OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 48, Iss. 18 — Jun. 20, 2009
  • pp: 3407–3423

Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions

Vladimir Katkovnik, Artem Migukin, and Jaakko Astola  »View Author Affiliations

Applied Optics, Vol. 48, Issue 18, pp. 3407-3423 (2009)

View Full Text Article

Enhanced HTML    Acrobat PDF (1705 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We consider reconstruction of a wave field distribution in an input/object plane from data in an output/diffraction (sensor) plane. We provide digital modeling both for the forward and backward wave field propagation. A novel algebraic matrix form of the discrete diffraction transform (DDT) originated in Katkovnik et al. [ Appl. Opt. 47, 3481 (2008)] is proposed for the forward modeling that is aliasing free and precise for pixelwise invariant object and sensor plane distributions. This “matrix DDT” is a base for formalization of the object wave field reconstruction (backward propagation) as an inverse problem. The transfer matrices of the matrix DDT are used for calculations as well as for the analysis of conditions when the perfect reconstruction of the object wave field distribution is possible. We show by simulation that the developed inverse propagation algorithm demonstrates an improved accuracy as compared with the standard convolutional and discrete Fresnel transform algorithms.

© 2009 Optical Society of America

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.3190) Image processing : Inverse problems
(070.2025) Fourier optics and signal processing : Discrete optical signal processing

ToC Category:
Image Processing

Original Manuscript: December 15, 2008
Revised Manuscript: April 14, 2009
Manuscript Accepted: April 17, 2009
Published: June 11, 2009

Vladimir Katkovnik, Artem Migukin, and Jaakko Astola, "Backward discrete wave field propagation modeling as an inverse problem: toward perfect reconstruction of wave field distributions," Appl. Opt. 48, 3407-3423 (2009)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. Th. Kreis, Handbook of Holographic Interferometry (Optical and Digital Methods) (Wiley-VCH, 2005).
  2. V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481-3493 (2008). [CrossRef] [PubMed]
  3. G. S. Sherman, “Integral-transform formulation of diffraction theory,” J. Opt. Soc. Am. 57, 1490-1498 (1967). [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  5. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359-367 (2007). [CrossRef]
  6. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh--Sommerfeld diffraction formula,” Appl. Opt. 45, 1102-1110 (2006). [CrossRef] [PubMed]
  7. L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557-2563 (2004). [CrossRef]
  8. 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]
  9. M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003). [CrossRef]
  10. 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]
  11. L. Yaroslavsky, “Discrete transforms, fast algorithms and point spread functions of numerical reconstruction of digitally recorded holograms,” in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., Vol. 7 of EURASIP Book Series on Signal Processing and Communications (Hindawi Publishing, 2007), pp. 93-141.
  12. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177-6186 (2001). [CrossRef]
  13. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed., Signal Processing Series (Prentice-Hall, 1999).
  14. H. Theil, “Linear algebra and matrix methods in econometrics, in Handbook of Econometrics, Z. Griliches and M. D. Intriligator, eds. (North-Holland, 1983), Vol. 1. [CrossRef]
  15. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).
  16. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998). [CrossRef]
  17. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992). [CrossRef]
  18. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006). [CrossRef]
  19. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006). [CrossRef]
  20. V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” Proceedings of the 2008 European Signal Processing Conference (EUSIPCO 2008) (Wiley, 2008).
  21. V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006). [CrossRef]
  22. L. L. Scharf, Statistical Signal Processing (Prentice-Hall, 1991).

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