Abstract

This paper investigates the use of photometric stereo (PS) with reflectance functions that are diffuse but not Lambertian. We show that, for the special case where light sources are arranged at 90° intervals around the optical axis, standard PS is not limited to Lambertian surfaces, and we define criteria for its use. A series of rough test surfaces are used as models for surface microstructure—we found that the Oren Nayar (ON) reflectance model accurately predicted the surfaces’ reflectance functions. The ON model does not meet our theoretical criteria for using PS, but PS performs well in simulations if the microroughness is moderate (rms slope <0.3). When PS was applied to real surfaces, the estimated and actual slopes were highly correlated, but there were significant errors in the slope estimates for the rougher samples.

© 2012 Optical Society of America

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References

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  1. R. Woodham, “Photometric methods for determining surface orientation from multiple images,” Opt. Eng. 19, 139–144(1980).
  2. L. B. Wolff, “Diffuse-reflectance model for smooth dielectric surfaces,” J. Opt. Soc. Am. A 11, 2956–2968 (1994).
    [CrossRef]
  3. M. Oren and S. K. Nayar, “Generalization of Lambert’s reflectance model,” in SIGGRAPH ’94—Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1994), pp. 239–246.
  4. H. Tagare and R. J. P. deFigueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
    [CrossRef]
  5. H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann–Kirchhoff model against radiance data,” Comp. Vision Image Understand. 102, 145–168 (2006).
    [CrossRef]
  6. H. Ragheb and E. R. Hancock, “The modified Beckmann–Kirchhoff scattering theory for rough surface analysis,” Patt. Recogn. 40, 2004–2020 (2007).
    [CrossRef]
  7. H. Ragheb and E. R. Hancock, “A light scattering model for layered dielectrics with rough surface boundaries,” Int. J. Comput. Vis. 79, 179–207 (2008).
    [CrossRef]
  8. B. van Ginneken, M. Stavridi, and J. J. Koenderink, “Diffuse and specular reflectance from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
    [CrossRef]
  9. J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
    [CrossRef]
  10. S. K. Nayar and M. Oren, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
    [CrossRef]
  11. H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Patt. Recogn. 38, 1574–1595(2005).
    [CrossRef]
  12. http://www.cs.columbia.edu/CAVE/software/curet/html/brdfp.html .
  13. http://www.cs.columbia.edu/CAVE/software/curet/html/brdfm.html .

2008

H. Ragheb and E. R. Hancock, “A light scattering model for layered dielectrics with rough surface boundaries,” Int. J. Comput. Vis. 79, 179–207 (2008).
[CrossRef]

2007

H. Ragheb and E. R. Hancock, “The modified Beckmann–Kirchhoff scattering theory for rough surface analysis,” Patt. Recogn. 40, 2004–2020 (2007).
[CrossRef]

2006

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann–Kirchhoff model against radiance data,” Comp. Vision Image Understand. 102, 145–168 (2006).
[CrossRef]

2005

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Patt. Recogn. 38, 1574–1595(2005).
[CrossRef]

1999

J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
[CrossRef]

1998

1995

S. K. Nayar and M. Oren, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

1994

1991

H. Tagare and R. J. P. deFigueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

1980

R. Woodham, “Photometric methods for determining surface orientation from multiple images,” Opt. Eng. 19, 139–144(1980).

Dana, K. J.

J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
[CrossRef]

deFigueiredo, R. J. P.

H. Tagare and R. J. P. deFigueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

Hancock, E. R.

H. Ragheb and E. R. Hancock, “A light scattering model for layered dielectrics with rough surface boundaries,” Int. J. Comput. Vis. 79, 179–207 (2008).
[CrossRef]

H. Ragheb and E. R. Hancock, “The modified Beckmann–Kirchhoff scattering theory for rough surface analysis,” Patt. Recogn. 40, 2004–2020 (2007).
[CrossRef]

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann–Kirchhoff model against radiance data,” Comp. Vision Image Understand. 102, 145–168 (2006).
[CrossRef]

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Patt. Recogn. 38, 1574–1595(2005).
[CrossRef]

Koenderink, J. J.

J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
[CrossRef]

B. van Ginneken, M. Stavridi, and J. J. Koenderink, “Diffuse and specular reflectance from rough surfaces,” Appl. Opt. 37, 130–139 (1998).
[CrossRef]

Nayar, S. K.

J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
[CrossRef]

S. K. Nayar and M. Oren, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

M. Oren and S. K. Nayar, “Generalization of Lambert’s reflectance model,” in SIGGRAPH ’94—Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1994), pp. 239–246.

Oren, M.

S. K. Nayar and M. Oren, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

M. Oren and S. K. Nayar, “Generalization of Lambert’s reflectance model,” in SIGGRAPH ’94—Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1994), pp. 239–246.

Ragheb, H.

H. Ragheb and E. R. Hancock, “A light scattering model for layered dielectrics with rough surface boundaries,” Int. J. Comput. Vis. 79, 179–207 (2008).
[CrossRef]

H. Ragheb and E. R. Hancock, “The modified Beckmann–Kirchhoff scattering theory for rough surface analysis,” Patt. Recogn. 40, 2004–2020 (2007).
[CrossRef]

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann–Kirchhoff model against radiance data,” Comp. Vision Image Understand. 102, 145–168 (2006).
[CrossRef]

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Patt. Recogn. 38, 1574–1595(2005).
[CrossRef]

Stavridi, M.

Tagare, H.

H. Tagare and R. J. P. deFigueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

Van Doorn, A. J.

J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
[CrossRef]

van Ginneken, B.

Wolff, L. B.

Woodham, R.

R. Woodham, “Photometric methods for determining surface orientation from multiple images,” Opt. Eng. 19, 139–144(1980).

Appl. Opt.

Comp. Vision Image Understand.

H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann–Kirchhoff model against radiance data,” Comp. Vision Image Understand. 102, 145–168 (2006).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

H. Tagare and R. J. P. deFigueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
[CrossRef]

Int. J. Comput. Vis.

J. J. Koenderink, A. J. Van Doorn, K. J. Dana, and S. K. Nayar, “Bidirectional reflection distribution function of thoroughly pitted surfaces,” Int. J. Comput. Vis. 31, 129–144 (1999).
[CrossRef]

S. K. Nayar and M. Oren, “Generalization of the Lambertian model and implications for machine vision,” Int. J. Comput. Vis. 14, 227–251 (1995).
[CrossRef]

H. Ragheb and E. R. Hancock, “A light scattering model for layered dielectrics with rough surface boundaries,” Int. J. Comput. Vis. 79, 179–207 (2008).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

R. Woodham, “Photometric methods for determining surface orientation from multiple images,” Opt. Eng. 19, 139–144(1980).

Patt. Recogn.

H. Ragheb and E. R. Hancock, “Surface radiance correction for shape from shading,” Patt. Recogn. 38, 1574–1595(2005).
[CrossRef]

H. Ragheb and E. R. Hancock, “The modified Beckmann–Kirchhoff scattering theory for rough surface analysis,” Patt. Recogn. 40, 2004–2020 (2007).
[CrossRef]

Other

http://www.cs.columbia.edu/CAVE/software/curet/html/brdfp.html .

http://www.cs.columbia.edu/CAVE/software/curet/html/brdfm.html .

M. Oren and S. K. Nayar, “Generalization of Lambert’s reflectance model,” in SIGGRAPH ’94—Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1994), pp. 239–246.

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Figures (13)

Fig. 1.
Fig. 1.

(a) Imaging geometry of a facet. (b) Facet oriented and lit from the opposite direction.

Fig. 2.
Fig. 2.

A test object is lit with a collimated source with a zenith λ. Self-shadowing occurs for facets with gradient >tanγ, where γ=90°λ. We have reversed the p axis so that the two diagrams correspond.

Fig. 3.
Fig. 3.

Experimental setup.

Fig. 4.
Fig. 4.

Cross polarizers suppress the specular lobe.

Fig. 5.
Fig. 5.

Test surfaces: row 1, flat 1, flat 2, fracture 1, fracture 2; row 2, cork, fiber, sandpaper, gravel; row 3, shallow pits, deep pits, sparse spheres, dense spheres.

Fig. 6.
Fig. 6.

Testing reciprocity for our samples.

Fig. 7.
Fig. 7.

(a) Fracture 2 sample with p=0.84 and (b) estimated amount of shadowing as a function of average slope.

Fig. 8.
Fig. 8.

Measured, Lambertian, and Oren Nayar reflectance functions.

Fig. 9.
Fig. 9.

(a) Reflectance functions predicted by Oren’s model and (b) slope estimated from the reflection model.

Fig. 10.
Fig. 10.

(a) Slope error as a function of slope and (b) error in slope estimate of facets with given slopes rendered with ON model).

Fig. 11.
Fig. 11.

Slopes estimated from samples.

Fig. 12.
Fig. 12.

Error in estimated slopes.

Fig. 13.
Fig. 13.

Geometry.

Tables (1)

Tables Icon

Table 1. Standard Deviation of Slope Estimated Using PS and the ON Model

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

i=L.N|N|.
i(τ,λ)=ρ·pcosτsinλqsinτsinλ+cosλp2+q2+1,
ptanλ=i(180,λ)i(0,λ)i(180,λ)+i(0,λ),
qtanλ=(i(0,λ)+i(180,λ))2i(90,λ)i(180,λ)+i(0,λ).
Rλ(p)=a0+a1p+a2p2+a3p3+a4p4+..
Rλ(p)=(ptanγ)Cλ(p),
Rλ(p)=Rλ(p).
ptanλ=ρRλ(p)ρRλ(p)ρRλ(p)+ρRλ(p)=ρRλ(p)ρRλ(p)ρRλ(p)+ρRλ(p),
ptanλ=(ptanγ)Cλ(p,λ)(ptanγ)Cλ(p,λ)(ptanγ)Cλ(p,λ)+(ptanγ)Cλ(p,λ).
[p+e(p)]tanλ=iλ(p)iλ(p)iλ(p)+iλ(p).
[p+e(p)]tanλ=iλ(p)iλ(p)iλ(p)+iλ(p).
LON(p)=LLa(p)[A+Bp2+1(psinλp2cosλ)].
LON(p)ptanλ=LNL(p)[A+B(psinλ)p2+1B(p2cosλ)p2+1].
LON(p,λ)=LLa(p,λ)[A+Bmax(0,cos(ϕrϕi))sinαtanβ],
A=10.5σON2σON2+0.33,
B=0.45σON2σON2+0.09,
α=max(θi,θr),
β=min(θi,θr).
α=θi,
β=θr.
max(0,cos(ϕiϕr)=1,
θi=λθr=λarctan(p),
LON(p,λ)=LLa(p,λ)[A+Bpsin(λarctan(p))],
LON(p,λ)=LLa(p,λ)[A+Bp(sin(arctan(p))cosλcos(arctan(p))sinλ)].
cos(arctan(p))=1(p2+1),
sin(arctan(p))=p(p2+1),
LON(p,λ)=LLa(p,λ)[A+Bpp2+1(pcosλsinλ)].

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