## On the single point resolution of on-axis digital holography

JOSA A, Vol. 27, Issue 8, pp. 1856-1862 (2010)

http://dx.doi.org/10.1364/JOSAA.27.001856

Acrobat PDF (298 KB)

### Abstract

On-axis digital holography (DH) is becoming widely used for its time-resolved three-dimensional (3D) imaging capabilities. A 3D volume can be reconstructed from a single hologram. DH is applied as a metrological tool in experimental mechanics, biology, and fluid dynamics, and therefore the estimation and the improvement of the resolution are current challenges. However, the resolution depends on experimental parameters such as the recording distance, the sensor definition, the pixel size, and also on the location of the object in the field of view. This paper derives resolution bounds in DH by using estimation theory. The single point resolution expresses the standard deviations on the estimation of the spatial coordinates of a point source from its hologram. Cramér–Rao lower bounds give a lower limit for the resolution. The closed-form expressions of the Cramér–Rao lower bounds are obtained for a point source located on and out of the optical axis. The influences of the 3D location of the source, the numerical aperture, and the signal-to-noise ratio are studied.

© 2010 Optical Society of America

## 1. INTRODUCTION

2. M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. **190**, 87–94 (2001). [CrossRef]

3. A. Stern and B. Javidi, “Improved-resolution digital holography using the generalized sampling theorem for locally band-limited fields,” J. Opt. Soc. Am. A **23**, 1227–1235 (2006). [CrossRef]

4. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. **45**, 836–850 (2006). [CrossRef] [PubMed]

5. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) **48**, 095801 (2009). [CrossRef]

6. A. J. Den Dekker and A. Van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A **14**, 547–557 (1997). [CrossRef]

*Ω*, classical Rayleigh resolution limits (lateral

6. A. J. Den Dekker and A. Van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A **14**, 547–557 (1997). [CrossRef]

12. C. W. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am. **59**, 164–175 (1969). [CrossRef]

13. S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. **17**, 27–57 (2006). [CrossRef]

14. S. Van Aert, D. Van Dirk, and A. J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—Classical criteria and a statistical alternative,” Opt. Express **14**, 3830–3839 (2006). [CrossRef] [PubMed]

15. M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory **52**, 3411–3437 (2006). [CrossRef]

16. P. Réfrégier, J. Fade, and M. Roche, “Estimation precision of the degree of polarization from a single speckle intensity image,” Opt. Lett. **32**, 739–741 (2007). [CrossRef] [PubMed]

17. A. Sentenac, C. A. Guérin, P. C. Chaumet, F. Drsek, H. Giovannini, N. Bertaux, and M. Holschneider, “Influence of multiple scattering on the resolution of an imaging system: a Cramér–Rao analysis,” Opt. Express **15**, 1340–1347 (2007). [CrossRef] [PubMed]

*x*,

*y*,

*z*) and a model of noise, the single point resolution is estimated by means of the variance on the parameters. Cramér–Rao lower bound (CRLB) [18, 19] gives a lower bound on the variance of the estimators. This bound is reached asymptotically (for large data) by the maximum likelihood estimator. Let us note that the maximum likelihood estimator has already been used with success in DH to detect objects of simple geometrical form, leading to enhanced accuracy [20

20. F. Soulez, L. Denis, C. Fournier, E. Thiebaut, and C. Goepfert, “Inverse problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A **24**, 1164–1171 (2007). [CrossRef]

21. S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express **15**, 18275–18282 (2007). [CrossRef] [PubMed]

22. F. Soulez, L. Denis, E. Thiebaut, C. Fournier, and C. Goepfert, “Inverse problem approach for particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A **24**, 3708–3716 (2007). [CrossRef]

## 2. STATISTICAL ESTIMATION OF SINGLE POINT RESOLUTION

### 2A. Single Point Hologram Model

*L*(sampling and quantization effects are neglected; see Fig. 1 ). For a point

*z*and lateral point source location

*λ*being the wavelength of the light, omitting the proportionality factor and the offset level. Due to noise, the measured image is a perturbed version of the model. We consider in the first approximation the noise as white Gaussian.

### 2B. Cramér–Rao Lower Bound

*d*represents the data (pixel values),

**θ**, and

*C*is a constant. Using Eqs. (4, 5) and neglecting sampling and quantization effects, the Fisher matrix becomes

## 3. SINGLE POINT RESOLUTION ON THE OPTICAL AXIS

### 3A. Analytical Form of the Fisher Information Matrix

**can be calculated from Eq. (6). For a square sensor of side**

*I**L*,

**has the formUsing a symbolic computation software, the integral (7) can be calculated with the expression of the model (2). It gives an analytical expression composed of many terms that can be significantly simplified assumingThe simple expressions obtained areLet us note that the assumption (8) is always verified in classical conditions. It means that more than two oscillations of the chirp function are recorded on the sensor. Due to the parity of**

*I*### 3B. Analytical Form of the Covariance Matrix

## 4. SINGLE POINT RESOLUTION MAP

### 4A. Analytical Form of the Fisher Information Matrix

### 4B. Analytical Form of the Covariance Matrix

### 4C. Standard Deviation Maps

*x*error map the maximum is 2.0 times the minimum, while on the

*z*error map the maximum is 3.7 times the minimum. By using non-dimensional variables

*z*or

*L*changes. These analytical results have been compared with numerical integrations of Eq. (11) (see Section 5) to make sure that hypothesis (8) is valid. The relative error on the error maps is negligible (lower than 0.3%).

## 5. INFLUENCE OF SAMPLING AND PIXEL INTEGRATION

22. F. Soulez, L. Denis, E. Thiebaut, C. Fournier, and C. Goepfert, “Inverse problem approach for particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A **24**, 3708–3716 (2007). [CrossRef]

23. L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. **25**, 115017 (2009). [CrossRef]

*x*or

*y*location at infinity, the formulas give a null variance due to the absence of signal attenuation and sampling considerations.

*N*is the number of rows and

*M*is number of columns of the sensor. Equation (17) can be computed for all

*L*) these differences decrease: for

*z*and

*L*contrary to the ideal sensor case of Sections 3, 4 (where sampling is neglected).

## 6. CONCLUSION

*a posteriori*(MAP) approaches have been shown to lead to reconstructions with a few artifacts [24

24. S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A **21**, 737–750 (2004). [CrossRef]

25. L. Denis, D. Lorenz, E. Thiebaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. **34**, 3475–3477 (2009). [CrossRef] [PubMed]

26. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**, 13040–13049 (2009). [CrossRef] [PubMed]

15. M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory **52**, 3411–3437 (2006). [CrossRef]

## ACKNOWLEDGMENT

1. | T. M. Kreis, |

2. | M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. |

3. | A. Stern and B. Javidi, “Improved-resolution digital holography using the generalized sampling theorem for locally band-limited fields,” J. Opt. Soc. Am. A |

4. | J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. |

5. | D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) |

6. | A. J. Den Dekker and A. Van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A |

7. | J. W. Goodman, |

8. | A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. (Bellingham) |

9. | A. Stern and B. Javidi, “Space-bandwidth conditions for efficient phase-shifting digital holographic microscopy,” J. Opt. Soc. Am. A |

10. | S. M. Kay, |

11. | P. Réfrégier, |

12. | C. W. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am. |

13. | S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. |

14. | S. Van Aert, D. Van Dirk, and A. J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—Classical criteria and a statistical alternative,” Opt. Express |

15. | M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory |

16. | P. Réfrégier, J. Fade, and M. Roche, “Estimation precision of the degree of polarization from a single speckle intensity image,” Opt. Lett. |

17. | A. Sentenac, C. A. Guérin, P. C. Chaumet, F. Drsek, H. Giovannini, N. Bertaux, and M. Holschneider, “Influence of multiple scattering on the resolution of an imaging system: a Cramér–Rao analysis,” Opt. Express |

18. | C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. |

19. | H. Cramér, |

20. | F. Soulez, L. Denis, C. Fournier, E. Thiebaut, and C. Goepfert, “Inverse problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A |

21. | S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express |

22. | F. Soulez, L. Denis, E. Thiebaut, C. Fournier, and C. Goepfert, “Inverse problem approach for particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A |

23. | L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. |

24. | S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A |

25. | L. Denis, D. Lorenz, E. Thiebaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. |

26. | D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: March 15, 2010

Manuscript Accepted: May 14, 2010

Published: July 28, 2010

**Virtual Issues**

August 20, 2010 *Spotlight on Optics*

**Citation**

Corinne Fournier, Loïc Denis, and Thierry Fournel, "On the single point resolution of on-axis digital holography," J. Opt. Soc. Am. A **27**, 1856-1862 (2010)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-8-1856

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### References

- T. M. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).
- M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001). [CrossRef]
- A. Stern and B. Javidi, “Improved-resolution digital holography using the generalized sampling theorem for locally band-limited fields,” J. Opt. Soc. Am. A 23, 1227–1235 (2006). [CrossRef]
- J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006). [CrossRef] [PubMed]
- D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009). [CrossRef]
- A. J. Den Dekker and A. Van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A 14, 547–557 (1997). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
- A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. (Bellingham) 43, 239–250 (2004). [CrossRef]
- A. Stern and B. Javidi, “Space-bandwidth conditions for efficient phase-shifting digital holographic microscopy,” J. Opt. Soc. Am. A 25, 736–741 (2008). [CrossRef]
- S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 2005).
- P. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer Verlag, 2004).
- C. W. Helstrom, “Detection and resolution of incoherent objects by a background-limited optical system,” J. Opt. Soc. Am. 59, 164–175 (1969). [CrossRef]
- S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–57 (2006). [CrossRef]
- S. Van Aert, D. Van Dirk, and A. J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—Classical criteria and a statistical alternative,” Opt. Express 14, 3830–3839 (2006). [CrossRef] [PubMed]
- M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory 52, 3411–3437 (2006). [CrossRef]
- P. Réfrégier, J. Fade, and M. Roche, “Estimation precision of the degree of polarization from a single speckle intensity image,” Opt. Lett. 32, 739–741 (2007). [CrossRef] [PubMed]
- A. Sentenac, C. A. Guérin, P. C. Chaumet, F. Drsek, H. Giovannini, N. Bertaux, and M. Holschneider, “Influence of multiple scattering on the resolution of an imaging system: a Cramér–Rao analysis,” Opt. Express 15, 1340–1347 (2007). [CrossRef] [PubMed]
- C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–89 (1945).
- H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, 1946).
- F. Soulez, L. Denis, C. Fournier, E. Thiebaut, and C. Goepfert, “Inverse problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A 24, 1164–1171 (2007). [CrossRef]
- S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007). [CrossRef] [PubMed]
- F. Soulez, L. Denis, E. Thiebaut, C. Fournier, and C. Goepfert, “Inverse problem approach for particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A 24, 3708–3716 (2007). [CrossRef]
- L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. 25, 115017 (2009). [CrossRef]
- S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004). [CrossRef]
- L. Denis, D. Lorenz, E. Thiebaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009). [CrossRef] [PubMed]
- D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009). [CrossRef] [PubMed]

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