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  • Vol. 6, Iss. 4 — May. 4, 2011
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Dependency and precision of the refocusing criterion based on amplitude analysis in digital holographic microscopy

Ahmed El Mallahi and Frank Dubois  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6684-6698 (2011)
http://dx.doi.org/10.1364/OE.19.006684


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Abstract

We analyze the dependency and the accuracy of the refocusing criterion based on the integrated modulus amplitude in the case of amplitude object. Analytical dependencies on the defocus distance and the numerical aperture are found. This theoretical prediction for the refocusing criterion is well supported by simulation. We study also the robustness of the refocusing criterion by adding salt and pepper and speckle-type noises. We demonstrate that the refocusing criterion is robust up to an significant level of noise that can perturb the holograms.

© 2011 Optical Society of America

1. Introduction

Digital holographic microscopy has been applied and demonstrated in many applications as in observation of biological samples [6

6. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai, R. R. Dasari, and M. S. Feld, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004). [CrossRef] [PubMed]

,8

8. L. Yu, S. Mohanty, J. Zhang, S. Genc, M. K. Kim, M. W. Berns, and Z. Chen, “Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery,” Appl. Opt. 17, 12031–12038 (2009).

], refractometry [9

9. M. Seesta and M. Gustafsson, “Object characterization with refractometric digital Fourier holography,” Opt. Lett. 30, 471–473 (2005). [CrossRef]

,10

10. M. Gustafsson and M. Sebesta, “Refractometry of Microscopic Objects with Digital Holography,” Appl. Opt. 43, 4796–4801 (2005). [CrossRef]

], analysis of living cells [11

11. D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high-resolution living-cell analysis,” Appl. Opt. 43, 6536–6544 (2004). [CrossRef]

,12

12. N. Warnasooriya, F. Joud, P. Bun, G. Tessier, M. Coppey-Moisan, P. Desbiolles, M. Atlan, M. Abboud, and M. Gross, “Imaging gold nanoparticles in living cell environments using heterodyne digital holographic microscopy,” Opt. Express 18, 3264–3273 (2010). [CrossRef] [PubMed]

] and velocimetry [13

13. D. Lebrun, A. Benkouider, S. Cotmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express 11, 224–229 (2003). [CrossRef] [PubMed]

]. This technique allows for example to implement powerful processing to improve the digital holographic reconstruction [13

13. D. Lebrun, A. Benkouider, S. Cotmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express 11, 224–229 (2003). [CrossRef] [PubMed]

, 14

14. N. Pandey and B. Hennelly, “Fixed-point numerical-reconstruction for digital holographic microscopy,” Opt. Lett. 35, 1076–1078 (2010). [CrossRef] [PubMed]

], to study concentration profiles inside confined deformable bodies flowing in microchannels [15

15. C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, “Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography,” Appl. Opt. 45, 5305–5314 (2008). [CrossRef]

], to perform 3D pattern recognition [16

16. F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, “Pattern recognition with digital holographic microscope working in partially coherent illumination,” Appl. Opt. 41, 4108–4119 (2002). [CrossRef] [PubMed]

, 17

17. B. Javidi, I. Moon, S. Yeom, and E. Carapezza, “Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography,” Opt. Express 13, 4492–4506 (2005). [CrossRef] [PubMed]

], to control the image size as a function of the distance and the wavelength [18

18. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital hologram,” Opt. Lett. 29, 854–856 (2004). [CrossRef] [PubMed]

], to achieve quantitative phase contrast imaging [19

19. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. 24, 291–293 (1999). [CrossRef]

, 20

20. F. Charrire, A. Marian, F. Montford, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178–180 (2006). [CrossRef]

], to process border artifacts [21

21. F. Dubois, O. Monnom, C. Yourassowski, and J.-C. Legros, “Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies,” Appl. Opt. 41, 2621–2626 (2002). [CrossRef] [PubMed]

] and aberration compensation [22

22. T. Colomb, E. Cuche, F. Charrire, J. Khn, N. Aspert, F. Monfort, P. Marquet, and C. Despeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45, 851–863 (2006). [CrossRef] [PubMed]

].

In this paper, we analyze the dependency of our refocusing criterion based on the integrated amplitude modulus for pure amplitude objects. The analysis is performed thanks to the propagation on simulated data. The reason of this choice is that in this case the data can be accurately prepared with well-defined out of focus distances. Indeed with real data, it can be always objected that we cannot guarantee that the refocus criterion selects the actual best focus plane. We obtain an analytical dependency for this criterion that is very well confirmed for small defocus distances. We also study quantitatively the robustness of our criterion by simulating two different types of noise and demonstrate the precision of the refocusing criterion even if an important level of noise is added. It permits to prove the capability of the refocusing criterion to retrieve the focus plane for practice situations where recorded holograms are noisy and blurred.

In section 2, we describe the theoretical aspects for the dependency of the criterion and in section 3, we confirm by simulations the analytical expressions found. In section 4, the robustness of the refocusing criterion is analyzed. Conclusions are given in section 5.

2. Mathematical development

In order to determine the refocus distance of objects in digital holography, we have developed in [31

31. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowski, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006). [CrossRef] [PubMed]

] a method based on a specific focus measure equal to the integration of the amplitude modulus. This method is built on two physical invariants, the energy and the amplitude, that makes the method built on a well-defined theoretical background. It has been shown that this focus measure is minimized for pure amplitude objects while it is maximized for pure phase ones. We analyze now the refocusing criterion for a pure amplitude object in order to determine its dependency.

3. Dependency of the refocusing criterion: simulation and experimental results

To verify the dependency on ɛ2 and on NA6 of the refocusing criterion, we simulate the effect of a numerical aperture on a 5 μm particle for different value of NA (from 0.05 to 1 with step of 0.05). We then propagate this particle in order to obtain a defocus particle seen through a numerical aperture. This simulates a recorded defocus particle depending on the numerical aperture. On Fig. 1(a), we see the initial simulated particle of 5μm diameter where 1μm = 10 pixels. We filter it by a NA of 0.3 on Fig. 1(b) and defocus it to −50μm from the focus plane in Fig. 1(c). We then apply the digital holographic propagation with incremental steps and we compute, for each step, the refocusing criterion ∫ |tɛ(x,y′)| dxdy′ − ∫|t(x,y)|dxdy in order to verify the dependency on ɛ2 and NA6.

Fig. 1 (a) Simulated particle of 5μm, (b) filtered by a numerical aperture of 0.3 and (c) defocused to −50μm. Sampling step : 1μm = 10 pixels. Image size: 256 × 256 pixels.

3.1. The ɛ2 dependency

First, we fix the numerical aperture and analyze the evolution of the refocusing criterion as a function of ɛ. For small values of numerical aperture (0.05, 0.1 and 0.15), the ɛ2 dependency is very well verified as shown in Fig. 2. In order to point out the dependency of the refocusing criterion, we use a logarithmic representation to extract directly the slope and confirm the second-power dependency on ɛ. It is so expected to obtain a straight line with a slope equal to 2. Before computing the logarithm, we translate the curve to the right in order to put the focus plane (50μm) at 0 and perform the analysis on the right positive part thanks to the symmetry of the curve.

Fig. 2 Evolution of the refocusing criterion as a function of ɛ (log-log) for NA=0.05, 0.1 and 0.15, showing the ɛ2-dependency.

For bigger value of NA, the dependency is slightly lower than the second order of ɛ. On Fig. 3, the curves for the numerical apertures 0.2, 0.3, 0.4 and 0.5 are shown where we see that we obtain a good linearity although we observe a deviation for the higher values. The deviation with respect to the linearity is originated from the fact that the ɛ2 dependency is valid for small value of ɛ.

Fig. 3 Evolution of the refocusing criterion as a function of ɛ (log-log) for NA=0.2, 0.3, 0.4 and 0.5.

In the Fig. 4, we plotted the evolution of the refocusing criterion (renormalized) as a function of ɛ for the different values of numerical aperture where we can observe the typical parabolic shapes of the criterion for different numerical apertures of the lens.

Fig. 4 Evolution of the refocusing criterion as a function of ɛ for different values of NA.

In order to verify this ɛ2 dependency experimentally, we perform a test on a digital hologram of an opaque polyethylene spherical particle of 220 μm diameter. The digital holographic microscope used for this experiment is a Mach-Zehnder interferometer in a microscope configuration working with a partial spatial coherent source, described in [15

15. C. Minetti, N. Callens, G. Coupier, T. Podgorski, and F. Dubois, “Fast measurements of concentration profiles inside deformable objects in microflows with spatial coherence digital holography,” Appl. Opt. 45, 5305–5314 (2008). [CrossRef]

]. The field of view is 720 μm × 720 μm recorded on 1024 × 1024 pixels. The x10 microscope lenses have a numerical aperture of 0.10. The particle is recorded in a defocused plane of approximately −80μm, as shown in the Fig. 5(a).

Fig. 5 (a) Intensity image of a particle of 220μm in the recorded plane (defocused by a distance of approximately −80 μm). (b) Reconstructed intensity image at a distance of 79 μm determined by the minimum of the refocusing criterion.

The evolution of the refocusing criterion (renormalized) with respect to the distance of propagation is provided by Fig. 6 illustrating that the minimum occurs when the focus distance is reached. The parabolic shape of the refocusing criterion as a function of ɛ is also observed. In Fig. 7, the ɛ2 dependency of the criterion is highlighted using a log-log representation where a slope of 1.9798 is obtained verifying the ɛ2 dependency of the refocusing criterion.

Fig. 6 Experimental evolution of the refocusing criterion as a function of ɛ for NA = 0.10, showing the parabolic shape of the criterion and the minimum centered on the focus plane.
Fig. 7 Experimental evolution of the refocusing criterion as a function of ɛ (log-log) for NA=0.10, showing the ɛ2-dependency.

3.2. The NA6 dependency

In order to verify the NA6-dependency of the criterion, we fix a ɛ distance from the focus plane and analyze the evolution of the refocusing criterion versus NA. If we take for ɛ small value (0.1 to 0.5μm), meaning that we stay very close to the focus plane, the NA6-dependency is very well verified for values of NA from 0.05 to 0.25 as we show in the Fig. 8 for ɛ = 0.1, 0.3 and 0.5μm.

Fig. 8 Evolution of the refocusing criterion as a function of NA for ɛ = 0.1, 0.3 and 0.5μm.

The expression for the NA6-dependency of the refocusing criterion is well verified close to the focus plane and also for small value of numerical aperture. This sixth-order dependency in NA decreases quickly as soon as we deviate from the focus plane. For bigger distance, the NA6-dependency is not verified even if we restrict the value of NA as we can see in the Fig. 9 for ɛ = 1, 2 and 5μm. Due to the strong assumptions made to establish the dependency at the sixth order dependency, it is not surprising that significant deviations occur when we deviate from the focus plane. The important point is that the shape of the refocusing criterion is rapidly varying with the numerical aperture as observed in Fig. 4.

Fig. 9 Evolution of the refocusing criterion as a function of NA for ɛ = 1, 2 and 5μm.

4. Analysis of the precision of the refocusing criterion

In this section we analyze the robustness of the refocusing criterion. The accuracy of the refocusing criterion is limited by the presence of noise. If we have a perfectly noise free hologram, it is possible to determine exactly the position when the refocus criterion is minimum. However, in realistic cases, there is noise that induces fluctuations of the refocusing criterion leading to a limited accuracy. To evaluate these effects, we simulate two different types of noise that can perturb the recorded holograms. The first one is a salt and pepper noise, uniformly distributed, which is completely incoherent and corresponds to noise that could be originated from the electronic devices. The second type of noise simulated is a noise inherent to the coherent nature of the laser, leading to a speckle-type noise. This speckle noise aims to test the behavior of the criterion when there is a noise source in the sample volume that is currently under analysis. Indeed, if the sample is scattering the light, it results, due to the filtering process operated by the lens aperture, a speckle field in the detector plane. Therefore, if salt and pepper is modeling the electronic noise, the speckle noise is significant to simulate the optical noise created by the optical system itself.

4.1. Salt and pepper noise

Salt and pepper noise is a noise type that could be originated from the electronic devices and can corrupt the holograms during the recording step. First, we simulate a 5 μm particle seen through a numerical aperture and we defocus it to a distance of −50μm. Then, we add a complex random noise of zero mean to the complex amplitude of the defocused filtered particle to obtain a simulation of a recorded noisy defocused particle. As this type of noise could come from the electronic device during the recording step, it is meaningful to consider it as an additive noise. To quantify the level of salt and pepper noise, we assume that a 100% of salt and pepper noise has a maximum range, for both the real and imaginary parts equal to the maximum modulus amplitude of the particle field without noise. We simulate in this way different levels of noise for different values of the numerical aperture that filtered the particle.

In a first step, we study the effect of the different levels of noise on the evolution of the refocusing criterion as a function of the distance in order to see how evolves the minimum of the curve as the noise increases (in comparison with Fig. 4 without noise). We start by adding 5% of noise and increase by step of 5% up to the evolution of the criterion is completely disturbed and no minimum is actually visible. We estimate that the maximum level of noise that can perturb the recorded image while keeping a global meaningful minimum on the focus plane is 25%. On Fig. 10 is shown the evolution of the criterion for NA = 0.3 and for different levels of noise where we can observe that the global minimum of the criterion is situated around the focus plane (ɛ = 50μm) even if an important level of noise (up to 25%) perturbed the image before the refocusing process.

Fig. 10 Evolution of the refocusing criterion as a function of ɛ for NA = 0.3 and for different levels of salt and pepper noise.

In a second step, we quantify this robustness by estimating the deviation that the criterion gives from the initial known focus plane which is at 50μm. We simulate 50 realizations of several levels of noise added to the original hologram and we plot the mean deviations from the best focus plane and the associated standard deviations. In the Fig. 11 is shown the evolution of this deviation for NA = 0.3 with the error bars corresponding to the standard deviations. We see that up to 25% of added noise the deviation is just about a few μm. Upwards 25% of noise added, the deviation from the focus plane becomes important. The refocusing criterion is too disturbed and the global minimum can no longer be associated with precision to the focus plane.

Fig. 11 Evolution of the deviation from the best focus plane as a function of level noise added for NA = 0.3 (and associated standard deviation - error bars), showing the robustness of the refocusing criterion up to 35% of noise added - A zoom is performed to show more clearly the zone corresponding to the 5–20% level of noise added.

The simulations are also performed for other values of the numerical aperture see Table 1 where the deviations from the best focus plane and their associated standard deviations are shown for several levels of speckle noise. It is shown that the refocusing criterion is robust up to about 25% of salt and pepper noise for the different numerical apertures and allows to determine the focus plane with a good accuracy. The deviation from the best focus plane is less than 1μm with a associated standard deviation of around half a micron for the numerical aperture higher than 0.30. For NA = 0.30, the deviation from the focus plane is around 2μm with a standard deviation of around 1μm which determines the focus plane with a acceptable precision. Above a level of 25% of salt and pepper noise added, the deviation from the focus plane becomes too important. These simulations demonstrates the robustness of the refocusing criterion on noisy images (see Fig. 12 for an illustration of 10, 20 and 30% of level noise added).

Fig. 12 (a) Illustration of 10%, (b) 20% and (c) 30% of noise added on a 5μm particle filtered by a numerical aperture of 0.3 and defocused to −50μm.

Table 1. Average Deviations from the Focus Plane (μ) for Different Values of Numerical Aperture and Different Levels of Salt and Pepper Noisea

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4.2. Speckle noise

Speckle noise is inherent to the coherent nature of the laser light and can corrupt the holograms. A light amplitude a illuminating a particle of transparency, in its best focus plane, gives rise, t just after t, to the amplitude at. We assume that the illumination amplitude is a constant A with an additional noise n due, for instance, to the light scattered in the optical set up before the particle. Therefore the emerging amplitude out of the best focus particle plane is expressed by at = At+nt. We observe that it results the particle transparency with an added noise multiplied by the transparency of the particle.

In order to simulate this process and its influence on the refocus criterion, we added to the transparency amplitude of the object a complex noise multiplied by the transparency amplitude. Several strengths of noise are used in order to simulate the robustness of the criterion. We apply the filtering process to introduce the numerical aperture effect and a defocus distance. The amount of applied noise is determined by computing the ratio between the averaged amplitude modulus of the filtered noise without particle (Fig. 13(a) - NA = 0.60) and the particle without noise (Fig. 13(b) - NA = 0.60, 5 μm particle). The corresponding noisy particle filtered by the numerical aperture focus and defocus to −50μm are, respectively, shown by the Fig. 13(c) and Fig. 13(d).

Fig. 13 (a) Filtered noise by a numerical aperture of NA = 0.60 (level of speckle noise = 24%). (b) Particle of 5μm filtered by a numerical aperture of 0.60. (c) Corresponding noisy particle. (d) Defocused by a distance of −50μm.

First, we study the evolution of the refocusing criterion as a function of the refocusing distance for different values of speckle noise level and with different values of NA. We illustrate it for NA = 0.60 in Fig. 14 up to a level of speckle noise of 36%. We see that for this numerical aperture, the refocusing criterion gives a global minimum located around the focus plane (50μm) up to a level of speckle noise of around 35%. Above this level, the curve becomes very noisy and the global minimum does no more correspond to the focus plane. We compute the simulations for the others numerical apertures and obtain also that the maximum of speckle noise level that can corrupt the holograms while keeping a global meaningful minimum is around 30%, which shows that under this type of noise, the refocusing criterion is robust.

Fig. 14 Evolution of the refocusing criterion as a function of ɛ for NA = 0.60 and for different levels of speckle noise.

In a second time, we quantify this robustness by simulating for each NA 50 realizations of different levels of speckle noise. We then calculate the deviations from the best focus plane for each numerical aperture and for each level of speckle noise, with the associated standard deviations (illustrated for NA = 0.60 in Fig. 15). These simulations show that when a speckle type noise of different levels corrupt the hologram, the refocusing criterion permits to obtain the focus plane with a precision of a few μm up to a speckle noise level of 35%.

Fig. 15 Evolution of the deviation from the best focus plane as a function of speckle noise level for NA = 0.60 (and associated standard deviation - error bars), showing the robustness of the refocusing criterion up to 35% of noise added - A zoom is performed to show more clearly the zone corresponding to the 12 – 36% level of speckle noise.

The simulations are also performed for other values of the numerical aperture see Table 2 where the deviations from the best focus plane and their associated standard deviations are reported for different levels of speckle noise. It is shown that the refocusing criterion is robust up to around 36% of speckle noise for the numerical apertures equals to 0.45, 0.60 and 0.75 with a precision of around 1 – 3μm. For NA = 0.30, the maximum level of speckle noise that can corrupt the holograms is around 24%. Above this level of speckle noise, the deviation from the focus plane becomes too important and no global minimum is actually visible on the refocusing criterion curve.

Table 2. Average Deviations from the Focus Plane (μ) for Different Values of Numerical Aperture and Different Levels of Speckle Noisea

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These simulations show that when the holograms are corrupted with a speckle type noise, the refocusing criterion permits to obtain the focus plane with a precision of a few μm up to a speckle noise level of around 24%. This level of speckle noise that can corrupt the holograms can be increased up to 36% for the tested numerical aperture higher than 0.30.

5. Conclusion

In this article, we study the dependency of the refocusing criterion presented in [31

31. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowski, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006). [CrossRef] [PubMed]

] in the case of amplitude object. We obtain analytical expressions showing a ɛ2 and a NA6 dependencies of the criterion. These theoretical dependencies are confirmed by simulations. The ɛ2 behavior is confirmed for small values of numerical aperture with a slightly deviation for bigger values of NA. The NA6 dependency is verified close to the focus plane but decreases rapidly when we deviates from the focus plane. It is due to the strong assumptions of the theoretical model.

We also demonstrate the robustness of this criterion when corrupted by two types of noise, a salt-pepper one and a speckle-type one. For the first type, we show that the focus plane can be reconstructed with a precision of a few microns up to a noise added level of 25%, which corrupts largely the recorded holograms. The study of the influence of different numerical apertures show that for the tested NA higher than 0.30, the deviation from the focus plane is less than a micron. For NA = 0.30, the deviation is around 2μm with a precision of around one micron which permits to refocus particles with a acceptable accuracy. For the speckle-type noise, we study the effect of the numerical aperture on the refocusing criterion and demonstrate that we can reconstruct particles that are corrupted up to 36% of speckle noise type with a precision of around one micron for the tested numerical aperture higher than 0.30. For NA = 0.30, the acceptable level of speckle noise that can corrupt the holograms is around 24%. These simulations have shown that the refocusing criterion is a robust and precise tool to reconstruct digitally the best focus plane in the case of amplitude objects.

Acknowledgments

The authors acknowledge financial support from l’Institut Bruxellois pour la Recherche et l’Innovation (IRSIB) in the frame of the Holoflow Impulse 2008 project.

References and links

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Y. J. Choo and B. S. Kang, “The characteristics of the particle position along an optical axis in particle holography,” Meas. Sci. Technol. 17, 761–770 (2006). [CrossRef]

28.

Y. Yang, B. S. Kang, and Y. J. Choo, “Application of the correlation coefficient method for determination of the focus plane to digital particle holography,” Appl. Opt. 47, 817–824 (2008). [CrossRef] [PubMed]

29.

M. Liebling and M. Unser, “Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion,” J. Opt. Soc. Am. A 21, 2424–2430 (2004). [CrossRef]

30.

P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47, 176–182 (2008). [CrossRef]

31.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowski, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006). [CrossRef] [PubMed]

32.

M. Antkowiak, N. Callens, C. Yourassowski, and F. Dubois, “Extended focused imaging of a microparticle field with digital holographic microscopy,” Opt. Lett. 33, 1626–1628 (2008). [CrossRef] [PubMed]

33.

W. Li, N. C. Loomis, Q. Hu, and C. S. Davis, “Focus detection from digital in-line holograms based on spectral l1 norms,” J. Opt. Soc. Am. A 24, 3054–3062 (2007). [CrossRef]

34.

M. Nazarathy and J. Shamir, “Fourier optics described by operator,” J. Opt. Soc. Am. 70, 150–159 (1980). [CrossRef]

OCIS Codes
(090.1760) Holography : Computer holography
(090.2880) Holography : Holographic interferometry
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: December 3, 2010
Revised Manuscript: February 4, 2011
Manuscript Accepted: February 9, 2011
Published: March 24, 2011

Virtual Issues
Vol. 6, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Ahmed El Mallahi and Frank Dubois, "Dependency and precision of the refocusing criterion based on amplitude analysis in digital holographic microscopy," Opt. Express 19, 6684-6698 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-7-6684


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References

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  28. Y. Yang, B. S. Kang, and Y. J. Choo, "Application of the correlation coefficient method for determination of the focus plane to digital particle holography," Appl. Opt. 47, 817-824 (2008). [CrossRef] [PubMed]
  29. M. Liebling, and M. Unser, "Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion," J. Opt. Soc. Am. A 21, 2424-2430 (2004). [CrossRef]
  30. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, "Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging," Appl. Opt. 47, 176-182 (2008). [CrossRef]
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  34. M. Nazarathy, and J. Shamir, "Fourier optics described by operator," J. Opt. Soc. Am. 70, 150-159 (1980). [CrossRef]

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