Radiometry & Shape from Shading · 2009-12-15 · • Lambert’s Cosine law ... Input images...
Transcript of Radiometry & Shape from Shading · 2009-12-15 · • Lambert’s Cosine law ... Input images...
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Shape acquisition recap
• stereo
• observe objects from multiple positions
• establish correspondences
• triangulation gives depth (“Z”) maps, meshes
• structure from motion
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Shape acquisition recap
• passive stereo has problems finding correspondences
• active stereo has resolution limits
• multi-megapixel cameras: cheap + available
• scan line techniques take lots of time for hi-res models
• multi-megapixel projectors: expensive
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Shape from appearance
• material appearance gives cues for shape
• can be precise locally (per-pixel)
• challenging to get full geometry
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Normal maps
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Normal maps
http://en.wikipedia.org/wiki/File:Normal_map_example.pngAuthor: Paolo Cignoni, License: Creative Commons Attribution Share-Alike 1.0
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Normal maps
http://en.wikipedia.org/wiki/File:Normal_map_example.pngAuthor: Paolo Cignoni, License: Creative Commons Attribution Share-Alike 1.0
• normal maps can be measured in every surface point
• with active methods
• by looking at local appearance
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Today’s Menu
• Radiometry
• Lambert’s Cosine law
• Photometric Stereo
• Shape from Shading
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Solid Angle
length, unit
area, unit
1 meter = 1 m
1 m2
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Solid Angle
angle = arc length of projection on unit circle,unit
solid angle = area of projection on unit sphere, unit
1 radian = 1 rad
1 steradian = 1 sr
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Radiometry
Name
Symbol
Unit
Energy
Q
[Q] = 1 Joule = 1 J = 1 kgm2 s−2
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Radiometry
Name
Symbol
Unit
radiant power, flux
[P ] = 1 J s−1 = 1Watt = 1WP = dQ
dt , Φ
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Radiometry
Name
Symbol
Unit
area A
radiant power, flux
[P ] = 1 J s−1 = 1Watt = 1WP = dQ
dt , Φ
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Radiometry
Name
Symbol
Unit
radiant intensity (in direction, of a point light source)
dω
I = dPdω
[I] = 1W sr−1
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Radiometry
Name
Symbol
Unit
irradiance (in a point on the surface)
dA
E = dPdA
[E] = 1W m−2
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Radiometry
Name
Symbol
Unit
radiant exitance, radiosity (from a point)
dA
[B] = 1W m−2B = dP
dA
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Radiometry
Name
Symbol
Unit
radiance
dA
dω
L = d2Pdω dA
[L] = 1 W sr−1m−2
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Radiometry
In vacuum,
radiance is constant along rays!
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Radiometry
Pinhole cameras record radiance.
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Lambertian Surfacesappear equally bright from all directions
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Lambertian Surfacesappear equally bright from all directions
L =ρ
πE
Albedo ρ ranges from 0 (perfect black) to 1 (perfect white).
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How is E computed?
dA
r
E ∼ I
r2
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How is E computed?
dA
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How is E computed?
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Lambert’s Cosine LawE ∼ cos θ · I
�nθ
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Lambert’s Cosine Law
�n�l
����l��� = ��n� = 1
E ∼��l,�n
�· I
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Lambertian Surfaces
L =ρ
πE
L =ρ
�l,�n
�· I
r2
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Lambertian Surfaces
L ∼ ρ ·��l,�n
�
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Photometric Stereo
ρ ·�
lx ly lz�
·
nx
ny
nz
= αL
αρ �n subsumes camera properties, light source brightness, and are unknown
r2
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Photometric Stereo
�lx ly lz
�·
ρ · nx
ρ · ny
ρ · nz
= αL
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Photometric Stereo
1
23
l1,x l1,y l1,z
l2,x l2,y l2,z
l3,x l3,y l3,z
·
ρ · nx
ρ · ny
ρ · nz
= α
L1
L2
L3
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Photometric Stereo
• In practice, use more light sources
• discard darkest observations: shadowed
• discard brightest observations: not Lambertian
• use least-squares solution for over-determined system
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Photometric Stereo
[Rushmeier et al., 1997]33Monday, December 14, 2009
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Photometric Stereo
[Rushmeier et al., 1997]
Estimated albedo
Estimated normals (lit)Input images
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Photometric Stereo
Practical considerations:
• illumination: flashes vs. steady light sources
• subtract dark frame
• calibrate: light source positions
• either build or measure (mirror balls)
• calibrate: light source intensities
• move same light source: very precise
• observe on known reflectance target: more flexible
• some pre-computation possible
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Photometric Stereo• good:
• fast (real-time)
• high resolution (one normal vector per pixel!)
• bad:
• normals only
• active light only
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Shape from Shading
• like photometric stereo, but with one image and one light source only!
• useful for astronomy etc.
• underconstrained problem. Assume:
• uniform, known albedo
• known illumination
• smooth variation in surface normal
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Shape from Shading
• Further simplifications:
• Camera far from object: in image = in world
• denotes the depth in pixel
(x, y) (x, y)
z(x, y)
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Shape from ShadingDefinitions:
Let
Then, and are tangents, and the normal is
p :=∂
∂xz, q :=
∂
∂yz
10p
01q
�n =1�
1 + p2 + q2
10p
×
01q
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Shape from ShadingDefinitions:
Let
Then, and are tangents, and the normal is
p :=∂
∂xz, q :=
∂
∂yz
10p
01q
�n =1�
1 + p2 + q2
−p−q1
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Shape from ShadingDefinitions:
Let
be the radiance for a hypothesis of and , and
be the radiance in the captured picture.
p q
L
R(p, q) := ρ ·�
�l,1�
1 + p2 + q2
−p−q1
�
· 1α
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Shape from Shading
Minimize the functional:
for a control parameter
�
A(L−R(p, q))2 + λ(|∇p|2 + |∇q|2) dx dy
Data term Smoothness Term
λ
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Shape from Shading
Solve by iteration:• initialize with an estimate for and • after step , set
for local averages and • iterate
�
A(L−R(p, q))2 + λ(|∇p|2 + |∇q|2) dx dy
p q
k
pk+1 = pk +14λ
(L−R(pk, qk))∂R(pk, qk)
∂p
qk+1 = qk +14λ
(L−R(pk, qk))∂R(pk, qk)
∂q
p q
see [Trucco and Verri, 1998] for Details43Monday, December 14, 2009
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Shape from Shading• Problems:
• end of computation not necessarily a global optimum
• boundary conditions need to be chosen (enforce after each iteration)
• for a consistent surface,
but this is not necessarily the case
∂2z
∂x ∂y=
∂2z
∂y ∂x⇒ ∂p
∂y=
∂q
∂x
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Shape from Shading• Enforce constraint in functional: add
to the energy functional (makes things difficult)
• Maintain constraint throughout optimization
• after each iteration, project p and q to closest integrable function pair (in Fourier space)
�∂p
∂y− ∂q
∂x
�2
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Shape form shadingWhen consistent, p and q can be integrated to yield z:
[Trucco and Verri, 1998]46Monday, December 14, 2009
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Shape form shadingWhen consistent, p and q can be integrated to yield z:
[Trucco and Verri, 1998]47Monday, December 14, 2009
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Limitations
• material properties
• occlusions ( shadows )
• how to know albedo and light intensity?
• further assumptions needed
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Conclusions
• appearance can be used to estimate shape
• assumptions required, Lambertian common
• photometric stereo is precise, but active light required
• shape from shading can be useful where stereo is impossible
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Knowing more ...
[Blanz and Vetter, 1999]
• with more assumptions, you can know more about the shape
• a statistical model + a single image can give you
• shape
• texture + colors
• additional semantics ...
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Knowing more ...
[Blanz and Vetter, 1999]
Video source: http://mi.informatik.uni-siegen.de/movies/siggraph99.mpg
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Questions ?
52Monday, December 14, 2009