## Principal components analysis on the spectral bidirectional reflectance distribution function of ceramic colour standards |

Optics Express, Vol. 19, Issue 20, pp. 19199-19211 (2011)

http://dx.doi.org/10.1364/OE.19.019199

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### Abstract

The Bidirectional Reflectance Distribution Function (BRDF) is essential to characterize an object’s reflectance properties. This function depends both on the various illumination-observation geometries as well as on the wavelength. As a result, the comprehensive interpretation of the data becomes rather complex. In this work we assess the use of the multivariable analysis technique of Principal Components Analysis (PCA) applied to the experimental BRDF data of a ceramic colour standard. It will be shown that the result may be linked to the various reflection processes occurring on the surface, assuming that the incoming spectral distribution is affected by each one of these processes in a specific manner. Moreover, this procedure facilitates the task of interpolating a series of BRDF measurements obtained for a particular sample.

© 2011 OSA

## 1. Introduction

13. J. L. Simonds, “Application of characteristic vector analysis to photographic and optical response data,” J. Opt. Soc. Am. **53**, 968–971 (1963). [CrossRef]

15. A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. **46**, 9–17 (2007). [CrossRef]

## 2. Principal Components Analysis

*A*, to the variance of a selected original spectrum,

_{j}*F*, is given by

_{i}*i*due only to the eigenspectrum

*j*, is given by: Due to the fact that

**E2**represent the relative contribution to the variance of the original spectra of the variance of the eigenspectra obtained by the PCA method.

## 3. Results

16. A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” *Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010* (CIE, Vienna, Austria, 2010), pp. 79–84.

*θ*,

_{i}*ϕ*) and the two observation ones (

_{i}*θ*,

_{s}*ϕ*). These spherical coordinates are defined relative to the Sample Coordinate System, whose z axis is parallel to the sample’s normal direction (Fig. 1). Moreover, thanks to its periscopic system with beamsplitter, it is possible to measure under retro-reflection conditions. This configuration yields important information about the BRDF, as will be shown further on. For the data acquisition a CS-2000 Konica Minolta spectroradiometer is used. This device operates within the visible range [380 nm–780 nm], performing spectral measurements with a 1 nm spectral sampling interval and a 4 nm bandwidth. The relative uncertainty of the BRDF measurement with k=2 is less than 0.02 (limited by the linearity correction when the measurement ranges within more than 5 decades). This uncertainty is comparable to other instruments with similar features [17

_{s}17. 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]

19. 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**(29), 5454–5467 (2008). [PubMed]

### 3.1. BRDF data

*θ*and

_{i}*θ*(which take the values 0°, 10°, 20°, 30°, 40°, 50°, 60° and 70°),

_{s}*ϕ*(which take only the value 0°) and

_{i}*ϕ*(which take the values 0°, 30°, 60°, 90°, 120°, 150°, 180°).

_{s}*θ*. Figures 3 and 4 show that the bigger the angle

_{i}*θ*, the higher the resulting BRDF (with the exception of the 0°-configuration). Furthermore, these figures clearly reveal the effect of

_{i}*θ*upon the spectral distribution: Whereas the spectral distributions at 40°, 50°, 60° and 70° are very similar, those recorded at 10° and 20° are slightly different from the rest, and also different from each other. On the other hand, within the diffuse region all spectra are rather similar (Fig. 2). The high frequency noise in the spectra (e.g. between 450 nm and 500 nm and around 770 nm) is an artefact produced by the division of the measured radiance and the measured irradiance on the sample to obtain the BRDF, and it corresponds to the narrow lines of the Xe lamp used in GEFE.

_{i}*λ*= 500 nm measured within the incidence plane. This plane comprises two half-planes, each of which is characterized by a particular angle

*ϕ*. We will say that, by definition, the half-plane containing the incoming light is

_{s}*ϕ*= 0°, whereas the half-plane containing the specularly reflected light is

_{s}*ϕ*= 180°.

_{s}*θ*= 0° has a higher value that those spectra at

_{i}*θ*= 10°, 20° and 30°. On the other hand, an increase of the BRDF at retro-reflection conditions is observe in Fig. 5. Both effects might be explained by the coherent backscattering of light (CBS) [20

_{i}20. R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. **63**, 561–564 (1995). [CrossRef]

21. T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” *Proc. SPIE*6682, 66820E (2007). [CrossRef]

*θ*for a specific illumination polar angle

_{s}*θ*. A negative

_{i}*θ*indicates, by definition, that the observation is carried out in the first half-plane.

_{s}*λ*= 500 nm) when observing from outside the incidence plane (half-planes 30:150), for

*θ*= 20°, 30°, 40°, 50°, 60° and 70°. This figure reveals more clearly the presence of the BRDF’s diffuse component.

_{i}*θ*) or the observation (

_{i}*θ*) angle is large (grazing angles), whereas for smaller angles it remains relatively constant. There is also a certain rise in BRDF with

_{s}*θ*within the 150-half-plane (close to the half-plane that contains the specular direction); in fact, the larger

_{s}*θ*is, the steeper the increase is. Therefore, it can be concluded that the diffuse BRDF is more uniform within the small-angle range (applicable both for illumination (

_{i}*θ*) as well as for observation (

_{i}*θ*) angles). The weak point of this type of representation is that it is not possible to assess the spectral dependency with the different angular configurations. We will see that this type of representations can be used to represent and highlight precisely this spectral variation, resorting for this purpose to PCA.

_{s}### 3.2. PCA on BRDF data

*N*= 401 wavelengths and

*M*= 448 BRDF spectra, with a 448 × 448 covariant matrix). The resulting eigenspectra are shown in Fig. 7. These are the spectra associated with the four highest eigenvalues (in descending order) which are, consequently, the most relevant ones when it comes to computing the total variance of the data (their joint contribution accounts for 99.6% of this variance: eigenvalues 0.9864, 0.0066, 0.0018 and 0.0013 for these four eigenspectra and 0.0003 for the first neglected eigenspectrum). We can see that the eigenspectrum PC1 (Fig. 7(a)) looks very similar to the specular spectra in Figs. 3 and 4. Similarly, there seems to be a large similarity between PC2 (Fig. 7(b)) and diffuse spectra in Fig. 2.

*A*, the simplest way is to locate an spectrum

_{j}*i*where this eigenspectrum has a substantial contribution to the total variance; then one only needs to look at the sign of

*e*. If the sign is negative then the eigenspectrum

_{i,j}*A*has to be inverted in the representation, but not in subsequent calculations.

_{j}## 4. Discussion

### 4.1. Physical interpretation of the eigenspectra of the BRDF

- Eigenspectrum PC1. This component always makes a contribution in specular-reflection scenarios (Fig. 8). It never contributes significantly in other configuration scenarios (Fig. 9). It is the only component that always contributes significantly in specular conditions; that is why it can be stated that eigenspectrum PC1 is influenced by Fresnel reflection.
- Eigenspectrum PC2. It predominates over the other components, except for the case where specular reflection is involved. It represents the BRDF’s diffuse component.
- Eigenspectrum PC3. It contributes too to the BRDF’s diffuse component. The reason why PCA renders them as two separate components is that, under specular reflection conditions at low incidence angles, eigenspectrum PC3’s contribution to the variance is clearly more relevant than that of eigenspectrum PC2. Taking into account that they have a rather different behavior under specular reflection conditions, it can be said that the spectral distributions associated to eigenspectra PC2 and PC3 originate from different physical phenomena.
- Eigenspectrum PC4. Even though this component’s contribution clearly exceeds that of eigenspectrum PC1 under non-specular observation conditions, its contribution is nonetheless much smaller than that of eigenspectra PC2 and PC3. It is under specular conditions with
*θ*= 10° and_{i}*θ*= 20° where this component’s contribution becomes really significant (see also Fig. 3). Perhaps this could be explained by the presence in the material of a second flat layer, where small incidence angles could be reflected and transmitted, but where large incidence angles would be trapped due to total reflection._{i}

### 4.2. PCA’s usefulness for the development of BRDF models

*F̄*) that interest us: where

_{i,j}*F̄*is the resulting reconstruction of

_{i,j}*F̄*when only

_{i}*A*is used. If we recall that, as stated in a previous section,

_{j}*F̄*=

_{i}*F*– 〈

_{i}*F*〉, then it can be inferred that the

_{i}*F*corresponding to the principal component

_{i}*j*can be computed as

*F*=

_{i}*F̄*+

_{i}*κ*〈

_{i,j}*F*〉, where

_{i}*κ*represents the relative contribution associated with

_{i,j}*A*to the

_{j}*i*spectrum of the BRDF. If we assume that this fraction is proportional to this component’s contribution, then under this approximation it can be said that:

*κ*=

_{i,j}**E2**

*. This approximation can be used to perform the model’s fitting, taking*

_{i,j}*κ*as free parameter with an initial value

_{i,j}**E2**

*.*

_{i,j}*ε*resulting from the reconstruction of the angular configurations

_{r,i}*i*that relies, respectively, on the four and five eigenspectra associated with the highest eigenvalues. The average error is computed as follows: where

*F̄*represents the reconstruction of

_{r,i}*F̄*.

_{i}## 5. Conclusions

## Acknowledgments

## References and links

1. | F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160. |

2. | C. Bordier, C. Andraud, and J. Lafait, “Model of light scattering that includes polarization effects by multilayered media,” J. Opt. Soc. Am. A |

3. | L. Simonot, “Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. |

4. | R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol. |

5. | B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM |

6. | G. J. Ward, “Measuring and modelling anisotropic reflection,” Comput. Graphics |

7. | X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol. |

8. | J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graphics |

9. | E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” |

10. | S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” |

11. | A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” |

12. | I. G. E. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express |

13. | J. L. Simonds, “Application of characteristic vector analysis to photographic and optical response data,” J. Opt. Soc. Am. |

14. | J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002). [CrossRef] [PubMed] |

15. | A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. |

16. | A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” |

17. | T. A. Germer and C. C. Asmail, “Goniometric optical scatter instrument for out-of-plane ellipsometry measurements,” Rev. Sci. Instrum. |

18. | D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia |

19. | 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. |

20. | R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. |

21. | T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” |

**OCIS Codes**

(030.5630) Coherence and statistical optics : Radiometry

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(290.1483) Scattering : BSDF, BRDF, and BTDF

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: June 27, 2011

Revised Manuscript: August 19, 2011

Manuscript Accepted: August 23, 2011

Published: September 19, 2011

**Citation**

A. Ferrero, J. Campos, A. M. Rabal, A. Pons, M. L. Hernanz, and A. Corróns, "Principal components analysis on the spectral bidirectional reflectance distribution function of ceramic colour standards," Opt. Express **19**, 19199-19211 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19199

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### References

- F. E. Nicodemus, J. C. Richmond, and J. J. Hsia, “Geometrical considerations and nomenclature for reflectance,” National Bureau of Standards Monograph (National Bureau of Standards, 1977), Vol. 160.
- C. Bordier, C. Andraud, and J. Lafait, “Model of light scattering that includes polarization effects by multilayered media,” J. Opt. Soc. Am. A 25, 1406–1419 (2008). [CrossRef]
- L. Simonot, “Photometric model of diffuse surfaces described as a distribution of interfaced Lambertian facets,” Appl. Opt. 48, 5793–5801 (2009). [CrossRef] [PubMed]
- R. L. Cook and K. E. Torrance, “A reflectance model for computer graphics,” Technical report, Computer Graphics (ACM, 1981), Vol. 15, No. 3.
- B. T. Phong, “Illumination for computer generated pictures,” Commun. ACM 18(6), 311–317 (1975). [CrossRef]
- G. J. Ward, “Measuring and modelling anisotropic reflection,” Comput. Graphics 26, 265–272 (1992). [CrossRef]
- X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Technical report, Computer Graphics (1991), Vol. 25, No. 4.
- J. F. Blinn, “Models of light reflection for computer synthesized pictures,” Comput. Graphics 11, 192–198 (1977). [CrossRef]
- E. P. F. Lafortune, S. C. Foo, K. E. Torrance, and D. P. Greenberg, “Non-linear approximation of reflectance functions,” Technical report (Cornell University, 1997).
- S. H. Westin, H. Li, and K. E. Torrance, “A comparison of four BRDF models,” Technical report PCG-04-02, Program of Computer Graphics (Cornell University, April2004).
- A. Ngan, F. Durand, and W. Matusik, “Experimental analysis of BRDF models,” Eurographics Symposium on Rendering, K. Bala and P. Dutre, eds. (2005).
- I. G. E. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008). [CrossRef] [PubMed]
- J. L. Simonds, “Application of characteristic vector analysis to photographic and optical response data,” J. Opt. Soc. Am. 53, 968–971 (1963). [CrossRef]
- J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal-component characterization of noise for infrared images,” Appl. Opt.41, 320–331 (2002). [CrossRef] [PubMed]
- A. Ferrero, J. Alda, J. Campos, J. M. López-Alonso, and A. Pons, “Principal components analysis of the photoresponse nonuniformity of a matrix detector,” Appl. Opt. 46, 9–17 (2007). [CrossRef]
- A. M. Rabal, A. Ferrero, J. L. Fontecha, A. Pons, J. Campos, A. Corróns, and A. M. Rubio, “Gonio-spectrophotometer for low-uncertainty measurements of bidirectional scattering distribution function (BSDF),” Proceedings of CIE Expert Symposium on “Spectral and Imaging Methods for Photometry and Radiometry,” Publication CIE x036:2010 (CIE, Vienna, Austria, 2010), pp. 79–84.
- 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]
- D. Hünerhoff, U. Grusemann, and A Höpe, “New robot-based gonioreflectometer for measuring spectral diffuse reflection.” Metrologia 43, S11–S16 (2006). [CrossRef]
- 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(29), 5454–5467 (2008). [PubMed]
- R. Corey, M. Kissner, and P. Saulnier, “Coherent backscattering of light,” Am. J. Phys. 63, 561–564 (1995). [CrossRef]
- T. J. Papetti, W. E. Walker, C. E. Keffer, and B. E. Johnson, “Coherent backscatter: measurement of the retroreflective BRDF peak exhibited by several surfaces relevant to ladar applications,” Proc. SPIE6682, 66820E (2007). [CrossRef]

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