Abstract

Complex object wave recovery from a single-shot interference pattern is an important practical problem in interferometry and digital holography. The most popular single-shot interferogram analysis method involves Fourier filtering of the cross term, but this method suffers from poor resolution. To obtain full pixel resolution, it is necessary to model the object wave recovery as an optimization problem. The optimization approach typically involves minimizing a cost function consisting of a data consistency term and one or more constraint terms. Despite its potential performance advantages, this method is not used widely due to several tedious and difficult tasks such as empirical tuning of free parameters. We introduce a new optimization approach, mean gradient descent (MGD), for single-shot interferogram analysis that is simple to implement. MGD does not have any free parameters whose empirical tuning is critical to the object wave recovery. The MGD iteration does not try to achieve minimization of a cost function but instead aims to reach a solution point where the data consistency and the constraint terms balance each other. This is achieved by iteratively progressing the solution in the direction that bisects the descent directions associated with the error and constraint terms. Numerical illustrations are shown for recovery of a step phase object from its corresponding off-axis as well as on-axis interferograms simulated with multiple noise levels. Our results show full pixel resolution as evident from the recovery of the phase step and excellent rms phase accuracy relative to the ground truth phase map. The concept of MGD as presented here can potentially find applications to a wider class of optimization problems.

© 2019 Optical Society of America

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References

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2019 (2)

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

Y. Baek, K. Lee, S. Shin, and Y. Park, “Kramers-Kronig holographic imaging for high-space-bandwidth product,” Optica 6, 45–51 (2019).
[Crossref]

2017 (1)

2016 (2)

M. Singh and K. Khare, “Accurate efficient carrier estimation for single-shot digital holographic imaging,” Opt. Lett. 41, 4871–4874 (2016).
[Crossref]

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single shot digital holography using iterative reconstruction with alternating updates of amplitude and phase,” Electron. Imaging 19, 1–6 (2016).
[Crossref]

2015 (2)

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement with classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

C. Gaur, B. Mohan, and K. Khare, “Sparsity-assisted solution to the twin image problem in phase retrieval,” J. Opt. Soc. Am. A 32, 1922–1927 (2015).
[Crossref]

2014 (1)

2013 (2)

Y. Shi and Q. Chang, “Efficient algorithm for isotropic and anisotropic total variation deblurring and denoising,” J. Appl. Math. 2013, 797239 (2013).
[Crossref]

K. Khare, P. T. S. Ali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Exp. 21, 2581–2591 (2013).
[Crossref]

2008 (1)

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[Crossref]

2006 (2)

2004 (1)

1997 (2)

1988 (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1986 (1)

1983 (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” Proc. IEEE 130, 11–16 (1983).
[Crossref]

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[Crossref]

Ahlawat, S.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

Ali, P. T. S.

K. Khare, P. T. S. Ali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Exp. 21, 2581–2591 (2013).
[Crossref]

Baek, Y.

Blu, T.

Bouman, C. A.

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single shot digital holography using iterative reconstruction with alternating updates of amplitude and phase,” Electron. Imaging 19, 1–6 (2016).
[Crossref]

Brandwood, D. H.

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” Proc. IEEE 130, 11–16 (1983).
[Crossref]

Brito-Loeza, C.

Chang, Q.

Y. Shi and Q. Chang, “Efficient algorithm for isotropic and anisotropic total variation deblurring and denoising,” J. Appl. Math. 2013, 797239 (2013).
[Crossref]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Cuevas, F. J.

Dinda, A. K.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

Espinosa-Romero, A.

Galvan, C.

Gaur, C.

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[Crossref]

Jha, A. K.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement with classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Joseph, J.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

K. Khare, P. T. S. Ali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Exp. 21, 2581–2591 (2013).
[Crossref]

Katz, J.

Khare, K.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

M. Singh and K. Khare, “Single-shot interferogram analysis for accurate reconstruction of step phase objects,” J. Opt. Soc. Am. A 34, 349–355 (2017).
[Crossref]

M. Singh and K. Khare, “Accurate efficient carrier estimation for single-shot digital holographic imaging,” Opt. Lett. 41, 4871–4874 (2016).
[Crossref]

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement with classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

C. Gaur, B. Mohan, and K. Khare, “Sparsity-assisted solution to the twin image problem in phase retrieval,” J. Opt. Soc. Am. A 32, 1922–1927 (2015).
[Crossref]

K. Khare, P. T. S. Ali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Exp. 21, 2581–2591 (2013).
[Crossref]

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[Crossref]

Kreis, T.

Lee, D. J.

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single shot digital holography using iterative reconstruction with alternating updates of amplitude and phase,” Electron. Imaging 19, 1–6 (2016).
[Crossref]

Lee, K.

Legarda-Saenz, R.

Liebling, M.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).

Malkiel, E.

Mangal, J.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

Marroquin, J. L.

Mathur, S. R.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2007).

Mohan, B.

Monga, R.

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

Padilla, J. M.

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

Pan, X.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[Crossref]

Park, Y.

Prabhakar, S.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement with classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Quiroga, J. A.

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

Rivera, M.

Servin, M.

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[Crossref]

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).

Sheng, J.

Shi, Y.

Y. Shi and Q. Chang, “Efficient algorithm for isotropic and anisotropic total variation deblurring and denoising,” J. Appl. Math. 2013, 797239 (2013).
[Crossref]

Shin, S.

Sidky, E. Y.

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[Crossref]

Singh, M.

Singh, R. P.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement with classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
[Crossref]

Unser, M.

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2007).

Weiner, A. M.

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single shot digital holography using iterative reconstruction with alternating updates of amplitude and phase,” Electron. Imaging 19, 1–6 (2016).
[Crossref]

Yamaguchi, I.

Zhang, T.

Appl. Opt. (4)

Electron. Imaging (1)

D. J. Lee, C. A. Bouman, and A. M. Weiner, “Single shot digital holography using iterative reconstruction with alternating updates of amplitude and phase,” Electron. Imaging 19, 1–6 (2016).
[Crossref]

J. Appl. Math. (1)

Y. Shi and Q. Chang, “Efficient algorithm for isotropic and anisotropic total variation deblurring and denoising,” J. Appl. Math. 2013, 797239 (2013).
[Crossref]

J. Biophoton. (1)

J. Mangal, R. Monga, S. R. Mathur, A. K. Dinda, J. Joseph, S. Ahlawat, and K. Khare, “Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy,” J. Biophoton. 12, e201800409 (2019).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Exp. (1)

K. Khare, P. T. S. Ali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Exp. 21, 2581–2591 (2013).
[Crossref]

Opt. Lett. (2)

Optica (1)

Phys. Med. Biol. (1)

E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008).
[Crossref]

Phys. Rev. A (1)

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement with classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Proc. IEEE (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” Proc. IEEE 130, 11–16 (1983).
[Crossref]

Prog. Opt. (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Other (4)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).

P. Ferraro, A. Wax, and Z. Zalevsky, eds., Coherent Light Microscopy: Imaging and Quantitative Phase Analysis (Springer, 2011).

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley, 2014).

D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed. (Springer, 2007).

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Figures (5)

Fig. 1.
Fig. 1. (a) Phase map of square-shaped object with a step phase of $ 2\pi /3 $ radians over $ 250 \times 250 $ pixels defined on a $ 500 \times 500 $ pixel grid. (b) Hologram of the object in (a) with a tilted plane reference beam simulated with Poisson noise corresponding to average light level of $ {10^4} $ photons/pixel. (c) Zoomed-in version of Fourier magnitude of hologram showing the circular filter of radius 0.5 times the distance between dc and cross-term peaks. For display purpose, the square root of Fourier magnitude of the hologram is shown. (d) Phase image reconstructed using the Fourier filtering method.
Fig. 2.
Fig. 2. (a) Phase and (b) amplitude of solution for object wave after 200 MGD iterations with step size $ t $ kept fixed.
Fig. 3.
Fig. 3. Behavior of (a) logarithm of relative hologram error, $ {C_{\rm{err}}}/{\Vert H\Vert_2^2} $ , with iteration number, and (b) logarithm of $ {C_{\rm{TV}}} $ with iteration number for three light levels of $ {10^3} $ , $ {10^4} $ , and $ {10^5} $ photons/pixel. The magenta line shows the ground-truth TV for the ideal step-phase object shown in Fig. 1(a). (c) Variation of angle ( $ \theta $ ) between $ {\hat{\textbf{u}}_1} $ and $ {\hat{\textbf{u}}_2} $ with number of iterations corresponding to the average light levels of $ {10^3} $ , $ {10^4} $ , and $ {10^5} $ photons/pixel.
Fig. 4.
Fig. 4. Progression of solution with MGD algorithm for Poisson noise realization with light level of $ {10^4} $ photons/pixel. Amplitude of the solution after (a) 500 and (b) 2000 iterations. Phase reconstruction after (c) 500 and (d) 2000 iterations. (e) Phase profile of the resultant solution in (d) along the dotted line. Note that the solution contains sharp edges as compared to FTM solution shown in Fig. 1(d).
Fig. 5.
Fig. 5. Hologram of step-phase object with (a) on-axis and (b) near on-axis spherical reference wave simulated with Poisson noise with an average light level of $ {10^4} $ photons/pixel. (c), (d) Fourier magnitudes of the holograms in (a) and (b), respectively, showing overlap between dc and cross terms. (e), (g) and (f), (h) Reconstructed phase maps and their corresponding phase profiles along the dotted lines, respectively, after 2500 MGD iterations.

Tables (1)

Tables Icon

Table 1. Phase rms Error Values after 2000 Iterations of MGD Algorithm Corresponding to Three Cases of Noise Levels Added to the Hologram Data a

Equations (13)

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

H = | R | 2 + | O | 2 + R O + R O ,
C ( O , O ) = H | O + R | 2 2 2 + α T V ( O , O ) ,
= C e r r ( O , O ) + α C T V ( O , O ) ,
T V ( O , O ) = i = a l l p i x e l s [ | x O i | + | y O i | ] ,
u ^ 1 = O C e r r ( O , O ) O C e r r ( O , O ) 2
u ^ 2 = O C T V ( O , O ) O C T V ( O , O ) 2 .
O C e r r ( O , O ) = 2 ( H | O + R | 2 ) ( O + R )
O C T V ( O , O ) = [ x O | x O | x ^ + y O | y O | y ^ ] .
u = u ^ 1 + u ^ 2 2 ,
O ( n + 1 ) = O ( n ) t O ( n ) 2 [ u ] O = O ( n ) .
D 1 , n = t 2 O ( n ) 2 [ u ^ 1 ] O = O ( n ) 2 , D 2 , n = t 2 O ( n ) 2 [ u ^ 2 ] O = O ( n ) 2
v 1 = [ r e a l ( u ^ 1 j ) , i m a g ( u ^ 1 j ) ] , v 2 = [ r e a l ( u ^ 2 j ) , i m a g ( u ^ 2 j ) ] .
θ = arccos [ v 1 v 2 v 1 2 v 2 2 ] .

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