3-D Shape from Shading

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3-D Shape from Shading

• Image formation and the shape from shading problem

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3D shape from shading
Prados Faugeras, 2005
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Image intensities depend on

• 3-D surface shape
• Surface reflectance properties
• Illumination in the scene
• Viewing geometry

Compute explicitly
General assumptions
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From Woodham, 1984 (images courtesy of Merle
Norman Cosmetics)
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Viewing geometry
What fraction of the incident light is reflected
toward the viewer?
radiance
irradiance
Reflectance Function F(i,e,g)
surface radiance surface irradiance
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Reflectance functions
Fmirror(i,e,g) 1 if i e i e g
0 otherwise
Fmatte(i,e,g) ? cos i if i lt 90 0
otherwise ? albedo
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More reflectance functions
Fmoon(i,e,g) ? cos i if i lt 90 cos
e 0 otherwise
FSEM(i,e,g) 1 /cos e
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Representing surface orientation using
stereographic projection
sphere has radius 1
g
f
(f,g) (0, 2)
(f,g) (2, 0)
viewer
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Given surface brightness, can we determine
surface orientation?
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Reflectance Map R(f,g)
Given (1) light source (2) viewer position (3)
surface reflectance properties
R(f,g) relates brightness to surface orientation
Image Irradiance Equation I(x,y) I0 ?(x,y)
R(f,g)
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Ikeuchi Horn shape-from-shading algorithm

• Three sources of constraint
• (1) image intensity I(x,y)
• (2) surface smoothness
• e.g. minimize total variation in surface
• (3) points of known surface orientation
• e.g. occluding boundaries, shadow boundaries

(known viewer direction, light source direction,
surface reflectance properties)
I(x,y) I0 ?(x,y) R(f,g)
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Boundaries influence shape perception
Ramachandran, 1988