## Solving inverse problems for optical scanning holography using an adaptively iterative shrinkage-thresholding algorithm |

Optics Express, Vol. 20, Issue 6, pp. 5942-5954 (2012)

http://dx.doi.org/10.1364/OE.20.005942

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### Abstract

Optical scanning holography (OSH) records a three-dimensional object into a two-dimensional hologram through two-dimensional optical scanning. The recovery of sectional images from the hologram, termed as an inverse problem, has been previously implemented by conventional methods as well as the use of

© 2012 OSA

## 1. Introduction

1. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A **2**(4), 521–527 (1985). [CrossRef]

2. B. D. Duncan and T.-C. Poon, “Gaussian beam analysis of optical scanning holography,” J. Opt. Soc. Am. A **9**(2), 229–236 (1992). [CrossRef]

3. J. Swoger, M. Martínez-Corral, J. Huisken, and E. H. Stelzer, “Optical scanning holography as a technique for high-resolution three-dimensional biological microscopy,” J. Opt. Soc. Am. A **19**(9), 1910–1918 (2002). [CrossRef] [PubMed]

4. E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. **48**(34), H113–H119 (2009). [CrossRef] [PubMed]

5. G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A **23**(7), 1699–1707 (2006). [CrossRef] [PubMed]

6. G. Indebetouw, A. El Maghnouji, and R. Foster, “Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,” J. Opt. Soc. Am. A **22**(5), 892–898 (2005). [CrossRef] [PubMed]

7. T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. **45**(5), 872–879 (2006). [CrossRef] [PubMed]

8. H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. **47**(19), D164–D175 (2008). [CrossRef] [PubMed]

9. X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express **16**(22), 17215–17226 (2008). [CrossRef] [PubMed]

10. X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett. **34**(20), 3098–3100 (2009). [CrossRef] [PubMed]

11. S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. **1**(4), 606–617 (2007). [CrossRef]

12. X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A **27**(7), 1630–1637 (2010). [CrossRef] [PubMed]

14. I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. **45**(3), 600–616 (1997). [CrossRef]

15. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express **17**(10), 8062–8080 (2009). [CrossRef] [PubMed]

17. X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express **18**(24), 24825–24841 (2010). [CrossRef] [PubMed]

18. D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express **18**(8), 8630–8646 (2010). [CrossRef] [PubMed]

## 2. Theory and method

### 2.1. OSH formulation

### 2.2. Computation simplification

### Mode 1: Matrix multiplication

### Mode 2: Matrix and vector multiplication

### 2.3. Adaptively iterative shrinkage-thresholding (AIST) algorithm

21. S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. **43**(1), 129–159 (2001). [CrossRef]

22. E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. **59**(8), 1207–1223 (2006). [CrossRef]

25. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. **57**(11), 1413–1457 (2004). [CrossRef]

**Algorithm 1**gives the pseudo-code of AIST algorithm to reconstruct the sectional image of OSH.

Algorithm 1 Implementation of adaptive iterative shrinkage-thresholding (AIST) algorithm |
---|

Requirement: matrix |

Initializing: Adaptive coefficient M. |

1: repeat |

2: |

3: Shrinkage of the initial Lagrangian multiplier by |

4: Update |

5: Update solution |

6: until . |

7: Output initial Lagrangian multiplier |

8: repeat |

9: |

10: Shrinkage of iterating Lagrangian multiplier by |

11: repeat |

12: Update |

13: Update solution |

14: until |

15: until the stopping criteria are met. |

## 3. Experiments and results

28. Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. **9**(3), 81–84 (2002). [CrossRef]

### 3.1. Sectional image reconstruction by a half data

12. X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A **27**(7), 1630–1637 (2010). [CrossRef] [PubMed]

### 3.2. Multi-planar object reconstruction

### 3.3. DOF analysis

## 4. Conclusion

## Acknowledgments

## References and links

1. | T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A |

2. | B. D. Duncan and T.-C. Poon, “Gaussian beam analysis of optical scanning holography,” J. Opt. Soc. Am. A |

3. | J. Swoger, M. Martínez-Corral, J. Huisken, and E. H. Stelzer, “Optical scanning holography as a technique for high-resolution three-dimensional biological microscopy,” J. Opt. Soc. Am. A |

4. | E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. |

5. | G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A |

6. | G. Indebetouw, A. El Maghnouji, and R. Foster, “Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,” J. Opt. Soc. Am. A |

7. | T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. |

8. | H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. |

9. | X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express |

10. | X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett. |

11. | S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. |

12. | X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A |

13. | S. Wei and H. Xu, “Staircasing reduction model applied to total variation based image reconstruction,”in |

14. | I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. |

15. | Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express |

16. | Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. |

17. | X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express |

18. | D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express |

19. | T.-C. Poon, |

20. | X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sectional image reconstruction in optical scanning holography,” in |

21. | S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. |

22. | E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. |

23. | A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Fast L1-minimization algorithms and an application in robust face recognition: a review,” in |

24. | S. J. Wright, R. D. Nowak, and M. A. Figueiredo, “Sparse Reconstruction by Separable Approximation,” IEEET, Signal Process. |

25. | I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. |

26. | P. Combettes and V. Wajs, “Signal recovery by proximal forward backward splitting,” Multiscale Model. Simul. |

27. | E. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for L1-regularized minimization with applications to compressed sensing,” |

28. | Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. |

**OCIS Codes**

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

(180.6900) Microscopy : Three-dimensional microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: December 19, 2011

Revised Manuscript: February 14, 2012

Manuscript Accepted: February 20, 2012

Published: February 27, 2012

**Virtual Issues**

Vol. 7, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Fengjun Zhao, Xiaochao Qu, Xin Zhang, Ting-Chung Poon, Taegeun Kim, You Seok Kim, and Jimin Liang, "Solving inverse problems for optical scanning holography using an adaptively iterative shrinkage-thresholding algorithm," Opt. Express **20**, 5942-5954 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-5942

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### References

- T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A2(4), 521–527 (1985). [CrossRef]
- B. D. Duncan and T.-C. Poon, “Gaussian beam analysis of optical scanning holography,” J. Opt. Soc. Am. A9(2), 229–236 (1992). [CrossRef]
- J. Swoger, M. Martínez-Corral, J. Huisken, and E. H. Stelzer, “Optical scanning holography as a technique for high-resolution three-dimensional biological microscopy,” J. Opt. Soc. Am. A19(9), 1910–1918 (2002). [CrossRef] [PubMed]
- E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt.48(34), H113–H119 (2009). [CrossRef] [PubMed]
- G. Indebetouw and W. Zhong, “Scanning holographic microscopy of three-dimensional fluorescent specimens,” J. Opt. Soc. Am. A23(7), 1699–1707 (2006). [CrossRef] [PubMed]
- G. Indebetouw, A. El Maghnouji, and R. Foster, “Scanning holographic microscopy with transverse resolution exceeding the Rayleigh limit and extended depth of focus,” J. Opt. Soc. Am. A22(5), 892–898 (2005). [CrossRef] [PubMed]
- T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt.45(5), 872–879 (2006). [CrossRef] [PubMed]
- H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt.47(19), D164–D175 (2008). [CrossRef] [PubMed]
- X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express16(22), 17215–17226 (2008). [CrossRef] [PubMed]
- X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett.34(20), 3098–3100 (2009). [CrossRef] [PubMed]
- S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process.1(4), 606–617 (2007). [CrossRef]
- X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A27(7), 1630–1637 (2010). [CrossRef] [PubMed]
- S. Wei and H. Xu, “Staircasing reduction model applied to total variation based image reconstruction,”in 17th European Signal Processing Conference (EUSIPCO, Glasgow, Scotland, 2009), pp. 2579–2583.
- I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process.45(3), 600–616 (1997). [CrossRef]
- Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17(10), 8062–8080 (2009). [CrossRef] [PubMed]
- Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun.284(24), 5871–5876 (2011). [CrossRef]
- X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express18(24), 24825–24841 (2010). [CrossRef] [PubMed]
- D. Han, J. Tian, S. Zhu, J. Feng, C. Qin, B. Zhang, and X. Yang, “A fast reconstruction algorithm for fluorescence molecular tomography with sparsity regularization,” Opt. Express18(8), 8630–8646 (2010). [CrossRef] [PubMed]
- T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, New York, 2007).
- X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sectional image reconstruction in optical scanning holography,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2009), paper DMA3.
- S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev.43(1), 129–159 (2001). [CrossRef]
- E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math.59(8), 1207–1223 (2006). [CrossRef]
- A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Fast L1-minimization algorithms and an application in robust face recognition: a review,” in proceedings of IEEE International Conference on Image Processing, (Institute of Electrical and Electronics Engineers, California, 2010), pp. 1849–1852.
- S. J. Wright, R. D. Nowak, and M. A. Figueiredo, “Sparse Reconstruction by Separable Approximation,” IEEET, Signal Process.57, 2479–2493 (2009).
- I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math.57(11), 1413–1457 (2004). [CrossRef]
- P. Combettes and V. Wajs, “Signal recovery by proximal forward backward splitting,” Multiscale Model. Simul.4(4), 1168–1200 (2005). [CrossRef]
- E. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for L1-regularized minimization with applications to compressed sensing,” Tech. Rep. TR07–07 (Rice Univ., Houston, TX, 2007).
- Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002). [CrossRef]

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