Obtaining the Transfer Function of optical instruments using large calibrated reference objects

A. J. Henning; J. M. Huntley; C. L. Giusca

doi:10.1364/OE.23.016617

1. Introduction

Deconvolution of images obtained by 3D optical microscopes is a well documented technique which attempts to counteract the effect of blur introduced by the instrument itself. The success of the method relies on accurate knowledge of the Point Spread Function (PSF), which can be either generated from the theoretical behaviour of the instrument [1] or can be determined experimentally [1–5 ]. The experimental determination has the advantage that it should capture the true behaviour of the instrument, though great care is needed in order to obtain an accurate measurement. There are several ways to measure the PSF, such as by imaging a point like object [3], through Blind Deconvolution [1,3], and via the use of a Shack-Hartmann Wavefront Sensor [5].

Coherence Scanning Interferometers (CSI) are interferometric microscopes that use a broadband light source to limit the region in space over which interference fringes are clearly seen in order to improve the determination of the topography of the scattering surface from the measurement result [6–9 ]. Recent work has proposed that the PSF of an instrument such as a CSI could be determined from a single measurement of a strongly scattering spherical reference artefact with a diameter of around 50 μm [4, 10, 11]. In [4] a spherical cap was measured and, using knowledge of the form of the surface, an attempt to determine the Transfer Function (TF) from the measurement result was made. The PSF was then calculated from the TF via a Fourier Transform (FT). In [4], Mandal et al introduced the model of their surface into their calculations numerically, and as a result needed to artificially smooth the model to prevent aliasing artefacts [12]. Subsequent correction was thus required for the resulting attenuation of the higher frequency components of the model.

In this paper we consider analytically the Fourier transform of a portion of the spherical shell, i.e. the ’foil’ model presented in [4]. The portion is defined by excluding those parts of the the spherical surface from which, according to geometrical optics, the light would not be collected. Two interesting consequences of the analysis are as follows:

The artificial smoothing in both radial and polar directions of the model used in [4] is completely avoided, which should thus allow a more accurate measurement of the PSF.
Points of zero amplitude are present at spatial frequencies in the FT where we would not intuitively expect the TF to be zero valued. This implies that if this ’windowed foil model’ is a true representation of the sphere used in the measurement, then a single measurement of such an object would be insufficient to determine the entire TF.

The structure of the paper is as follows. In section 2 we present the analytical three-dimensional FT of the windowed foil model, which demonstrates the presence of zeros in the TF. In section 3 we consider the effect of sphere diameter on the location of the zeros. Numerical solutions to the equations in section 4 allow the FT to be examined more generally, including the case of a small spherical cap and non-spherical shell which are considered in sections 5 and 6 respectively. We conclude by considering the size range in which the object should lie, and identify measurements that would potentially allow the entirety of the TF to be measured when zeros are present in the FT.

2. The 3-D Fourier transform of a spherical shell

In [13] it is stated that, within certain limits, the result of a measurement of a strongly scattering object by a CSI is given by convolving the PSF with a function that defines the surface i.e. one which only takes a value at the air/material interface. This result is used in the following work, however we slightly modify the definition of the surface, defining it as Δ(r) = 4πiδ(r;S)W, where, as in [13], W is a window function and S is a surface in three dimensional space that has the value 1 if the position vector r lies on S and is zero elsewhere [14]. This change in definition has no great effect on the ideas presented there, and the end result is equivalent apart from the moving of a couple of terms out of the Inverse FT of the TF, H, and into the new Δ.Transforming the results into spatial frequency space via FTs gives

\tilde{O} (k) = \tilde{Δ} (k) \tilde{H} (k)

where Õ(k) is the FT of the output of the measurement,

\tilde{Δ} (k)

is the FT of the surface, and

\tilde{H} (k)

is the TF of the instrument. In order to calculate the TF of the instrument from the result of this measurement

\tilde{Δ} (k)

needs to be known, and it must not be equal to zero at any point where the TF is to be evaluated.

For a strongly scattering object, the measurement will not be of the entire surface of the object and in [4] an approximation of the top surface of a spherical object is created and then Fourier transformed to obtain $\tilde{Δ} (k)$ . In order to allow the surface to be included in a space represented by a discrete array of data points, the surface is given a Gaussian form in the direction corresponding to the optical axis of the measurement system. A phase ramp is applied in real space in order to shift the spatial frequency components into the region where non-zero values of the TF are found and the resulting surface is cropped so that only a section that lies within a given angle of the optical axis is retained, leaving a section of surface corresponding to the top of the ball. Finally the surface is multiplied by function that takes the form of a Gaussian in the directions perpendicular to the optical axis which removes the hard cut off of the surface. This Gaussian function perpendicular to the optical axis is the window function referred to in the paper [12]. The drawback of replacing the delta function with a smooth Gaussian however is the attenuation of the higher spatial frequencies of $\tilde{Δ} (k)$ which thus need to be boosted by a subsequent step in the analysis.

In the following we take a more analytical approach to determine the Fourier transform of the surface, thereby removing some of the numerical artefacts that the previous method introduces. In section 2.1 we provide the mathematical derivation of a full spherical shell of infinitesimally thin wall thickness. In sections 2.2 and 2.3 two alternative approaches to deal with the case of a partial spherical shell are presented.

2.1. The Fourier transform of a full spherical shell

The derivation of the FT of a spherical shell is obtained as follows. F(k) denotes the FT of f (r) = f (r), i.e. there is no variation in f with angle due to the spherical symmetry of the shell. r represents the position vector of a point in real space which has Cartesian coordinates (x, y, z) and spherical polar coordinates (r,θ,φ), where θ is the polar angle measured to the z axis, and φ is the azimuth angle measuring rotation about the z axis. The equivalent coordinate systems for k are represented here as (k_x,k_y,k_z) and (k,α,β), respectively. We then have

F (k) = \int \int \int f (r) \exp (i r . k) d^{3} r

The r.k term can be written in spherical polars as

r . k = r k [\sin (α) \cos (β) \sin (θ) \cos (ϕ) + \sin (α) \sin (β) \sin (θ) \sin (ϕ) + \cos (α) \cos (θ)]

Then

\begin{array}{l} F (k) = \int_{0}^{\infty} f (r) d r \int_{0}^{π} r^{2} \sin (θ) d θ \int_{0}^{2 π} d ϕ \exp [i r k \sin (α) \sin (θ) \cos (β - ϕ)] \times \\ \exp [i r k \cos (α) \cos (θ)] \\ = 2 π \int_{0}^{\infty} f (r) d r \int_{0}^{π} r^{2} \sin (θ) d θ J_{0} (r k \sin (α) \sin (θ)) \exp [i r k \cos (α) \cos (θ)] \\ = 2 π \int_{0}^{\infty} f (r) r^{2} d r \int_{- 1}^{1} d \cos (θ) J_{0} (r k \sin (α) {(1 - \cos^{2} (θ))}^{1 / 2}) \exp [i r k \cos (α) \cos (θ)] \\ = 4 π \int_{0}^{\infty} f (r) r^{2} d r \int_{0}^{1} d y J_{0} (r k \sin (α) {(1 - y^{2})}^{1 / 2}) \cos [r k \cos (α) y] \end{array}

where y = cos(θ). Using Eq. 6.677.6 in [15]

\begin{array}{l} F (k) = 4 π \int_{0}^{\infty} f (r) r^{2} d r \frac{\sin ({(r^{2} k^{2} \sin^{2} (α) + r^{2} k^{2} \cos^{2} (α))}^{1 / 2})}{{(r^{2} k^{2} \sin^{2} (α) + r^{2} k^{2} \cos^{2} (α))}^{1 / 2}} \\ = 4 π \int_{0}^{\infty} f (r) r d r \frac{\sin (r k)}{k} \end{array}

For an infinitesimally thin spherical shell of radius r ₀

f (r) = δ (r - r_{0})

so

F (k) = 4 π r_{0}^{2} \frac{\sin (r_{0} k)}{r_{0} k}

and we obtain the result that the FT of the spherical shell with infinitesimally thin walls is a sinc function.

2.2. The Fourier transform of a portion of a spherical shell - method 1

There are at least two ways that one can modify the results of section 2.1 to deal with the case of a portion of a spherical shell. The first, outlined in the current section, involves restricting the range of integration of the theta variable in Eq. (4). The second (section 2.3) involves the application of a window function in real space, equivalent to convolution in k space, and allows one to prove certain results analytically. Both approaches are equivalent in that the window function (or effective window function in the case of the first method) that we use takes a value of one in the region of the surface where the microscope measures the surface and is zero elsewhere. This means that the interaction of the PSF with the surface that is retained is the same at all points, but does lead to the surface having a sharp cut off. The result of a measurement by a true instrument may differ from this at the points where the surface cuts off, however this limiting case is instructive, and the shift invariance needed for Fourier Optics to be applicable is maintained.

The Fourier transform of the partial sphere no longer has full spherical symmetry, but does have rotational symmetry about the k_z axis. The transform evaluated on any plane containing the k_z axis therefore specifies the full 3-D transform. For simplicity we choose this plane to be the k_x,k_z plane which corresponds to the case β = 0. Using the spherical polar to Cartesian transformations k_x = k sin(α), k_z = k cos(α), Eq. (5) becomes

F (k_{x}, k_{y}) = 2 π \int_{0}^{\infty} f (r) r^{2} d r \int_{y_{0}}^{1} d y J_{0} (r k_{x} {(1 - y^{2})}^{1 / 2})) \exp (i r k_{z} y)

where y ₀ = cos(θ ₀) and θ ₀ is the polar angle that defines the edge of the partial spherical shell. Inserting Eq. (6) into (8) gives the following result:

F (k_{x}, k_{y}) = 2 π r_{0}^{2} \int_{y_{0}}^{1} d y J_{0} (r_{0} k_{x} {(1 - y^{2})}^{1 / 2}) \exp (i r_{0} k_{z} y)

It is convenient to work in the non-dimensional variables $k_{x}^{*} = r_{0} k_{x},$ $k_{z}^{*} = r_{0} k_{z}$ and $F^{*} (k_{x}^{*}, k_{y}^{*}) = F (k_{x}, k_{y}) / 4 π r_{0}^{2}$ , so that Eq. (9) simplifies to

F^{*} (k_{x}^{*}, k_{z}^{*}) = \frac{1}{2} \int_{y 0}^{1} d y J_{0} (k_{x}^{*} {(1 - y^{2})}^{1 / 2}) \exp (i k_{z}^{*} y)

We know of no analytical solution to Eq. (10), apart from the special case y ₀ = −1 considered in section (2.1). However, it is straightforward to integrate numerically. The result of doing so for the four cases θ ₀ = π,π/2,π/3,π/4 is shown in Fig. 1 (a)–(d), respectively. The integration was implemented using MATLAB function integral. Fig. 1(a) is the sinc function from the full shell considered in section (2.1). The partial shell results ((b)–(d)) are more complex patterns. A microscope working in reflection has a transfer function that is centred on a point on the k_z axis. It is clear from Figs. 1(b)–(d) that $| F^{*} (k_{x}^{*}, k_{z}^{*}) |$ for a partial shell has a string of minima along this axis, the frequency of which decreases with decreasing θ ₀. While this is an intuitive, and reasonably computationally efficient, method by which to obtain the FT, it is not possible to tell whether these points are true zeros from the numerical results; this particular point is therefore considered further in the next section.

Fig. 1 Slice through the 3D Fourier transform $| F^{*} (k_{x}^{*}, k_{Z}^{*}) |$ of a full spherical shell (a) and partial spherical shell ((b)–(d)), respectively. The logarithm of the absolute values is shown to allow visualisation of the full dynamic range.

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2.3. The Fourier transform of a portion of a spherical shell - method 2

In this section the Fourier transform of a portion of a spherical shell is obtained by multiplying the solution for the infinitesimal shell A(r) from section 2.1 by a function, B(r), that takes the value of 1 in the region of the sphere that is to be retained, and is equal to zero in the region to be discarded. B(r) thus takes the role of the window function, W, mentioned earlier. This concept can be applied to any two objects, and would allow the FT of sections of more complex objects to be obtained should it be desired.

The Fourier transform of the windowed spherical shell is obtained from the convolution theorem

F . T . {A (r) . B (r)} = F . T . {A (r)} \otimes F . T {B (r)}

We consider for now the special case of a hemispherical shell. For convenience B(r) is taken to be invariant in the x and y directions, have a value of 1 between z = 0 and z = 2r ₀, and be zero elsewhere. The FT of this is zero everywhere except on the line k_x = 0, k_y = 0 where it is,

F . T . {B (z)} = 2 \frac{\sin (r_{0} . k_{z})}{k_{z}} \exp (i r_{0} . k_{z})

Convolving this function with Eq. (7) will give the FT of an infinitesimally thin hemispherical shell, A.B,

{F . T . {A . B} = (4 π r_{0} \frac{\sin (r_{0} k)}{k}) \otimes (2 \frac{\sin (r_{0} . k_{z})}{k_{z}} \exp (i r_{0} . k_{z})) |}_{k_{x}, k_{y} = 0}

As observed numerically in section 2.2 there are points where this convolution gives F.T.{A.B} = 0. To demonstrate that these points are present, we start by looking at the convolution of the two functions for values along the k_z axis.

F (k_{z}) = 8 π r_{0}^{3} \int_{- \infty}^{\infty} (sinc (r_{0} κ)) (sinc (r_{0} . (k_{z} - κ)) \exp (i r_{0} . (k_{z} - κ))) d κ

When this integral is evaluated for k_z = 2nπ/r ₀, where n is any integer other than 0, it can be found that the result is equal to zero. The integral can be carried out in mathematica if the exp(iα) function is replaced by cos(α) + i sin(α) and the integration of the real and imaginary parts is carried out separately. When the result is evaluated for the limits of the integral being ∞ and −∞ both the real and imaginary parts simultaneously equal zero. For a sphere with a diameter of 53 μm, as used in [4], the separation of the zeros will be 1.19 × 10⁵ m⁻¹. This is twice the spacing that is found for the FT of the entire infinitesimal spherical shell, as can be found from Eq. (7). This separation is far less than the length along the k_z axis throughout which the TF has non-zero values when a lens of any significant numerical aperture is used, as will be demonstrated in the following sections. This would imply that the measurement of the entire TF would not be possible from the measurement of this artefact, if the foil model is an accurate representation of the physical system.

Before we continue two things should be noted; firstly, in [4] only a spherical cap was measured which will have a different FT, but the result for the hemispherical shell presented here should help with interpreting the numerical data that follows. The measurement of the entire hemispherical surface would, in any case, introduce a further problem: the strongly scattering object would block some of the illuminating light reaching areas of the surface, and stop some of the scattered light from being collected, thus removing shift invariance from the measurement. Secondly, the set of spatial frequencies that are present in the Discrete FT of the measured data are dependent upon the size of the space that is measured, as the data is often recorded at points a fixed distance apart. Should the size of the measurement region be changed then different spatial frequencies will be present, and a calibration for this set will also be needed.

3. The regime in which the radius of the sphere should lie

Now that the presence of zeros in the FT of the infinitesimal hemispherical shell have been shown, numerical methods are used to examine the solution of Eq. (13) more carefully. The volume of spatial frequency space in which the TF lies according to the construction in [16] is the region we will focus on. A schematic slice through this region is shown in Fig. 2, with the red area corresponding to the slice through the volume in which non-zero values of the TF are expected to be found. k_max and k_min are the magnitudes of the wavevectors corresponding to the shortest and longest wavelength of the illuminating light respectively. The volume is symmetric through a revolution about the k_z axis, and the maximum spatial frequency is given by 2k_max. The non-zero solutions lie within a cone with a half angle equal to that of the objective lens, and above the two regions marked with the dashed white lines, that are circles of radius k_min whose centre lies a distance k_min along the lines at ±θ. Along the k_z axis, the maximum and minimum values for which the TF is non-zero are therefore 2k_max and 2k_min cos(θ) respectively. For a spherical object, the relevant FT will be that of the cap of the infinitesimally thin spherical shell that lies within a cone with a half angle equal to that of the objective lens.

Fig. 2 The construction that demonstrates the region in which the TF will take non-zero values, shown in red. It is symmetric throughout a rotation about the k_z axis. The TF will be taken to be zero in the blue region.

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If a microscope with a numerical aperture of 0.55 is considered, and with an illumination that spans from 400 nm to 700 nm then, on the z axis, 2k_min cos(θ) is approximately 15 × 10⁶ m⁻¹, while 2k_max is approximately 31.4 × 10⁶ m⁻¹. In order to have a separation between the zeros of greater than this size the hemisphere would have to have a radius of no more than 382 nm, and that does not take into account the actual location of the TF which can possibly have a zero in it even if the spacing of the zeros was greater than its height. In order to avoid zeros in a real experiment, a far smaller object would be needed. Unfortunately, one of the criteria for the results in [13] to be applicable is that the surface is slowly varying. The radius of curvature for a sphere of less than a micron violates this condition. As such, in the regime that it is applicable more than one reference artefact would be needed in order to measure the full TF, with the artefacts being chosen carefully so that the zeros do not coincide.

If this limiting case for the representation of the surface is an accurate description of what is measured than it can be seen that a measurement of a single spherical reference artefact is insufficient to measure the entire transfer function. The accuracy to which we wish to know the surface of the artefact suggests that we would want to limit the calibration to the measurement of two artefacts. By carefully choosing their sizes so that the location of the zeros in each measurement does not overlap, it is possible to measure the entire TF. However, even if there are no points where the FTs of the object are both zero, there may be points where the term is small for both objects which would leave the measurement of the TF very susceptible to noise. The use of further calibrated objects, or very careful selection of two spherical objects should allow this problem to be avoided.

4. Numerical solutions

A slice through the absolute value of the three dimensional FT of an infinitesimally thin hemispherical shell is shown in Fig. 3. This plot was generated using arrays of discrete data, with the solution for the FT of the spherical shell and of the sheet being given by Eq’s (7) and (12) respectively. These functions were then convolved numerically. Eq. (1) shows that in order to obtain the outcome of a measurement this result should be multiplied by $\tilde{H}$ , the TF of the instrument.

Fig. 3 The absolute value of the FT of a hemispherical shell in a region where non-zero values of the TF would typically be found. This function has zeros periodically along the z axis.

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It is clear from the above discussion that, in order to keep the zeros separated by as great a distance as possible that we would like the object to be as small as possible, however, if we wish to apply the method described in [13] then we are limited to objects where the radius of curvature is not too great [17]. Therefore, a compromise must be struck in this method, the ball should be as small as it can be without leading to the approximations used in [13] becoming too inaccurate. As such, the authors suggest that spheres around 10 μm to 20 μm in diameter give a suitable balance of these criteria.

The results of the numerical convolution appear to show that the FT of the hemisphere is only exactly zero for points on the k_z axis, and for points on the k_z = 0 plane. However the magnitude of the FT does approach zero at points throughout the spatial frequency space. It can be seen on Fig. 3, that the magnitude within the region shown has the greatest values along the k_z axis, and the peak value within the local area drops away both with increasing values of k_x and k_z.

5. The cap of a spherical object

In a measurement, only a limited portion of the sphere will actually be measured. It can be seen from geometrical arguments that, if the reflection is specular, once the angle between the tangent of the surface and a plane perpendicular to the optical axis is greater than the half angle of the lens then the light illuminating the object through the objective lens will be scattered into directions that mean it is not collected. This cap of the sphere corresponds to the surface that lies within a cone with a half angle equal to that of the lens. The FT of this cap can be obtained using the same method as was applied to the hemisphere, by defining the function B to have a value of 1 between z ₀ = r ₀ cos(θ) and z ₁ = 2r ₀ + r ₀ cos(θ), changing Eq. (13) to

{F . T . {A . B} = (4 π r_{0}^{2} sinc (r_{0} k)) \otimes (2 r_{0} sinc (r_{0} k_{z}) \exp {i r_{0} [1 + \cos (θ)] k_{z}}) |}_{k_{x}, k_{y} = 0}

In Fig. 4 the result of the convolution is shown within the region where the TF would be non-zero, according to the construction given above, for an objective with a numerical aperture of 0.55. The wavelengths present in the source are in the range 500 nm – 650 nm and the cap is from a sphere of radius 53 μm. There are still points where the FT is equal to zero on the z axis and the spacing between the zeros on the z axis becomes 2π/(r ₀ [1 − cos(θ)]). It can be seen, as shown earlier in Fig. 1, that the smaller the cap, the greater the distance between the zeros, however the region around the k_z axis where the FT has significant magnitude is reduced. The decrease in the magnitude of the FT away from the k_z axis can be clearly seen in Fig. 4(a).

Fig. 4 (a) the absolute value of the FT of the cap of a sphere that is given by the section of the surface of a 53 μm ball within an intersecting cone with a half angle of 33.37 degrees starting at its centre. A region around the point where the TF is non-zero is shown. (b) A slice through the section of the FT that coincides with the TF according to the construction given in the text above.

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6. Non-spherical objects

The FT for an oblate or prolate spheroid is easy to calculate from the result for a spherical shell by stretching or shrinking one of the axes. A rotation of this object may change the section of the surface that is imaged and break the symmetry. This allows the line of zeros to be shifted from the z- axis as can be seen in Fig. 5. The section of the surface that is imaged is that where its angle between the tangent and the x, y plane is less than the half angle of the lens, and the corresponding section of the surface on a slice through the object is shown by the solid red line in Fig. 5(a). The surface of this object is given by r ² = x ² /a ² + y ² /a ² + z ², where a = 1.1 and r = 26.5 μm. This object is then rotated about the y axis by -3 degrees. It may be found that the line on which the zeros are found lies perpendicular to the base of the cap that is imaged. This line will not be rotated by the -3 degrees that the object was, however, this shift means that by taking more than one measurement with the object tilted in between, the entire TF may be measured using a single object. In addition the flattening of the sphere leads to a smaller drop off of the magnitude of the FT with distance from the line where the zeros are found.

Fig. 5 Part (a) The cap of an oblate spheroid that is imaged after a rotation of -3 degrees about the y- axis and, part (b) the section of its FT that lies within the region defined above as containing non-zero values of the TF. The breaking of the symmetry of the object, combined with the rotation leads to the location of the zero valued points in the FT being shifted from the k_z- axis

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7. Conclusion

A theoretical analysis of the Fourier transform of both a full and partial spherical shell has been presented. The results have direct relevance to the determination of the Transfer Function when calibrating a coherence scanning interferometer with a spherical artefact. The analytical approach presented here provides a better estimate of the Fourier transform of the standard ’foil’ model of the artefact than numerical schemes described previously in the literature and therefore offers the potential for improved determination of an instrument’s performance.

Furthermore, in the case of the cap of a hemispherical shell, the amplitude of some of the spatial frequency components lying along the axis of rotation of the shell have been shown to be identically equal to zero. The practical importance of this observation is dependent on the validity of the windowed foil model (in particular that the light scattering occurs from a zone having zero thickness in the radial direction and with sharp cut-off in the θ direction, where θ is the polar angle measured to the z axis). In situations where the model is valid then a single measurement of a large spherical object does not provide all the spatial frequencies within the Transfer Function of the instrument. Possible solutions proposed to address this problem include the measurement of: (i) a smaller artefact (approximately 10 μm to 20 μm in diameter) to reduce the number of zeros; (ii) two or more artefacts of different sizes for which the zeros will occur in different locations; (iii) a non-spherical artefact, the rotation of which will change the direction of the k space axis containing the zeros.

Acknowledgments

This work was funded by the NMS Engineering & Flow Metrology Programme 2011 – 2014 and the EMRP project Microparts. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. The authors would like to thank Daniel O’Connor (NPL) and Richard Leach (The University of Nottingham) for useful discussions.

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Obtaining the Transfer Function of optical instruments using large calibrated reference objects

Abstract

1. Introduction

2. The 3-D Fourier transform of a spherical shell

2.1. The Fourier transform of a full spherical shell

2.2. The Fourier transform of a portion of a spherical shell - method 1

2.3. The Fourier transform of a portion of a spherical shell - method 2

3. The regime in which the radius of the sphere should lie

4. Numerical solutions

5. The cap of a spherical object

6. Non-spherical objects

7. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (15)

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