Shape from Shading for Lambertian Surfaces

We fi rst apply this technique for diff use refl ecting opaque objects. For the sake of simplicity, we assume that the surface of a Lambertian object is illuminated by parallel light. The radiance L of a Lambertian surface (Section 6.4.3) does not depend on the viewing angle and is given by:

=

L ρ (λ ) E cos γ,                                              (8.10)

π

T

1

2

T

where E is the irradiance and γ the angle between the surface normal and the illumination direction. The relation between the surface normal and the incident and exitant radiation can most easily be understood in the gradient space. This space is spanned by the gradient of the surface height a(X, Y ):

∂ X

∂ Y

s = ∇ a =  ∂ a, ∂ a

 = Σ s, s

Σ  .                  (8.11)

8.5 Shape from Shading†                                                                                    219

 

 

x2 -s2   surface normal   j s   light di   -s1 x1

rection

 

tanq i

 

Figure 8.8: Radiance computation illustrated in the gradient space for a Lam- bertian surface illuminated by a distant light source with an incidence angle θ _i and an azimuthal angle φ _i of zero.

 

This gradient is directly related to the surface normal n by

∂ X

∂ Y

T

n =  − ∂ a, − ∂ a, 1   = Σ − s₁, − s₂, 1 Σ .                           (8.12)

=

This equations shows that the gradient space can be understood as a plane parallel to the XY plane at a height Z 1 if we invert the direc- tions of the X and Y axes. The X and Y coordinates where the surface normal vector and other directional vectors intersect this plane are the corresponding coordinates in the gradient space.

=

The geometry of Lambertian refl ection in the gradient space is illus- trated in Fig. 8.8. Without loss of generality, we set the direction of the light source as the x direction. Then, the light direction is given by the vector l (tan θ _i, 0, 1)T, and the radiance L of the surface can be expressed as

L ρ (λ )  n T l

ρ (λ )            − s₁ tan θ _i + 1                

            

= π E | n || l | =

E                                            .          (8.13)

i

1

2

π       1 + tan2 θ    1 + s2 + s2

 

1

2

Contour plots of the radiance distribution in the gradient space are shown in Fig. 8.9a for a light source with an incidence angle of θ _i = 0°. In the case of the light source at the zenith, the contour lines of equal radiance mark lines with constant absolute slope s = (s2 + s2)1/2. How-

−

ever, the radiance changes with surface slope are low, especially for low surface inclinations. An oblique illumination leads to a much higher con- trast in the radiance (Fig. 8.9b). With an oblique illumination, however, the maximum surface slope in the direction opposite to the light source is limited to π /2 θ when the surface normal is perpendicular to the light direction.

With a single illumination source, the information about the surface normal is incomplete even if the surface refl ectivity is known. Only the component of the surface normal in the direction of the illumination

220                                                                                                         8 3-D Imaging

 

A                                                                    b

1                                                                                                           1

 

s2                                                                                                       s2

 

0.5                                                                   0.5

 

 

0                                                                     0

 

 

-0.5

 

-0.5

 

 

-1

-1              -0.5              0                0.5  s1      1

-1

-1               -0.5              0                0.5

s1   1

 

=

−

Figure 8.9: Contour plot of the radiance of a Lambertian surface with homoge- neous refl ectivity illuminated by parallel light shown in the gradient space for surface slopes between 1 and 1. The radiance is normalized to the radiance for a fl at surface. a Zero incidence angle θ _i 0°; the spacing of the contour lines is 0.05. b Oblique illumination with an incidence angle of 45° and an azimuthal angle of 0°; the spacing of the contour lines is 0.1.

 

change is given. Thus surface reconstruction with a single illumination source constitutes a complex mathematical problem that will not be con- sidered further here. In the next section we consider how many illumi- nations from diff erent directions are required to solve the shape from shading problem in a unique way. This technique is known as photomet- ric stereo.

 

Photogrammetric Stereo

The curved contour lines in Fig. 8.9 indicate that the relation between surface slope and radiance is nonlinear. This means that even if we take two diff erent illuminations of the same surface (Fig. 8.10), the surface slope may not be determined in a unique way. This is the case when the curved contour lines intersect each other at more than one point. Only a third exposure with yet another illumination direction would make the solution unique.

Using three exposures also has the signifi cant advantage that the re- fl ectivity of the surface can be eliminated by the use of ratio imaging. As an example, we illuminate a Lambertian surface with the same light source from three diff erent directions

l ₁ = (0, 0, 1)

l ₂ = (tan θ _i, 0, 1)

l ₃ = (0, tan θ _i, 1).

(8.14)

8.5 Shape from Shading†                                                                                    221

1

s2

0.5

0

-0.5

 

 

-1

-1              -0.5              0                0.5  s1      1

Figure 8.10: Superimposed contour plots of the radiance of a Lambertian surface with homogeneous refl ectivity illuminated by a light source with an angle of incidence of 45° and an azimuthal angle of 0° and 90°, respectively.

Then

L₂/L₁ =

− s₁ tan θ _i + 1

 1 + tan2 θ _i

, L₃/L₁ =

− s₂ tan θ _i + 1

 1 + tan2 θ _i

.       (8.15)

Now the equations are linear in s₁ and s₂ and — even better — they are decoupled: s₁ and s₂ depend only on L₂/L₁ and L₃/L₁, respectively (Fig. 8.11). In addition, the normalized radiance in Eq. (8.15) does not depend on the refl ectivity of the surface. The refl ectivity of the surface is contained in Eq. (8.10) as a factor and thus cancels out when the ratio of two radiance distributions of the same surface is computed.

 

8.5.3 Shape from Refraction for Specular Surfaces‡

For specular surfaces, the shape from shading techniques discussed in Sec- tion 8.5.1 do not work at all as light is only refl ected towards the camera when the angle of incidence from the light source is equal to the angle of refl ectance. Thus, extended light sources are required. Then, it turns out that for transpar- ent specular surfaces, shape from refraction techniques are more advantageous than shape from refl ection techniques because the radiance is higher, steeper surface slopes can be measured, and the nonlinearities of the slope/radiance relationship are lower.

A shape from refraction technique requires a special illumination technique, as no signifi cant radiance variations occur, except for the small fraction of light refl ected at the surface. The base of the shape from refraction technique is the telecentric illumination system which converts a spatial radiance distribution into an angular radiance distribution. Then, all we have to do is to compute the relation between the surface slope and the angle of the refracted beam and to use a light source with an appropriate spatial radiance distribution.

222                                                                                                         8 3-D Imaging

 

a

1

 

s2

 

 

0.5

 

 

 

 

0

 

 

 

 

-0.5

b

1

 

s2

 

 

0.5

 

 

 

 

0

 

 

 

 

-0.5

 

                                                                                                                                                                                         

 

 

                                                                                                                                                                                         

-1

-1               -0.5              0                0.5

s1   1

-1

-1               -0.5              0                0.5

s1 1

 

Figure 8.11: Contour plots of the radiance of a Lambertian surface illuminated by parallel light with an incidence angle of 45° and an azimuthal angle of 0° ( a ) and 90° ( b ), respectively, and normalized by the radiance of the illumination at 0° incidence according to Eq. (8.15). The step size of the contour lines is 0.1. Note the perfect linear relation between the normalized radiance and the x and y surface slope components.

 

Figure 8.12 illustrates the optical geometry for the simple case when the camera is placed far above and a light source below a transparent surface of a medium with a higher index of refraction. The relation between the surface slope s and the angle γ is given by Jä hne et al. [84] as

s = tan α =          n tan γ        ≈ 4 tan γ  1 + 3 tan² γ                       (8.16)

n −  1 + tan2 γ

with n = n₂/n₁. The inverse relation is

tan γ  = s , n2 + (n2 − 1)s2 − 1

2

 

 

≈ 1 s.1 +  3 s2Σ  .                    (8.17)

, n2 + (n2 − 1)s2 + s2                      4          32

,

In principle, the shape from refraction technique works for slopes up to infi nity (vertical surfaces). In this limiting case, the ray to the camera grazes the surface (Fig. 8.12b) and

tan γ = n2 − 1.                                                  (8.18)

The refraction law thus causes light rays to be inclined in a certain direction relative to the slope of the water surface. If we make the radiance of the light source dependent on the direction of the light beams, the water surface slope becomes visible. The details of the construction of such a system are described by Jä hne et al. [84]. Here we just assume that the radiance of the light rays is proportional to tan γ in the x₁ direction. Then we obtain the relation

L ∝ s₁

 

n2 + (n2 − 1)s2 − 1

,                                 .                               (8.19)

, n2 + (n2 − 1)s2 + s2

8.5 Shape from Shading†                                                                                    223

A                                               b

  

Figure 8.12: Refraction at an inclined surface as the basis for the shape from refraction technique. The camera is far above the surface. a Rays emitted by the light source at an angle γ are refracted in the direction of the camera. b Even for a slope of infi nity (vertical surface, α = 90 °), rays from the light source meet the camera.

 

1

 

s2

 

0.5

 

 

0

 

 

-0.5

 

 

-1

-1              -0.5              0                0.5

s1   1

 

Figure 8.13: Radiance map for the shape from refraction technique where the radiance in a telecentric illumination source varies linearly in the x₁ direction.

 

Of course, again we have the problem that from a scalar quantity such as the radiance no vector component such as the slope can be inferred. The shape from refraction technique, however, comes very close to an ideal setup. If the radiance varies only linearly in the x₁ direction, as assumed, the radiance map in the gradient space is also almost linear (Fig. 8.13). A slight infl uence of the cross slope (resulting from the nonlinear terms in Eq. (8.19) in s2) becomes apparent only at quite high slopes.

Ratio imaging can also be used with the shape from refraction technique. Color images have three independent primary colors: red, green, and blue. With a

224                                                                                                         8 3-D Imaging

 

total of three channels, we can identify the position in a telecentric illumination system — and thus the inclination of the water surface — uniquely and still have one degree of freedom left for corrections. With color imaging we also have the advantage that all three illuminations are taken simultaneously. Thus moving objects can also be observed.

A unique position coding with color can be achieved, for example, with the following color wedges:

 

G( s ) = (1/2 + cs₁)E₀( s )

R( s )  = [1/2 − c/2(s₁ + s₂)]E₀( s )

B( s )  = [1/2 − c/2(s₁ − s₂)]E₀( s ).

(8.20)

We have again assumed a linear relation between one component of the slope and the radiance, with nonlinear isotropic corrections of the form s₁E₀( s ); c is a calibration factor relating the measured radiance to the surface slope.

We now have three illuminations to determine two slope components. Thus, we can take one to compensate for unwanted spatial variation of E₀. This can be done by normalizing the three color channels by the sum of all channels G + R + B:

  G   = 2 . 1 + cs₁Σ  ,

G + R + B            3 2

 

(8.21)

B − R  G + R + B

2

= 3 cs₂.

Then the position on the wedge from which the light originates is given as

s₁ = 1 2G − R − B ,         s₂ =  3    B − R  .                       (8.22)

2c G + R + B                 2c G + R + B

From these position values, the x and y components of the slope can be com- puted according to Eq. (8.19).

 

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