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n̂q
dA
dI
azf
Surface Radiance and Image Irradiance
a
22 )cos/(
cos
)cos/(
cos
aad
aqd
f
I
z
A=
2
cos
cos˜̃¯
ˆÁÁË
Ê=
f
z
I
A
qa
dd
Same solid angle
PinholeCameraModel
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n̂dA
dI
azf
d
Surface Radiance and Image Irradiance
a
ap
aap 3
2
2
2
cos4)cos/(
cos
4˜¯
ˆÁË
Ê==Wz
d
z
d
Solid angle subtended by the lens, as seen by the patch dA
qap
dqdd coscos4
cos 32
˜¯
ˆÁË
Ê=W=z
dALALP
Power from patch dA through the lens
W
ap
qap
dd
dd 4
2
32
cos4
coscos4 ˜̃
¯
ˆÁÁË
Ê=˜
¯
ˆÁË
Ê==f
dL
z
d
I
AL
I
PE
Thus, we conclude
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n̂dA
dI
azf
d
Surface Radiance and Image Irradiance
aW
ap
qap
dd
dd 4
2
32
cos4
coscos4 ˜̃
¯
ˆÁÁË
Ê=˜
¯
ˆÁË
Ê==f
dL
z
d
I
AL
I
PE
†
E µ LeImage intensity isproportional to ExidentRadiance
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Radiance emitted by point sources• small, distant sphere radius e
and uniform radiance E, whichis far away and subtends asolid angle of about
†
W = ped
Ê
Ë Á
ˆ
¯ ˜
2
†
Le = rd x( ) Li x,w( )cosqidwW
Ú= rd x( ) Lidw cosqi
W
Ú
ª rd x( ) er(x)
Ê
Ë Á
ˆ
¯ ˜
2
E cosqi
= krd x( )cosqi
r x( )2
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Standard nearby point source model
• N is the surface normal• rho is diffuse albedo• S is source vector - a vector
from x to the source, whoselength is the intensity term– works because a dot-product is
basically a cosine
†
rd x( ) N x( ) •S x( )r x( )2
Ê
Ë Á
ˆ
¯ ˜
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Standard distant point source model
• Issue: nearby point source getsbigger if one gets closer– the sun doesn’t for any
reasonable binding of closer• Assume that all points in the
model are close to each otherwith respect to the distance tothe source. Then the sourcevector doesn’t vary much, andthe distance doesn’t vary mucheither, and we can roll theconstants together to get:
†
rd x( ) N x( )• Sd x( )( )
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Line sources
radiosity due to line source varies with inverse distance, if the source is long enough
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Area sources
• Examples: diffuser boxes, whitewalls.
• The radiosity at a point due toan area source is obtained byadding up the contribution overthe section of view hemispheresubtended by the source– change variables and add up
over the source
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Area Source Shadows
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Shading models
• Local shading model– Surface has radiosity due only
to sources visible at each point– Advantages:
• often easy to manipulate,expressions easy
• supports quite simpletheories of how shapeinformation can beextracted from shading
• Global shading model– surface radiosity is due to
radiance reflected from othersurfaces as well as fromsurfaces
– Advantages:• usually very accurate
– Disadvantage:• extremely difficult to infer
anything from shadingvalues
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Photometric stereo
• Assume:– a local shading model– a set of point sources that are infinitely distant– a set of pictures of an object, obtained in exactly the same
camera/object configuration but using different sources– A Lambertian object (or the specular component has been
identified and removed)
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Projection model for surface recovery - usually calleda Monge patch
†
z = f (x, y)
r n =∂z /∂x∂z /∂y
1
È
Î
Í Í Í
˘
˚
˙ ˙ ˙
ˆ n = r n / r n †
ˆ n
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Image model
• For each point source, we knowthe source vector (byassumption). We assume weknow the scaling constant of thelinear camera. Fold the normaland the reflectance into onevector g, and the scalingconstant and source vector intoanother Vj
• Out of shadow:
• In shadow:
†
I j(x,y) = 0
†
I j (x, y) = kB(x, y)
= kr(x, y) N(x, y) •S j( )= g(x, y)• Vj
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Dealing with shadows
†
I12(x, y)
I22(x, y)
..In
2(x, y)
Ê
Ë
Á Á Á Á
ˆ
¯
˜ ˜ ˜ ˜
=
I1(x, y) 0 .. 00 I2(x, y) .. .... .. .. 00 .. 0 In(x, y)
Ê
Ë
Á Á Á Á
ˆ
¯
˜ ˜ ˜ ˜
V1T
V2T
..Vn
T
Ê
Ë
Á Á Á Á
ˆ
¯
˜ ˜ ˜ ˜
g(x, y)
Known Known Known Unknown
†
r b = Ar x General form:
n x 3n x 1
For each x,y point
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Recovering normal and reflectance
• Given sufficient sources, we can solve the previousequation (most likely need a least squares solution) for
g(x, y) = r(x,y) N(x, y)• Recall that N(x, y) is the unit normal• This means that r(x,y) is the magnitude of g(x, y)• This yields a check
– If the magnitude of g(x, y) is greater than 1, there’s a problem• And N(x, y) = g(x, y) / r(x,y)
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Example figures
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Recovered reflectance
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Recovered normal field
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Recovering a surface from normals - 1
• Recall the surface is written as
• This means the normal has theform:
• If we write the known vector gas
• Then we obtain values for thepartial derivatives of thesurface:
†
(x, y, f (x, y))
†
g(x, y) =
g1(x, y)g2 (x, y)g3(x, y)
Ê
Ë
Á Á
ˆ
¯
˜ ˜
†
fx (x, y) = g1(x, y) g3(x, y)( )
†
fy(x, y) = g2(x, y) g3(x, y)( )
†
N(x, y) =1
fx2 + fy
2 +1Ê
Ë Á
ˆ
¯ ˜
- fx
- fy
1
Ê
Ë
Á Á
ˆ
¯
˜ ˜
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Recovering a surface from normals - 2
• Recall that mixed secondpartials are equal --- this givesus a check. We must have:
(or they should be similar, atleast)
• We can now recover the surfaceheight at any point byintegration along some path,e.g.
†
∂ g1(x, y) g3(x, y)( )∂y
=
∂ g2(x, y) g3(x, y)( )∂x
†
f (x, y) = fx (s, y)ds0
x
Ú +
fy (x, t)dt0
y
Ú + c
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Surface recovered by integration
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Shape from Shading with ambiguity
What to do if light source unknown?
Only one image?
Ans: Use prior knowledge in the form of constraints(e.g. regularization methods, Bayesian methods), takeadvantage of the habits (regularities) of images.
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Horn’s approach
• Want to estimate scene parameters (surface slopes fx(x,y) and fy(x,y) atevery image position, (x,y).
• Have a rendering function that takes you from some given set of sceneparameters to observation data (e.g. r(x,y) n(x,y) gives image intensity for any(x,y)).
• Could try to find the parameters fx(x,y) & fy(x,y) that minimize thedifference from the observations I(x,y).
• But the problem is “ill-posed”, or underspecified from that constraint alone.So add-in additional requirements that the scene parameters must satisfy (thesurface slopes fx(x,y) & fy(x,y) must be smooth at every point).
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RegularizationFor each normal, compute the distance fromthe normal to its neighbors:
†
Prior/regularizer
s(i, j) =r n i + k, j + l( ) -
r n (i, j)( )2
k={-1,1}Âl={-1,1}Â
Intensity Error
r(i, j) = I i, j( ) - Ipred (i, j)( )2
Err = r(i, j) + ls(i, j)i, jÂ
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Computer Vision - A Modern ApproachSet: Sources, shadows and shading
Slides by D.A. Forsyth
Shape from Shading Ambiguity
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Computer Vision - A Modern ApproachSet: Sources, shadows and shading
Slides by D.A. Forsyth
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Curious Experimental Fact
• Prepare two rooms, one with white walls and whiteobjects, one with black walls and black objects
• Illuminate the black room with bright light, the white roomwith dim light
• People can tell which is which (due to Gilchrist)
• Why? (a local shading model predicts they can’t).
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Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
A view of awhite room,under dim light.Below, we see across-section ofthe imageintensitycorresponding tothe line drawnon the image.
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Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
A view of a blackroom, underbright light.Below, we see across-section ofthe imageintensitycorresponding tothe line drawnon the image.
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What’s going on here?
• local shading model is a poor description of physicalprocesses that give rise to images– because surfaces reflect light onto one another
• This is a major nuisance; the distribution of light (inprinciple) depends on the configuration of every radiator;big distant ones are as important as small nearby ones(solid angle)
• The effects are easy to model• It appears to be hard to extract information from these
models
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Interreflections - a global shading model
• Other surfaces are now area sources - this yields:
• Vis(x, u) is 1 if they can see each other, 0 if they can’t
†
Radiosity at surface = Exitance + Radiosity due to other surfaces
B x( ) = E x( )+ rd x( ) B u( )cosqi cosqs
pr(x,u)2 Vis x,u( )dAuall othersurfaces
Ú
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What do we do about this?
• Attempt to build approximations– Ambient illumination
• Study qualitative effects– reflexes– decreased dynamic range– smoothing
• Try to use other information to control errors
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Shadows cast by a point source
• A point that can’t see the sourceis in shadow
• For point sources, the geometryis simple
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Ambient Illumination
• Two forms– Add a constant to the radiosity at every point in the scene to
account for brighter shadows than predicted by point source model• Advantages: simple, easily managed (e.g. how would you
change photometric stereo?)• Disadvantages: poor approximation (compare black and white
rooms– Add a term at each point that depends on the size of the clear
viewing hemisphere at each point (see next slide)• Advantages: appears to be quite a good approximation, but
jury is out• Disadvantages: difficult to work with
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At a point inside a cube or room, the surface sees light in alldirections, so add a large term. At a point on the base of a groove,the surface sees relatively little light, so add a smaller term.
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Reflexes
• A characteristic feature of interreflections is little brightpatches in concave regions– Examples in following slides– Perhaps one should detect and reason about reflexes?– Known that artists reproduce reflexes, but often too big and in the
wrong place
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Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
At the top, geometry of asemi-circular bump on aplane; below, predictedradiosity solutions, scaledto lie on top of each other,for different albedos ofthe geometry. Whenalbedo is close to zero,shading follows a localmodel; when it is close toone, there are substantialreflexes.
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Radiosity observed in animage of this geometry;note the reflexes, whichare circled.
Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
![Page 40: Surface Radiance and Image Irradiance - Vision Labsvision.psych.umn.edu/.../courses/CompVis03/Shape-from-shading.pdf · Surface Radiance and Image Irradiance a a p a p a 3 2 2 2 cos](https://reader030.vdocuments.site/reader030/viewer/2022040605/5eaaab9d9c5bfa135b19150b/html5/thumbnails/40.jpg)
Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
At the top, geometry of agutter with triangularcross-section; below,predicted radiositysolutions, scaled to lie ontop of each other, fordifferent albedos of thegeometry. When albedois close to zero, shadingfollows a local model;when it is close to one,there are substantialreflexes.
![Page 41: Surface Radiance and Image Irradiance - Vision Labsvision.psych.umn.edu/.../courses/CompVis03/Shape-from-shading.pdf · Surface Radiance and Image Irradiance a a p a p a 3 2 2 2 cos](https://reader030.vdocuments.site/reader030/viewer/2022040605/5eaaab9d9c5bfa135b19150b/html5/thumbnails/41.jpg)
Radiosity observed in animage of this geometry;above, for a black gutterand below for a white one
Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
![Page 42: Surface Radiance and Image Irradiance - Vision Labsvision.psych.umn.edu/.../courses/CompVis03/Shape-from-shading.pdf · Surface Radiance and Image Irradiance a a p a p a 3 2 2 2 cos](https://reader030.vdocuments.site/reader030/viewer/2022040605/5eaaab9d9c5bfa135b19150b/html5/thumbnails/42.jpg)
Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
At the top, geometry of agutter with triangularcross-section; below,predicted radiositysolutions, scaled to lie ontop of each other, fordifferent albedos of thegeometry. When albedois close to zero, shadingfollows a local model;when it is close to one,there are substantialreflexes.
![Page 43: Surface Radiance and Image Irradiance - Vision Labsvision.psych.umn.edu/.../courses/CompVis03/Shape-from-shading.pdf · Surface Radiance and Image Irradiance a a p a p a 3 2 2 2 cos](https://reader030.vdocuments.site/reader030/viewer/2022040605/5eaaab9d9c5bfa135b19150b/html5/thumbnails/43.jpg)
Radiosity observed in animage of this geometryfor a white gutter.
Figure from “Mutual Illumination,” by D.A. Forsyth and A.P. Zisserman, Proc. CVPR, 1989, copyright 1989 IEEE
![Page 44: Surface Radiance and Image Irradiance - Vision Labsvision.psych.umn.edu/.../courses/CompVis03/Shape-from-shading.pdf · Surface Radiance and Image Irradiance a a p a p a 3 2 2 2 cos](https://reader030.vdocuments.site/reader030/viewer/2022040605/5eaaab9d9c5bfa135b19150b/html5/thumbnails/44.jpg)
Smoothing
• Interreflections smooth detail– E.g. you can’t see the pattern of a stained glass window by looking
at the floor at the base of the window; at best, you’ll see colouredblobs.
– This is because, as I move from point to point on a surface, thepattern that I see in my incoming hemisphere doesn’t change allthat much
– Implies that fast changes in the radiosity are local phenomena.
![Page 45: Surface Radiance and Image Irradiance - Vision Labsvision.psych.umn.edu/.../courses/CompVis03/Shape-from-shading.pdf · Surface Radiance and Image Irradiance a a p a p a 3 2 2 2 cos](https://reader030.vdocuments.site/reader030/viewer/2022040605/5eaaab9d9c5bfa135b19150b/html5/thumbnails/45.jpg)
Fix a small patch near alarge radiator carrying aperiodic radiosity signal;the radiosity on the surfaceis periodic, and its amplitudefalls very fast with thefrequency of the signal. Thegeometry is illustratedabove. Below, we show agraph of amplitude as afunction of spatialfrequency, for differentinclinations of the smallpatch. This means that ifyou observe a highfrequency signal, it didn’tcome from a distant source.