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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 16735–16744
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3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation

Laurence Wilson and Rongjing Zhang  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 16735-16744 (2012)
http://dx.doi.org/10.1364/OE.20.016735


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Abstract

The Rayleigh-Sommerfeld back-propagation method is a fast and highly flexible volume reconstruction scheme for digital holographic microscopy. We present a new method for 3D localization of weakly scattering objects using this technique. A well-known aspect of classical optics (the Gouy phase shift) can be used to discriminate between objects lying on either side of the holographic image plane. This results in an unambiguous, model-free measurement of the axial coordinate of microscopic samples, and is demonstrated both on an individual colloidal sphere, and on a more complex object — a layer of such particles in close contact.

© 2012 OSA

1. Introduction

In comparison, the Rayleigh-Sommerfeld back-propagation scheme [11

11. J. W. Goodman, Introduction to Fourier Optics3rd Ed., (Roberts & Company, 2005).

, 12

12. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007). [CrossRef] [PubMed]

] is highly flexible and fast but has not been widely used, in part because a rigorous criterion for localizing an object’s position in three dimensions has not been established. Various localization schemes have been put forward for use with DHM (both with Rayleigh-Sommerfeld and other reconstruction methods). Most methods so far employed are heuristic in some sense, finding peaks in the axial intensity profile [13

13. G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003). [CrossRef] [PubMed]

15

15. J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45, 3893–3901 (2006). [CrossRef] [PubMed]

], or using in-plane Laplacian derivatives to locate the edges of objects [5

5. J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. B. amnd, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. USA 104, 17512–17517 (2007). [CrossRef] [PubMed]

]. Instead, our approach is based on a well-known physical principle – the Gouy phase shift – and Rayleigh-Gans (RG) scattering theory. It is applicable to a wide variety of weakly scattering microscopic objects such as colloidal suspensions or swimming planktonic bacteria.

2. Theory and methods

Fig. 1 (a) Optical layout; an LED is placed behind the condenser aperture, which is closed as far as possible to approximate a point source. (b) The scattering geometry. (c) Calculated RG hologram with pixel values rescaled to the range 0–255. The scale bar indicates 10μm.

We first consider a small spherical particle, with radius a, in a typical DHM setup (Fig. 1(b)). A plane wave with amplitude E0 is incident on a particle located at the origin. The scattered field Es interferes with the unscattered light E0, resulting in a total intensity at a point r [29

29. F. Cheong, B. Krishnatreya, and D. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18, 13563–13573 (2010). [CrossRef] [PubMed]

]:
I(r)=|E0(r)|2+[E0*(r)Es(r)]+|Es(r)|2.
(2)
For a weak scatterer, the scattered intensity |Es(r)|2 is small, and so can be ignored. Dividing through by the unscattered intensity we obtain the ‘normalized hologram’ b(r):
b(r)1+[E0*(r)Es(r)|E0(r)|2].
(3)
Each dipole element in the scatterer gives a separate contribution to the scattered field; these contributions may be integrated over the volume of the scatterer to give the total scattered field [19

19. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons., 1983).

]. The incident, unpolarized field may be represented as the superposition of two orthogonally polarized beams. This leads to a scattering pattern which is azimuthally symmetric about the z-axis,
b(r)1+[k2exp(ikr)2πr(m1)Vf(θ)(1+cosθ)],
(4)
where V is the volume of the scatterer, and f(θ) is the form factor for a sphere [20

20. B. Berne and R. Pecora, Dynamic Light Scattering (John Wiley & Sons, Inc., 1976).

],
f(θ)=3(qa)3(sinqaqacosqa),whereq=2ksinθ2.
(5)
Throughout, we assume that the scatterer is far enough from the scattering plane that geometric rotation of polarization is small [29

29. F. Cheong, B. Krishnatreya, and D. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18, 13563–13573 (2010). [CrossRef] [PubMed]

]. We use Eq. (4) to calculate the RG hologram of a spherical particle as shown in Fig. 1(c). The hologram was calculated on a plane located 40μm from the scatterer (i.e. the scatterer was located at zp = −40μm in our coordinate system, as we consider the center of the hologram plane to be the origin), and measures 512 pixels in each dimension. The other relevant parameters were λ = 0.505μm, a = 0.1μm, the particle and surrounding medium refractive indices np = 1.55 and nm = 1.33 respectively, and the sampling frequency in the x, y plane, 10 pixels/μm. Based on this simulated data, we reconstruct the light field associated with the particle using the Rayleigh-Sommerfeld scheme. This method has been described in detail elsewhere [11

11. J. W. Goodman, Introduction to Fourier Optics3rd Ed., (Roberts & Company, 2005).

, 12

12. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007). [CrossRef] [PubMed]

], but briefly, we recover the electric field at a height −z′ with the use of the Rayleigh-Sommerfeld propagator
h(x,y,z)=12πzexp(ikr)r
(6)
where r′ = (x2 + y2 + z2)1/2. Note the use of primed coordinates to indicate a position in the reconstructed (as opposed to physical) volume. We can reconstruct the field (and from there, the image) at any height above the hologram plane by the convolution
Es(x,y,z)=Es(x,y,0)h(x,y,z).
(7)
To demonstrate this, a stack of 512 images were numerically reconstructed, at a spacing along the optical axis of Δz′ = −0.156μm, where the sign of this increment indicates a reconstruction upstream (from hologram to scatterer).

The central region of the plane y′ = 0 (i.e. through the center of the scatterer) is shown in Fig. 2(a). Note the small bright region below the midpoint and the small dark region just above the midpoint; this feature is due to the phase shift on passing through the geometrical focus. This image was then convolved with a kernel based on the Sobel filter [30

30. E. L. Hall, Computer Image Processing and Recognition (Academic Press, 1979).

] to extract the intensity gradient in the negative z′-direction only, which we denote g(x′, z′):
g(x,z)=zI(x,z)I(x,z)Sz,
(8)
where the filter kernel Sz is given by
Sz=(010000010).
(9)
For the approximation in Eq. (8) to be valid, the optical field must vary slowly compared to the sampling frequency in the axial direction. This condition is easily satisfied, as Δz′ can be made arbitrarily small. For clarity, we retain only the values of g(x′, z′) greater than zero, setting all negative values to zero. The result of this process can be seen in Fig. 2(b). This gradient operation provides a very efficient method of locating the center of the scatterer, with sub-pixel resolution. To demonstrate this, we extracted the intensity profiles along lines through the center vertical (red) and horizontal (black) directions of each of the images. These plots are shown in Fig. 2(c) and Fig. 2(d). In both panels, the axial intensity profile is shifted by +40μm, to place the scatterer’s center at the origin. Note also that this method has yielded the center position of a scatterer whose z-coordinate is negative; the field, imaged onto the camera sensor is diverging. If a particle’s z-coordinate is positive, the field is converging when it arrives at the image sensor and has not yet passed through its geometrical focus. This allows us to separate objects lying on different sides of the hologram plane unambiguously. Whether the particles have a physical position zp above or below the focal plane, they have twin images at ±zp in the reconstructed space. However, the twin images are not identical. If we only reconstruct the half-space ‘upstream’ of the focal plane (i.e. z′ < 0), we find that g(zp) is a maximum for particles with z < 0, and a minimum for those with z > 0. As a check of this method, we also generated a hologram using the more accurate Mie scattering solution for the scattered field [19

19. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons., 1983).

, 29

29. F. Cheong, B. Krishnatreya, and D. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express 18, 13563–13573 (2010). [CrossRef] [PubMed]

], using the parameters outlined above. The reconstructed axial coordinate obtained by analyzing the Mie hologram was z = −39.92μm, an error of 0.2%.

Fig. 2 Analysis of simulated data for a single weakly scattering sphere (see text for details) — Scale bars represent 2μm. (a) Intensity as a function of x and z in a plane through the center of the scatterer. Note the transition from bright to dark upon passing through the center of the particle in the vertical direction. (b) g(x′, z′) > 0 taken from the previous image. (c) Intensity profiles sampled through the center of the first panel, in the horizontal direction (black) and vertical direction (red). (d) Intensity profiles sampled through the center of the second panel (the gradient image g(x′, z′)). Note that the axial profiles in this panel and the previous one have been shifted by 40 μm to place the scatterer at the origin. Here and elsewhere, we use primed coordinates to indicate a position in the reconstructed volume.

Fig. 3 Example data from a single particle. Scale bars represent 2μm in all cases. (a) Vertical slice through the center of an image stack created by physically translating the sample (see text). (b) Image of a particle located at z ≈ 9μm (‘downstream’ of the focal plane in the illumination path). (c) Optical field reconstructed from the previous panel. The hologram plane (z′ = 0) would be located below the bottom of the image. (d) Intensity gradient g(x′, z′) < 0. The dark central spot is azimuthally symmetric about the z′-axis and gives the particle location in all three dimensions. (e,f) The companion images to (b,c), for a particle located at z ≈ −9μm (‘upstream’ in the illumination path). (g) Intensity gradient g(x′, z′) > 0. The particle location is specified by a maximum of g(x′, z′) for those scatterers with z < 0. The intensity in Panels (c) and (f) have been rescaled for clarity, but the shape of the optical field is unchanged.

Imaging was performed using a Mikrotron MC-1362 camera attached to an EPIX PIXCI E4 frame grabber, with an exposure time of 19.9 ms per frame. Sample chambers were constructed from microscope coverslips with internal dimensions 0.2 × 5 × 20mm. One such chamber was filled by capillary action, with an aqueous suspension of polystyrene particles (Bangs Labs, manufacturer-specified diameter 535 nm). This sample was allowed to dry, leaving particles adhered to the bottom surface of the chamber. The chamber was then refilled with immersion oil (Nikon ‘type A’, nd = 1.515 at 23°C). This was done to minimize the effects of aberrations incurred when imaging through water into glass and to limit any multiple reflections between the particles and the wall of the sample chamber. The stepper motor used to position the objective lens turret was used to defocus the image of a single isolated particle by a defined amount. Although the calibrated motor readout implied a precision of ±25 nm, this did not accurately reflect our ability to reproducibly place a particle in ‘perfect focus’. The slight flexibility of the sample chamber wall and the viscosity of the immersion oil (which causes the coverslip to deform as the objective lens is moved) place a more conservative uncertainty on the true z position of the particle. Furthermore, the selection of a perfectly in-focus plane (i.e. placing a particle at z = 0μm) was performed by eye, and was hence a little subjective. Based on several cycles of defocusing then refocusing the image of a layer of particles, we estimate the total uncertainty in z to be ±150 nm.

Once a suitable particle had been selected, a stack of images was acquired by moving the focus of the objective lens in steps of 1μm across a range of defocus, giving a range of axial particle coordinates from zp = −50μm to +50μm (zp = 0μm implies that the particle lies in the focal plane). Next, a patch of closely-spaced particles was found. Images of this particle layer were acquired at different defocus distances. This second experiment demonstrates the technique’s use for measuring the z-coordinate of a flat, extended object. In both cases, the filter Sz was convolved with every reconstructed xz′ plane. Due to the chosen form of the convolution kernel, an identical stack of gradient images g(x′, y′, z′) would be obtained by convolution with the yz′ planes. In both the single- and multiple-particle cases, images were taken with and without the sample in place. This allowed the removal of unwanted background artifacts caused by imperfections in the optical system [12

12. S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007). [CrossRef] [PubMed]

].

3. Results

Figure 3(a) shows an xz slice through a stack of images of a single particle, created by manually refocusing the stage at Δz = 1μm intervals (a similar process was implemented by Elliot and Poon [34

34. M. Elliot and W. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Coll. Interf. Sci. 92, 133–194 (2001). [CrossRef]

] who compared different schemes for imaging colloids: Phase contrast, DIC etc.) If the particle was in focus in the microscope (z′ = z = 0), this would be a map of the intensity close to the camera’s image sensor. Figure 3(b) and Fig. 3(e) show example xy images of an isolated particle at z = 9μm and z = −9μm respectively. From each of these images, a stack similar to that in Fig. 3(a) can be reconstructed (Figs. 3(c) and 3(f)), using Eq.s (3) and (7).

Note that the pixel values of each calculated stack have been rescaled to fill 256 gray levels for clarity; slight differences in background illumination level lead to the apparent intensity difference between the two xz′ images, but the important qualitative features are the same. In both cases, the particle is located in the center of the panel. Figure 3(d) shows g(x′, z′) < 0 from Fig. 3(b), and Fig. 3(g) shows g(x′, z′) > 0 from Fig. 3(f), clearly showing the position of the particle in both cases. These intensity features may be fitted in three dimensions in order to accurately localize the particle. We find z = 8.9±0.15μm for Fig. 3(d) and z = −9.1±0.15μm for Fig. 3(g), with uncertainties dominated by the ±150 nm instrumental error.

Next, we used our method to reconstruct the axial position of an extended object. To do this, we imaged a layer of particles resting on a coverslip, as described above. Images were acquired at a range of defocus distances (positions relative to the focal plane) from z = 0μm to z = −80μm in steps of Δz = −4μm. Similarly accurate results were obtained from data in the range z = 0μm → 80μm (data not shown). From each of these stage displacements, a full stack of images was calculated, and from these we obtained the gradient g(x′, y′, z′) as before. As opposed to fitting subregions corresponding to individual particles, we integrated over all x′, y′ values at constant z′, for each actual stage displacement z. This yielded a single value which is a function of actual (z) and reconstructed (z′) defocus, and which we denote G(z, z′) = ∫ (g z;x′, y′, z′)dx′dy′. These one-dimensional quantities are plotted in Fig. 4, where each row of colors corresponds to an actual defocus distance z. The horizontal axis shows reconstructed defocus distance, and the colors indicate the magnitude of G(z, z′). The data have been rescaled such that the maximum value of G(z, z′) is unity for each actual defocus z. Although the peaks in G(z, z′) broaden considerably as a function of z, the peak values are well-defined and easy to fit in all cases.

Fig. 4 2D map comparing actual (z) and reconstructed (z′) defocus distances of a layer of closely-spaced particles. Around 2000 closely-spaced particles resting on the surface of a glass slide were imaged with varying amounts of defocus (see text). The focal plane was translated through the sample in order to image the particle layer with an axial position in the range z = 0 μm to z = −80 μm, in steps of Δz = −4 μm (vertical axis). For each actual displacement, a stack of 400 images was reconstructed, spaced 200 nm apart. The color bar represents the total intensity gradient G(z, z′) = ∫ g(z′;z)dx′dy′, again including only values of g(x′, y′, z′) > 0.

The results of this fitting procedure as well as some sample images of the 2D layer of particles can be seen in Fig. 5. The first panel, Fig. 5(a) shows the layer of particles with a defocus of z = −4μm; Fig. 5(b) shows the layer with a defocus of −12μm. Even though the individual particle images are indistinct at −12μm, the reconstruction accurately reproduces the optical field in the plane of the particles. As stated in the introduction, we consider each particle to be composed of independent scattering centers; we therefore find the distance between particles to be unimportant. The maximum value of G(z, z′) indicates the axial position of the layer of particles. We plot these reconstructed defocus values as a function of actual defocus in Fig. 5(c), where excellent correspondence is found. Figure 5(d) shows the absolute error as a function of physical stage displacement. There is a systematic error of around −150 nm in the reconstructed particle positions. This seems most likely to be dominated by uncertainty in the z = 0 (‘in-focus’) position, which was chosen by eye, and would lead to the same offset in all measurements. The scatter of points around this value is more likely due to the mechanical considerations outlined above, as some tuning of the focus position was required. Lastly, although we have not investigated the matter in detail here, preliminary results show that this method is applicable to particles located at close lateral (x′, y′) positions, but separated axially (data not shown). This implies that the tracking of multiple weakly-scattering objects (such as colloids or bacteria) in 3D should be reasonably straightforward, at least in the dilute limit. Special treatment may be required to uniquely identify objects in very close proximity — a limitation shared by conventional 2D video tracking.

Fig. 5 (a) A layer of particles resting on the surface of a coverslip, defocus +4μm. (b) The same particles, defocus +12μm. The scale bars denote 5μm in both panels. (c) Displacement of the particle layer obtained from the reconstructed optical field, as a function of actual stage displacement. The solid line shows z = z′. (d) The absolute error in position measurements from the previous panel. The data show a systematic error of approximately 150 nm, consistent with the estimate of focus positioning uncertainty. Note that the vertical scale in panel (d) has a much higher resolution than in panel (c).

4. Conclusion

We have outlined a novel method for 3D object localization in digital holographic microscopy. This technique draws on a well-known aspect of classical wave optics, the Gouy phase anomaly, to localize objects in three dimensions with high precision. The Rayleigh-Gans theory that describes the scattering of light by phase objects is underpinned by the concept that small volumes of the same body can be treated as independent scattering centers. Each scattering center is the source of secondary spherical wavelets which, when numerically refocused, demonstrate the Gouy phase shift. The interference between scattered and unscattered fields results in a contrast inversion about their geometrical focus. We anticipate that this technique will find many applications in microbiology, where the subjects are well described as weak phase objects. Not only can single, small scatterers be tracked with high resolution in space and time, but the structure of larger objects, which may be considered as a distributed collection of scattering centers, can also be studied.

References and links

1.

S. 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]

2.

J. Fung, K. E. Martin, R. W. Perry, D. M. Katz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express 19, 8051–8065 (2011). [CrossRef] [PubMed]

3.

W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt. 31, 4973–4978 (1992). [CrossRef] [PubMed]

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W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA 98, 11301–11305 (2001). [CrossRef] [PubMed]

5.

J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. B. amnd, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. USA 104, 17512–17517 (2007). [CrossRef] [PubMed]

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Z. Frentz, S. Kuehn, D. Hekstra, and S. Leiber, “Microbial population dynamics by digital in-line holographic microscopy,” Rev. Sci. Inst. 81, 084301 (2010). [CrossRef]

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A. van Blaaderen and P. Wiltzius, “Real-space structure of colloidal hard-sphere glasses,” Science 270, 1177–1179 (1995). [CrossRef]

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S.-H. Lee and D. G. Grier, “Holographic microscopy of holographically trapped three-dimensional structures,” Opt. Express 15, 1505–1512 (2007). [CrossRef] [PubMed]

13.

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827–833 (2003). [CrossRef] [PubMed]

14.

C. Fournier, C. Ducottet, and T. Fournel, “Digital in-line holography: influence of the reconstruction function on the axial profile of a reconstructed particle image,” Meas. Sci. Technol. 15, 686–693 (2004). [CrossRef]

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J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45, 3893–3901 (2006). [CrossRef] [PubMed]

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M. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010). [CrossRef]

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OCIS Codes
(180.6900) Microscopy : Three-dimensional microscopy
(090.1995) Holography : Digital holography

ToC Category:
Microscopy

History
Original Manuscript: June 18, 2012
Revised Manuscript: June 28, 2012
Manuscript Accepted: June 28, 2012
Published: July 9, 2012

Virtual Issues
Vol. 7, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Laurence Wilson and Rongjing Zhang, "3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation," Opt. Express 20, 16735-16744 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16735


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References

  1. S. 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. Express15, 18275–18282 (2007). [CrossRef] [PubMed]
  2. J. Fung, K. E. Martin, R. W. Perry, D. M. Katz, R. McGorty, and V. N. Manoharan, “Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy,” Opt. Express19, 8051–8065 (2011). [CrossRef] [PubMed]
  3. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, “Fourier-transform holographic microscope,” Appl. Opt.31, 4973–4978 (1992). [CrossRef] [PubMed]
  4. W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. USA98, 11301–11305 (2001). [CrossRef] [PubMed]
  5. J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. B. amnd, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. USA104, 17512–17517 (2007). [CrossRef] [PubMed]
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