OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 10 — Oct. 1, 2013
  • pp: 2012–2020

Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography

Nikhil Chacko, Michael Liebling, and Thierry Blu  »View Author Affiliations

JOSA A, Vol. 30, Issue 10, pp. 2012-2020 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (1248 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(100.2000) Image processing : Digital image processing
(090.1995) Holography : Digital holography
(070.7345) Fourier optics and signal processing : Wave propagation
(110.7410) Imaging systems : Wavelets

ToC Category:
Image Processing

Original Manuscript: May 10, 2013
Manuscript Accepted: August 8, 2013
Published: September 18, 2013

Nikhil Chacko, Michael Liebling, and Thierry Blu, "Discretization of continuous convolution operators for accurate modeling of wave propagation in digital holography," J. Opt. Soc. Am. A 30, 2012-2020 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  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/ .

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited