Transcript Document
Computer Vision
Radiometry, lights and surfaces
Marc Pollefeys COMP 256 Slides from David Forsyth,,…
Computer Vision
Last class
• Camera Models
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' – Pinhole Perspective Projection – Affine Projection • Camera with Lenses • Sensing • The Human Eye
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x z y
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Computer Vision
Radiometry
• Questions: – how “bright” will surfaces be? – what is “brightness”?
• measuring light • interactions between light and surfaces • Core idea - think about light arriving at a surface • around any point is a hemisphere of directions • Simplest problems can be dealt with by reasoning about this hemisphere 3
Computer Vision
Lambert’s wall
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Computer Vision
More complex wall
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Foreshortening
• Principle: two sources that look the same to a receiver must have the same effect on the receiver.
• Principle: two receivers that look the same to a source must receive the same amount of energy.
• “look the same” means produce the same input hemisphere (or output hemisphere) • Reason: what else can a receiver know about a source but what appears on its input hemisphere? (ditto, swapping receiver and source) • Crucial consequence: a big source (resp. receiver), viewed at a glancing angle, must produce (resp. experience) the same effect as a small source (resp. receiver) viewed frontally.
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Computer Vision
Measuring Angle
• To define radiance, we require the concept of solid angle • The solid angle sub-
tended by an object from a point P is the area of the projection of the object onto the unit sphere centered at P
• Measured in steradians, sr • Definition is analogous to projected angle in 2D • If I’m at P, and I look out, solid angle tells me how much of my view is filled with an object 7
Computer Vision
Solid Angle of a Small Patch
• Later, it will be important to talk about the solid angle of a small piece of surface
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dA
cos
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2 A
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sin
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Computer Vision
Measuring Light in Free Space
• Desirable property: in a vacuum, the relevant unit does not go down along a straight line.
• We have
total power leaving s to r = total power arriving at r from s
• Also: • How do we get a unit with this property? Think about the power transferred from an infinitesimal source to an infinitesimal receiver.
Power arriving at r is proportional to:
– solid angle subtended by s at r (because if s looked bigger from r, there’d be more) – foreshortened area of r (because a bigger r will collect more power) 9
Computer Vision light
Radiance
• All this suggests that the light transferred from source to receiver should be measured as:
Radiant power per unit foreshortened area per unit solid angle
• This is radiance • Units: watts per square meter per steradian (wm -2 sr -1 ) • Usually written as: , , surface • Crucial property: In a vacuum, radiance leaving p in the direction of q is the same as radiance arriving at q from p – which was how we got to the unit 10
Computer Vision
Radiance is constant along straight lines
• Power 1->2, leaving 1: 1 , ,
dA
1 cos 1
dA
2 cos
r
2 2 • Power 1->2, arriving at 2: 2 , ,
dA
2 cos 2
dA
1 cos
r
2 1 • But these must be the same, so that the two radiances are equal 11
Computer Vision
Spectral Quantities
• To handle color properly, it is important to talk about spectral radiance • Defined at a particular wavelength, per unit wavelength: L
(x,
,
)
• To get total radiance, integrate over spectrum:
L
(
x
, , ) 0
L
(
x
, , )
d
More about color later… 12
Computer Vision
Irradiance, E
light • How much light is arriving at a surface?
• Sensible unit is
Irradiance
• Incident power per unit area not foreshortened • This is a function of incoming angle. • A surface experiencing radiance L(x ,, ) coming in from d experiences irradiance
E
• Crucial property: Total power arriving at the surface is given by adding irradiance over all incoming angles -- this is why it’s a natural unit • Total power is
L E
x
,
i
,
i
x
,
i
,
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cos
i d
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, , cos light sin
d
d
surface surface 13
Computer Vision
Example: Radiometry of thin lenses
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z
A
' ' / cos cos
z
A
/ cos cos 2
A A
' cos cos 4
z d
/ 2 cos cos 2 4
d z
2 cos 3
P E
L
A
cos
P A
' 4
d z
d
2
L
A
cos 3 cos 4
z
2
L
A
A
' cos 3 cos
z z
' 2
E
4
d z
' 2 cos 4
L I
f
E
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I
kE
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Computer Vision
Reflectance
• We have all the things we need dealing with the transport of light • Reflectance is all about the way light interacts with surfaces • It is an entire field of study on its own • The most important quantity is the BRDF… 15
Computer Vision
Light at surfaces
• Many effects when light strikes a surface -- could be: – absorbed – transmitted • skin – reflected • mirror – scattered • milk – travel along the surface and leave at some other point • sweaty skin • Assume that – surfaces don’t fluoresce • e.g. scorpions, washing powder – surfaces don’t emit light (i.e. are cool) – all the light leaving a point is due to that arriving at that point 16
Computer Vision
The BRDF
• Assuming that – surfaces don’t fluoresce – surfaces don’t emit light (i.e. are cool) – all the light leaving a point is due to that arriving at that point • Can model this situation with the Bidirectional Reflectance Distribution Function (BRDF) • the ratio of the radiance in the outgoing direction to the incident irradiance for an incoming direction 17
Computer Vision
BRDF
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BRDF
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L o E i
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, ,
i o
L i
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L o
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x
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, cos
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i d
Computer Vision
BRDF
• Units: inverse steradians (sr -1 ) wm -2 sr 2 -1 wm • Symmetric in incoming and outgoing directions - this is the Helmholtz reciprocity principle
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• Radiance leaving a surface in a particular direction: – add contributions from every incoming direction
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Computer Vision
Intermezzo - Helmholtz stereo
• Classic stereo assumption: same appearance from all viewpoints (=Lambertian) • Doesn’t hold for general BRDF • Idea (Zickler et al. ECCV’02), exploit reciprocity!
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Computer Vision
Suppressing Angles - Radiosity
• In many situations, we do not really need angle coordinates – e.g. cotton cloth, where the reflected light is not dependent on angle • Appropriate radiometric unit is radiosity – total power leaving a point on the surface, per unit area on the surface (Wm -2 ) – note that this is independent of the direction
B
• Radiosity from radiance?
– sum radiance leaving surface over all exit directions, multiplying by a cosine because this is per unit area not per unit foreshortened area
L o
x
, , cos
d
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Computer Vision
Radiosity
• Important relationship: – radiosity of a surface whose radiance is independent of angle (e.g. that cotton cloth)
B
L o
x
, , cos
L o
cos
L o
L o
0 2 2 0
d
cos
d
sin
d
d
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Computer Vision
Suppressing the angles in the BRDF
• BRDF is a very general notion – some surfaces need it (underside of a CD; tiger eye; etc) – very hard to measure • ,illuminate from one direction, view from another, repeat – very unstable • minor surface damage can change the BRDF • e.g. ridges of oil left by contact with the skin can act as lenses • for many surfaces, light leaving the surface is largely independent of exit angle – surface roughness is one source of this property 23
Computer Vision
Directional hemispheric reflectance
• Directional hemispheric reflectance: – the fraction of the incident irradiance in a given direction that is reflected by the surface (whatever the direction of reflection) – unitless, range is 0-1 • Note that DHR varies with incoming direction – e.g. a ridged surface, where left facing ridges are absorbent and right facing ridges reflect.
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L o
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cos
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Computer Vision
Lambertian surfaces and albedo
• For some surfaces, the DHR is independent of illumination direction too – cotton cloth, carpets, matte paper, matte paints, etc.
• For such surfaces, radiance leaving the surface is independent of angle • Called Lambertian surfaces (same Lambert) or ideal
diffuse surfaces
• Use radiosity as a unit to describe light leaving the surface • DHR is often called diffuse reflectance, or albedo • for a Lambertian surface, BRDF is independent of angle, too.
• Useful fact:
brdf
d
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Specular surfaces
• Another important class of surfaces is specular, or mirror-like.
– radiation arriving along a direction leaves along the specular direction – reflect about normal – some fraction is absorbed, some reflected – on real surfaces, energy usually goes into a lobe of directions – can write a BRDF, but requires the use of funny functions
Computer Vision
Phong’s model
• There are very few cases where the exact shape of the specular lobe matters.
• Typically: – very, very small -- mirror – small -- blurry mirror – bigger -- see only light sources as “specularities” – very big -- faint specularities • Phong’s model – reflected energy falls off with cos
n
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Computer Vision
Lambertian + specular
• Widespread model – all surfaces are Lambertian plus specular component • Advantages – easy to manipulate – very often quite close true • Disadvantages – some surfaces are not • e.g. underside of CD’s, feathers of many birds, blue spots on many marine crustaceans and fish, most rough surfaces, oil films (skin!), wet surfaces – Generally, very little advantage in modelling behaviour of light at a surface in more detail -- it is quite difficult to understand behaviour of L+S surfaces 28
Computer Vision
Diffuse + Specular example
29 cos n ( ), =2,10,100,1000
www.exaflop.org/docs/ lca/lca1.html
Computer Vision
Next class: Sources Shadows and Shading
30 F&P Chapter 5 upcoming assignment: photometric stereo