Abstract

We propose a method for shape reconstruction from color shades produced by multiple chromatic light sources. The linear relation between surface-normal vectors and three-dimensional response vectors for a uniformly colored and illuminated region of a surface can be reconstructed in two steps. In the first step a quadratic form of metric in response space induced from a natural metric in normal space is reconstructed. At this stage proper image segmentation can be obtained. In the second step an exact mapping from response space into the space of surface normals is reconstructed. The matrix for this mapping is one of the square roots of the quadratic-form matrix that satisfies the integrability constraint. The method is in all respects much simpler than existing methods for solving the depth-from-shading task for monochromatic images.

© 1994 Optical Society of America

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References

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  1. B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. H. Winston, ed. (McGraw-Hill, New York,1975), pp. 115–155.
  2. B. A. Wandell, “The synthesis and analysis of color images,” IEEE Pattern Anal. Mach. Intell. PAMI-A9, 2–13 (1987).
    [CrossRef]
  3. B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graphics Image Process. 33, 174–208 (1986).
    [CrossRef]
  4. K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
    [CrossRef]
  5. P. P. Nikolaev, “Some algorithms for surface color recognition,” in Simulation of Learning and Behavior, M. S. Smirnov, ed. (Nauka, Moscow, 1975), pp. 121–151 (in Russian).
  6. P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988) (in Russian).
    [PubMed]
  7. M. H. Brill, “Image segmentation by object color: a unifying framework and connection to color constancy,” J. Opt. Soc. Am. A 7, 2041–2047 (1990).
    [CrossRef] [PubMed]
  8. A. P. Petrov, “Light, color, and shape,” in Intellectual Processes and Their Simulation, E. P. Velikhov, ed. (Nauka, Moscow, 1987), pp. 350–358 (in Russian).
  9. L. L. Kontsevich, “Computation of surface shape for uniformly colored and illuminated-by-chromatic-light-sources convex object from its two-dimensional projection,” in Data Processing in Information Systems (Academy of Sciences of the USSR, Moscow, 1986), pp. 16–19 (in Russian).

1990 (1)

1988 (1)

P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988) (in Russian).
[PubMed]

1987 (1)

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Pattern Anal. Mach. Intell. PAMI-A9, 2–13 (1987).
[CrossRef]

1986 (1)

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graphics Image Process. 33, 174–208 (1986).
[CrossRef]

1981 (1)

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Brill, M. H.

Brooks, M. J.

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graphics Image Process. 33, 174–208 (1986).
[CrossRef]

Horn, B. K. P.

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graphics Image Process. 33, 174–208 (1986).
[CrossRef]

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. H. Winston, ed. (McGraw-Hill, New York,1975), pp. 115–155.

Ikeuchi, K.

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Kontsevich, L. L.

L. L. Kontsevich, “Computation of surface shape for uniformly colored and illuminated-by-chromatic-light-sources convex object from its two-dimensional projection,” in Data Processing in Information Systems (Academy of Sciences of the USSR, Moscow, 1986), pp. 16–19 (in Russian).

Nikolaev, P. P.

P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988) (in Russian).
[PubMed]

P. P. Nikolaev, “Some algorithms for surface color recognition,” in Simulation of Learning and Behavior, M. S. Smirnov, ed. (Nauka, Moscow, 1975), pp. 121–151 (in Russian).

Petrov, A. P.

A. P. Petrov, “Light, color, and shape,” in Intellectual Processes and Their Simulation, E. P. Velikhov, ed. (Nauka, Moscow, 1987), pp. 350–358 (in Russian).

Wandell, B. A.

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Pattern Anal. Mach. Intell. PAMI-A9, 2–13 (1987).
[CrossRef]

Artif. Intell. (1)

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Biofizika (1)

P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988) (in Russian).
[PubMed]

Comput. Vis. Graphics Image Process. (1)

B. K. P. Horn, M. J. Brooks, “The variational approach to shape from shading,” Comput. Vis. Graphics Image Process. 33, 174–208 (1986).
[CrossRef]

IEEE Pattern Anal. Mach. Intell. (1)

B. A. Wandell, “The synthesis and analysis of color images,” IEEE Pattern Anal. Mach. Intell. PAMI-A9, 2–13 (1987).
[CrossRef]

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

Other (4)

A. P. Petrov, “Light, color, and shape,” in Intellectual Processes and Their Simulation, E. P. Velikhov, ed. (Nauka, Moscow, 1987), pp. 350–358 (in Russian).

L. L. Kontsevich, “Computation of surface shape for uniformly colored and illuminated-by-chromatic-light-sources convex object from its two-dimensional projection,” in Data Processing in Information Systems (Academy of Sciences of the USSR, Moscow, 1986), pp. 16–19 (in Russian).

B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. H. Winston, ed. (McGraw-Hill, New York,1975), pp. 115–155.

P. P. Nikolaev, “Some algorithms for surface color recognition,” in Simulation of Learning and Behavior, M. S. Smirnov, ed. (Nauka, Moscow, 1975), pp. 121–151 (in Russian).

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Figures (5)

Fig. 1
Fig. 1

Arbitrary surface illuminated by two light sources ( p ̂ 1 and p ̂ 2). The region outlined by the heavy black line is illuminated by both sources simultaneously. In this region M = M ( p ̂ 1 ) + M ( p ̂ 2 ).

Fig. 2
Fig. 2

Rotation of the basis in response space causes a change of normal field. The relative orientations of the normals remain invariable.

Fig. 3
Fig. 3

Reconstructed shape of a spherical surface from the computer-generated image for different levels of noise: (a) σ = 0, (b) σ = 0.2, (c) σ = 0.4.

Fig. 4
Fig. 4

(a) Top, high-resolution image of an egg in the red channel and bottom, three images for all channels with the low resolution used in the processing; (b) reconstructed shape of the egg (viewed from the side).

Fig. 5
Fig. 5

Partial image of a child’s face. The details are the same as for Fig. 4.

Tables (1)

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Table 1 Results of Shape Reconstruction

Equations (15)

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r i = n ̂ p ̂ S ( λ ) ρ ( λ ) ν i ( λ ) d λ ,
r = M n ̂ ,
m i , j = p j S ( λ ) ρ ( λ ) ν i ( λ ) d λ
n ̂ p ̂ 0 .
M = n ̂ p ̂ 0 . M ( p ̂ ) ,
Q = ( M 1 ) T M 1 .
| n | = | r | Q = r T Q r = 1 .
[ ( M O ) 1 ] T ( M O ) 1 = ( M ) 1 O O T M 1 = ( M 1 ) T M 1 .
r 1 2 q 1 , 1 + r 2 2 q 2 , 2 + r 3 2 q 3 , 3 + 2 r 1 r 2 q 1 , 2 + 2 r 1 r 3 q 1 , 3 + 2 r 2 r 3 q 2 , 3 = 1 .
1 ɛ 1 < | r | Q < 1 + ɛ 2 .
d z = n 1 n 3 d x n 2 n 3 d y .
p i , j = [ ( z / x ) i , j + ( z / x ) i + 1 , j ] / 2 , q i , j = [ ( z / y ) i , j + ( z / y ) i , j + 1 ] / 2 .
p i , j + q i + 1 , j p i , j + 1 p i , j = 0 .
i , j ( p i , j + q i + 1 , j p i , j + 1 p i , j ) 2
[ ( z x p ) 2 + ( z y q ) 2 ] d x d y ,

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