Radiometric concepts
Slide from Srinivas Narasimhan
dω (solid angle subtended by dA )
source
R
(1) Solid Angle : dω =
dA '
θi
(foreshortened area)
dA (surface area)
dA cos θ i
dA'
=
R2
R2
( steradian )
dP
dω

E=
dP
dA
(4) Surface Radiance:
d P
2
L=
(watts / m steradian )
 dA cos θ r  dω
(watts/steradian)
Power emitted per unit solid angle
“Flux” synonymous with power
(3) Surface Irradiance :
dA
2
Solid angle subtended by hemisphere = 2
(half surface area of sphere)
(2) Radiant Intensity of Source :
dω
θr
( watts / m*m )
Power incident per unit surface area.
• Flux emitted per unit foreshortened area
per unit solid angle.
• L depends on direction
θr
• Surface can radiate into whole hemisphere.
• L depends on reflectance properties of surface.
Light Transport Assumption
●
Radiance is constant as it propagates along ray
∂P
L=
from surface to camera
∂  ∂ A cos 
∂ P 1=∂ P 2
L1 ∂ 1 ∂ A1 = L 2 ∂ 2 ∂ A2
∂ 1 =
∂ A2
r
2
∂  2=
L1 = L 2
∂ A1
r2
for
=0
Relation between Image Irradiance E and Scene Radiance L
image plane
surface patch
θ
dA s
dω s
α
dω i
α
image patch
dω L
dAi
z
f
• Solid angles of the double cone (orange and green):
dω i = dω s
dAi cos α
 f /cos α 2
=
dA s cos θ
dAs
 z /cos α 2
dAi
=
cos α
cos θ
 
z
f
• Solid angle subtended by lens:
(1)
2
dω L =
π d cos α
4  z /cos α2
(2)
2
Relation between Image Irradiance E and Scene Radiance L
image plane
θ
surface patch
dω s
dA s
α
dω i
α
image patch
dω L
dAi
z
f
• Flux received by lens from
dA s=
Flux projected onto image
L  dAs cos θ  dω L = E dAi
• From (1), (2), and (3):
π
E=L
4
(3)
2
 
d
f
cos α 4
• Image irradiance is proportional to Scene Radiance
• Small field of view  Effects of 4th power of cosine are small.
dAi
Specular Reflection and Mirror BRDF
source intensity
I
specular/mirror
direction
s , s
incident
direction
normal
i ,i
surface
element
n
viewing
direction
e , e
• Very smooth surface.
v r
• All incident light energy reflected in a SINGLE direction. (only when = )
• Mirror BRDF is simply a double-delta function :
• Surface Radiance :
L = I ρs δ θ i −θ v  δ  φi π −φ v 
Gonioreflectometers
Specular Reflections in Nature
Compare sizes of objects and their
reflections!
The reflections when seen from a lower
view
point are always longer than when
viewed
from a higher view point.
It's surprising how long
the reflections are when
viewed sitting on the river
bank.
White-out Conditions from an Overcast Sky
CAN’T perceive the shape of the snow covered terrain!
CAN perceive shape in regions
lit by the street lamp!!
WHY?
Phong Model: An Empirical Approximation
• How to model the angular falloff of highlights:
N
N H
R
-S
E
L=I s  R⋅E 
nshiney
R = −S  2 N⋅S  N
L=I s  N⋅H 
nshiney
H =  ES / 2
Phong Model
Blinn-Phong Model
• Sort of works, easy to compute
• But not physically based (no energy conservation and
reciprocity).
• Very commonly used in computer graphics.
Glossy Surfaces
• Delta Function too harsh a BRDF model
(valid only for highly polished mirrors and metals).
• Many glossy surfaces show broader highlights in addition to mirror reflection.
•
• Surfaces are not perfectly smooth – they show micro-surface geometry (roughness).
• Example Models : Phong model
Torrance Sparrow model
Blurred Highlights and Surface Roughness
Roughness
Phong Examples
• These spheres illustrate the Phong model as lighting
direction and nshiny are varied: