iπm^c2)γc*γ¯c]mc,α¯r,β=nr[(iπn^r2)βr*β¯r]nr,α¯c,β=nc[(iπn^c2)βc*β¯c]nc.
α¯c,p=pc=0Mc1{iπ[pcΔmc+Δnc]2}z^c,q*z¯^c,q,α¯c,q=qc=LcNcLc1{iπ[qcLcΔmc+Δnc]2}z^c,q*z¯^c,q,α¯r,p=pr=0Mr1{iπ[prΔmr+Δnr]2}z^r,q*z¯^r,q,α¯r,q=qc=LrNrLr1{iπ[qrLrΔmr+Δnr]2}z^r,q*z¯^r,q.
α¯r=α¯r,β+α¯r,γ+α¯r,p+α¯r,q,α¯c=α¯c,β+α¯c,γ+α¯c,p+α¯c,q.
α¯=α¯r+α¯c.
X=CZT(x,α).
x¯=CZT¯x(x,α;X¯),α¯=CZT¯α(x,α;X¯),
α¯=CZT¯α(g,α;G¯).

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