Basic Principles of Surface Reflectance: Thanks To Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan
Basic Principles of Surface Reflectance: Thanks To Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan
Basic Principles of Surface Reflectance: Thanks To Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan
source sensor
Need to consider
normal
light propagation in
a cone
surface
element
dA'
R (foreshortened area)
r d
i
dA (surface area) dA
dA' dA cos i
(1) Solid Angle : d 2
( steradian ) (4) Surface Radiance (tricky) :
R R2
d
What is the solid angle subtended by a hemisphere? L 2
(watts / m steradian )
(dA cos r ) d
d
(2) Radiant Intensity of Source : J ( watts / steradian )
d Flux emitted per unit foreshortened area
per unit solid angle.
Light Flux (power) emitted per unit solid angle
L depends on direction r
Lighting
No Change in
Radiance
Surface Camera
Radiance Properties
Radiance is constant as it propagates along ray
Derived from conservation of flux
Fundamental in Light Transport.
d 1 L1d1dA1 L2 d 2 dA2 d 2
d1 dA2 r 2 d 2 dA1 r 2
dA1dA2
d1dA1 2
d 2 dA2
r
L1 L2
Relationship between Scene and Image Brightness
Scene Image
Scene Lens
Radiance L Irradiance E
Linear Mapping!
Non-linear Mapping!
image plane
surface patch
d s dAs
di
image patch
d L
dAi
f z
di
image patch
dAi d L
f z
Flux received by lens from dAs = Flux projected onto image dAi
2
d
From (1), (2), and (3): E L cos 4
4 f
The camera response function relates image irradiance at the image plane
to the measured pixel intensity values.
g:E I
g 1 : I E
Use a color chart with precisely known reflectances.
255
g 1 ?
Pixel Values
g
0
0 ? 1
90% 59.1% 36.2% 19.8% 9.0% 3.1%
Irradiance = const * Reflectance
(Hood 1986)
Combine the calibrated images (for example, using averaging weighted by exposures).
(Mitsunaga)
(Debevec)
Images taken with a fish-eye lens of the sky show the wide range of brightnesses.
Computer Vision: Building Machines that See
Lighting
Camera
Physical Models
Computer
Scene
Scene Interpretation
Lighting
Camera
Physical Models
Computer
Scene
Scene Generation
source sensor
normal
surface
element
source
z
incident
direction viewing
direction
( i , i ) ( r , r )
normal
y
surface
x element
Lsurface ( r , r )
BRDF : f ( i , i ; r , r )
E surface ( i , i )
Important Properties of BRDFs
source
z
incident
direction viewing
direction
( i , i ) ( r , r )
normal
y
surface
x element
Rotational Symmetry (Isotropy):
Appearance does not change when surface is rotated about the normal.
Appearance does not change when source and viewing directions are swapped.
f ( i , i ; r , r ) f ( r , r ; i , i )
Differential Solid Angle and Spherical Polar Coordinates
Derivation of the Scene Radiance Equation Important!
From the definition of BRDF:
Lsurface ( r , r ) E surface ( i , i ) f ( i , i ; r , r )
Write Surface Irradiance in terms of Source Radiance:
L surface
( r , r ) L ( i , i ) f ( i , i ; r , r ) cos i di
src
Lsurface ( r , r ) (i ,i ) f (i , i ; r ,r ) cosi di
src
L
2
Convert from solid angle to theta-phi representation:
/2
Lsurface ( r , r ) (i , i ) f (i , i ; r , r ) cosi sin i di di
src
L
0
Mechanisms of Surface Reflection
source
incident
direction surface
reflection
body
reflection
surface
source intensity I
incident
direction s normal n
i viewing
direction v
surface
element
WHY?
Diffuse Reflection from Uniform Sky
/2
( r , r ) L ( i , i ) f ( i , i ; r , r ) cos i sin i d i di
surface src
L
0
Lsurface ( r , r ) Lsky
incident
direction s normal n
( i , i )
viewing
surface direction v ( v , v )
element
f ( i , i ; v , v ) s ( i v ) (i v )
Surface Radiance : L I s ( i v ) (i v )
BRDFs of Glossy Surfaces
Example Models : Phong Model (no physical basis, but sort of works (empirical))
nshiny
L I s (cos )
Phong Examples
Klinker-Shafer-Kanade 1988
Color of Source
(Specular reflection)
Does not specify any specific model for
Diffuse/specular reflection G
Color of Surface
(Diffuse/Body Reflection)
B
Dror, Adelson, Wilsky
Specular Reflection and Mirror BRDF - RECALL
source intensity I specular/mirror
direction r ( r , r )
incident
direction s normal n
( i , i )
viewing
surface direction v ( v , v )
element
f ( i , i ; v , v ) s ( i v ) (i v )
Surface Radiance : L I s ( i v ) (i v )
Glossy Surfaces
Surfaces are not perfectly smooth they show micro-surface geometry (roughness).
Roughness
Phong Model: An Empirical Approximation
How to model the angular falloff of highlights:
N N H
R
-S
E
nshiny nshiny
L I s ( R.E ) L I s ( N .H )
Phong Model Blinn-Phong Model
R S 2( N .S ) N H (E S ) / 2
Sort of works, easy to compute
But not physically based (no energy conservation and reciprocity).
Very commonly used in computer graphics.
Phong Examples