Dependency and precision of the refocusing criterion based on amplitude analysis in digital holographic microscopy

Ahmed El Mallahi; Frank Dubois

doi:10.1364/OE.19.006684

1. Introduction

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 [13]. 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 [15], to perform 3D pattern recognition [16, 17], to control the image size as a function of the distance and the wavelength [18], to achieve quantitative phase contrast imaging [19, 20], to process border artifacts [21] and aberration compensation [22].

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 [23], 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 [24]. A criterion based on the maximization of the intensity local variance has also been proposed in [25]. Ferraro and al. have proposed in [26] 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 [27] 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 [28]. Liebling and Unser have applied in [29] 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 [30] four concepts for autofocusing in digital holographic microscopic investigations on pure phase objects.

In [31], 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 [32] 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 [33], 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 [31] 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 [34]):

\begin{array}{l} t_{ɛ} (x^{'}, y^{'}) = R [ɛ] t (x, y) = exp (j k ɛ) F^{- 1} Q [- λ^{2} ɛ] F t (x, y) \\ = exp (j k ɛ) F^{- 1} exp (- \frac{j k λ^{2} ɛ}{2} (ν_{x}^{2} + ν_{y}^{2})) F t (x, y) \end{array}

where (x,y) and (ν_x,ν_y) are respectively spatial coordinates and spatial frequencies, F and Q are respectively the Fourier transform operator and the quadratic phase operator defined by the relation Q[a]f(ρ) = exp [j(k/2)aρ²]f(ρ) with k = 2π/λ, where λ is the wavelength and

j = \sqrt{- 1}

. Considering a small propagation distance ɛ, we develop Q[−λ²ɛ] until the first order and we obtain :

t_{ɛ} (x^{'}, y^{'}) \approx exp (j k ɛ) (t - \frac{j k λ^{2} ɛ}{2} \underset{\equiv t^{'}}{\underset{︸}{F^{- 1} (ν_{x}^{2} + ν_{y}^{2}) F t (x, y)}})

According that t is real, t′ is also real. We integrate now the amplitude modulus of the previous expression in order to obtain the refocusing criterion from [31] where we use k = 2π/λ and we consider that ɛ is small:

\int | t_{ɛ} (x^{'}, y^{'}) | d x^{'} d y^{'} \approx \int t \sqrt{1 + π^{2} λ^{2} ɛ^{2} \frac{{t^{'}}^{2}}{t^{2}}} d x d y \approx \int t (1 + \frac{π^{2} λ^{2} ɛ^{2}}{2} \frac{t^{' 2}}{t^{2}} d x d y)

where it is assumed that t(x,y) is never equal to zero.

We obtain at this step the following expression:

\int | t_{ɛ} (x^{'}, y^{'}) | d x^{'} d y^{'} = \int | t (x, y) | d x d y + \frac{π^{2} λ^{2} ɛ^{2}}{2} \int \frac{{| t^{'} |}^{2}}{t} d x d y

The term on the left side is the integrated amplitude at the propagate plane P′ depending on ɛ while the first term on the right is the integrated amplitude of the object in the initial plane P. We can show here the dependency in ɛ² of the refocusing criterion since the integral no longer depends on ɛ. We have now to evaluate the second right term where the integral is strongly dependant of the object t(x,y) in order to show how this term can influence the refocusing criterion. A simplifying assumption is to consider the maximum value of the object amplitude t_M = max {t(x,y)}. In this case, we obtain an inequality:

\int | t_{ɛ} (x^{'}, y^{'}) | d x^{'} d y^{'} - \int | t (x, y) | d x d y \geq \frac{π^{2} λ^{2} ɛ^{3}}{2 t_{M}} \int {| t^{'} |}^{2} d x d y

The evaluation of the integral on the right part of the inequality (noted I) will give an analytical expression for the dependency of the refocusing criterion whose the variation will be faster than I. Using the Parseval’s theorem, we have

I = \int {| T^{'} |}^{2} d ν_{x} d ν_{y} = \int {(ν_{x}^{2} + ν_{y}^{2})}^{2} {| F t (x, y) |}^{2} d ν_{x} d ν_{y}

where T′ = Ft′. We will now investigate the effect of the numerical aperture on the refocusing criterion. For that purpose, we assume that the object t(x,y) is on the front focal plane of a lens which the focal distance f is limited by an aperture p. The amplitude of the object that will be actually imaged by the system is the front focal plane of the lens, noted t_f and is given by:

t_{f} (x, y) = A F p (λ f ν_{x}, λ f ν_{y}) T (ν_{x}, ν_{y})

where A is a constant that does not play a significant role. The filtering process described by 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 (ν_x,ν_y)| is constant. That could seem a hard assumption but it is basically what it is assumed when evaluating the classical depth of field [34]. Therefore, we assume that :

t_{f} (x, y) \approx c F p (λ f ν_{x}, λ f ν_{y})

where c is a constant. Now, we can replace Eq. (8) in Eq. (6) to evaluate the integral I and we obtain :

I \approx c \int {(ν_{x}^{2} + ν_{y}^{2})}^{2} {p (λ f ν_{x}, λ f ν_{y})}^{2} d ν_{x} d ν_{y}

To compute this expression, we consider a circular aperture pupil function of diameter D. According that the numerical aperture NA is given by NA = D/f, Eq. (9) computed in polar coordinate system, leads to :

I \approx 2 c π \int_{0}^{NA / 2 λ} r^{5} d r \propto {NA}^{6}

Here we can show that after our assumptions, a NA⁶ dependency of the refocusing criterion is found. We have analyzed the refocusing criterion and have found a ɛ² and a NA⁶ dependency. In the next section, we simulate the effect of a numerical aperture on a 5 μm particle and then propagate it in order to obtain a defocus particle seen through a numerical aperture. In this way, we simulate a recorded defocus particle which can be refocus using the criterion in order to verify the dependency in ɛ² and NA⁶.

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 [32]. 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 ∫ |t_ɛ(x′,y′)| dx′dy′ − ∫|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.

Level of Salt and Pepper noise	NA: 0.30		NA: 0.45		NA: 0.60		NA: 0.75
Level of Salt and Pepper noise	μ	σ	μ	σ	μ	σ	μ	σ
15 %	0.828	0.631	0.416	0.312	0.404	0.269	0.168	0.134
25 %	1.866	1.127	0.752	0.595	0.664	0.451	0.356	0.326
35 %	6.244	8.940	1.656	1.176	14.072	20.399	18.152	21.299

Level of Speckle noise	NA: 0.30		NA: 0.45		NA: 0.60		NA: 0.75
Level of Speckle noise	μ	σ	μ	σ	μ	σ	μ	σ
12 %	0.072	0.053	0.044	0.049	0.092	0.056	0.028	0.044
24 %	0.521	0.875	0.228	0.186	0.442	0.435	0.396	0.686
36 %	6.448	13.086	0.896	1.170	1.728	1.548	1.084	1.072

Abstract

1. Introduction

2. Mathematical development

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

3.1. The ɛ2 dependency

3.2. The NA6 dependency

4. Analysis of the precision of the refocusing criterion

4.1. Salt and pepper noise

4.2. Speckle noise

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (15)

Tables (2)

Equations (10)

Optics Express

3.1. The ɛ² dependency

3.2. The NA⁶ dependency