## Reconstruction of sectional images in holography using inverse imaging

Optics Express, Vol. 16, Issue 22, pp. 17215-17226 (2008)

http://dx.doi.org/10.1364/OE.16.017215

Acrobat PDF (492 KB)

### Abstract

This paper discusses the reconstruction of sectional images from a hologram generated by optical scanning holography. We present a mathematical model for the holographic image capture, which facilitates the use of inverse imaging techniques to recover individual sections. This framework is much more flexible than existing work, in the sense that it can handle objects with multiple sections, and possibly corrupted with white Gaussian noise. Simulation results show that the algorithm is capable of recovering a prescribed section while suppressing the other ones as defocus noise. The proposed algorithm is applicable to on-axis holograms acquired by conventional holography as well as phase-shifting holography.

© 2008 Optical Society of America

## 1. Introduction

1. B. D. Duncan and T.-C. Poon, “Gaussian Beam Analysis of Optical Scanning Holography,” J. Opt. Soc. Am. A **9**, 229–236 (1992). [CrossRef]

2. B. W. Schilling and G. C. Templeton, “Three-dimensional Remote Sensing by Optical Scanning Holography,” Applied Optics **40**, 5474–5481 (2001). [CrossRef]

3. T. Kim, T.-C. Poon, and G. Indebetouw, “Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography,” Opt. Eng. **41**, 1331–1338 (2002). [CrossRef]

4. P. P. Banerjee and R. M. Misra, “Dependence of Photorefractive Beam Fanning on Beam Parameters,” Optics Communications **100**, 166–172 (1993). [CrossRef]

5. T.-C. Poon, “Recent Progress in Optical Scanning Holography,” J. Holography Speckle **1**, 6–25 (2004). [CrossRef]

6. G. Indebetouw and W. Zhong, “Scanning Holographic Microscopy of Three-dimensional Fluorescent Specimens,” J. Opt. Soc. Am. A **23**, 1699–1707 (2006). [CrossRef]

7. T.-C. Poon, “Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis,” J. Opt. Soc. Am. A **2**, 521–527 (1985). [CrossRef]

7. T.-C. Poon, “Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis,” J. Opt. Soc. Am. A **2**, 521–527 (1985). [CrossRef]

8. C. J. Kuo, “Electronic Holography,” Opt. Eng. **35**, 1528 (1996). [CrossRef]

9. K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, “Multiple Multiple-exposure Hologram,” Applied Optics **24**, 4467–4472 (1985). [CrossRef] [PubMed]

10. G. Indebetouw, “Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning,” J. Mod. Opt. **49**, 1479–1500 (2002). [CrossRef]

11. T. Kim, “Optical Sectioning by Optical Scanning Holography and a Wiener Filter,” Applied Optics **45**, 872–879 (2006). [CrossRef] [PubMed]

12. 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,” Applied Optics **47**, 164–175 (2008). [CrossRef]

## 2. Imaging Model

### 2.1. Principles of Optical Scanning Holography

7. T.-C. Poon, “Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis,” J. Opt. Soc. Am. A **2**, 521–527 (1985). [CrossRef]

*ω*+

*α*and

*w*pass through two pupils

*p*and

_{1}*p*respectively, and they are then combined together by a beam splitter (BS). The combined beam is projected onto a section of an object, which is at a distance

_{2}*z*away from the scanner in the system. A photodetector (PD) collects transmitted and scattered light from the object and converts it to an electronic signal. The demodulation part that follows, including a bandpass filter (BPF) centred at frequency

*a*, an electronic multiplexing detection and low pass filters (LPF), processes the signal and generates two electronic holograms in a computer.

14. T.-C. Poon, *Optical Scanning Holography with MATLAB*, 1st ed. (Springer-Verlag, New York, 2007). [CrossRef]

15. J. Swoger, M. Martínez-Corral, J. Huisken, and E. Stelzer, “Optical Scanning Holography as a Technique for High-resolution Three-dimensional Biological Microscopy,” J. Opt. Soc. Am. A **19**, 1910–1918 (2002). [CrossRef]

*k*is a constant and

_{0}*z*is the distance as shown in Fig. 1. The corresponding spatial impulse response is

*ϕ*(

*x,y;z*), its complex hologram by OSH in an incoherent mode of operation is given by

*z*can be denoted by

*z*

_{1},

*z*

_{2}, …,

*z*. Then, the hologram can be represented by a summation,

_{n}*z*

_{1}. A reconstruction of this sectional image means to recover |

*ϕ*(

*x,y,z1*)|

^{2}from a hologram

*g*(

_{c}*x,y*).

### 2.2. Two-Sectional Images: An Example

*g*(

_{c}*x,y*) in Eq. (4) is a general expression with

*n*sections. As mentioned above, the current literature has demonstrated methods only with two-sectional images. In what follows, we set

*n*=2 to simplify the description, but we note here that generalizing to other values of

*n*is straightforward and is done for the simulation with three sections.

16. G. Indebetouw, W. Zhong, and D. Chamberlin-Long, “Point-spread Function Synthesis in Scanning Holographic Microscopy,” J. Opt. Soc. Am. A **23**, 1708–1717 (2006). [CrossRef]

*g*(

_{c}*x,y*).

*h*

_{1}(

*x,y*) and

*h*(

_{2}*x,y*) can be expressed as follows. First, we use a lexico-graphical ordering to convert |

*ϕ*(

*x,y;z*)|

_{1}^{2}and |

*ϕ*(

*x,y;z*)|

_{2}^{2}into vectors, which we denote as

*ψ*and

_{1}*ψ*respectively. If the original hologram is of size

_{2}*N*×

*N*, then these vectors are of a length

*N*. Similarly, we also convert

^{2}*g*(

_{c}*x,y*) into the length-

*N*vector, which we call

^{2}*γ*. We then form two

_{c}*N*

^{2}×

*N*

^{2}matrices,

*H*

_{1}and

*H*

_{2}, using values from

*h*

_{1}(

*x,y*) and

*h*(

_{2}*x,y*) respectively, so that Eq. (5) becomes

*H*

_{1}and

*H*

_{2}are block-circulant-circulant-block (BCCB) if we assume periodic extension on the boundary [17

17. M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Processing Magazine **14**, 24–41 (1997). [CrossRef]

*H*=[

*H*

_{1}

*H*

_{2}] and ψ=[

*ψ*

^{T}

_{1}

*ψ*

^{T}

_{2}]

*. Note that*

^{T}*H*is the matrix representing the operation in OSH, which scans an object denoted by

*ψ*, and generates

*γ*. To reconstruct sectional images from a hologram, we need to compute

_{c}*ψ*from the observed

*γ*, which is an inverse imaging problem.

_{c}*γ*=

_{c}*H*is also applicable to multiple sections in OSH, provided we let

_{y}*H*=[

*H*

_{1}…

*H*

_{n}] and

*ψ*=[

*ψ*

^{T}

_{1}…

*ψ*]

^{T}_{n}*. Thus, the inverse imaging approach is applicable to more than two sections, which goes beyond the methods described in [11*

^{T}11. T. Kim, “Optical Sectioning by Optical Scanning Holography and a Wiener Filter,” Applied Optics **45**, 872–879 (2006). [CrossRef] [PubMed]

12. 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,” Applied Optics **47**, 164–175 (2008). [CrossRef]

## 3. Computational Considerations

*H*and

*γ*in Eq. (9) are complex-valued in general. It is however possible to work on real-valued quantities, by first pre-processing the signals similar to the conventional reconstructionmethod. This also allows us to eliminate the imaginary part of the defocus noise.

_{c}*z*

_{1}. From Eq. (5), we can convolve both sides with the complex conjugate free-space impulse response

*h*

_{1}(

*x,y*), somewhat like a matched filter. Thus,

*ϕ*(

*x,y;z*)|

_{2}^{2}*

*h*

_{21}(

*x,y*) in Eq. (10) is the defocus noise imposed on the focused section at

*z*

_{1}. It should be noted that the convolution with the conjugate impulse response

*h**

_{1}(

*x,y*) keeps the focused section purely real and the defocused section complex. Then we separate out the imaginary part of the defocused section by preserving the real-valued part of

*g*(

_{c}*x,y*)*

*h**

_{1}(

*x,y*).

*ψ*is real, the imaging equation becomes

*β*=Re[

_{c}*H**

_{1}γc] and A=Re[

*H**

_{1}

*H*]. The inverse imaging equation we will solve is

*ν*above to denote random Gaussian noise that is present in a typical imaging system.

*ψ*in the equation above because of the ill-posed nature of the inverse imaging problem [18]. The regularization method is employed to reach the solution [19]. We formulate the image reconstruction as a minimization problem [20

20. F. Natterer and F. Wübbeling, *Mathematical Methods in Image Reconstruction*, 1st ed. (SIAM, Philadelphia, 2001). [CrossRef]

*L*norm,

^{2}*C*stands for the Laplacian operator, and

*λ*is the regularization parameter. The first term ‖

*Aψ*-

*β*‖

_{c}^{2}measures the fidelity of

*ψ*to the observed data

*β*, and the second term ‖

_{c}*Cψ*‖

^{2}is a penalty arising from the prior knowledge of

*ψ*. Since sectional images of a hologram present the same characteristic of smoothness as common images,

*C*is chosen as a high pass filter to suppress noise at higher frequencies.

21. L. Vese, “A Study in the BV Space of a Denoising-deblurring Variational Problem,” Applied Mathematics and Optimization **44**, 131–161 (2001). [CrossRef]

22. G. Aubert and P. Kornprobst, *Mathematical Problems in Image Processing: Partial Differential Equations and Calculus of Variations*, 2nd ed. (Springer-Verlag, New York, 2006). [PubMed]

*A*and

*C*matrices are either huge or rank deficient, or both. We use the CG method to solve the set of linear equations, which has a symmetric positive definite coefficient matrix,

*A*+

^{T}A*λC*, for

^{T}C*λ*>0 [23]. The kth step of the CG process takes the form

*ψ*(

*)=*

^{k}*ψ*+

^{(k-1)}*a*

_{kp}*, where*

^{(k)}*ψ*is the approximation to

^{(k)}*ψ*after

*k*iterations, and

*p*and

^{(k)}*q*are two auxiliary iteration vectors of length

^{(k)}*n*.

^{2}## 4. Results

11. T. Kim, “Optical Sectioning by Optical Scanning Holography and a Wiener Filter,” Applied Optics **45**, 872–879 (2006). [CrossRef] [PubMed]

12. 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,” Applied Optics **47**, 164–175 (2008). [CrossRef]

### 4.1. Hologram Containing 2-sectional Images

*z*

_{1}and

*z*

_{2}, away from the scanner of the OSH system. We suppose that the object is illuminated by a HeNe laser, whose wavelength is 0.63

*µ*m, and that two sections including an element in each are located at

*z*

_{1}=10mm and

*z*=11mm, respectively. The two sections are scanned to generate a hologram. During the scanning, the hologramis contaminated by additive white Gaussian noise with zero mean and variance of 0.01.

_{2}*z*

_{1}and

*z*

_{2}are shown in Fig. 2(b). The spatial rate of change in the phase varies along

*x*and

*y*directions. The rate at which the FZP varies is determined by the distance between the focus of L

_{1}and the plane to be reconstructed in the object. The shorter the distance is, the higher rate the FZP varies at. Optical scanning technique produces the complex hologram of the object. The real part of the hologram is referred to as the sine-FZP hologram, while the imaginary part is the cosine-FZP hologram [14

14. T.-C. Poon, *Optical Scanning Holography with MATLAB*, 1st ed. (Springer-Verlag, New York, 2007). [CrossRef]

*z*

_{1}, and Fig. 4(b) to the one at

*z*

_{2}. Relating them with the two original sections of the object in Fig. 2(a), the rectangular elements are recovered in the corresponding focused sections, and the defocus noise is weakened at the same time. The weakening effect results from the preservation of real-valued part, which eliminates the imaginary part of the defocus noise. However, we observe blurry residuals due to the remaining real part of the defocus noise.

*z*

_{1}and 28.85 dB at

*z*

_{2}. Since our algorithm is iterative in nature, a stopping criterion is needed. Our experience through various experiments is that the total number of iterations is effective as a simple rule, with around 50 iterations sufficient to obtain an acceptable reconstruction. Since the core of our algorithm is conjugate gradient, which is frequently employed in image processing, it can be expected that this would not induce a heavy computation in a reasonable implementation. It is also possible to take advantage of the structure of the

*H*matrix in Eq. (9) to improve the speed of the algorithm. The experiment demonstrates that the reconstruction can recover sectional images without visible influence of defocus noise, and with efficient suppression of additive white Gaussian noise.

*a priori*knowledge of the power spectrum of the noise. Here, the spectrum is estimated by the imaginary part of the defocus noise. The method is suitable for objects containing elements with regular density distributions. To those including either complicated elements or multiple sections, the imaginary part is comprised of the information of various elements. The error of the estimator increases significantly, and the reconstruction can become complicated and error-prone.

*et al*. [12

**47**, 164–175 (2008). [CrossRef]

### 4.2. Hologram Containing 3-sectional Images

*z*

_{1},

*z*

_{2}and

*z*

_{3}sections. They are shown in Fig. 7(a). The object is scanned by the same laser as used in the first experiment. Three sections are

*z*

_{1}=7mm,

*z*

_{2}=8mm and

*z*

_{3}=9mm away from the scanner. White Gaussian noise corrupts the hologram, with a resulting SNR equal to 28 dB. FZPs of a point source in three section are shown in Fig. 7(b). The noisy hologram is shown in Fig. 8. It combines the sectional images at

*z*

_{1},

*z*

_{2}and

*z*

_{3}together. The hologram is more mixed than a two-section hologram and consequently it is more difficult to acquire any sectional information by reviewing it directly.

*λ*and shown in Fig. 10. Regularization parameter affects the image quality of the reconstruction. We can yield similar results in terms of SNR by choosing l from 15 to 4000. A rather large range of

*λ*demonstrates that the reconstruction is not sensitive to the choice of the regularization parameter.

## 5. Conclusions

## Acknowledgment

## References and links

1. | B. D. Duncan and T.-C. Poon, “Gaussian Beam Analysis of Optical Scanning Holography,” J. Opt. Soc. Am. A |

2. | B. W. Schilling and G. C. Templeton, “Three-dimensional Remote Sensing by Optical Scanning Holography,” Applied Optics |

3. | T. Kim, T.-C. Poon, and G. Indebetouw, “Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography,” Opt. Eng. |

4. | P. P. Banerjee and R. M. Misra, “Dependence of Photorefractive Beam Fanning on Beam Parameters,” Optics Communications |

5. | T.-C. Poon, “Recent Progress in Optical Scanning Holography,” J. Holography Speckle |

6. | G. Indebetouw and W. Zhong, “Scanning Holographic Microscopy of Three-dimensional Fluorescent Specimens,” J. Opt. Soc. Am. A |

7. | T.-C. Poon, “Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis,” J. Opt. Soc. Am. A |

8. | C. J. Kuo, “Electronic Holography,” Opt. Eng. |

9. | K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, “Multiple Multiple-exposure Hologram,” Applied Optics |

10. | G. Indebetouw, “Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning,” J. Mod. Opt. |

11. | T. Kim, “Optical Sectioning by Optical Scanning Holography and a Wiener Filter,” Applied Optics |

12. | 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,” Applied Optics |

13. | H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, |

14. | T.-C. Poon, |

15. | J. Swoger, M. Martínez-Corral, J. Huisken, and E. Stelzer, “Optical Scanning Holography as a Technique for High-resolution Three-dimensional Biological Microscopy,” J. Opt. Soc. Am. A |

16. | G. Indebetouw, W. Zhong, and D. Chamberlin-Long, “Point-spread Function Synthesis in Scanning Holographic Microscopy,” J. Opt. Soc. Am. A |

17. | M. R. Banham and A. K. Katsaggelos, “Digital Image Restoration,” IEEE Signal Processing Magazine |

18. | J. M. Blackledge, |

19. | A. Tikhonov and V. Arsenin, |

20. | F. Natterer and F. Wübbeling, |

21. | L. Vese, “A Study in the BV Space of a Denoising-deblurring Variational Problem,” Applied Mathematics and Optimization |

22. | G. Aubert and P. Kornprobst, |

23. | C. R. Vogel, |

**OCIS Codes**

(090.1760) Holography : Computer holography

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

(180.6900) Microscopy : Three-dimensional microscopy

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Holography

**History**

Original Manuscript: June 19, 2008

Revised Manuscript: September 3, 2008

Manuscript Accepted: October 6, 2008

Published: October 13, 2008

**Virtual Issues**

Vol. 3, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Xin Zhang, Edmund Y. Lam, and Ting-Chung Poon, "Reconstruction of sectional images in holography using inverse imaging," Opt. Express **16**, 17215-17226 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17215

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### References

- B. D. Duncan and T.-C. Poon, "Gaussian Beam Analysis of Optical Scanning Holography," J. Opt. Soc. Am. A 9, 229-236 (1992). [CrossRef]
- B. W. Schilling and G. C. Templeton, "Three-dimensional Remote Sensing by Optical Scanning Holography," Appl. Opt. 40, 5474-5481 (2001). [CrossRef]
- T. Kim, T.-C. Poon, and G. Indebetouw, "Depth Detection and Image Recovery in Remote Sensing by Optical Scanning Holography," Opt. Eng. 41, 1331-1338 (2002). [CrossRef]
- P. P. Banerjee and R. M. Misra, "Dependence of Photorefractive Beam Fanning on Beam Parameters," Opt. Commun. 100, 166-172 (1993). [CrossRef]
- T.-C. Poon, "Recent Progress in Optical Scanning Holography," J. Holography Speckle 1, 6-25 (2004). [CrossRef]
- G. Indebetouw and W. Zhong, "Scanning Holographic Microscopy of Three-dimensional Fluorescent Specimens," J. Opt. Soc. Am. A 23, 1699-1707 (2006). [CrossRef]
- T.-C. Poon, "Scanning Holography and Two-dimensional Image Processing by Acousto-optic Two-pupil Synthesis," J. Opt. Soc. Am. A 2, 521-527 (1985). [CrossRef]
- C. J. Kuo, "Electronic Holography," Opt. Eng. 35, 1528 (1996). [CrossRef]
- K. M. Johnson, M. Armstrong, L. Hesselink, and J. W. Goodman, "Multiple Multiple-exposure Hologram," Applied Optics 24, 4467-4472 (1985). [CrossRef] [PubMed]
- G. Indebetouw, "Properties of a Scanning Holographic Microscopy: Improved Resolution, Extended Depth-offocus, and/or Optical Sectioning," J. Mod. Opt. 49, 1479-1500 (2002). [CrossRef]
- T. Kim, "Optical Sectioning by Optical Scanning Holography and a Wiener Filter," Appl. Opt. 45, 872-879 (2006). [CrossRef] [PubMed]
- H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, "Optical Sectioning for Optical Scanning Holography Using Phasespace Filtering with Wigner Distribution Functions," Appl. Opt. 47, 164-175 (2008). [CrossRef]
- H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing, 1st ed. (Wiley, Chichester, 2001).
- T.-C. Poon, Optical Scanning Holography with MATLAB, 1st ed. (Springer-Verlag, New York, 2007). [CrossRef]
- J. Swoger, M. Martínez-Corral, J. Huisken, and E. Stelzer, "Optical Scanning Holography as a Technique for High-resolution Three-dimensional Biological Microscopy," J. Opt. Soc. Am. A 19, 1910-1918 (2002). [CrossRef]
- G. Indebetouw, W. Zhong, and D. Chamberlin-Long, "Point-spread Function Synthesis in Scanning Holographic Microscopy," J. Opt. Soc. Am. A 23, 1708-1717 (2006). [CrossRef]
- M. R. Banham and A. K. Katsaggelos, "Digital Image Restoration," IEEE Signal Processing Magazine 14, 24-41 (1997). [CrossRef]
- J.M. Blackledge, Digital Image Processing: Mathematical and Computational Methods, 1st ed. (Horwood,West Sussex, 2005).
- A. Tikhonov and V. Arsenin, Solutions of Ill-posed Problems, 1st ed. (V.H. Winston and Sons, Washington, 1977).
- F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, 1st ed. (SIAM, Philadelphia, 2001). [CrossRef]
- L. Vese, "A Study in the BV Space of a Denoising-deblurring Variational Problem," Appl. Math. Optimization 44, 131-161 (2001). [CrossRef]
- G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and Calculus of Variations, 2nd ed. (Springer-Verlag, New York, 2006). [PubMed]
- C. R. Vogel, Computational Methods for Inverse Problems, 1st ed. (SIAM, Philadelphia, 2002). [CrossRef]

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