Abstract

In this paper, we propose a new geometric super-resolving approach that overcomes the geometric resolution reduction caused by the spatially large pixels of the detector array. The improvement process is obtained by applying an axial scanning procedure. In the scanning process, several images are captured corresponding to focus applied at several axial planes. By applying an iterative Gerchberg–Saxton-based algorithm, we managed to retrieve the phase and to reconstruct the original high-resolution image from the captured set of low-resolution images. In addition, the paper also presents a numerically efficient algorithm to compute the free space Fresnel integral.

© 2014 Optical Society of America

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  1. M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851–866 (1999).
    [CrossRef]
  2. C. Xiao, J. Yu, and Y. Xue, “A high-efficiency super-resolution reconstruction algorithm from image/video sequences,” in Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (IEEE, 2007), pp. 573–580.
  3. L. L. Huang, L. Xiao, and Z. H. Wei, “Efficient and effective total variation image super-resolution: a preconditioned operator splitting approach,” Math. Probl. Eng. 2011, 1–20 (2011).
    [CrossRef]
  4. N. K. Bose, S. Lertrattanapanich, and J. Koo, “Advances in superresolution using L-curve,” in IEEE International Symposium on Circuits and Systems (IEEE, 2001), pp. 433–436.
  5. W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis–Gerchberg method in image super-resolution and inpainting,” Comp. J. 52, 80–89 (2009).
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    [CrossRef]
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  13. V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions,” Appl. Opt. 47, 3481–3493 (2008).
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  14. I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis,” IEEE Trans. Signal Process. 54, 4261–4270 (2006).
    [CrossRef]
  15. S. B. Tucker, J. Ojeda-Castañeda, and W. T. Cathey, “Matrix description of near-field diffraction and the fractional Fourier transform,” J. Opt. Soc. Am. A 16, 316–322 (1999).
    [CrossRef]
  16. Y. Rivenson and A. Stern, “Conditions for practicing compressive Fresnel holography,” Opt. Lett. 36, 3365–3367 (2011).
    [CrossRef]
  17. L. P. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic, 2004).
  18. J. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
    [CrossRef]
  19. J. Li and C. Li, “Algorithm study of Collins formula and inverse Collins Formula,” Appl. Opt. 47, A97–A102 (2008).
    [CrossRef]
  20. J. Li, Z. Fan, and Y. Fu, “FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality,” Proc. SPIE 4915, 180 (2002).
    [CrossRef]
  21. J. Li, C. Yuan, P. Tankam, and P. Picart, “The calculation research of classical diffraction formulas in convolution form,” Opt. Commun. 284, 3202–3206 (2011).
    [CrossRef]
  22. P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” in Advances in Neural Information Processing Systems (2013), pp. 2796–2804.
  23. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605, (2006).
    [CrossRef]
  24. A. Migukin, V. Katkovnik, and J. Astola, “Wave field reconstruction from multiple plane intensity-only data: augmented Lagrangian algorithm,” J. Opt. Soc. Am. A 28, 993–1002 (2011).
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  25. L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
    [CrossRef]
  26. K. Ishizuka, “Phase retrieval from image intensities: why does exit wave restoration using IWFR work so well?” Microscopy 62, S109–S118 (2013).
    [CrossRef]

2013

K. Ishizuka, “Phase retrieval from image intensities: why does exit wave restoration using IWFR work so well?” Microscopy 62, S109–S118 (2013).
[CrossRef]

2011

2010

2009

2008

2007

J. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

2006

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis,” IEEE Trans. Signal Process. 54, 4261–4270 (2006).
[CrossRef]

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605, (2006).
[CrossRef]

2005

W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).

2004

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef]

2002

J. Li, Z. Fan, and Y. Fu, “FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality,” Proc. SPIE 4915, 180 (2002).
[CrossRef]

1999

S. B. Tucker, J. Ojeda-Castañeda, and W. T. Cathey, “Matrix description of near-field diffraction and the fractional Fourier transform,” J. Opt. Soc. Am. A 16, 316–322 (1999).
[CrossRef]

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851–866 (1999).
[CrossRef]

Aizenberg, I.

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis,” IEEE Trans. Signal Process. 54, 4261–4270 (2006).
[CrossRef]

Allen, L. J.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef]

Almoro, P.

Astola, J.

Borkowski, A.

Bose, N. K.

N. K. Bose, S. Lertrattanapanich, and J. Koo, “Advances in superresolution using L-curve,” in IEEE International Symposium on Circuits and Systems (IEEE, 2001), pp. 433–436.

Cathey, W. T.

Chan, R. H.

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851–866 (1999).
[CrossRef]

Chatterjee, P.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis–Gerchberg method in image super-resolution and inpainting,” Comp. J. 52, 80–89 (2009).

Chaudhuri, S.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis–Gerchberg method in image super-resolution and inpainting,” Comp. J. 52, 80–89 (2009).

Ching, W. K.

W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).

Egiazarian, K.

Fan, Z.

J. Li, Z. Fan, and Y. Fu, “FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality,” Proc. SPIE 4915, 180 (2002).
[CrossRef]

Fixler, O.

Fu, Y.

J. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

J. Li, Z. Fan, and Y. Fu, “FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality,” Proc. SPIE 4915, 180 (2002).
[CrossRef]

Huang, L. L.

L. L. Huang, L. Xiao, and Z. H. Wei, “Efficient and effective total variation image super-resolution: a preconditioned operator splitting approach,” Math. Probl. Eng. 2011, 1–20 (2011).
[CrossRef]

Ishizuka, K.

K. Ishizuka, “Phase retrieval from image intensities: why does exit wave restoration using IWFR work so well?” Microscopy 62, S109–S118 (2013).
[CrossRef]

Jain, P.

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” in Advances in Neural Information Processing Systems (2013), pp. 2796–2804.

Javidi, B.

Katkovnik, V.

Koo, J.

N. K. Bose, S. Lertrattanapanich, and J. Koo, “Advances in superresolution using L-curve,” in IEEE International Symposium on Circuits and Systems (IEEE, 2001), pp. 433–436.

Lertrattanapanich, S.

N. K. Bose, S. Lertrattanapanich, and J. Koo, “Advances in superresolution using L-curve,” in IEEE International Symposium on Circuits and Systems (IEEE, 2001), pp. 433–436.

Li, C.

Li, J.

J. Li, C. Yuan, P. Tankam, and P. Picart, “The calculation research of classical diffraction formulas in convolution form,” Opt. Commun. 284, 3202–3206 (2011).
[CrossRef]

J. Li and C. Li, “Algorithm study of Collins formula and inverse Collins Formula,” Appl. Opt. 47, A97–A102 (2008).
[CrossRef]

J. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

J. Li, Z. Fan, and Y. Fu, “FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality,” Proc. SPIE 4915, 180 (2002).
[CrossRef]

Marom, E.

McBride, W.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef]

Migukin, A.

Mudassar, A. A.

Mukherjee, S.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis–Gerchberg method in image super-resolution and inpainting,” Comp. J. 52, 80–89 (2009).

Netrapalli, P.

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” in Advances in Neural Information Processing Systems (2013), pp. 2796–2804.

Ng, M. K.

W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851–866 (1999).
[CrossRef]

O’Leary, N. L.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef]

Ojeda-Castañeda, J.

Osten, W.

Oxley, M. P.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef]

Pedrini, G.

Peng, Z.

J. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

Picart, P.

J. Li, C. Yuan, P. Tankam, and P. Picart, “The calculation research of classical diffraction formulas in convolution form,” Opt. Commun. 284, 3202–3206 (2011).
[CrossRef]

Rivenson, Y.

Sanghavi, S.

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” in Advances in Neural Information Processing Systems (2013), pp. 2796–2804.

Seetharaman, G.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis–Gerchberg method in image super-resolution and inpainting,” Comp. J. 52, 80–89 (2009).

Shengyi, L.

Stern, A.

Sze, K. N.

W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).

Tang, W. C.

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851–866 (1999).
[CrossRef]

Tankam, P.

J. Li, C. Yuan, P. Tankam, and P. Picart, “The calculation research of classical diffraction formulas in convolution form,” Opt. Commun. 284, 3202–3206 (2011).
[CrossRef]

Tucker, S. B.

ul Haq, I.

Wei, Z. H.

L. L. Huang, L. Xiao, and Z. H. Wei, “Efficient and effective total variation image super-resolution: a preconditioned operator splitting approach,” Math. Probl. Eng. 2011, 1–20 (2011).
[CrossRef]

Xiao, C.

C. Xiao, J. Yu, and Y. Xue, “A high-efficiency super-resolution reconstruction algorithm from image/video sequences,” in Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (IEEE, 2007), pp. 573–580.

Xiao, L.

L. L. Huang, L. Xiao, and Z. H. Wei, “Efficient and effective total variation image super-resolution: a preconditioned operator splitting approach,” Math. Probl. Eng. 2011, 1–20 (2011).
[CrossRef]

Xiaojun, H.

Xue, Y.

C. Xiao, J. Yu, and Y. Xue, “A high-efficiency super-resolution reconstruction algorithm from image/video sequences,” in Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (IEEE, 2007), pp. 573–580.

Yaroslavsky, L. P.

L. P. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic, 2004).

Yau, A. C.

W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).

Yu, J.

C. Xiao, J. Yu, and Y. Xue, “A high-efficiency super-resolution reconstruction algorithm from image/video sequences,” in Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (IEEE, 2007), pp. 573–580.

Yuan, C.

J. Li, C. Yuan, P. Tankam, and P. Picart, “The calculation research of classical diffraction formulas in convolution form,” Opt. Commun. 284, 3202–3206 (2011).
[CrossRef]

Yulie, W.

Zalevsky, Z.

Appl. Opt.

Comp. J.

P. Chatterjee, S. Mukherjee, S. Chaudhuri, and G. Seetharaman, “Application of Papoulis–Gerchberg method in image super-resolution and inpainting,” Comp. J. 52, 80–89 (2009).

IEEE Trans. Signal Process.

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis,” IEEE Trans. Signal Process. 54, 4261–4270 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Math. Probl. Eng.

L. L. Huang, L. Xiao, and Z. H. Wei, “Efficient and effective total variation image super-resolution: a preconditioned operator splitting approach,” Math. Probl. Eng. 2011, 1–20 (2011).
[CrossRef]

Microscopy

K. Ishizuka, “Phase retrieval from image intensities: why does exit wave restoration using IWFR work so well?” Microscopy 62, S109–S118 (2013).
[CrossRef]

Numerical Linear Algebra with Applications

W. K. Ching, M. K. Ng, K. N. Sze, and A. C. Yau, “Super‐resolution image reconstruction using multisensors,” Numerical Linear Algebra with Applications 12, 271–281 (2005).

Opt. Commun.

J. Li, Z. Peng, and Y. Fu, “Diffraction transfer function and its calculation of classic diffraction formula,” Opt. Commun. 280, 243–248 (2007).
[CrossRef]

J. Li, C. Yuan, P. Tankam, and P. Picart, “The calculation research of classical diffraction formulas in convolution form,” Opt. Commun. 284, 3202–3206 (2011).
[CrossRef]

Opt. Lett.

Proc. SPIE

J. Li, Z. Fan, and Y. Fu, “FFT calculation for Fresnel diffraction and energy conservation criterion of sampling quality,” Proc. SPIE 4915, 180 (2002).
[CrossRef]

SIAM J. Sci. Comput.

M. K. Ng, R. H. Chan, and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput. 21, 851–866 (1999).
[CrossRef]

Ultramicroscopy

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[CrossRef]

Other

C. Xiao, J. Yu, and Y. Xue, “A high-efficiency super-resolution reconstruction algorithm from image/video sequences,” in Third International IEEE Conference on Signal-Image Technologies and Internet-Based System (IEEE, 2007), pp. 573–580.

N. K. Bose, S. Lertrattanapanich, and J. Koo, “Advances in superresolution using L-curve,” in IEEE International Symposium on Circuits and Systems (IEEE, 2001), pp. 433–436.

P. Netrapalli, P. Jain, and S. Sanghavi, “Phase retrieval using alternating minimization,” in Advances in Neural Information Processing Systems (2013), pp. 2796–2804.

L. P. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic, 2004).

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Figures (16)

Fig. 1.
Fig. 1.

Propagation images at different planes in high resolution. Starting with the top left image and proceeding clockwise, the images are from the first, eighth, sixteenth, and the thirty-second planes, respectively.

Fig. 2.
Fig. 2.

Same images as Fig. 1 but in lower resolution four times each axis.

Fig. 3.
Fig. 3.

Propagation phases at different planes in high resolution for the complicated phase, which was chosen to be the Lena image. Starting with the top left image and proceeding clockwise, the images are from the first, eighth, sixteenth, and the thirty-second planes, respectively.

Fig. 4.
Fig. 4.

Graphs of the RMS error of the amplitude and phase changes, depending upon the most distant plane defined relative to zf.

Fig. 5.
Fig. 5.

Graphs of the RMS error of the amplitude and phase, depending upon the desired resolution factor.

Fig. 6.
Fig. 6.

Graphs of the normalized RMS error of the amplitude for super-resolved and low-resolution images depending upon the desired resolution factor.

Fig. 7.
Fig. 7.

Examples of the reconstruction ability of the algorithm to several different resolutions for Lena image. (a) Reference image intensity. (b1) (c1) (d1) Low resolution image intensity at the z0 plane for resolution factor two, four, and eight times at each axis. (b2) (c2) (d2) Reconstructed images intensity at z0 plane, for resolution factor two, four, and eight times at each axis, respectively.

Fig. 8.
Fig. 8.

Examples of the reconstruction ability of the algorithm to several different resolutions for resolution target. (a) Reference image intensity. (b1) (c1) (d1) Low-resolution images intensity at the z0 plane for resolution factor two, four, and eight times at each axis. (b2) (c2) (d2) Reconstructed images intensity at the z0 plane for resolution factor two, four, and eight times at each axis, respectively.

Fig. 9.
Fig. 9.

Phase reconstruction ability of the algorithm to recover the high-resolution phase. (a) Reference image phase. (b)–(d) Reconstructed image phase for a resolution factor of two, four, and eight in each axis at the z0 plane, respectively.

Fig. 10.
Fig. 10.

Graphs of the RMS error of the amplitude and phase changes depending on the number of planes that took place at the reconstruction.

Fig. 11.
Fig. 11.

Graphs of the RMS error of the amplitude and phase changes, depending on the increasing noise in gray level percentage.

Fig. 12.
Fig. 12.

Graph of the RMS error of the difference between the original and estimated images as a function of the standard deviation of the error of the plane location in the lateral axis in terms of detector pixel size.

Fig. 13.
Fig. 13.

Logarithmic graph of the RMS error of the difference between the original and estimated images as a function of the standard deviation of the error of the plane location in the longitudinal axis, in terms of detector’s pixel size.

Fig. 14.
Fig. 14.

Images of the reconstructed phase (proceeding clockwise) for the first, tenth, twentieth, and fortieth iterations, respectively.

Fig. 15.
Fig. 15.

Reconstruction process according to the number of iterations. Starting with the top left image and proceeding clockwise, the reconstructed images after the first, tenth, twentieth, and fortieth iterations are shown.

Fig. 16.
Fig. 16.

Graphs of the RMS error of the amplitude and the elapsed time, depending on the number of iterations.

Equations (46)

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

u2(x;z)=exp(jπz/λ)jλzu1(ζ)exp(jπλz(xζ)2)dζ,
u2[lΔx;z]=exp(jπz/λ)jλzs=L/2L/21u1[sΔx]exp(jπλzΔx2(ls)2),
u2[lΔx;z]=s=L/2L/21u1[sΔx]hz[lΔxsΔx]=u1[lΔx]*hz[lΔx],
hz[l]=exp(jπz/λ)jλzexp(jπλzΔx2l2).
[678]*[12345]=[60000760008760008760008760008700008][12345]=[6000076000876000876000876000870000887080000006076additionalcolumns][12345{00zeropadding]=[6780000][1234500]=iFFT{FFT{[6780000]}·FFT{[1234500]}},
u2[lΔx;z]=iFFT{FFT{u1[lΔx]}·Hz[sΔf]},
Hz[sΔf]=FFT{hz[lΔx]}=exp(jπzλ(1λ2s2Δf2)),
STF;z=diag{Hz[sΔf]},
u2[lΔx;z]=FTF;zu1[lΔx],
FTF;z=DF1STF;zDF.
FFT;z*FTF;z=(DF1STF;zDF)*(DF1STF;zDF)=DF1STF;z*DFDF1STF;zDF=I
DFFTF;zDF1=DFDF1STF;zDFDF1=STF;z.
FFT;z1=(DF1STF;zDF)1=DF1STF;z1DF=DF1STF;z*DF=FFT;z*=FTF;z.
U2(x,y;z)=exp(j2πz/λ)jλzU1(ζ,η)exp(jπλz[(xζ)2+(yη)2])dζdη,
U2[kΔy,lΔx;z]=FTF;z[k]U1[kΔy,lΔx]FFT;zT[l]
vec(U2)=(FTF;z[l]FTF;z[k])vec(U1),
u2[lΔx;z]=iFFT{FFT{u1[lΔx]}·FFT{hz[lΔx]}},
FIR;z=ΔxDF1SIR;zDF.
SIR;z=diag{DFhz[lΔx]}.
u2[lΔx2;z]=Δx1jλz·exp(jπz/λ)hz[lΔx2]FFT{u1[lΔx1]hz[lΔx1]}=FFT;zu1[lΔx1],
FFT;z=μSFT2;zDFSFT1;z.
SFT1;z=diag{hz[lΔx1]},SFT2;z=diag{hz[lΔx2]},
|l{πλzl2Δx12}|l=L/2πand|l{πλzl2Δx22}|l=L/2π.
FFR;z=μSFT2;zDF.
u2(x)=exp(jπL2/λ)jλBu1(ζ)exp(jπλB(Aζ22xζ+Dx2))dζ.
u2[lΔx]=CTFu1A[lAΔx],
CTF=SCTF,2DF1SCTF,1DF,SCTF,1=diag{exp(jπλBAΔf2s2)},SCTF,2=diag{exp(jπλ[L2+CAΔx2l2])},
u2[lΔx2]=CFTu1[sΔx1],
CFT=μCSCFT,2DFSCFT,1,SCFT,1=diag{exp(jπλBAΔx12s2)},SCTF,2=diag{exp(jπλBDΔx22l2)},
|F{b0eiθ0}|2=|b^1eiθ1|2.
|F1{b1eiθ1}|2=|b^0eiθ0|2,
|Frzb0eiθ0|2=|b^1eiθ1|2;
|Frzb1eiθ1|2=|b^0eiθ0|2,
|[Frz1Frzm]b0eiθ0|2=|[b^1eiθ1b^meiθm]|2;
1m|mFrzmbmeiθm|2=|b^0eiθ0|2,
|cm|2=D|bm|2,
gm=cm(1M1M).
VariablesEquations=2L(m+1)2Lm+(m+1)L/M1.
2M1m.
|[Frz11Frz12Frzm1Frzm2]b0eiθ0|2=|[b^1eiθ1b^meiθm]|2.
|[Frz12B0eiΘ0Frz11TFrzm2B0eiΘ0Frzm1T]|2=|[B^1eiΘ1B^meiΘm]|2,
1m|m(Frzm1Frzm2)bmeiθm|2=|b^0eiθ0|2
1m|mFrzm2BmeiΘmFrzm1T|2=|B^0eiΘ0|2.
|Cm|2=D|Bm|2DT,
Gm=Cm1MN1M,N,
zf=NΔx2λ.

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