Abstract

Flake-based parameters were recently introduced as a physical concept to predict a series of measurement geometries producing similar reflection data for effect paints. We derive expressions to calculate these so-called isochromatic lines, connecting the two Helmholtz-reciprocal in-plane geometries with a series of out-of-plane geometries. Thus isochromatic lines can be regarded as an extension of the Helmholtz reciprocity principle, which is valid for effect paints. We experimentally studied seven effect paint samples with large angular color variation along the length of four isochromatic lines. A change in illumination angles by up to 75° while following isochromatic lines led to a standard deviation in color parameters of less than two units. When isochromatic lines were not followed, these colorimetric parameters varied by more than 10 units already by change in detection angle of 10°. Therefore the concept of isochromatic lines works well for effect paints.

© 2014 Optical Society of America

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References

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  1. American Society for Testing and Material Test Method E2194-03, “Standard practice for multi-angle color measurement of metal flake pigmented materials” (ASTM, 2009).
  2. American Society for Testing and Material Test Method E2539-08, “Standard practice for multi-angle color measurement of interference pigments” (ASTM, 2009).
  3. F. LeLoup, P. Hanselaer, M. Pointer, and J. Versluys, “Characterization of gonio-apparent colours,” in Proceedings of the AIC Color Conference (AIC, 2005), pp. 515–519.
  4. T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
    [CrossRef]
  5. D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
    [CrossRef]
  6. F. B. Leloup, S. Forment, P. Dutré, M. R. Pointer, and P. Hanselaer, “Design of an instrument for measuring the spectral bidirectional scatter distribution function,” Appl. Opt. 47, 5454–5467 (2008).
  7. R. Baribeau, W. S. Neil, and É. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurement,” J. Mod. Opt. 56, 1497–1503 (2009).
    [CrossRef]
  8. A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
    [CrossRef]
  9. A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
    [CrossRef]
  10. A. Takagi, S. Sato, and G. Baba, “Prediction of spectral reflectance factor distribution of color-shift paint finishes,” Color Res. Appl. 32, 378–387 (2007).
    [CrossRef]
  11. N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
    [CrossRef]
  12. GonioSpectrophotometric Colour Measurement System GCMS-4 from Murakami Color Research Laboratory, http://www.mcrl.co.jp/english/GCMS3B.html .
  13. SOC-200 BDR Bidirectional Reflectometer from Surface Optics Corp., http://surfaceoptics.com/products/reflectometers-emissometers/soc200/ .
  14. SMS CASI Scatterometer from Schmitt Industries, http://www.schmitt-ind.com/products-services-measurement-systems-casi.shtml .
  15. So-called scatterometers available from Eckhardt Optics LLC, http://eckop.com/optical-instruments/scatterometers/ .
  16. A. Ferrero, J. Campos, A. Rabal, A. Pons, M. L. Hernanz, and A. Corróns, “Variables separation of the spectral BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
    [CrossRef]
  17. A. Ferrero, A. Rabal, J. Campos, F. Martínez-Verdú, E. Chorro, E. Perales, A. Pons, and M. L. Hernanz, “Spectral BRDF-based determination of proper measurement geometries to characterize color shift of special effect coatings,” J. Opt. Soc. Am. A 30, 206–214 (2013).
    [CrossRef]
  18. E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
    [CrossRef]
  19. T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43, 1266–1274 (2004).
    [CrossRef]
  20. E. Kirchner and J. Houweling, “Measuring flake orientation for metallic coatings,” Progress in organic coatings 64, 287–293 (2009).
  21. E. Kirchner, “Derivation of flake-based parameters,” (2012). Available on request to the author.
  22. M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).
  23. E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
    [CrossRef]

2013 (3)

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).

A. Ferrero, A. Rabal, J. Campos, F. Martínez-Verdú, E. Chorro, E. Perales, A. Pons, and M. L. Hernanz, “Spectral BRDF-based determination of proper measurement geometries to characterize color shift of special effect coatings,” J. Opt. Soc. Am. A 30, 206–214 (2013).
[CrossRef]

2012 (4)

A. Ferrero, J. Campos, A. Rabal, A. Pons, M. L. Hernanz, and A. Corróns, “Variables separation of the spectral BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

2009 (3)

E. Kirchner and J. Houweling, “Measuring flake orientation for metallic coatings,” Progress in organic coatings 64, 287–293 (2009).

R. Baribeau, W. S. Neil, and É. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurement,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

2008 (1)

2007 (1)

A. Takagi, S. Sato, and G. Baba, “Prediction of spectral reflectance factor distribution of color-shift paint finishes,” Color Res. Appl. 32, 378–387 (2007).
[CrossRef]

2006 (1)

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

2004 (1)

1999 (1)

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Asmail, C. C.

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Atamas, T.

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

Baba, G.

A. Takagi, S. Sato, and G. Baba, “Prediction of spectral reflectance factor distribution of color-shift paint finishes,” Color Res. Appl. 32, 378–387 (2007).
[CrossRef]

Baribeau, R.

R. Baribeau, W. S. Neil, and É. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurement,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

Campos, J.

A. Ferrero, A. Rabal, J. Campos, F. Martínez-Verdú, E. Chorro, E. Perales, A. Pons, and M. L. Hernanz, “Spectral BRDF-based determination of proper measurement geometries to characterize color shift of special effect coatings,” J. Opt. Soc. Am. A 30, 206–214 (2013).
[CrossRef]

A. Ferrero, J. Campos, A. Rabal, A. Pons, M. L. Hernanz, and A. Corróns, “Variables separation of the spectral BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

Chorro, E.

A. Ferrero, A. Rabal, J. Campos, F. Martínez-Verdú, E. Chorro, E. Perales, A. Pons, and M. L. Hernanz, “Spectral BRDF-based determination of proper measurement geometries to characterize color shift of special effect coatings,” J. Opt. Soc. Am. A 30, 206–214 (2013).
[CrossRef]

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

Corróns, A.

A. Ferrero, J. Campos, A. Rabal, A. Pons, M. L. Hernanz, and A. Corróns, “Variables separation of the spectral BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Côté, É.

R. Baribeau, W. S. Neil, and É. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurement,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

Cramer, W.

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

Delafosse, D.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Dietz, R.

M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).

Dutré, P.

Faucheu, J.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Ferrero, A.

Flury, M.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Fontecha, J. L.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Forment, S.

Germer, T. A.

T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43, 1266–1274 (2004).
[CrossRef]

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Grusemann, U.

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Hanselaer, P.

F. B. Leloup, S. Forment, P. Dutré, M. R. Pointer, and P. Hanselaer, “Design of an instrument for measuring the spectral bidirectional scatter distribution function,” Appl. Opt. 47, 5454–5467 (2008).

F. LeLoup, P. Hanselaer, M. Pointer, and J. Versluys, “Characterization of gonio-apparent colours,” in Proceedings of the AIC Color Conference (AIC, 2005), pp. 515–519.

Hauer, K. O.

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

Hernanz, M. L.

Höpe, A.

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Houweling, J.

E. Kirchner and J. Houweling, “Measuring flake orientation for metallic coatings,” Progress in organic coatings 64, 287–293 (2009).

Huber, A.

M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).

Hünerhoff, D.

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Kirchner, E.

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

E. Kirchner and J. Houweling, “Measuring flake orientation for metallic coatings,” Progress in organic coatings 64, 287–293 (2009).

E. Kirchner, “Derivation of flake-based parameters,” (2012). Available on request to the author.

LeLoup, F.

F. LeLoup, P. Hanselaer, M. Pointer, and J. Versluys, “Characterization of gonio-apparent colours,” in Proceedings of the AIC Color Conference (AIC, 2005), pp. 515–519.

Leloup, F. B.

Martínez-Verdú, F.

Martínez-Verdú, F. M.

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

Marx, E.

Matsapey, N.

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Mezger, N.

M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).

Neil, W. S.

R. Baribeau, W. S. Neil, and É. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurement,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

Perales, E.

A. Ferrero, A. Rabal, J. Campos, F. Martínez-Verdú, E. Chorro, E. Perales, A. Pons, and M. L. Hernanz, “Spectral BRDF-based determination of proper measurement geometries to characterize color shift of special effect coatings,” J. Opt. Soc. Am. A 30, 206–214 (2013).
[CrossRef]

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

Pointer, M.

F. LeLoup, P. Hanselaer, M. Pointer, and J. Versluys, “Characterization of gonio-apparent colours,” in Proceedings of the AIC Color Conference (AIC, 2005), pp. 515–519.

Pointer, M. R.

Pons, A.

A. Ferrero, A. Rabal, J. Campos, F. Martínez-Verdú, E. Chorro, E. Perales, A. Pons, and M. L. Hernanz, “Spectral BRDF-based determination of proper measurement geometries to characterize color shift of special effect coatings,” J. Opt. Soc. Am. A 30, 206–214 (2013).
[CrossRef]

A. Ferrero, J. Campos, A. Rabal, A. Pons, M. L. Hernanz, and A. Corróns, “Variables separation of the spectral BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

Rabal, A.

Rabal, A. M.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Rösler, M.

M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).

Rubiño, A. M.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Sato, S.

A. Takagi, S. Sato, and G. Baba, “Prediction of spectral reflectance factor distribution of color-shift paint finishes,” Color Res. Appl. 32, 378–387 (2007).
[CrossRef]

Takagi, A.

A. Takagi, S. Sato, and G. Baba, “Prediction of spectral reflectance factor distribution of color-shift paint finishes,” Color Res. Appl. 32, 378–387 (2007).
[CrossRef]

Teichert, S.

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

Versluys, J.

F. LeLoup, P. Hanselaer, M. Pointer, and J. Versluys, “Characterization of gonio-apparent colours,” in Proceedings of the AIC Color Conference (AIC, 2005), pp. 515–519.

Appl. Opt. (2)

Color Res. Appl. (2)

A. Takagi, S. Sato, and G. Baba, “Prediction of spectral reflectance factor distribution of color-shift paint finishes,” Color Res. Appl. 32, 378–387 (2007).
[CrossRef]

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

J. Mod. Opt. (2)

E. Chorro, E. Perales, F. M. Martínez-Verdú, J. Campos, and A. Pons, “Colorimetric and spectral evaluation of the optical anisotropy of metallic and pearlescent samples,” J. Mod. Opt. 56, 1457–1465 (2009).
[CrossRef]

R. Baribeau, W. S. Neil, and É. Côté, “Development of a robot-based gonioreflectometer for spectral BRDF measurement,” J. Mod. Opt. 56, 1497–1503 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

N. Matsapey, J. Faucheu, M. Flury, and D. Delafosse, “Design of a gonio-spectro-photometer for optical characterization of gonio-apparent materials,” Meas. Sci. Technol. 24, 065901 (2013).
[CrossRef]

Metrologia (2)

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “BRDF measurement automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

D. Hünerhoff, U. Grusemann, and A. Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection,” Metrologia 43, S11–S16 (2006).
[CrossRef]

Polym. Paint Colour J. (1)

M. Rösler, N. Mezger, R. Dietz, and A. Huber, “Isoline concepts to find measurement geometries for effect coatings full Gonio color characterization,” Polym. Paint Colour J. 203, 48–54 (2013).

Progress in organic coatings (1)

E. Kirchner and J. Houweling, “Measuring flake orientation for metallic coatings,” Progress in organic coatings 64, 287–293 (2009).

Rev. Sci. Instrum. (2)

A. Höpe, T. Atamas, D. Hünerhoff, S. Teichert, and K. O. Hauer, “ARGon3: 3D appearance robot-based gonioreflectometer at PTB,” Rev. Sci. Instrum. 83, 045102 (2012).
[CrossRef]

T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. 70, 3688–3695 (1999).
[CrossRef]

Other (8)

E. Kirchner, “Derivation of flake-based parameters,” (2012). Available on request to the author.

GonioSpectrophotometric Colour Measurement System GCMS-4 from Murakami Color Research Laboratory, http://www.mcrl.co.jp/english/GCMS3B.html .

SOC-200 BDR Bidirectional Reflectometer from Surface Optics Corp., http://surfaceoptics.com/products/reflectometers-emissometers/soc200/ .

SMS CASI Scatterometer from Schmitt Industries, http://www.schmitt-ind.com/products-services-measurement-systems-casi.shtml .

So-called scatterometers available from Eckhardt Optics LLC, http://eckop.com/optical-instruments/scatterometers/ .

American Society for Testing and Material Test Method E2194-03, “Standard practice for multi-angle color measurement of metal flake pigmented materials” (ASTM, 2009).

American Society for Testing and Material Test Method E2539-08, “Standard practice for multi-angle color measurement of interference pigments” (ASTM, 2009).

F. LeLoup, P. Hanselaer, M. Pointer, and J. Versluys, “Characterization of gonio-apparent colours,” in Proceedings of the AIC Color Conference (AIC, 2005), pp. 515–519.

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Figures (4)

Fig. 1.
Fig. 1.

Definition of angles for the concept of flake-based parameters.

Fig. 2.
Fig. 2.

Geometries on the same isochromatic line IL01 as the standard 25° aspecular geometry, (θill=45°, θdet=20°, ϕdet=180°). All these geometries correspond to θflake=7.5° and θinc=20.7°. Markers with equal value of ϕdet belong to the same measurement geometry. For example, at ϕdet=180° the plus-solution of Eq. (14) is θill=20° (square marker) and θdet=45° (triangular marker). The minus solution at ϕdet=180° has θill=45° (circular marker) and θdet=20° (diamond marker).

Fig. 3.
Fig. 3.

Geometries on the isochromatic line IL04, which includes the out-of-plane 45as60az125.3 geometry.

Fig. 4.
Fig. 4.

Characteristics of the four isochromatic lines discussed in this article.

Tables (4)

Tables Icon

Table 1. Selection of Geometries That Are on the Same Isochromatic Lines Il01, Il02, and Il03a

Tables Icon

Table 2. Some Geometries on the Same Isochromatic Line as the 45as60az125.3 Out-of-Plane Geometrya

Tables Icon

Table 3. Details of Seven Samples Used for Experimental Verification of Concepts Developed in This Article

Tables Icon

Table 4. Standard Deviations in Colorimetric Parametersa

Equations (18)

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cosθflake=cosθ1+cosθ2[2+2sinθ1sinθ2cosϕdet+2cosθ1cosθ2]1/2,
cos(2θinc)=sinθ1sinθ2cosϕdet+cosθ1cosθ2,
nairsinθill=ncoatsinθ1,
nairsin|θdet|=ncoatsinθ2.
n^flake=(xflakeyflakezflake)=(sinθflakecosϕflakesinθflakesinϕflakecosθflake).
r⃗1=(sinθ10cosθ1),
r⃗2=(sinθ2cosϕdetsinθ2sinϕdetcosθ2),
(r⃗1·n⃗flake)=cosθinc.
xflakesinθ1+zflakecosθ1=cosθinc,
sinθ1=cosθinczflakecosθ1xflake.
cos2θ1+(cosθinczflakecosθ1xflake)2=1,
cos2θ1{xflake2+zflake2}+cosθ1{2zflakecosθinc}+{cos2θincxflake2}=0.
cosθ1=cosθflakecosθinc±sinθflakecosϕflakesin2θincsin2θflakesin2ϕflake1sin2θflakesin2ϕflake,
r⃗2=2(r⃗1·n⃗flake)n⃗flaker⃗1.
cosθ2=cosθ1+2(sinθflakecosϕflakesinθ1+cosθflakecosθ1)cosθflake.
tanϕflake=sinθ2sinϕdetsinθ2cosϕdet+sinθ1.
cos(2θinc)=r⃗1·r⃗2.
cosϕdet=cos(2θinc)cosθ1cosθ2sinθ1sinθ2.

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