## A tomographic approach to inverse Mie particle characterization from scattered light

Optics Express, Vol. 15, Issue 19, pp. 12217-12229 (2007)

http://dx.doi.org/10.1364/OE.15.012217

Acrobat PDF (281 KB)

### Abstract

The problem of computing the internal electromagnetic field of a homogeneous sphere from the observation of its scattered light field is explored. Using empirical observations it shown that, to good approximation for low contrast objects, there is a simple Fourier relationship between a component of the internal E-field and the scattered light in a preferred plane. Based on this relationship an empirical algorithm is proposed to construct a spherically symmetric particle of approximately the same diameter as the original, homogeneous, one. The size parameter (*ka*) of this particle is then estimated and shown to be nearly identical to that of the original particle. The size parameter can then be combined with the integrated power of the scatter in the preferred plane to estimate refractive index. The estimated values are shown to be accurate in the presence of moderate noise for a class of size parameters.

© 2007 Optical Society of America

## 1. Introduction

3. P. J. Wyatt, “Differential Light Scattering: a Physical Method for Identifying Living Bacterial Cells,” Appl. Opt. **7**, 1879–1896 (1968). [CrossRef] [PubMed]

5. P. C. Chaumet, K. Belkebir, and A. Sentenac, “Three-dimensional sub wavelength optical imaging using the coupled dipole method,” Phys. Rev. B , **69** (245405): 1–7 (2004). [CrossRef]

*a*and complex index of refraction N

_{1}in a medium with refractive index N and hence, relative index of refraction of

6. A. C. Kak and M. Slaney, *Principles of Computerized Tomographic Imaging*, (Society of Industrial and Applied Mathematics, 2001). [CrossRef]

7. V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosc. Radiat. Transfer **100**, 55–63 (2006). [CrossRef]

8. J. Everitt and I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem Soc. Trans. **19**, 504–5 (1991). [PubMed]

9. K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. **43**, 5110–5115 (2004). [CrossRef] [PubMed]

10. P. H. Faye, “Spatial light-scattering analysis as a means of characterizing and classifying non-spherical particles,” Meas. Sci. Technol. **9**, 141–149 (1998). [CrossRef]

11. Y. L. Pan, K. B. Aptowicz, R. K. Chang, M. Hart, and J. D. Eversole, “Characterizing and monitoring aerosols by light scattering,” Opt. Lett. **28**, 589–591 (2003). [CrossRef] [PubMed]

*et al.*, [12

12. B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, “Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements,” Opt. Express **14**, 12473–12484 (2006). [CrossRef] [PubMed]

## 2. The forward electromagnetic scattering problem

**ê**

*,*

_{x}**ê**

*,*

_{y}**ê**

*). Defining the observed scattering direction as*

_{z}**ê**

*a scattering plane can be defined as that plane that contains both the incident*

_{r}**ê**

*and the scattered vector*

_{i}**ê**

*. The incident electromagnetic field, assumed to be propagating along the z-axis has electric field*

_{r}**E**

*that can be regarded as a superposition of components parallel and perpendicular to the scattering plane:*

_{i}_{inc}=2

*π*N

_{1}/

*λ*is the wave number in the internal medium, and

*λ*is the wavelength of the incident light

*in vacuo*.

*r*,

*θ*,

*φ*) are defined in terms of a conventional spherical coordinate system relative to the scattering plane and the direction of propagation. This notation permits the relation between incident and scattered fields to be written (omitting the time dependence) in compact matrix form as

*θ*and

*φ*.

**E**

*[2]. Examples are the T-Matrix [15], the use of FTDT [16] methods as well as the Discrete Dipole Approximation [17*

_{s}17. B. T. Draine, “The Discrete-dipole approximation and its application to the interstellar graphite grains,” Astrophys. J. **333**, 848–872 (1988). [CrossRef]

**E**

*=E*

_{i}_{0}exp

^{ikr cosθ}

**ê**

*.*

_{x}*with n=1 denotes the spherical Bessel function*

_{n}*j*

*(*

_{n}*k*

_{1}

*r*) and with n=3 the spherical Hankel function

*h*

^{(1)}

*(*

_{n}*kr*). The functions

*π*

*and*

_{n}*τ*

*are derived from the Legendre polynomials P*

_{n}*via the relationships*

_{n}_{⊥i}=E

*and E*

_{x}_{‖i}=0 whereby E

_{‖s}(0,y,

*z*)=0 and E

_{⊥s}=S

_{1}E

_{⊥i}=S

_{1}E

*. As such, pure S*

_{x}_{1}can be observed in the y-z plane as there is no incident parallel component in this case. Observations in the x-y plane can be similarly viewed as pure S

_{2}as there is no perpendicular component.

*r*

*with polarizations*

_{j}**P**

*due to the electric field*

_{j}**E**

*where*

_{j}**E**

*=*

_{j}*α*

_{j}**P**

*and*

_{j}*α*is the polarizability of the medium at these locations [18

18. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. **11**, 1491–1499 (1994). [CrossRef]

**d**with unit vector d̂ and magnitude

**d**as a superposition of individual fields as

**k**

_{obs}=(k

_{ox}, k

_{oy}, k

_{oz}), centered on the spherical particle so that

**k**̂

_{obs}=

**d**̂, this equation can be rewritten as

**k**

_{obs}=(0, k

_{oy}, k

_{oz}), and that the d distance is implicit in the treatment the x component of this sum can be written as

*α*is diagonal, and does not depend on location “j”, because the sphere is homogeneous, one can substitute

**k**

_{obs}

**k**

_{obs}=(k

_{oy}, k

_{oz})multiplied by the constant phase and amplitude factor

_{s⊥}=E

_{sx}and also since E

_{s‖}=E

_{sy}+E

_{sz}=0, it implies that E

_{sy}=0

*and*E

_{sz}=0.

## 3. The inverse electromagnetic scattering problem

### 3.1 Inverse Mie theory

*Given the amplitude scattering functions*

*S*

_{1}

*and*

*S*

_{2},

*invert for radius*“

*a*”

*and refractive index*N

_{1}.

8. J. Everitt and I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem Soc. Trans. **19**, 504–5 (1991). [PubMed]

7. V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosc. Radiat. Transfer **100**, 55–63 (2006). [CrossRef]

9. K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. **43**, 5110–5115 (2004). [CrossRef] [PubMed]

19. I. K. Ludlow and J. Everitt, “Inverse Mie Problem,” J. Opt. Soc. Am. A. **17**, 2229–2235 (2000). [CrossRef]

*a*

*and*

_{n}*b*

*can be obtained from the scattered data using the integral relationships:*

_{n}## 3.2 Diffraction tomography

*V*. The DWBA has been successfully used in a number of areas to provide a simplified option for formulating scatter [22

22. A. C. Lavery, T. K. Stanton, D. E. McGehee, and D. Z Chu, “Three-dimensional modeling of acoustic backscattering from fluid-like zooplankton,” **111**, J. Acous. Soc. Am 1197–1210 (2002). [CrossRef]

23. P. L. D. Roberts and J. S. Jaffe, “Multiple angle acoustic classification of zooplankton,” J. Acous Soc. Am. **121**, 2060–2070 (2007). [CrossRef]

*heuristic*, are well known and usually imply that both |

*m*-1|≪1 and kd|

*m*-1|≪1. Here d is some characteristic dimension of the particle and k is the wave number, presumably inside the object. Although this theory has proven useful in some situations [24

24. H. R. Gordon, “Rayeigh-Gans scattering approximation: surprisingly useful for understanding backscatter from disk-like particles,” Opt. Express **15**, 5572–5588 (2007). [CrossRef] [PubMed]

25. A. Katz, A. Alimova, M. Xu, E. Rudolph, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R, R. Alfano, “Bacteria Size Determination by Elastic Light Scattering,” IEEE J. Sel. Top. Quantum Electron. **9**, 277–287 (2003). [CrossRef]

**k**

_{inc}is taken to be that of the interior. In this case, the observed complex scattering coefficients can be rewritten as

## 4. Numerical simulations to investigate the linear phase assumption via Mie theory.

*µm*were used with an incident light wavelength of .5

*µm*. The size parameters for these particles are therefore =π, 2π, 4π, 8π, and 16π.

_{x-int}=E

_{x}(

**r**)exp(ik

_{z}z). We refer to this assumption as a Reduced Distorted Wave Born Approximation (RDWBA) as the assumption of an internal field with linear phase is more general than the DWBA. This was accomplished by simply examining the z-dependent phase and noting its linearity as well as by computing the standard deviation of the superposition of all of the z dependent phases at a given z coordinate as a function of particle size and contrast for the particles.

*a*,

*m*)=(1

*µm*, 1.05) and (1

*µm*, 1.58). The graph depicts this phase for a lattice of 21 points as a function of the z dependence where the error bars denote one standard deviation. Note that the graphs span the range of phases from -π to π so that the phase wraps. The results indicate that a linear phase assumption for the E

_{x}field is clearly warranted in the lower contrast case. This was true in all of the lower contrast cases. This is certainly not true in the higher contrast case and was also found to be unreasonable in all of the higher contrast ones.

## 5. A proposed algorithm for particle size and refractive index estimation

### 5.1 Motivation

## 5.2 The numerical implementation of the algorithm

**r**vector can be written as (x,y,z) and the s vector as (s

_{x},s

_{y},s

_{z}). Evaluating the Fourier Transform along the s

_{x}axis leads to

_{x},0,0) can be viewed as the one dimensional Fourier Transform of the doubly projected structure E

_{x}(

**r**). Via the uniqueness of the Fourier relationship this then implies that E

_{x}(

**r**) can be retrieved from 𝔽(s

_{x},0,0) via an Inverse Fourier Transform. However, since this function has been assumed to be spherically symmetric, any line through the origin of Fourier space will do. The algorithm therefore takes an inverse transform of 𝔽(s

_{x},0,0) in order to estimate

_{x}(|

**r**|). The above discussion defines the “projection slice theorem” that is the basis for most tomographic theory in this linear case. These algorithms are standard now and good explanations of how they work can be found in several texts [6

6. A. C. Kak and M. Slaney, *Principles of Computerized Tomographic Imaging*, (Society of Industrial and Applied Mathematics, 2001). [CrossRef]

*S*

_{1}amplitude function, and then performed an inversion of this procedure to obtain an image of a slice through the 3-dimensional particle. Excellent reconstruction of homogeneous spheres resulted from this procedure. The Inverse Radon Transform was implemented in MATLAB (Mathworks, MA) using the “iradon” function as was all of the numerical studies at this stage.

## 5.3 Numerical results

30. C. Matzler, “Matlab codes for Mie Scattering and Absorption,” http://diogenes.iwt.unibremen. de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html

*S*

_{1}component as a function of size and refractive index using the Mie model. This provided a reference data set. Next, given the values of the size index a set of noisy complex angular scatter data was produced as above. An estimate of refractive index was then obtained by computing the closest index of refraction for the given size index that was experimentally obtained (as listed in Table 1) and the predicted, noisy, observed power estimate from the simulated

*S*

_{1}component. This was done 100 times for each size class and the average and standard deviation for the refractive index were computed. The values are listed in Table 2.

## 6. Discussion and conclusion

31. W. Rysakov and M. Ston, “Light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer **69**, 651–665 (2001). [CrossRef]

32. V. M. Rysakov, “light scattering by “soft” particles of arbitrary shape and size: II-Arbitrary orientation of particles in the space,” J. Quant. Spectrosc. Radiat. Transfer **98**, 85–100 (2006). [CrossRef]

33. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

34. L. M. Shulman, “Analysis of polarimetric data by solving the inverse scattering problem,” Quant. Spectrosc. Radiat. Transfer **88**, 243–256 (2004). [CrossRef]

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic Press, 2000). |

3. | P. J. Wyatt, “Differential Light Scattering: a Physical Method for Identifying Living Bacterial Cells,” Appl. Opt. |

4. | D. Carlton and R. Kress, |

5. | P. C. Chaumet, K. Belkebir, and A. Sentenac, “Three-dimensional sub wavelength optical imaging using the coupled dipole method,” Phys. Rev. B , |

6. | A. C. Kak and M. Slaney, |

7. | V. V. Berdnik and V. A. Loiko, “Particle sizing by multiangle light-scattering data using the high-order neural networks,” J. Quant. Spectrosc. Radiat. Transfer |

8. | J. Everitt and I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem Soc. Trans. |

9. | K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. |

10. | P. H. Faye, “Spatial light-scattering analysis as a means of characterizing and classifying non-spherical particles,” Meas. Sci. Technol. |

11. | Y. L. Pan, K. B. Aptowicz, R. K. Chang, M. Hart, and J. D. Eversole, “Characterizing and monitoring aerosols by light scattering,” Opt. Lett. |

12. | B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, “Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements,” Opt. Express |

13. | D. R. Bohren and D. R. Huffman, |

14. | H. C. Van de Hulst, |

15. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

16. | A. Taflove, |

17. | B. T. Draine, “The Discrete-dipole approximation and its application to the interstellar graphite grains,” Astrophys. J. |

18. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. |

19. | I. K. Ludlow and J. Everitt, “Inverse Mie Problem,” J. Opt. Soc. Am. A. |

20. | T. L. Blundell and L. N. Johnson, |

21. | A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. |

22. | A. C. Lavery, T. K. Stanton, D. E. McGehee, and D. Z Chu, “Three-dimensional modeling of acoustic backscattering from fluid-like zooplankton,” |

23. | P. L. D. Roberts and J. S. Jaffe, “Multiple angle acoustic classification of zooplankton,” J. Acous Soc. Am. |

24. | H. R. Gordon, “Rayeigh-Gans scattering approximation: surprisingly useful for understanding backscatter from disk-like particles,” Opt. Express |

25. | A. Katz, A. Alimova, M. Xu, E. Rudolph, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R, R. Alfano, “Bacteria Size Determination by Elastic Light Scattering,” IEEE J. Sel. Top. Quantum Electron. |

26. | A. Lompado, “Light Scattering by a Spherical Particle,” http://diogenes.iwt.unibremen. de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html |

27. | C. C. Dobson and J. W. L. Lewis, “Survey of the Mie Problem Source Function,” J. Opt. Soc. Am. A |

28. | Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. |

29. | A. V. Oppenheim, R. W. Schaefer, and J. R. BuckA. V. Oppenheim, |

30. | C. Matzler, “Matlab codes for Mie Scattering and Absorption,” http://diogenes.iwt.unibremen. de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html |

31. | W. Rysakov and M. Ston, “Light scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transfer |

32. | V. M. Rysakov, “light scattering by “soft” particles of arbitrary shape and size: II-Arbitrary orientation of particles in the space,” J. Quant. Spectrosc. Radiat. Transfer |

33. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

34. | L. M. Shulman, “Analysis of polarimetric data by solving the inverse scattering problem,” Quant. Spectrosc. Radiat. Transfer |

**OCIS Codes**

(110.6960) Imaging systems : Tomography

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(290.3200) Scattering : Inverse scattering

(290.4020) Scattering : Mie theory

**ToC Category:**

Scattering

**History**

Original Manuscript: June 21, 2007

Revised Manuscript: August 27, 2007

Manuscript Accepted: September 1, 2007

Published: September 11, 2007

**Virtual Issues**

Vol. 2, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Jules S. Jaffe, "A tomographic approach to inverse mie particle characterization from scattered light," Opt. Express **15**, 12217-12229 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-19-12217

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### References

- M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, Seventh Edition, 2003).
- M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles, (Academic Press, 2000).
- P. J. Wyatt, "Differential light scattering: a physical method for identifying living bacterial cells," Appl. Opt. 7, 1879-1896 (1968). [CrossRef] [PubMed]
- D. Carlton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, (Springer-Verlag, Berlin, 1998).
- P. C. Chaumet, K. Belkebir, and A. Sentenac, "Three-dimensional sub wavelength optical imaging using the coupled dipole method," Phys. Rev. B, 69 (245405): 1-7 (2004). [CrossRef]
- A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, (Society of Industrial and Applied Mathematics, 2001). [CrossRef]
- V. V. Berdnik and V. A. Loiko, "Particle sizing by multiangle light-scattering data using the high-order neural networks," J. Quant. Spectrosc. Radiat. Transf. 100, 55-63 (2006). [CrossRef]
- J. Everitt and I. K. Ludlow, "Particle sizing using methods of discrete Legendre analysis," Biochem Soc. Trans. 19, 504-5 (1991). [PubMed]
- K. A. Semyanov, P. A. Tarasov A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, "Single-particle sizing from light scattering by spectral decomposition," Appl. Opt. 43, 5110-5115 (2004). [CrossRef] [PubMed]
- P. H. Faye, "Spatial light-scattering analysis as a means of characterizing and classifying non-spherical particles," Meas. Sci. Technol. 9, 141-149 (1998). [CrossRef]
- Y. L. Pan, K. B. Aptowicz, R. K. Chang, M. Hart, and J. D. Eversole, "Characterizing and monitoring aerosols by light scattering," Opt. Lett. 28, 589-591 (2003). [CrossRef] [PubMed]
- B. Shao, J. S. Jaffe, M. Chachisvilis, and S. C. Esener, "Angular resolved light scattering for discriminating among marine picoplankton: modeling and experimental measurements," Opt. Express 14, 12473-12484 (2006). [CrossRef] [PubMed]
- D. R. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-VCH, (1983).
- H. C. Van de Hulst, Light Scattering by Small Particles, (Dover, 1981).
- M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge University Press, Cambridge, 2002).
- A. Taflove, Computational Electrodynamics: The Finite-difference Time-domain Method, (Artech House, Boston, MA, 1995).
- B. T. Draine, "The Discrete-dipole approximation and its application to the interstellar graphite grains," Astrophys. J. 333, 848-872 (1988). [CrossRef]
- B. T. Draine and P. J. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. 11, 1491-1499 (1994). [CrossRef]
- I. K. Ludlow and J. Everitt, "Inverse Mie Problem," J. Opt. Soc. Am. A. 17, 2229 - 2235 (2000). [CrossRef]
- T. L. Blundell and L. N. Johnson, Protein Crystallography, (Academic Press, 1976).
- A. J. Devaney, "Inversion formula for inverse scattering within the Born approximation," Opt. Lett. 7, 111-112 (1982). [CrossRef] [PubMed]
- A. C. Lavery, T. K. Stanton, D. E. McGehee, and D. Z Chu, "Three-dimensional modeling of acoustic backscattering from fluid-like zooplankton," J. Acous. Soc. Am 111, 1197-1210 (2002). [CrossRef]
- P. L. D. Roberts and J. S. Jaffe, "Multiple angle acoustic classification of zooplankton," J. Acoust. Soc. Am. 121, 2060-2070 (2007). [CrossRef]
- H. R. Gordon, "Rayeigh-Gans scattering approximation: surprisingly useful for understanding backscatter from disk-like particles," Opt. Express 15, 5572-5588 (2007). [CrossRef] [PubMed]
- A. Katz, A. Alimova, M. Xu, E. Rudolph, M. K. Shah, H. E. Savage, R. B. Rosen. S. A. McCormick and R, R. Alfano, "Bacteria Size Determination by Elastic Light Scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003). [CrossRef]
- A. Lompado, "Light Scattering by a Spherical Particle," http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.html>
- C. C. Dobson and J. W. L. Lewis, "Survey of the Mie Problem Source Function," J. Opt. Soc. Am. A 6, 463-466 (1989). [CrossRef]
- Q. Fu and W. Sun, "Mie theory for light scattering by a spherical particle in an absorbing medium," Appl. Opt. 9, 1354-1361 (2001). [CrossRef]
- A. V. Oppenheim, R. W. Schaefer, and J. R. Buck, Discrete-Time Signal Processing A. V. Oppenheim, ed., (Prentice Hall Signal Processing Series, 1999) 2nd Edition.
- C. Matzler, "Matlab codes for Mie Scattering and Absorption," http://diogenes.iwt.uni-bremen.de/vt/laser/wriedt/Mie_Type_Codes/body_mie_type_codes.htmla>
- W. Rysakov and M. Ston, "Light scattering by spheroids," J. Quant. Spectrosc. Radiat. Transf. 69, 651-665 (2001). [CrossRef]
- V. M. Rysakov, "light scattering by "soft" particles of arbitrary shape and size: II-Arbitrary orientation of particles in the space," J. Quant. Spectrosc. Radiat. Transf. 98, 85-100 (2006). [CrossRef]
- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- L. M. Shulman, "Analysis of polarimetric data by solving the inverse scattering problem," Quant. Spectrosc. Radiat. Transf. 88, 243-256 (2004). [CrossRef]

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