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 OSA
In optical microscopy, a classical limitation is related to small depths of focus that are due to the high numerical aperture of lenses when the high magnification ratios are requested. Consequently, mechanical scanning along the optical axis has to be performed to investigate sample thicker than the depth of focus. Digital holography allows us to overcome this limitation [1,3]. It provides a numerical investigation of the third dimension by performing a plane-by-plane refocusing. The digitally recorded hologram on CCD camera [4,5] is processed to compute the phase and intensity information in order to obtain the complex amplitude signal. By a discrete implementation of the Kirchkoff-Fresnel propagation equation, this complex amplitude signal is then propagated to perform a digital holographic refocusing of the sample image, slide by slide.
Digital holographic microscopy has been applied and demonstrated in many applications as in observation of biological samples [6,8], refractometry [9,10], analysis of living cells [11,12] and velocimetry . This technique allows for example to implement powerful processing to improve the digital holographic reconstruction [13, 14], to study concentration profiles inside confined deformable bodies flowing in microchannels , to perform 3D pattern recognition [16, 17], to control the image size as a function of the distance and the wavelength , to achieve quantitative phase contrast imaging [19, 20], to process border artifacts  and aberration compensation .
To refocus an object using digital holography, a numerical criterion is required to decide at which distance the investigated object is focus. Different criteria have been proposed in the literature. In , Yu and Cai investigated a refocus criterion based on a gradient computation while a sharpness metric based on the self-entropy has been described in . A criterion based on the maximization of the intensity local variance has also been proposed in . Ferraro and al. have proposed in  a focus tracking method where an operator focuses the object during the recording of a sequence of holograms and by measuring its phase shift, the sample displacement in real time is obtained. Choo and Kang have presented in  the correlation coefficient (CC) method that uses the fact that the CC is maximum at the focus plane. This method has been applied recently in digital particle holography to locate the focus plane of particles . Liebling and Unser have applied in  the theory of Fresnelet to compute a new sharpness metric related to the sparity of the wavelet coefficients and their energies. Langehanenberg and al. have presented and compared in  four concepts for autofocusing in digital holographic microscopic investigations on pure phase objects.
In , we have developed a new focus determination method based on the invariances of both energy and amplitude. Those invariance properties allow to build two focus criteria, respectively for pure amplitude and pure phase objects, based on the score of the integrated amplitude modulus. It has been shown that this measure is minimized for pure amplitude objects while it is maximized for pure phase ones. We have also recently proposed in  a local focus criterion normalized by the total light intensity for cases where a number of objects are placed at different refocusing planes. In their recent paper in , Li et al. proposed a technique, closely related to our refocusing criterion where is developed a focus detection technique for objects with real amplitude based on the spectral content of digital holograms.
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  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.
Consider a pure amplitude positively defined object t(x,y) located in a plane P. We propagate this object up to a distance ɛ in a plane P′ parallel to P using the Fresnel free-space propagation operator (FPO) for the paraxial approximation of the Kirchhoff-Fresnel equation (thanks to the operational formalism described in ):31] where we use k = 2π/λ and we consider that ɛ is small:
We obtain at this step the following expression:Eq. (7) is effective if there are spatial frequencies of the object that are removed by the numerical aperture of the lens. We consider that we are in this situation and for the sake of simplicity, we assume that |T (νx,νy)| is constant. That could seem a hard assumption but it is basically what it is assumed when evaluating the classical depth of field . Therefore, we assume that : Eq. (8) in Eq. (6) to evaluate the integral I and we obtain : Eq. (9) computed in polar coordinate system, leads to :
We analyze in the next section the behavior of one particle in the field of view. This analysis can be extended to the case of more well-separated particles in the recorded plane. In this situation, to focus each particle, we have to compute the refocusing criterion in a region of interest around each particle in order to take into account the different influences of each particle separately and to compute precisely their focus planes . In the case of overlapping particles, an analysis will be proposed for a publication in a coming contribution.
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′)| dx′dy′ − ∫|t(x,y)|dxdy in order to verify the dependency on ɛ2 and NA6.
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.
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.
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.
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 . 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).
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.
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.
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.
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.
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.
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).
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).
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.
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%.
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.
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.
In this article, we study the dependency of the refocusing criterion presented in  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.
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.
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