Stefano Soatto (c) UCLA Vision Lab 1 Homogeneous representation Points Vectors Transformation...
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Stefano Soatto (c) UCLA Vision Lab 1
Homogeneous representation
Points
Vectors
Transformation
representation
Stefano Soatto (c) UCLA Vision Lab 3
Image Formation
Vision infers world properties form images. How do images depend on these properties? Two key elements
Geometry Radiometry We consider only simple models of these
Stefano Soatto (c) UCLA Vision Lab 6
Similar triangles <P’F’S’>,<ROF’> and <PSF><QOF>
(
fzz
11
'
1
2' ))(( ffzfz
Stefano Soatto (c) UCLA Vision Lab 10
Parallel lines meet
Common to draw image plane in front of the focal point. Moving the image plane merely scales the image.
(Forsyth & Ponce)
Stefano Soatto (c) UCLA Vision Lab 11
Vanishing points
• Each set of parallel lines meets at a different point– The vanishing point for this direction
• Sets of parallel lines on the same plane lead to collinear vanishing points. – The line is called the horizon for that plane
Stefano Soatto (c) UCLA Vision Lab 12
Properties of Projection
Points project to points Lines project to lines Planes project to the whole image or a half image Angles are not preserved Degenerate cases
Line through focal point projects to a point. Plane through focal point projects to line Plane perpendicular to image plane projects to part of
the image (with horizon).
Stefano Soatto (c) UCLA Vision Lab 17
Assumptions for thin lens equation Lens surfaces are spherical Incoming light rays make a small angle with
the optical axis The lens thickness is small compared to the
radii of curvature The refractive index is the same for the
media on both sides of the lens
Stefano Soatto (c) UCLA Vision Lab 19
Blur circle
A point at distance z is imaged at point 'z from the lens
fzz
11
'
1
and so
)()()(
'' zzfz
f
fz
fzz
|''|'
zzz
db
Points a t distance are brought into focus at distance z 'z
Thus points at distance will give rise to a blur circle of diameterz
with d the diameter of the lens
-zz’
-zz’
P
P’
Q
Q’db
Stefano Soatto (c) UCLA Vision Lab 20
Interaction of light with matter Absorption Scattering Refraction Reflection Other effects:
Diffraction: deviation of straight propagation in the presence of obstacles
Fluorescence:absorbtion of light of a given wavelength by a fluorescent molecule causes reemission at another wavelength
Stefano Soatto (c) UCLA Vision Lab 22
Solid Angle
2
0
2d
radian
2
]cos[2
sin2
sin
0
2
0
2
0
2
0
2
0
d
ddd
steradian (sr)
ddsind
hemisphere
Sphere: 4
Stefano Soatto (c) UCLA Vision Lab 24
Irradiance and Radiance
Irradiance Definition: power per unit area incident on a surface
W/m2 = luxdA
dE
ddA
dL
cos
2
iL
i
A
Radiance Definition: power per unit area and projected solid angle
W/m2sr
iii dLdA
ddE cos
2
2H
iii dLE cos
Stefano Soatto (c) UCLA Vision Lab 25
Radiant Intensity
Radiant flux W
Definition: flux per unit solid angle
W/sr = cd (candela)
d
dI
2S
dI
I
[ ]
Stefano Soatto (c) UCLA Vision Lab 27
Isotropic Point Source
Radiant flux WAll directions: solid angle 4
Radiant flux per
unit solid angle W/sr
d
dI
Radiant intensity
4
I
r
drdA 2
22 4
1
44 rdA
dA
rdA
d
dA
Id
dA
dE
• Note inverse square law fall off.
Stefano Soatto (c) UCLA Vision Lab 28
Isotropic Point Source
Radiant flux WAll directions: solid angle 4
Radiant flux per
unit solid angle W/sr
d
dI
Radiant intensity
4
I
r
drdAcos
12
2
3
2 444 hdA
dA
rdA
d
dA
Id
dA
dE
coscos
• Note cosine dependency.
h
cosrh
Stefano Soatto (c) UCLA Vision Lab 29
Isotropic Point Source
Point source at a finite distance
r
2
3
4 hE
cos
• Note cosine dependency.
h
• Note inverse square law fall off.
Stefano Soatto (c) UCLA Vision Lab 31
Hemispherical Source
L
dL
dL
ddLE
i
ii
iiii
2
0
2
020
2
2
0
2
0
2
sin2
sincosL
Stefano Soatto (c) UCLA Vision Lab 32
Reflectance
Reflectance: ratio of radiance to irradiance
dLr=fr dEi
Ei
i
Ei
Lr(x,)
Li(x,i)
irr dEfLi
rr dE
dLf
The surface becomes a light source
Stefano Soatto (c) UCLA Vision Lab 34
BRDF
i
rr
iii
irriirrriir
d
dd
dE
EdLf
/
,
;,;,,;,
2
iiriririr dEfEdL ;;;
i
iirirrr dEfL
;
ii
irirrir dE
EdLf
;;
;
iiiiiiiii dLdLdE cos
iiiiriri
dLf cos;
Stefano Soatto (c) UCLA Vision Lab 36
Perfectly Diffuse Reflectioni
Perfectly Diffuse Surface •Appears equally bright from all viewing directions (r, r)•Reflects all incident light, i.e.,
iiiiBi
dLL cos1
BrrrrBrrr LddLdLEr rr
sincos
LB(r, r) is constant for all directions (r, r)
1
s
BB E
Lf
Stefano Soatto (c) UCLA Vision Lab 37
Common Diffuse Reflectioni
Normal Diffuse Surface •Appears almost equally bright from most viewing directions (r, r), r << 90°•Reflects only a fraction of incident light, i.e.,
iiiiBBi
dLL cos
Bs
BB E
Lf
Reflectance : Albedo
Stefano Soatto (c) UCLA Vision Lab 38
Perfectly Diffuse Reflection
i
Lambertian cosine Law
s
sisiii sin
)()(E),(E
s
iiisisi
iiiiiii
iiiiiBr
E
ddE
ddL
dLfL
i i
i i
i
cos1
cos)()(1
sincos),(1
cos),(
Distant point light source
Stefano Soatto (c) UCLA Vision Lab 40
Perfectly Specular Reflection
),(),( iiirrr LL
ii
ririiirrsf
cossin
)()(),;,(
i i
iiiiiiiiirrsrrr ddLfL
sincos),(),;,(),(
From the definition of BRDF, the surface radiance is:
To satisfy:
i i
iiiiiririrri ddLL
),()()(),(
Stefano Soatto (c) UCLA Vision Lab 41
Lambertian Examples
Scene
(Oren and Nayar)
Lambertian sphere as the light moves.
(Steve Seitz)
Stefano Soatto (c) UCLA Vision Lab 42
Lambertian + Specular Model
)(cos),,()(
cos),,()(),,(
0
00
sn
sss
iiid
PLP
dPLPPL
Stefano Soatto (c) UCLA Vision Lab 43
Lambertian + specular
• Two parameters: how shiny, what kind of shiny.• 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 (but in graphics???)