Abstract

A photometric stereo technique is proposed that uses four extended sources and specular reflection to estimate surface topography. It is shown that if the intensity of incident light is weighted according to its zenith angle, then the radiance of surface facets will vary linearly with their slope. A simple system that approximates this lighting distribution is demonstrated. It is shown that surface slope in the range [0.5,0.5] can be recovered to within a multiplicative constant.

© 2010 Optical Society of America

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References

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  1. A. P. Pentland, “Linear shape from shading,” Int. J. Comput. Vis. 4, 153–162 (1990).
    [CrossRef]
  2. E. N. Coleman, Jr. and R. Jain, “Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry,” Comput. Vis. Graph. Image Process. 18, 309–328 (1982).
    [CrossRef]
  3. S. Barsky and M. Petrou, “The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1239–1252 (2003).
    [CrossRef]
  4. O. Drbohlav and R. Šára, “Specularities reduce ambiguity of uncalibrated photometric stereo,” in Proceedings of the European Conference on Computer Vision (2002), Vol. 2, pp. 46–62.
  5. H. D. Tagare and R. J. P. de Figueiredo, “A theory of photometric stereo for a class of diffuse non-Lambertian surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 133–152 (1991).
    [CrossRef]
  6. S. K. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
    [CrossRef]
  7. H. Ragheb and E. R. Hancock, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Underst. 102, 145–168 (2006).
    [CrossRef]
  8. 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]
  9. H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Proceedings of Computer Analysis of Images and Patterns, N.Petkov and M.Westenberg, eds. (Springer, 2003), pp. 98–106.
    [CrossRef]
  10. O. Drbohlav and R. Šára, “Using polarization to determine intrinsic surface properties,” Proc. SPIE 3826, 253–263 (1999).
    [CrossRef]
  11. G. Atkinson and E. R. Hancock, “Recovery of surface orientation from diffuse polarization,” IEEE Trans. Image Process. 15, 1653–1664 (2006).
    [CrossRef] [PubMed]
  12. R. J. Woodham, “Photometric method for determining surface orientation from multiple images,” Opt. Eng. (Bellingham) 19, 139–144 (1980).
  13. A. C. Sanderson, L. E. Weiss, and S. K. Nayar, “Structured highlight inspection of specular surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 44–55 (1988).
    [CrossRef]
  14. T. Chen, M. Goesele, and H. Seidel, “Mesostructure from specularity,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), Vol. 2, pp. 1825–1832.
  15. J. Blinn and M. Newell, “Texture and reflection in computer generated images,” Comm. ACM 19, 542–546 (1976).
    [CrossRef]
  16. J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.
  17. R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A 18, 2448–2459 (2001).
    [CrossRef]
  18. R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
    [CrossRef]
  19. R. Ramamoorthi, “Modeling illumination variation with spherical harmonics,” in Face Processing: Advanced Modeling Methods, W.Zhao and R.Chellappa, eds. (Elsevier, 2006), pp. 385–424.
  20. P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
    [CrossRef]
  21. A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?,” in Prooceedings of the European Conference on Computer Vision (2006), Vol. 1, pp. 578–591.

2008 (1)

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]

2006 (5)

G. Atkinson and E. R. Hancock, “Recovery of surface orientation from diffuse polarization,” IEEE Trans. Image Process. 15, 1653–1664 (2006).
[CrossRef] [PubMed]

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

T. Chen, M. Goesele, and H. Seidel, “Mesostructure from specularity,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), Vol. 2, pp. 1825–1832.

R. Ramamoorthi, “Modeling illumination variation with spherical harmonics,” in Face Processing: Advanced Modeling Methods, W.Zhao and R.Chellappa, eds. (Elsevier, 2006), pp. 385–424.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?,” in Prooceedings of the European Conference on Computer Vision (2006), Vol. 1, pp. 578–591.

2004 (1)

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

2003 (2)

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Proceedings of Computer Analysis of Images and Patterns, N.Petkov and M.Westenberg, eds. (Springer, 2003), pp. 98–106.
[CrossRef]

S. Barsky and M. Petrou, “The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1239–1252 (2003).
[CrossRef]

2002 (1)

O. Drbohlav and R. Šára, “Specularities reduce ambiguity of uncalibrated photometric stereo,” in Proceedings of the European Conference on Computer Vision (2002), Vol. 2, pp. 46–62.

2001 (1)

2000 (1)

J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.

1999 (1)

O. Drbohlav and R. Šára, “Using polarization to determine intrinsic surface properties,” Proc. SPIE 3826, 253–263 (1999).
[CrossRef]

1991 (1)

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

1990 (2)

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

A. P. Pentland, “Linear shape from shading,” Int. J. Comput. Vis. 4, 153–162 (1990).
[CrossRef]

1988 (2)

A. C. Sanderson, L. E. Weiss, and S. K. Nayar, “Structured highlight inspection of specular surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 44–55 (1988).
[CrossRef]

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

1982 (1)

E. N. Coleman, Jr. and R. Jain, “Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry,” Comput. Vis. Graph. Image Process. 18, 309–328 (1982).
[CrossRef]

1980 (1)

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

1976 (1)

J. Blinn and M. Newell, “Texture and reflection in computer generated images,” Comm. ACM 19, 542–546 (1976).
[CrossRef]

Agrawal, A.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?,” in Prooceedings of the European Conference on Computer Vision (2006), Vol. 1, pp. 578–591.

Atkinson, G.

G. Atkinson and E. R. Hancock, “Recovery of surface orientation from diffuse polarization,” IEEE Trans. Image Process. 15, 1653–1664 (2006).
[CrossRef] [PubMed]

Barsky, S.

S. Barsky and M. Petrou, “The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1239–1252 (2003).
[CrossRef]

Blinn, J.

J. Blinn and M. Newell, “Texture and reflection in computer generated images,” Comm. ACM 19, 542–546 (1976).
[CrossRef]

Chellappa, R.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?,” in Prooceedings of the European Conference on Computer Vision (2006), Vol. 1, pp. 578–591.

Chen, T.

T. Chen, M. Goesele, and H. Seidel, “Mesostructure from specularity,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), Vol. 2, pp. 1825–1832.

Coleman, E. N.

E. N. Coleman, Jr. and R. Jain, “Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry,” Comput. Vis. Graph. Image Process. 18, 309–328 (1982).
[CrossRef]

de Figueiredo, R. J. P.

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

Drbohlav, O.

O. Drbohlav and R. Šára, “Specularities reduce ambiguity of uncalibrated photometric stereo,” in Proceedings of the European Conference on Computer Vision (2002), Vol. 2, pp. 46–62.

O. Drbohlav and R. Šára, “Using polarization to determine intrinsic surface properties,” Proc. SPIE 3826, 253–263 (1999).
[CrossRef]

Goesele, M.

T. Chen, M. Goesele, and H. Seidel, “Mesostructure from specularity,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), Vol. 2, pp. 1825–1832.

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, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Underst. 102, 145–168 (2006).
[CrossRef]

G. Atkinson and E. R. Hancock, “Recovery of surface orientation from diffuse polarization,” IEEE Trans. Image Process. 15, 1653–1664 (2006).
[CrossRef] [PubMed]

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Proceedings of Computer Analysis of Images and Patterns, N.Petkov and M.Westenberg, eds. (Springer, 2003), pp. 98–106.
[CrossRef]

Hanrahan, P.

Heidrich, W.

J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.

Ikeuchi, K.

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

Jain, R.

E. N. Coleman, Jr. and R. Jain, “Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry,” Comput. Vis. Graph. Image Process. 18, 309–328 (1982).
[CrossRef]

Kanade, T.

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

Kautz, J.

J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.

Kube, P.

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

Nayar, S. K.

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

A. C. Sanderson, L. E. Weiss, and S. K. Nayar, “Structured highlight inspection of specular surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 44–55 (1988).
[CrossRef]

Newell, M.

J. Blinn and M. Newell, “Texture and reflection in computer generated images,” Comm. ACM 19, 542–546 (1976).
[CrossRef]

Pentland, A.

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

Pentland, A. P.

A. P. Pentland, “Linear shape from shading,” Int. J. Comput. Vis. 4, 153–162 (1990).
[CrossRef]

Petrou, M.

S. Barsky and M. Petrou, “The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1239–1252 (2003).
[CrossRef]

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, “Testing new variants of the Beckmann-Kirchhoff model against radiance data,” Comput. Vis. Image Underst. 102, 145–168 (2006).
[CrossRef]

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Proceedings of Computer Analysis of Images and Patterns, N.Petkov and M.Westenberg, eds. (Springer, 2003), pp. 98–106.
[CrossRef]

Ramamoorthi, R.

R. Ramamoorthi, “Modeling illumination variation with spherical harmonics,” in Face Processing: Advanced Modeling Methods, W.Zhao and R.Chellappa, eds. (Elsevier, 2006), pp. 385–424.

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

R. Ramamoorthi and P. Hanrahan, “On the relationship between radiance and irradiance: determining the illumination from images of a convex Lambertian object,” J. Opt. Soc. Am. A 18, 2448–2459 (2001).
[CrossRef]

Raskar, R.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?,” in Prooceedings of the European Conference on Computer Vision (2006), Vol. 1, pp. 578–591.

Sanderson, A. C.

A. C. Sanderson, L. E. Weiss, and S. K. Nayar, “Structured highlight inspection of specular surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 44–55 (1988).
[CrossRef]

Šára, R.

O. Drbohlav and R. Šára, “Specularities reduce ambiguity of uncalibrated photometric stereo,” in Proceedings of the European Conference on Computer Vision (2002), Vol. 2, pp. 46–62.

O. Drbohlav and R. Šára, “Using polarization to determine intrinsic surface properties,” Proc. SPIE 3826, 253–263 (1999).
[CrossRef]

Seidel, H.

T. Chen, M. Goesele, and H. Seidel, “Mesostructure from specularity,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), Vol. 2, pp. 1825–1832.

J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.

Tagare, H. D.

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

Vazquez, P.

J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.

Weiss, L. E.

A. C. Sanderson, L. E. Weiss, and S. K. Nayar, “Structured highlight inspection of specular surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 44–55 (1988).
[CrossRef]

Woodham, R. J.

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

ACM Trans. Graphics (1)

R. Ramamoorthi and P. Hanrahan, “A signal-processing framework for reflection,” ACM Trans. Graphics 23, 1004–1042 (2004).
[CrossRef]

Comm. ACM (1)

J. Blinn and M. Newell, “Texture and reflection in computer generated images,” Comm. ACM 19, 542–546 (1976).
[CrossRef]

Comput. Vis. Graph. Image Process. (1)

E. N. Coleman, Jr. and R. Jain, “Obtaining 3-dimensional shape of textured and specular surfaces using four-source photometry,” Comput. Vis. Graph. Image Process. 18, 309–328 (1982).
[CrossRef]

Comput. Vis. Image Underst. (1)

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

IEEE Trans. Image Process. (1)

G. Atkinson and E. R. Hancock, “Recovery of surface orientation from diffuse polarization,” IEEE Trans. Image Process. 15, 1653–1664 (2006).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (4)

A. C. Sanderson, L. E. Weiss, and S. K. Nayar, “Structured highlight inspection of specular surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 44–55 (1988).
[CrossRef]

S. Barsky and M. Petrou, “The 4-source photometric stereo technique for three-dimensional surfaces in the presence of highlights and shadows,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 1239–1252 (2003).
[CrossRef]

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

P. Kube and A. Pentland, “On the imaging of fractal surfaces,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 704–707 (1988).
[CrossRef]

IEEE Trans. Rob. Autom. (1)

S. K. Nayar, K. Ikeuchi, and T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

Int. J. Comput. Vis. (2)

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]

A. P. Pentland, “Linear shape from shading,” Int. J. Comput. Vis. 4, 153–162 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (Bellingham) (1)

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

Proc. SPIE (1)

O. Drbohlav and R. Šára, “Using polarization to determine intrinsic surface properties,” Proc. SPIE 3826, 253–263 (1999).
[CrossRef]

Other (6)

H. Ragheb and E. R. Hancock, “Rough surface correction and re-illumination using the modified Beckmann model,” in Proceedings of Computer Analysis of Images and Patterns, N.Petkov and M.Westenberg, eds. (Springer, 2003), pp. 98–106.
[CrossRef]

O. Drbohlav and R. Šára, “Specularities reduce ambiguity of uncalibrated photometric stereo,” in Proceedings of the European Conference on Computer Vision (2002), Vol. 2, pp. 46–62.

T. Chen, M. Goesele, and H. Seidel, “Mesostructure from specularity,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2006), Vol. 2, pp. 1825–1832.

J. Kautz, P. Vazquez, W. Heidrich, and H. Seidel, “Unified approach to prefiltered environment maps,” in Proceedings of the Eurographics Workshop on Rendering Techniques (Springer Computer Science, 2000), pp. 185–196.

R. Ramamoorthi, “Modeling illumination variation with spherical harmonics,” in Face Processing: Advanced Modeling Methods, W.Zhao and R.Chellappa, eds. (Elsevier, 2006), pp. 385–424.

A. Agrawal, R. Raskar, and R. Chellappa, “What is the range of surface reconstructions from a gradient field?,” in Prooceedings of the European Conference on Computer Vision (2006), Vol. 1, pp. 578–591.

Supplementary Material (10)

» Media 1: AVI (2959 KB)     
» Media 2: AVI (3068 KB)     
» Media 3: AVI (3330 KB)     
» Media 4: AVI (3272 KB)     
» Media 5: AVI (3182 KB)     
» Media 6: AVI (4266 KB)     
» Media 7: AVI (3368 KB)     
» Media 8: AVI (3557 KB)     
» Media 9: AVI (2556 KB)     
» Media 10: AVI (2927 KB)     

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Figures (21)

Fig. 1
Fig. 1

Lighting and viewing geometry of a surface facet.

Fig. 2
Fig. 2

Specularly reflecting facet, with slope expressed in polar angle form.

Fig. 3
Fig. 3

Relationship between slope and radiance for specular and Lambertian facets predicted by equations (8, 10).

Fig. 4
Fig. 4

Simulations: the effect of surface micro-roughness (left) and increasing the diffuse component (right) on the relationship between slope and radiance.

Fig. 5
Fig. 5

Measurement setup for analytical experiments. Facet and lighting zenith varied, azimuth held constant at zero.

Fig. 6
Fig. 6

Test cylinders, from left to right: polished aluminium, steel, roughened aluminium, anodized aluminium, PEEK and PA6.

Fig. 7
Fig. 7

Radiance plotted as a function of zenith and slope for the test cylinders.

Fig. 8
Fig. 8

Variation of integrated radiance with surface slope for the test cylinders.

Fig. 9
Fig. 9

Example surface (a 2-Euro coin): under ambient light (left), sum of images (center), sum of weighted images (right).

Fig. 10
Fig. 10

Achieving the azimuth lighting distribution (plan view).

Fig. 11
Fig. 11

Implementation of measurement setup.

Fig. 12
Fig. 12

Shading pattern for cone diffuser.

Fig. 13
Fig. 13

Arrangement of light sources used to obtain images A to D (plan view).

Fig. 14
Fig. 14

Variation of radiance with slope when lit by a desk lamp (left), for a combination of light sources AC (center) and BD (right).

Fig. 15
Fig. 15

Relationships between slope, the photometric stereo estimate of slope (PS) and the weighted integrated radiance (extended) for the test cylinders.

Fig. 16
Fig. 16

Estimated height functions (scaled) for the test cylinders obtained using the cone system.

Fig. 17
Fig. 17

Recovered surfaces: a 1-Euro coin (left, Media 1) and a ballbearing (right, Media 2). Note how the system’s inabilty to measure steep slopes results in the ballbearing having a bell-shaped surface.

Fig. 18
Fig. 18

Recovered surfaces: a pen tip (left, Media 3) and a gauge (right, Media 4).

Fig. 19
Fig. 19

Recovered surfaces: a section of a spanner (left, Media 5) and a beetle (right, Media 6).

Fig. 20
Fig. 20

Recovered surfaces: centipede (left, Media 7) and pill millipede (right, Media 8).

Fig. 21
Fig. 21

Recovered surfaces: Allen key (left, Media 9) and brass nozzle (right, Media 10).

Equations (12)

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

i ( p , q , τ i , σ i ) = i 0 R ( p , q , τ i , σ i ) ,
p = z x , q = z y .
i ( p , q ) = k 0 π 2 π + π i 0 ( τ i , σ i ) R ( p , q , τ i , σ i ) sin σ i d τ i d σ i .
i 0 ( τ i , σ i ) = σ i cos τ i ,
i 0 ( τ i , σ i ) = σ i sin τ i .
R ( τ f , σ f , τ i , σ i ) = δ ( τ i τ f , σ i 2 σ f ) .
i ( τ f , σ f ) = 2 σ f cos τ f sin 2 σ f .
i ( p , q ) 2 p sin 2 σ f .
R ( p , q , τ i , σ i ) = ρ p cos τ i sin σ i q sin τ i sin σ i + cos σ i p 2 + q 2 + 1 ,
i p ̂ ( p , q ) = π 3 ρ p 16 p 2 + q 2 + 1 .
R ( p , σ ) = k L p sin σ i + cos σ i ( p 2 + 1 ) + k S exp 1 2 ( p p a σ r ) 2 ,
k r = k S k S + k L .

Metrics