november 4, 19981 the reflectance map and shape-from-shading
TRANSCRIPT
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November 4, 1998 1
THE REFLECTANCE MAP AND SHAPE-FROM-SHADING
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November 4, 1998 2
REFLECTANCE MODELS
albedo Diffusealbedo
Specularalbedo
PHONG MODEL
E = L (aCOS bCOS )n
a=0.3, b=0.7, n=2 a=0.7, b=0.3, n=0.5
LAMBERTIAN MODEL
E = L COS
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November 4, 1998 3
REFLECTANCE MODELS
• Description of how light energy incident on an object is transferred from the object to the camera sensor
Surface
Surface Normal
HalfwayVector
IncidentLight L
ReflectedLight E
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November 4, 1998 4
REFLECTANCE MAP IS A VIEWER-CENTERED REPRESENTATION OF REFLECTANCE
Depth
SurfaceOrientation
Y
X
Z
IMAGE PLANE
z=f(x,y)
x y
dxdy
y(f , f , -1)
(f , f , -1)
(0,1,f )x
(0,1,f )x
(1,0,f )
(1,0,f )y=
x y
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November 4, 1998 5
REFLECTANCE MAP IS A VIEWER-CENTERED REPRESENTATION OF REFLECTANCE
(f x , f y , -1) = (p, q, -1)
p, q comprise a gradient or gradient space representation forlocal surface orientation.
Reflectance map expresses the reflectance of a material directly in terms of viewer-centered representation of local surface orientation.
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November 4, 1998 6
LAMBERTIAN REFLECTANCE MAP
LAMBERTIAN MODEL
E = L COS Y
XZ
(ps,qs,-1)
(p,q,-1)
COSpp qq
p q p qs s
s s
1
1 12 2 2 2
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November 4, 1998 7
LAMBERTIAN REFLECTANCE MAP
Grouping L and as a constant , local surface orientations that produce equivalent intensities under the Lambertian reflectance
map are quadratic conic section contours in gradient space.
E Lpp qq
p q p qs s
s s
1
1 12 2 2 2
Ipp qq
p q p qs s
s s
1
1 12 2 2 2
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November 4, 1998 8
LAMBERTIAN REFLECTANCE MAP
ps=0 qs=0
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November 4, 1998 9
LAMBERTIAN REFLECTANCE MAP
ps=0.7 qs=0.3
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November 4, 1998 10
LAMBERTIAN REFLECTANCE MAP
ps= -2 qs= -1
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November 4, 1998 11
PHOTOMETRIC STEREO
Derivation of local surface normal at each pixel creates the derived normal map.
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November 4, 1998 12
NORMAL MAP vs. DEPTH MAP
IMAGE PLANE
Depth
SurfaceOrientation
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November 4, 1998 13IMAGE PLANE
Depth
SurfaceOrientation
NORMAL MAP vs. DEPTH MAP
• Can determine Depth Map from Normal Map by integrating over gradients p,q across the image.
• Not all Normal Maps have a unique Depth Map. This happens when Depth Map produces different results depending upon image plane direction used to sum over gradients.
• Particularly a problem when there are errors in the Normal Map.
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November 4, 1998 14IMAGE PLANE
Depth
SurfaceOrientation
NORMAL MAP vs. DEPTH MAP
• A Normal Map that produces a unique Depth Map independent of image plane direction used to sum over gradients is called integrable.
• Integrability is enforced when the following condition holds:
p
y
q
x
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November 4, 1998 15
NORMAL MAP vs. DEPTH MAP
• A Normal Map that produces a unique Depth Map independent of image plane direction used to sum over gradients is called integrable.
• Integrability is enforced when the following condition holds:
p
y
q
x
( / / ) ( ) p y q x dxdy pdx qdy GREEN’S THEOREM
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November 4, 1998 16
NORMAL MAP vs. DEPTH MAP
VIOLATION OF INTEGRABILITY
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November 4, 1998 17
SHAPE FROM SHADINGCONSTANT INTENSITY
SHADING FROMLAMBERTIAN REFLECTANCE
From a monocular view with a single distant light source ofknown incident orientation upon an object with known
reflectance map, solve for the normal map.
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November 4, 1998 18
SHAPE FROM SHADING
• Formulate as solving the Image Irradiance equation for surface orientation variables p,q:
• Since this is underconstrained we can’t solve this equation directly
• What do we do ??.
I(x,y) = R(p,q)
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November 4, 1998 19
SHAPE FROM SHADING(Calculus of Variations Approach)
• First Attempt: Minimize error in agreement with Image Irradiance Equation over the region of interest:
( ( , ) ( , ))I x y R p q dxdyobject
2
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November 4, 1998 20
SHAPE FROM SHADING(Calculus of Variations Approach)
• Better Attempt: Regularize the Minimization of error in agreement with Image Irradiance Equation over the region of interest:
p p q q I x y R p q dxdyx y x y
object
2 2 2 2 2 ( ( , ) ( , ))