Abstract

Discretization of continuous (analog) convolution operators by direct sampling of the convolution kernel and use of fast Fourier transforms is highly efficient. However, it assumes the input and output signals are band-limited, a condition rarely met in practice, where signals have finite support or abrupt edges and sampling is nonideal. Here, we propose to approximate signals in analog, shift-invariant function spaces, which do not need to be band-limited, resulting in discrete coefficients for which we derive discrete convolution kernels that accurately model the analog convolution operator while taking into account nonideal sampling devices (such as finite fill-factor cameras). This approach retains the efficiency of direct sampling but not its limiting assumption. We propose fast forward and inverse algorithms that handle finite-length, periodic, and mirror-symmetric signals with rational sampling rates. We provide explicit convolution kernels for computing coherent wave propagation in the context of digital holography. When compared to band-limited methods in simulations, our method leads to fewer reconstruction artifacts when signals have sharp edges or when using nonideal sampling devices.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  3. L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).
  4. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
    [CrossRef]
  5. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
    [CrossRef]
  6. M. Unser, “Sampling-50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
    [CrossRef]
  7. M. Unser, “A general Hilbert space framework for the discretization of continuous signal processing operators,” Proc. SPIE 2569, 51–61 (1995).
    [CrossRef]
  8. S. Horbelt, M. Liebling, and M. Unser, “Discretization of the Radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21, 363–376 (2002).
    [CrossRef]
  9. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  10. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
    [CrossRef]
  11. A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
    [CrossRef]
  12. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
    [CrossRef]
  13. A. Stern and B. Javidi, “Sampling in the light of Wigner distribution,” J. Opt. Soc. Am. A 21, 360–366 (2004).
    [CrossRef]
  14. B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917–927 (2005).
    [CrossRef]
  15. J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
    [CrossRef]
  16. T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
    [CrossRef]
  17. D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
    [CrossRef]
  18. D. Mas, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
    [CrossRef]
  19. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
    [CrossRef]
  20. P. Ferraro, S. D. Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. 29, 854–856 (2004).
    [CrossRef]
  21. M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
    [CrossRef]
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).
  23. T. Blu and M. Unser, “Quantitative Fourier analysis of approximation techniques: part I-Interpolators and projectors,” IEEE Trans. Signal Process. 47, 2783–2795 (1999).
    [CrossRef]
  24. M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
    [CrossRef]
  25. E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59–69 (2000).
    [CrossRef]
  26. C. M. Brislawn, “Classification of nonexpansive symmetric extension transforms for multirate filter banks,” Appl. Comput. Harmon. Anal. 3, 337–357 (1996).
    [CrossRef]
  27. A. Fertner, “Computationally efficient methods for analysis and synthesis of real signals using FFT and IFFT,” IEEE Trans. Signal Process. 47, 1061–1064 (1999).
    [CrossRef]
  28. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
  29. 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]
  30. V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
    [CrossRef]
  31. Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
    [CrossRef]
  32. D. P. Kelly and D. Claus, “Filtering role of the sensor pixel in Fourier and Fresnel digital holography,” Appl. Opt. 52, A336–A345 (2013).
    [CrossRef]
  33. M. Liebling, “Fresnelab: sparse representations of digital holograms,” Proc. SPIE 8138, 81380I (2011).
    [CrossRef]
  34. A. F. Coskun, I. Sencan, T.-W. Su, and A. Ozcan, “Lensless wide-field fluorescent imaging on a chip using compressive decoding of sparse objects,” Opt. Express 18, 10510–10523 (2010).
    [CrossRef]
  35. M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, “Compressed sensing with off-axis frequency-shifting holography,” Opt. Lett. 35, 871–873 (2010).
    [CrossRef]
  36. Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Display Technol. 6, 506–509 (2010).
    [CrossRef]
  37. http://sybil.ece.ucsb.edu/ .

2013 (1)

2011 (1)

M. Liebling, “Fresnelab: sparse representations of digital holograms,” Proc. SPIE 8138, 81380I (2011).
[CrossRef]

2010 (3)

2009 (3)

2006 (1)

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]

2005 (1)

2004 (5)

2003 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

2002 (1)

S. Horbelt, M. Liebling, and M. Unser, “Discretization of the Radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21, 363–376 (2002).
[CrossRef]

2000 (3)

L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39, 5929–5935 (2000).
[CrossRef]

M. Unser, “Sampling-50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

1999 (4)

D. Mas, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

A. Fertner, “Computationally efficient methods for analysis and synthesis of real signals using FFT and IFFT,” IEEE Trans. Signal Process. 47, 1061–1064 (1999).
[CrossRef]

T. Blu and M. Unser, “Quantitative Fourier analysis of approximation techniques: part I-Interpolators and projectors,” IEEE Trans. Signal Process. 47, 2783–2795 (1999).
[CrossRef]

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

1997 (2)

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1996 (1)

C. M. Brislawn, “Classification of nonexpansive symmetric extension transforms for multirate filter banks,” Appl. Comput. Harmon. Anal. 3, 337–357 (1996).
[CrossRef]

1995 (1)

M. Unser, “A general Hilbert space framework for the discretization of continuous signal processing operators,” Proc. SPIE 2569, 51–61 (1995).
[CrossRef]

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

1949 (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Adams, M.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[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]

Alfieri, D.

Angelini, E.

Astola, J.

V. Katkovnik, A. Migukin, and J. Astola, “Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions,” Appl. Opt. 48, 3407–3423 (2009).
[CrossRef]

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]

Atlan, M.

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

T. Blu and M. Unser, “Quantitative Fourier analysis of approximation techniques: part I-Interpolators and projectors,” IEEE Trans. Signal Process. 47, 2783–2795 (1999).
[CrossRef]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Bovik, A.

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Brislawn, C. M.

C. M. Brislawn, “Classification of nonexpansive symmetric extension transforms for multirate filter banks,” Appl. Comput. Harmon. Anal. 3, 337–357 (1996).
[CrossRef]

Claus, D.

Coppola, G.

Coskun, A. F.

Cuche, E.

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

Depeursinge, C.

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

Ferraro, P.

Fertner, A.

A. Fertner, “Computationally efficient methods for analysis and synthesis of real signals using FFT and IFFT,” IEEE Trans. Signal Process. 47, 1061–1064 (1999).
[CrossRef]

Finizio, A.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gori, F.

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Healy, J. J.

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
[CrossRef]

Hennelly, B. M.

Horbelt, S.

S. Horbelt, M. Liebling, and M. Unser, “Discretization of the Radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21, 363–376 (2002).
[CrossRef]

Javidi, B.

Jueptner, W. P. O.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Katkovnik, V.

Kelly, D. P.

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Kreis, T. M.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Liebling, M.

M. Liebling, “Fresnelab: sparse representations of digital holograms,” Proc. SPIE 8138, 81380I (2011).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

S. Horbelt, M. Liebling, and M. Unser, “Discretization of the Radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21, 363–376 (2002).
[CrossRef]

Marim, M. M.

Marquet, P.

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

Mas, D.

D. Mas, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Matsushima, K.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Merzlyakov, N. S.

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).

Migukin, A.

Nicola, S. D.

Olivo-Marin, J.-C.

Onural, L.

Ozcan, A.

Pierattini, G.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Rivenson, Y.

Sencan, I.

Shannon, C. E.

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Sheikh, H.

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Sheridan, J. T.

Shimobaba, T.

Simoncelli, E.

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Stern, A.

Su, T.-W.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Unser, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

S. Horbelt, M. Liebling, and M. Unser, “Discretization of the Radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21, 363–376 (2002).
[CrossRef]

M. Unser, “Sampling-50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

T. Blu and M. Unser, “Quantitative Fourier analysis of approximation techniques: part I-Interpolators and projectors,” IEEE Trans. Signal Process. 47, 2783–2795 (1999).
[CrossRef]

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

M. Unser, “A general Hilbert space framework for the discretization of continuous signal processing operators,” Proc. SPIE 2569, 51–61 (1995).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

Wang, Z.

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Yamaguchi, I.

Yaroslavskii, L. P.

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).

Yaroslavsky, L. P.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Zhang, F.

Appl. Comput. Harmon. Anal. (1)

C. M. Brislawn, “Classification of nonexpansive symmetric extension transforms for multirate filter banks,” Appl. Comput. Harmon. Anal. 3, 337–357 (1996).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

IEEE Signal Process. Mag. (1)

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

IEEE Trans. Image Process. (2)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29–43 (2003).
[CrossRef]

Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[CrossRef]

IEEE Trans. Med. Imaging (1)

S. Horbelt, M. Liebling, and M. Unser, “Discretization of the Radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21, 363–376 (2002).
[CrossRef]

IEEE Trans. Signal Process. (3)

T. Blu and M. Unser, “Quantitative Fourier analysis of approximation techniques: part I-Interpolators and projectors,” IEEE Trans. Signal Process. 47, 2783–2795 (1999).
[CrossRef]

A. Fertner, “Computationally efficient methods for analysis and synthesis of real signals using FFT and IFFT,” IEEE Trans. Signal Process. 47, 1061–1064 (1999).
[CrossRef]

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. Display Technol. (1)

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

D. Mas, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

Opt. Eng. (1)

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Proc. IEEE (1)

M. Unser, “Sampling-50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Proc. IRE (1)

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Proc. SPIE (3)

M. Unser, “A general Hilbert space framework for the discretization of continuous signal processing operators,” Proc. SPIE 2569, 51–61 (1995).
[CrossRef]

M. Liebling, “Fresnelab: sparse representations of digital holograms,” Proc. SPIE 8138, 81380I (2011).
[CrossRef]

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Signal Process. (1)

J. J. Healy and J. T. Sheridan, “Sampling and discretization of the linear canonical transform,” Signal Process. 89, 641–648 (2009).
[CrossRef]

Other (6)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, 1980).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).

http://sybil.ece.ucsb.edu/ .

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

(a) Box signal formed with N=4096 samples where Δx1=10μm and aperture width w=5.15mm, (b) FrT computed using Fresnel integrals [22] with λ=632nm and z=5mm (only real values shown), (c) inverse FrT of (b) computed using CV-FFT, and (d) using IGCV-FFT where prior knowledge (φ1,φ2,Δx1) is exploited for filter design.

Fig. 2.
Fig. 2.

Two boundary conditions discussed for discrete FrT: (a) periodic boundaries; (b) propagation of a finite-sized object/field confined within a rectangular waveguide lined with mirrors on its four interior planar surfaces, analogous to using (c) mirror-symmetric boundaries for the discrete transform.

Fig. 3.
Fig. 3.

Discretization of continuous convolution operations based on generalized sampling theory. (a) The continuous convolution g(x)=fh(x) can be approximated using two suitable SI spaces, Vi=span{φi(/Δxik)}kZ, (i=1,2), as g˜V2(x)=(fV1h)V2(x), which in turn can be numerically computed by a discrete convolution without aliasing, even when f and h are not band-limited. (b), (c) When Δx2/Δx1=p/q, (p,qN), the expansion coefficients of fV1 and g˜V2 are related by digital filters for both (b) the forward and (c) the inverse convolution operation. (d), (e) Equivalent filters to (b) and (c) when the signals are defined by their discrete samples rather than expansion coefficients.

Fig. 4.
Fig. 4.

(a) f(x) composed of three types of basis functions (β0, β1, and β3); (b) f˜τ(x), where τ=1, (only real values shown) and its samples subsequently used for the recovery of f(x); (c) the reconstructed signal and samples in the band-limited space, obtained using CV-FFT; (d)–(f) the recovered signal in the three separate SI spaces, V1=span{βi(k)}kZ, i=0, 1, 3, using IGCV-FFT. Clover leaves indicate reconstruction artifacts (e.g., Gibbs oscillation) and hearts denote perfect reconstruction.

Fig. 5.
Fig. 5.

(a) Typical CCD with finite-size detector elements and (b) its corresponding family of 1D basis functions. (c) f˜(λ·z)0.5VCCD(kΔx2) (only absolute values shown) (λ=632nm, z=1cm, Δx1=Δx2=10μm, γ=0.7) for a square aperture, f(x) (not shown). (d) Reconstruction using CV-FFT and (e) its SSIM map [31] showing the presence of artifacts (white: SSIM=1, black: SSIM=0). (f) Reconstruction using IGCV-FFT, yielding (g) an SSIM map that is uniformly 1 (white, perfect reconstruction).

Fig. 6.
Fig. 6.

Comparison of methods to estimate f from f˜τV2 (not shown), where τ=2.5 and φ2=β1, using discrete-inverse fV1 with IGCV-FFT, and alternatively, using discretized-continuous-inverse (f˜τV2hFrT,τ1)V1 with GCV-FFT.

Equations (54)

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

g(x)=R2f(ξ)·h(xξ)dξfh(x).
g[k]=mZ2f[m]·hBL[km]f*hBL[k],
hBL(x)=1ΔxΔy[h(x)sinc(x/Δx)sinc(y/Δy)]
hRS,λ,z(x)=zjλ·exp(j2πλx2+z2)x2+z2,
h^RS,λ,z(ν)=exp(j2πz1λ2ν2),ν=(νx,νy).
hFrA,λ,z(x)=exp(j2πλz)jλz·exp(jπλzx2),
hFrA,λ,z(x)=jexp(j2πλz)·hFrT,τ(x)·hFrT,τ(y),
hFrT,τ(x)={exp(jπ4)·δ(x),τ=01τexp(jπx2τ2),otherwise
h^FrT,τ(ν)=exp(jπ4)·exp(jπτ2ν2),νR.
F˜τ{f}(x)=f˜τ(x)=fhFrT,τ(x),xR.
f˜τCVFFT[k]=FN1{FN(f)×UCVFFT[k0]}[k]
UCVFFT[k0]=rect(k0/N)×h^FrT,τ(k0/(NΔx)),
rect(ν)={1,|ν|<120,otherwise
V1={f|f(x)=kZc[k]·φ1(xΔx1k);c2<},
fV1(x)=1Δx1kZf,φ1(Δx1k)·φ1(xΔx1k)
=kZc[k]·φ1(xΔx1k),
φ1(m),φ1(n)=δ[mn],m,nZ.
φ^1(ν)=φ^1(ν)kZ|φ^1(ν+k)|2.
βn(x)=β0β0β0n+1terms(x),
g˜(x)=fV1h(x),
g˜V2(x)=kZd[k]·φ2(xΔx2k).
d[k]=Zc[]·u[pkq],
u(x)=1Δx2{φ1(xΔx1)h(x)φ2*(xΔx2)},
u[k]=u(kΔx2/p).
d[k]=(1/p)·FNq/p1{FN(c)×U}[k],0k<Nq/p,
U[k0]=qmZφ^1(k0Nmq)·h^(k0NΔx1mqΔx1)·φ^2*(pk0Nqmp),0k0<Nq.
g˜[k]=(1/p)·FNq/p1{FN(c)×Us}[k],0k<Nq/p,
Us[k0]=qmZφ^1(k0Nmq)·h^(k0NΔx1mqΔx1).
g˜V2[k]=(1/p)·FNq/p1{FN(f)×Uint}[k],0k<Nq/p,
Uint[k0]=qmZη^1(k0Nmq)·h^(k0NΔx1mqΔx1)·η^2*(pk0Nqmp),0k0<Nq
η^i(ν)=φ^i(ν)mZφ^i(ν+m),i=1,2,
η^i(ν)=φ^i(ν)·(mZφ^i*(ν+m))nZ|φ^i(ν+n)|2.
cm[k]={c[k],0k<Nc[2N1k],Nk<2N.
d[2k]=dm[2k]devenm[k],0k<N/2,
d[2k+1]=devenm[N1k],0k<(N1)/2,
devenm[k]=FN1{F2N{dm}[]+F2N{dm}[+N]2}[k],
F2N{dm}[k0]F2N{cm}[k0]×Um,0k0<2N,
F2N{cm}[k0]=FN{cevenm}[k0]+{exp(jπNk0)·FN{cevenm}[Nk0]},
Um[k0]=mZφ^1(k02Nm)·h^(k02NΔx1mΔx1)·φ^2*(k02Nm).
c=argminc2dlZc[l]·u[pql]2
c[k]=(1/q)·FN1{FNq/p(d)×V}[k],0k<N,
V[k0]=pqUk0modN[0,k0/N],0k0<Nq,
Ur[m,n]=U[r+Nm+Nqpn],0m<q,0n<p
URS[k0,l0]=qxqym,nZ{φ^1(k0mNxqxNx,l0nNyqyNy)·h^RS,λ,z(k0mNxqxNxΔx1,l0nNyqyNyΔy1)·φ^2*(k0mNxqxNxqx/px,l0nNyqyNyqy/py)},
UFrT[k0]=qmZ{φ^1(k0mNqN)·φ^2*(k0mNqNq/p)·exp(jπ4)·exp(jπτ2(k0mNqNΔx1)2)},
U(ej2πνΔx1/q)=qmZ{φ^1(Δx1νmq)·h^(νmqΔx1)·φ^2*(Δx2νmp)}.
d=A·c,
A=WNq/p1·U·WN,
U=[INq/pINq/p]·DU·[ININ],
WN[m,n]=exp(j2πmn/N),0m,n<N,
DU[m,n]=U[m]·δ[mn],0m,n<Nq,
IN[m,n]=δ[mn],0m,n<N,
rank(U)=r=0N/p1rank(Ur),
A=WN1·U·WNq/p,

Metrics