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 2π 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-2π 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

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References

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  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).
  5. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  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]
  9. V. Katkovnik and J. Astola, “Compressive sensing computational ghost imaging,” J. Opt. Soc. Am. A 29, 1556–1567 (2012).
    [CrossRef]
  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]
  13. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  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]

2012 (4)

2007 (3)

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]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

Astola, J.

Bioucas-Dias, J.

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.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

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 .

Dabov, K.

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]

Danielyan, A.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

Egiazarian, K.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

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]

Elad, M.

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

Foi, A.

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]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Hao, H.

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.

Hongxing, H.

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 .

Katkovnik, V.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

V. Katkovnik and J. Astola, “Compressive sensing computational ghost imaging,” J. Opt. Soc. Am. A 29, 1556–1567 (2012).
[CrossRef]

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]

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]

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]

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 .

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.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Q. Kemao, Windowed Fringe Pattern Analysis (SPIE, 2013).

Kreis, Th.

Th. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Valadão, G.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

IEEE Trans. Image Process. (3)

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[CrossRef]

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]

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lasers Eng. (1)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Other (8)

Q. Kemao, Windowed Fringe Pattern Analysis (SPIE, 2013).

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.

Th. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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.

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer, 2010).

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

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 .

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Figures (10)

Fig. 1.
Fig. 1.

Phase-shift interferometry: on-axis setup. In this Michelson configuration, a single laser beam of coherent light is split into two beams by a beam splitter. The path difference traveled by the reference (mirror) beam and object beam is reflected in the intensity of the aggregated beam u s . This intensity is measured by the sensor array. Shift of the mirror is used in order to introduce different phase shifts between the object and reference beams used in order to reconstruct the object wavefront from the intensity measurements.

Fig. 2.
Fig. 2.

SPAR algorithm: block scheme.

Fig. 3.
Fig. 3.

Wrapped phase for the Gaussian phase object. (a) The noisy raw data, (b) the initial WFT estimate, (c) SPAR reconstruction, and (d) the true image.

Fig. 4.
Fig. 4.

Absolute (unwrapped) phase for the Gaussian phase object. (a) The initial WFT estimate, (b) SPAR reconstruction, and (c) the true image.

Fig. 5.
Fig. 5.

Surfaces for the quadratic amplitude. (a) The noisy raw data, (b) the initial BM3D ampl estimate, (c) SPAR reconstruction, and (d) the true image.

Fig. 6.
Fig. 6.

Accuracy of the wavefront reconstruction (object with the Gaussian phase) versus the number of iterations, from left to right: ISNR exp for the exponential wavefront exp ( j φ ) , and ISNR ampl for the quadratic amplitude.

Fig. 7.
Fig. 7.

Wrapped phase for the truncated Gaussian phase object. (a) The noisy raw data, (b) the initial WFT estimate, (c) SPAR reconstruction, and (d) the true image.

Fig. 8.
Fig. 8.

Absolute (unwrapped) phase for the truncated Gaussian phase object. (a) The initial WFT estimate, (b) SPAR reconstruction, and (c) the true image.

Fig. 9.
Fig. 9.

Wrapped phase for the shear plane phase object. (a) The noisy raw data, (b) the initial WFT estimate, (c) SPAR reconstruction, and (d) the true image.

Fig. 10.
Fig. 10.

Absolute (unwrapped) phase for shear plane phase object. (a) The initial WFT estimate, (b) SPAR reconstruction, and (c) the true image.

Tables (2)

Tables Icon

Table 1 Sparse Phase Amplitude Reconstruction (SPAR) Algorithm

Tables Icon

Table 1. Accuracy Criteria for Phase and Amplitude Reconstructiona,b

Equations (20)

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B o = Ψ a , o θ a , o , φ o , abs = Ψ φ , o θ φ , o ,
θ a , o = Φ a , o B 0 , θ φ , o = Φ φ , o φ o , abs ,
u s = B o exp ( j φ o ) + A r exp ( j φ r s ) , s = 1 , , L ,
p ( Y s [ l ] = k ) = exp ( I s [ l ] χ ) ( I s [ l ] χ ) k k ! ,
I s = B o 2 + A r 2 + 2 B o A r cos ( φ o + φ r s ) .
( φ ˜ o , B ˜ o ) = arg min φ o , B o s = 1 4 Y s / χ B o 2 + A r 2 + 2 B o A r cos ( φ o + φ r s ) 2 2 ,
tan ( φ ˜ o ) = Y 4 Y 2 Y 1 Y 3 , B ˜ o = s = 1 4 Y s / 4 χ A r 2 .
u ^ s = arg min u s L 1 ( u s , v s ) , L 1 ( u s , v s ) = s = 1 L l [ | u s [ l ] | 2 χ Y s [ l ] log ( | u s [ l ] | 2 χ ) ] + 1 γ 1 s = 1 L u s v s 2 2 .
u ^ s [ l ] = b s [ l ] exp ( j · angle ( v s [ l ] ) ,
b s [ l ] = | v s [ l ] | / ( γ 1 χ ) + | v s [ l ] | 2 / ( γ 1 χ ) 2 + 4 Y s [ l ] ( 1 + 1 / ( γ 1 χ ) ) / χ 2 ( 1 + 1 / ( γ 1 χ ) ) .
u ^ s [ l ] Y s [ l ] / χ exp ( j · angle ( v s [ l ] ) , s = 1 , , L .
( φ ^ o , B ^ o ) = arg min φ o , B o L 2 ( B o , φ o , A r , u s ) ,
L 2 ( B o , φ o , A r , u s ) = s = 1 L u s ( B o exp ( j φ o ) + A r exp ( j φ r s ) ) 2 2 .
φ ^ o = angle ( s = 1 L u s ) , B ^ o = Re ( exp ( j φ ^ o ) s u s ) / 4 .
( θ ^ φ o , θ ^ B o ) = arg min θ φ o , θ B o L 3 ( θ φ o , θ B o , φ o , abs , B o ) ,
L 3 ( θ φ o , θ B o , φ o , abs , B o ) = τ a · θ B o 0 + τ φ · θ φ o 0 + 1 2 θ B o Φ a , o B o ) 2 2 + 1 2 θ φ o Φ φ , o φ o , abs 2 2 .
θ ^ B o = Φ a , o B o · H ( | Φ B o B o | T h a ) , θ ^ φ o = Φ φ , o φ o · H ( | Φ φ o φ o , abs | T h φ ) ,
v s = ( Ψ θ B o θ ^ B o ) exp ( j Ψ φ o θ ^ φ o ) + A r exp ( j φ r s ) .
( Ψ θ B o θ ^ B o ) = BM3D ampl ( B o , T h a ) , ( Ψ φ o θ ^ φ o ) = BM3D angle ( φ o , a b s , T h φ ) ,
ISNR amp = 10 log 10 B ˜ o B o 2 2 B ^ o 50 B o 2 2 , ISNR exp = 10 log 10 exp ( j φ ˜ o ) exp ( j φ ) 2 2 exp ( j φ ^ o 50 ) exp ( j φ ) 2 2 , ISNR phase , wrap = 10 log 10 W ( W ( φ ˜ o ) φ o ) 2 2 W ( W ( φ ^ o 50 ) φ o ) 2 2 , RMSE phase , abs = 1 N φ ^ o , abs 50 φ o 2 2 .

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