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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 8 — Aug. 1, 2014
  • pp: 1801–1810

Wavefront reconstruction in phase-shifting interferometry via sparse coding of amplitude and absolute phase

V. Katkovnik and J. Bioucas-Dias  »View Author Affiliations


JOSA A, Vol. 31, Issue 8, pp. 1801-1810 (2014)
http://dx.doi.org/10.1364/JOSAA.31.001801


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Abstract

Phase-shifting interferometry is a coherent optical method that combines high accuracy with high measurement speeds. This technique is therefore desirable in many applications such as the efficient industrial quality inspection process. However, despite its advantageous properties, the inference of the object amplitude and the phase, herein termed wavefront reconstruction, is not a trivial task owing to the Poissonian noise associated with the measurement process and to the 2 π phase periodicity of the observation mechanism. In this paper, we formulate the wavefront reconstruction as an inverse problem, where the amplitude and the absolute phase are assumed to admit sparse linear representations in suitable sparsifying transforms (dictionaries). Sparse modeling is a form of regularization of inverse problems which, in the case of the absolute phase, is not available to the conventional wavefront reconstruction techniques, as only interferometric phase modulo- 2 π is considered therein. The developed sparse modeling of the absolute phase solves two different problems: accuracy of the interferometric (wrapped) phase reconstruction and simultaneous phase unwrapping. Based on this rationale, we introduce the sparse phase and amplitude reconstruction (SPAR) algorithm. SPAR takes into full consideration the Poissonian (photon counting) measurements and uses the data-adaptive block-matching 3D (BM3D) frames as a sparse representation for the amplitude and for the absolute phase. SPAR effectiveness is documented by comparing its performance with that of competitors in a series of experiments.

© 2014 Optical Society of America

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(110.1650) Imaging systems : Coherence imaging
(070.2025) Fourier optics and signal processing : Discrete optical signal processing

ToC Category:
Imaging Systems

History
Original Manuscript: January 16, 2014
Manuscript Accepted: May 27, 2014
Published: July 24, 2014

Citation
V. Katkovnik and J. Bioucas-Dias, "Wavefront reconstruction in phase-shifting interferometry via sparse coding of amplitude and absolute phase," J. Opt. Soc. Am. A 31, 1801-1810 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-8-1801


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References

  1. Th. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).
  2. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  3. V. Katkovnik, J. Bioucas-Dias, and H. Hao, “Wavefront reconstruction from noisy fringe observations via sparse coding,” The 7th International Workshop on Advanced Optical Imaging and Metrology FRINGE 2013, Nürtingen, Germany, September8–11, 2013.
  4. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).
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  6. H. Hongxing, J. M. Bioucas-Dias, and V. Katkovnik, “Interferometric phase image estimation via sparse coding in the complex domain,” IEEE Trans. Geosci. Remote Sens. (in print). Available at http://www.lx.it.pt/~bioucas/files/ieee_tgrs_Sp_InPHASE_R2_double_5_5_2014.pdf .
  7. V. Katkovnik and J. Astola, “High-accuracy wavefield reconstruction: decoupled inverse imaging with sparse modeling of phase and amplitude,” J. Opt. Soc. Am. A 29, 44–54 (2012). [CrossRef]
  8. V. Katkovnik and J. Astola, “Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude,” J. Opt. Soc. Am. A 29, 105–116 (2012). [CrossRef]
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  10. To lighten the presentation, the symbols Ys, s=1,…,L are used both as random vectors and as sample vectors. The meaning should be clear from the context.
  11. A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012). [CrossRef]
  12. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007). [CrossRef]
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  14. Q. Kemao, Windowed Fringe Pattern Analysis (SPIE, 2013).
  15. J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007). [CrossRef]

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