Abstract

A shape dependent method for particle size distribution (PSD) estimation based on bulk scattering properties was elaborated. This method estimates the parameters of a particle size distribution with predefined shape from the bulk scattering spectra. The estimation routine was validated on simulated data of polystyrene in water suspensions. To investigate the effect of measurement errors on PSD estimates, a sensitivity analysis was performed. The influence of spectral resolution and range was rather limited. Good PSD estimations were obtained on noise-free spectra, spectra with limited random noise and for estimations on μs or μs in case of a multiplicative baseline. However, the PSD estimation deteriorated if an incorrect value for the refractive index of the particle relative to the medium was used as input parameter. Deviations caused by an incorrect distribution type were smaller for more narrow PSDs than for broader ones. Overall, this study showed the potential to estimate PSDs from bulk scattering spectra and indicated the factors affecting the accuracy.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

2015 (6)

R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 5(11), 27880–27898 (2015).
[Crossref]

H. Bayat, M. Rastgo, M. Mansouri Zadeh, and H. Vereecken, “Particle size distribution models, their characteristics and fitting capability,” J. Hydrol. 529, 872–889 (2015).
[Crossref]

Y. Ren, H. Qi, Q. Chen, L. Ruan, and H. Tan, “Simultaneous retrieval of the complex refractive index and particle size distribution,” Opt. Express 23(15), 19328–19337 (2015).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

Z. He, H. Qi, Y. Yao, and L. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng. 88, 306–314 (2015).
[Crossref]

2014 (2)

2013 (6)

2012 (1)

W. Liu, X. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[Crossref]

2011 (3)

R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model multicomponent system,” Anal. Chem. 83, 1931–1937 (2011).
[Crossref]

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simulat. 81, 1795–1841 (2011).
[Crossref]

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[Crossref]

2009 (2)

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol. A 116, 585–587 (2009).
[Crossref]

R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model two-component system,” Anal. Chem. 81, 7713–7723 (2009).
[Crossref]

2008 (5)

G. L. Frontini, F. Otero, M. G. Messineo, and G. E. Eliçabe, “Estimation of size distribution in concentrated particle systems from light scattering measurements,” Inverse Probl. Sci. Eng. 16(8), 995–1004 (2008).
[Crossref]

J. Champion, A. Walker, and S. Mitragotri, “Role of particle size in phagocytosis of polymeric microspheres,” Pharm. Res. 25(8), 1815–1821 (2008).
[Crossref] [PubMed]

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216–230 (2008).
[Crossref]

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 1094–4087 (2008).
[Crossref]

J. Jumelet, S. Bekki, C. David, and P. Keckhut, “Statistical estimation of stratospheric particle size distribution by combining optical modelling and lidar scattering measurements,” Atmos. Chem. Phys. 8, 8913–8949 (2008).
[Crossref]

2007 (2)

L. Ma, “Measurement of aerosol size distribution function using Mie scattering - Mathematical considerations,” J. Aerosol Sci. 38, 1150–1162 (2007).
[Crossref]

X. Sun, H. Tang, and J. Dai, “Retrieval of particle size distribution in the dependent model using the moment method,” Opt. Express 15(18), 11507–11516 (2007).
[Crossref] [PubMed]

2006 (2)

A. Malloy and B. Carr, “NanoParticle Tracking Analysis: the Halo system,” Part. Part. Syst. Charact. 23, 197–204 (2006).
[Crossref]

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

2004 (1)

M. T. Celis and L. H. Garcia-Rubio, “Stability of emulsions from multiwavelength transmission measurements,” Ind. Eng. Chem. Res. 43, 2067–2072 (2004).
[Crossref]

2002 (1)

C. Servais, R. Jones, and I. Roberts, “The influence of particle size distribution on the processing of food,” J. Food Eng. 51, 201–208 (2002).
[Crossref]

1999 (2)

M. Kandlikar and G. Ramachandran, “Inverse methods for estimating aerosol size distributions: a critical review,” J. Aerosol Sci. 30(4), 413–437 (1999).
[Crossref]

D. Müller, U. Wandinger, and A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. 38(12), 2346–2357 (1999).
[Crossref]

1995 (2)

F. Ferri, A. Bassini, and E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34(25), 5829–5839 (1995).
[Crossref] [PubMed]

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspension and scattering powders,” Wave Random Media 5, 413–426 (1995).
[Crossref]

1993 (1)

S. A. Prahl, M. J. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Optics 32(4), 559–598 (1993).
[Crossref]

1988 (1)

O. Glatter and M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122(2), 496–506 (1988).
[Crossref]

1958 (1)

J.K. Percus and G.J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110(1), 1–13 (1958).
[Crossref]

Aernouts, B.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 5(11), 27880–27898 (2015).
[Crossref]

B. Aernouts, R. Van Beers, R. Watté, J. Lammertyn, and W. Saeys, “Dependent scattering in Intralipid phantoms in the 600–1850 nm range,” Opt. Express 22(5), 6086–6098 (2014).
[Crossref] [PubMed]

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “Flexible tool for simulating the bulk optical properties of polydisperse spherical particles in an absorbing host: experimental validation,” Opt. Express 22(17), 20223–20238 (2014).
[Crossref] [PubMed]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500–2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref]

R. Watté, N. Nguyen Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. Nicolaï, and W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 21(26), 32630–32642 (2013).
[Crossref]

Ahn, S.

Ansmann, A.

Bassini, A.

Bauer, F.

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simulat. 81, 1795–1841 (2011).
[Crossref]

Bayat, H.

H. Bayat, M. Rastgo, M. Mansouri Zadeh, and H. Vereecken, “Particle size distribution models, their characteristics and fitting capability,” J. Hydrol. 529, 872–889 (2015).
[Crossref]

Bekki, S.

J. Jumelet, S. Bekki, C. David, and P. Keckhut, “Statistical estimation of stratospheric particle size distribution by combining optical modelling and lidar scattering measurements,” Atmos. Chem. Phys. 8, 8913–8949 (2008).
[Crossref]

Bouwman, W. G.

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

Cabassi, G.

Carr, B.

A. Malloy and B. Carr, “NanoParticle Tracking Analysis: the Halo system,” Part. Part. Syst. Charact. 23, 197–204 (2006).
[Crossref]

Cattaneo, T. M. P.

Celis, M. T.

M. T. Celis and L. H. Garcia-Rubio, “Stability of emulsions from multiwavelength transmission measurements,” Ind. Eng. Chem. Res. 43, 2067–2072 (2004).
[Crossref]

Champion, J.

J. Champion, A. Walker, and S. Mitragotri, “Role of particle size in phagocytosis of polymeric microspheres,” Pharm. Res. 25(8), 1815–1821 (2008).
[Crossref] [PubMed]

Chen, Q.

Chen, Y.

Cuppo, F.

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

Dai, J.

David, C.

J. Jumelet, S. Bekki, C. David, and P. Keckhut, “Statistical estimation of stratospheric particle size distribution by combining optical modelling and lidar scattering measurements,” Atmos. Chem. Phys. 8, 8913–8949 (2008).
[Crossref]

De Baerdemaeker, J.

De Block, J.

Delport, F.

Duc Nguyen, V.

Eliçabe, G. E.

G. L. Frontini, F. Otero, M. G. Messineo, and G. E. Eliçabe, “Estimation of size distribution in concentrated particle systems from light scattering measurements,” Inverse Probl. Sci. Eng. 16(8), 995–1004 (2008).
[Crossref]

Erkinbaev, C.

Faber, D.J.

Ferri, F.

Foschum, F.

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 1094–4087 (2008).
[Crossref]

Fricke, J.

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspension and scattering powders,” Wave Random Media 5, 413–426 (1995).
[Crossref]

Frontini, G. L.

G. L. Frontini, F. Otero, M. G. Messineo, and G. E. Eliçabe, “Estimation of size distribution in concentrated particle systems from light scattering measurements,” Inverse Probl. Sci. Eng. 16(8), 995–1004 (2008).
[Crossref]

Garcia-Rubio, L. H.

M. T. Celis and L. H. Garcia-Rubio, “Stability of emulsions from multiwavelength transmission measurements,” Ind. Eng. Chem. Res. 43, 2067–2072 (2004).
[Crossref]

Glatter, O.

O. Glatter and M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122(2), 496–506 (1988).
[Crossref]

Göbel, G.

G. Göbel, J. Kuhn, and J. Fricke, “Dependent scattering effects in latex-sphere suspension and scattering powders,” Wave Random Media 5, 413–426 (1995).
[Crossref]

Goddeeris, C.

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

Guan, J.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[Crossref]

He, Z.

Z. He, H. Qi, Y. Yao, and L. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng. 88, 306–314 (2015).
[Crossref]

Hofer, M.

O. Glatter and M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122(2), 496–506 (1988).
[Crossref]

Huybrechts, T.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

Jones, R.

C. Servais, R. Jones, and I. Roberts, “The influence of particle size distribution on the processing of food,” J. Food Eng. 51, 201–208 (2002).
[Crossref]

Jordens, J.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

Jumelet, J.

J. Jumelet, S. Bekki, C. David, and P. Keckhut, “Statistical estimation of stratospheric particle size distribution by combining optical modelling and lidar scattering measurements,” Atmos. Chem. Phys. 8, 8913–8949 (2008).
[Crossref]

Kalkman, J.

Kandlikar, M.

M. Kandlikar and G. Ramachandran, “Inverse methods for estimating aerosol size distributions: a critical review,” J. Aerosol Sci. 30(4), 413–437 (1999).
[Crossref]

Kasarova, S.

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol. A 116, 585–587 (2009).
[Crossref]

Keckhut, P.

J. Jumelet, S. Bekki, C. David, and P. Keckhut, “Statistical estimation of stratospheric particle size distribution by combining optical modelling and lidar scattering measurements,” Atmos. Chem. Phys. 8, 8913–8949 (2008).
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H. Keutelian, “The Kolmogorov-Smirnov test when parameters are estimated from data,” CDF Notes (1991).

Kienle, A.

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 1094–4087 (2008).
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Kuhn, J.

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Lammertyn, J.

Lee, H.

Lee, M.

Liu, W.

W. Liu, X. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[Crossref]

Lukas, M. A.

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simulat. 81, 1795–1841 (2011).
[Crossref]

Ma, L.

L. Ma, “Measurement of aerosol size distribution function using Mie scattering - Mathematical considerations,” J. Aerosol Sci. 38, 1150–1162 (2007).
[Crossref]

Mackowski, D.W.

D.W. Mackowski and M.I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 103–112 (2013).
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Malloy, A.

A. Malloy and B. Carr, “NanoParticle Tracking Analysis: the Halo system,” Part. Part. Syst. Charact. 23, 197–204 (2006).
[Crossref]

Mansouri Zadeh, M.

H. Bayat, M. Rastgo, M. Mansouri Zadeh, and H. Vereecken, “Particle size distribution models, their characteristics and fitting capability,” J. Hydrol. 529, 872–889 (2015).
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Marinoni, L.

Merchiers, M.

Merkus, H. G.

H. G. Merkus, Particle size measurements - Fundamentals, practice, quality (Springer, 2009), Chap. 2.

H. G. Merkus, Particle size measurements - Fundamentals, practice, quality (Springer, 2009), Chap. 9, 10 & 12.

Messineo, M. G.

G. L. Frontini, F. Otero, M. G. Messineo, and G. E. Eliçabe, “Estimation of size distribution in concentrated particle systems from light scattering measurements,” Inverse Probl. Sci. Eng. 16(8), 995–1004 (2008).
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Michels, R.

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 1094–4087 (2008).
[Crossref]

Mishchenko, M.I.

D.W. Mackowski and M.I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 103–112 (2013).
[Crossref]

Mitragotri, S.

J. Champion, A. Walker, and S. Mitragotri, “Role of particle size in phagocytosis of polymeric microspheres,” Pharm. Res. 25(8), 1815–1821 (2008).
[Crossref] [PubMed]

Müller, D.

Nguyen Do Trong, N.

Nicolaï, B.

Nikolov, I.

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol. A 116, 585–587 (2009).
[Crossref]

Otero, F.

G. L. Frontini, F. Otero, M. G. Messineo, and G. E. Eliçabe, “Estimation of size distribution in concentrated particle systems from light scattering measurements,” Inverse Probl. Sci. Eng. 16(8), 995–1004 (2008).
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Paganini, E.

Percus, J.K.

J.K. Percus and G.J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110(1), 1–13 (1958).
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Prahl, S. A.

S. A. Prahl, M. J. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Optics 32(4), 559–598 (1993).
[Crossref]

Profaizer, M.

Qi, H.

Y. Ren, H. Qi, Q. Chen, L. Ruan, and H. Tan, “Simultaneous retrieval of the complex refractive index and particle size distribution,” Opt. Express 23(15), 19328–19337 (2015).
[Crossref] [PubMed]

Z. He, H. Qi, Y. Yao, and L. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng. 88, 306–314 (2015).
[Crossref]

Ramachandran, G.

M. Kandlikar and G. Ramachandran, “Inverse methods for estimating aerosol size distributions: a critical review,” J. Aerosol Sci. 30(4), 413–437 (1999).
[Crossref]

Rastgo, M.

H. Bayat, M. Rastgo, M. Mansouri Zadeh, and H. Vereecken, “Particle size distribution models, their characteristics and fitting capability,” J. Hydrol. 529, 872–889 (2015).
[Crossref]

Ren, Y.

Reynaers, H.

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

Riefler, N.

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216–230 (2008).
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Rizzi, N.

Roberts, I.

C. Servais, R. Jones, and I. Roberts, “The influence of particle size distribution on the processing of food,” J. Food Eng. 51, 201–208 (2002).
[Crossref]

Ruan, L.

Y. Ren, H. Qi, Q. Chen, L. Ruan, and H. Tan, “Simultaneous retrieval of the complex refractive index and particle size distribution,” Opt. Express 23(15), 19328–19337 (2015).
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Z. He, H. Qi, Y. Yao, and L. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng. 88, 306–314 (2015).
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Saeys, W.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 5(11), 27880–27898 (2015).
[Crossref]

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “Flexible tool for simulating the bulk optical properties of polydisperse spherical particles in an absorbing host: experimental validation,” Opt. Express 22(17), 20223–20238 (2014).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, J. Lammertyn, and W. Saeys, “Dependent scattering in Intralipid phantoms in the 600–1850 nm range,” Opt. Express 22(5), 6086–6098 (2014).
[Crossref] [PubMed]

R. Watté, N. Nguyen Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. Nicolaï, and W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 21(26), 32630–32642 (2013).
[Crossref]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500–2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
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D. J. Segelstein, “The complex refractive index of water,” Ph.D. thesis, University of Missouri-Kansas City (1981).

Servais, C.

C. Servais, R. Jones, and I. Roberts, “The influence of particle size distribution on the processing of food,” J. Food Eng. 51, 201–208 (2002).
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Shen, J.

W. Liu, X. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[Crossref]

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
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Steponavicius, R.

R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model multicomponent system,” Anal. Chem. 83, 1931–1937 (2011).
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R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model two-component system,” Anal. Chem. 81, 7713–7723 (2009).
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Sultanova, N.

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol. A 116, 585–587 (2009).
[Crossref]

Sun, X.

W. Liu, X. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[Crossref]

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
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X. Sun, H. Tang, and J. Dai, “Retrieval of particle size distribution in the dependent model using the moment method,” Opt. Express 15(18), 11507–11516 (2007).
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Tan, H.

Tang, H.

Thennadil, S. N.

S. N. Thennadil and Y. Chen, “Alternative measurement configurations for extracting bulk optical properties using an integrating sphere setup,” Appl. Spectrosc. 71(2), 224–237 (2017).
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R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model multicomponent system,” Anal. Chem. 83, 1931–1937 (2011).
[Crossref]

R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model two-component system,” Anal. Chem. 81, 7713–7723 (2009).
[Crossref]

Tsuta, M.

Van Beers, R.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 5(11), 27880–27898 (2015).
[Crossref]

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “Flexible tool for simulating the bulk optical properties of polydisperse spherical particles in an absorbing host: experimental validation,” Opt. Express 22(17), 20223–20238 (2014).
[Crossref] [PubMed]

B. Aernouts, R. Van Beers, R. Watté, J. Lammertyn, and W. Saeys, “Dependent scattering in Intralipid phantoms in the 600–1850 nm range,” Opt. Express 22(5), 6086–6098 (2014).
[Crossref] [PubMed]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500–2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
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van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons, 1957), Chap. 14.

Van Den Mooter, G.

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

van der Pol, E.

van Gemert, M. J.

S. A. Prahl, M. J. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Optics 32(4), 559–598 (1993).
[Crossref]

Van Gerven, T.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

van Leeuwen, T.G.

Vereecken, H.

H. Bayat, M. Rastgo, M. Mansouri Zadeh, and H. Vereecken, “Particle size distribution models, their characteristics and fitting capability,” J. Hydrol. 529, 872–889 (2015).
[Crossref]

Vermeulen, D.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

Walker, A.

J. Champion, A. Walker, and S. Mitragotri, “Role of particle size in phagocytosis of polymeric microspheres,” Pharm. Res. 25(8), 1815–1821 (2008).
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Wandinger, U.

Wang, L.

Wang, X.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[Crossref]

Wang, Y.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[Crossref]

Watté, R.

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 5(11), 27880–27898 (2015).
[Crossref]

B. Aernouts, R. Van Beers, R. Watté, J. Lammertyn, and W. Saeys, “Dependent scattering in Intralipid phantoms in the 600–1850 nm range,” Opt. Express 22(5), 6086–6098 (2014).
[Crossref] [PubMed]

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “Flexible tool for simulating the bulk optical properties of polydisperse spherical particles in an absorbing host: experimental validation,” Opt. Express 22(17), 20223–20238 (2014).
[Crossref] [PubMed]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500–2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref]

R. Watté, N. Nguyen Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. Nicolaï, and W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 21(26), 32630–32642 (2013).
[Crossref]

Welch, A. J.

S. A. Prahl, M. J. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Optics 32(4), 559–598 (1993).
[Crossref]

Wriedt, T.

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216–230 (2008).
[Crossref]

Yao, Y.

Z. He, H. Qi, Y. Yao, and L. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng. 88, 306–314 (2015).
[Crossref]

Yevick, G.J.

J.K. Percus and G.J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Phys. Rev. 110(1), 1–13 (1958).
[Crossref]

Zamora-Rojas, E.

Zhu, X.

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[Crossref]

Acta Phys. Pol. A (1)

N. Sultanova, S. Kasarova, and I. Nikolov, “Dispersion properties of optical polymers,” Acta Phys. Pol. A 116, 585–587 (2009).
[Crossref]

Anal. Chem. (2)

R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model two-component system,” Anal. Chem. 81, 7713–7723 (2009).
[Crossref]

R. Steponavičius and S. N. Thennadil, “Extraction of chemical information of suspensions using radiative transfer theory to remove multiple scattering effects: application to a model multicomponent system,” Anal. Chem. 83, 1931–1937 (2011).
[Crossref]

Appl. Opt. (2)

Appl. Optics (1)

S. A. Prahl, M. J. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Optics 32(4), 559–598 (1993).
[Crossref]

Appl. Spectrosc. (1)

Appl. Therm. Eng. (1)

Z. He, H. Qi, Y. Yao, and L. Ruan, “Inverse estimation of the particle size distribution using the Fruit Fly Optimization Algorithm,” Appl. Therm. Eng. 88, 306–314 (2015).
[Crossref]

Atmos. Chem. Phys. (1)

J. Jumelet, S. Bekki, C. David, and P. Keckhut, “Statistical estimation of stratospheric particle size distribution by combining optical modelling and lidar scattering measurements,” Atmos. Chem. Phys. 8, 8913–8949 (2008).
[Crossref]

Colloids Surf. B (1)

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Jordens, D. Vermeulen, T. Van Gerven, J. Lammertyn, and W. Saeys, “Effect of ultrasonic homogenization on the Vis/NIR bulk optical properties of milk,” Colloids Surf. B 126, 510–519 (2015).
[Crossref]

Ind. Eng. Chem. Res. (1)

M. T. Celis and L. H. Garcia-Rubio, “Stability of emulsions from multiwavelength transmission measurements,” Ind. Eng. Chem. Res. 43, 2067–2072 (2004).
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Int. J. Pharm. (1)

C. Goddeeris, F. Cuppo, H. Reynaers, W. G. Bouwman, and G. Van Den Mooter, “Light scattering measurements on microemulsions: estimation of droplet sizes,” Int. J. Pharm. 312, 187–195 (2006).
[Crossref] [PubMed]

Inverse Probl. Sci. Eng. (1)

G. L. Frontini, F. Otero, M. G. Messineo, and G. E. Eliçabe, “Estimation of size distribution in concentrated particle systems from light scattering measurements,” Inverse Probl. Sci. Eng. 16(8), 995–1004 (2008).
[Crossref]

J. Aerosol Sci. (2)

M. Kandlikar and G. Ramachandran, “Inverse methods for estimating aerosol size distributions: a critical review,” J. Aerosol Sci. 30(4), 413–437 (1999).
[Crossref]

L. Ma, “Measurement of aerosol size distribution function using Mie scattering - Mathematical considerations,” J. Aerosol Sci. 38, 1150–1162 (2007).
[Crossref]

J. Colloid Interface Sci. (1)

O. Glatter and M. Hofer, “Interpretation of elastic light scattering data. III. Determination of size distributions of polydisperse systems,” J. Colloid Interface Sci. 122(2), 496–506 (1988).
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J. Dairy Sci. (1)

B. Aernouts, R. Van Beers, R. Watté, T. Huybrechts, J. Lammertyn, and W. Saeys, “Visible and near-infrared bulk optical properties of raw milk,” J. Dairy Sci. 98, 6727–6738 (2015).
[Crossref] [PubMed]

J. Food Eng. (1)

C. Servais, R. Jones, and I. Roberts, “The influence of particle size distribution on the processing of food,” J. Food Eng. 51, 201–208 (2002).
[Crossref]

J. Hydrol. (1)

H. Bayat, M. Rastgo, M. Mansouri Zadeh, and H. Vereecken, “Particle size distribution models, their characteristics and fitting capability,” J. Hydrol. 529, 872–889 (2015).
[Crossref]

J. Near Infrared Spectrosc. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

D.W. Mackowski and M.I. Mishchenko, “Direct simulation of extinction in a slab of spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 123, 103–112 (2013).
[Crossref]

Math. Comput. Simulat. (1)

F. Bauer and M. A. Lukas, “Comparing parameter choice methods for regularization of ill-posed problems,” Math. Comput. Simulat. 81, 1795–1841 (2011).
[Crossref]

Opt. Express (9)

B. Aernouts, R. Watté, R. Van Beers, F. Delport, M. Merchiers, J. De Block, J. Lammertyn, and W. Saeys, “Flexible tool for simulating the bulk optical properties of polydisperse spherical particles in an absorbing host: experimental validation,” Opt. Express 22(17), 20223–20238 (2014).
[Crossref] [PubMed]

B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500–2250 nm range,” Opt. Express 21(26), 32450–32467 (2013).
[Crossref]

X. Sun, H. Tang, and J. Dai, “Retrieval of particle size distribution in the dependent model using the moment method,” Opt. Express 15(18), 11507–11516 (2007).
[Crossref] [PubMed]

R. Watté, B. Aernouts, R. Van Beers, and W. Saeys, “Robust metamodel-based inverse estimation of bulk optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 5(11), 27880–27898 (2015).
[Crossref]

B. Aernouts, R. Van Beers, R. Watté, J. Lammertyn, and W. Saeys, “Dependent scattering in Intralipid phantoms in the 600–1850 nm range,” Opt. Express 22(5), 6086–6098 (2014).
[Crossref] [PubMed]

R. Watté, N. Nguyen Do Trong, B. Aernouts, C. Erkinbaev, J. De Baerdemaeker, B. Nicolaï, and W. Saeys, “Metamodeling approach for efficient estimation of optical properties of turbid media from spatially resolved diffuse reflectance measurements,” Opt. Express 21(26), 32630–32642 (2013).
[Crossref]

R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 1094–4087 (2008).
[Crossref]

Y. Ren, H. Qi, Q. Chen, L. Ruan, and H. Tan, “Simultaneous retrieval of the complex refractive index and particle size distribution,” Opt. Express 23(15), 19328–19337 (2015).
[Crossref] [PubMed]

V. Duc Nguyen, D.J. Faber, E. van der Pol, T.G. van Leeuwen, and J. Kalkman, “Dependent and multiple scattering in transmission and backscattering optical coherence tomography,” Opt. Express 21(24), 29145–29156 (2013).
[Crossref]

Opt. Laser Technol. (2)

X. Zhu, J. Shen, Y. Wang, J. Guan, X. Sun, and X. Wang, “The reconstruction of particle size distributions from dynamic light scattering data using particle swarm optimization techniques with different objective functions,” Opt. Laser Technol. 43, 1128–1137 (2011).
[Crossref]

W. Liu, X. Sun, and J. Shen, “A V-curve criterion for the parameter optimization of the Tikhonov regularization inversion algorithm for particle sizing,” Opt. Laser Technol. 44, 1–5 (2012).
[Crossref]

Opt. Lett. (1)

Part. Part. Syst. Charact. (2)

A. Malloy and B. Carr, “NanoParticle Tracking Analysis: the Halo system,” Part. Part. Syst. Charact. 23, 197–204 (2006).
[Crossref]

N. Riefler and T. Wriedt, “Intercomparison of inversion algorithms for particle-sizing using Mie scattering,” Part. Part. Syst. Charact. 25, 216–230 (2008).
[Crossref]

Pharm. Res. (1)

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Figures (13)

Fig. 1
Fig. 1 Schematic overview of flow between forward BOP simulations and inverse PSD estimations.
Fig. 2
Fig. 2 (a) & (c) Scattering coefficient spectra for particles of respectively 0.2 μm and 10 μm. (b) Absorption coefficient spectra of 0.5 μm, 4.5 μm and 12 μm particles with a medium width (2σ). (d) Anisotropy factor of particles with a 3 μm nominal diameter.
Fig. 3
Fig. 3 (a) Estimated volume fraction VF and (b) σ-parameter of lognormal PSDs estimated on noise-free μs spectra. Distribution widths: 1σ = full line, 2σ = dashed, 3σ = dotted.
Fig. 4
Fig. 4 Estimated lognormal, normal and Weibull distributions compared to the reference PSD for (left) 0.5 μm 3σ, (middle) 3 μm 2σ and (right) 12 μm 1σ. Estimations were made on noise-free μs spectra with a 10 nm resolution.
Fig. 5
Fig. 5 Lognormal distributions estimated based on spectra with a 1 nm and a 10 nm wavelength resolution. (left) 0.1 μm 1σ estimated on μ s spectra with standard random noise, (right) 3 μm 3σ estimated on g spectra with standard random noise.
Fig. 6
Fig. 6 Parameters of lognormal PSD’s estimated on spectra with different wavelength ranges. For every mode indicated on the figure, three PSD widths (1σ, 2σ, 3σ) are separated respectively by a dashed or dotted line. For every mode and σ combination there are three red dots indicating the wavelength range used for estimations: left = Vis (0.5–1.0 μm), middle = NIR (1.0–1.85 μm), right = VisNIR (0.5–1.85 μm). Mean and st. dev. of five estimates on scattering spectra with random noise are indicted in blue, while the corresponding estimated parameter values on a noise-free spectrum are indicated as red dots.
Fig. 7
Fig. 7 Estimated parameters of lognormal PSDs based on spectra with random noise. (left) Larger deviations for the smallest particles, (middle) increasing s.d. with increasing noise level, and (right) large deviations for medium sized particles estimated based on g. For every mode indicated, the three PSD widths are separated respectively by a dashed or dotted line. Every mode and σ combination contains three levels of random noise: 0.5×, 1× and 2×.
Fig. 8
Fig. 8 Scattering spectra subjected to multiplicative baseline effects and corresponding lognormal PSD estimates: (a)–(b) 2 μm 2σ based on μs, (c)–(d) 0.2 μm 2σ based on g, (e)–(f) 3 μm 2σ based on g, (g)–(h) 10 μm 2σ based on g.
Fig. 9
Fig. 9 Scattering spectra subjected to additive baseline effects and corresponding lognormal PSD estimates: (a)–(b) 0.2 μm 2σ based on μs, (c)–(d) 4.5 μm 2σ based on μs, (e)–(f) 0.2 μm 2σ based on g, (g)–(h) 1 μm 2σ based on g.
Fig. 10
Fig. 10 Scattering spectra simulated with a baseline error on the particle refractive index and corresponding lognormal PSD estimates: (a)–(b) 0.2 μm 2σ based on μs, (c)–(d) 10 μm 2σ based on μ s , (e)–(f) 0.2 μm 2σ based on g, (g)–(h) 10 μm 2σ based on g.
Fig. 11
Fig. 11 Scattering spectra and estimated PSDs for lognormal estimates based on bulk scattering spectra with the refractive index ’squeezed’ (0.5) and ’stretched’ (1.5): (top) 0.2 μm 3σ based on μs, (middle) 2 μm 1σ based on μ s , (bottom) 12 μm 3σ based on μ s .
Fig. 12
Fig. 12 (a) 0.1 μm 1σ estimated on μ s and (b) 2 μm 2σ on g as examples of poor estimations; (c) 12 μm 3σ showing small deviations in case of estimations based on μs or g; (d) 0.5 μm 2σ as example of PSD estimations performing equally well based on all three BOP types.
Fig. 13
Fig. 13 PSD estimates and corresponding scattering spectra for: (blue) lognormal PSD estimated on a μs spectrum with a −2% baseline on the refractive index and a +10% multiplicative baseline; (green) Weibull distribution estimated on a g spectrum with 1× random noise; (purple) normal distribution estimated on μ s with a +3% additive baseline.

Tables (5)

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Table 1 Probability density distribution types used as PSD shapes in the inverse estimations, together with the parameters to be estimated. Variable r is the particle radius in μm.

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Table 2 Distribution parameters of the lognormal reference PSDs. Subscript 1 indicates the narrowest PSDs (σ1 = 0.1), 2 is medium width (σ2 = 0.2) and 3 the widest PSDs (σ3 = 0.3).

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Table 3 Unscaled parameter boundaries for the PSD estimation: μ and σ are lognormal or normal distribution parameters, a and b are parameters for Weibull distributions.

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Table 4 Mean and st. dev. of difference between reference and estimated parameters, and average coefficient of variation over all PSDs for μ and σ, and over all VFs in case of VF.

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Table 5 Parameters estimated on input scattering spectra with and without noise. μ and σ are lognormal or normal distribution parameters, while a and b are for Weibull distributions.

Equations (5)

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m i n log 10 [ i = 1 N λ ( μ s , i μ s , i ^ μ s , i ) 2 ]
n max , n e w = n f i x e d + ( n max n f i x e d ) × f a c t o r
n min , n e w = n f i x e d ( n f i x e d n min ) × f a c t o r
n ( 0.5 1.17 μm ) = n f i x e d + n n f i x e d n max n f i x e d × ( n max , n e w n f i x e d )
n ( 1.17 1.85 μm ) = n f i x e d n f i x e d n n f i x e d n min × ( n f i x e d n min , n e w )

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