Basic Principles of Surface Reflectance
Thanks to Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan
Radiometry and Image Formation
Image Intensities
sensor
source
Need to consider
light propagation in
a cone
normal
surface
element
Image intensities = f ( normal, surface reflectance, illumination )
Note: Image intensity understanding is an under-constrained problem!
Differential Solid Angle and Spherical Polar Coordinates
Radiometric Concepts
d (solid angle subtended by dA
source
R
)
dA'
i
dA (surface area)
(1) Solid Angle :
d 
dA' dA cos  i

2
R
R2
( steradian )
What is the solid angle subtended by a hemisphere?
(2) Radiant Intensity of Source : J

d
d
dA
(4) Surface Radiance (tricky) :
L
d
(dA cos  r ) d
• Flux emitted per unit foreshortened area
per unit solid angle.
• L depends on direction
E
d
dA
2
(watts / m steradian )
( watts / steradian )
Light Flux (power) emitted per unit solid angle
(3) Surface Irradiance :
d
r
(foreshortened area)
( watts / m2 )
Light Flux (power) incident per unit surface area.
Does not depend on where the light is coming from!
r
• Surface can radiate into whole hemisphere.
• L depends on reflectance properties of surface.
The Fundamental Assumption in Vision
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  L1d1dA1  L2 d2 dA2  d  2
d1  dA2 r 2
d2  dA1 r 2
dA1dA2
d1dA1 
 d 2 dA2
2
r
 L1  L2
Relationship between Scene and Image Brightness
• Before light hits the image plane:
Scene
Scene
Radiance L
Lens
Image
Irradiance E
Linear Mapping!
• After light hits the image plane:
Image
Irradiance E
Camera
Electronics
Measured
Pixel Values, I
Non-linear Mapping!
Can we go from measured pixel value, I, to scene radiance, L?
Relation Between Image Irradiance E and Scene Radiance L
image plane
surface patch

dAs
ds
d i


image patch
dL
dAi
z
f
• Solid angles of the double cone (orange and green):
di  ds
dAi cos 
( f / cos  ) 2

dAs cos 
( z / cos  ) 2
dAs
dAi

cos   z 
 
cos   f 
• Solid angle subtended by lens:
d L 
d
4
2
cos 
( z / cos  ) 2
(1)
(2)
2
Relation Between Image Irradiance E and Scene Radiance L
image plane

surface patch
dAs
ds
d i


image patch
dL
dAi
z
f
• Flux received by lens from dAs
=
Flux projected onto image dAi
L (dAs cos  ) dL  E dAi
• From (1), (2), and (3):
E  L
 d
(3)
2
  cos  4
4 f
• Image irradiance is proportional to Scene Radiance!
• Small field of view  Effects of 4th power of cosine are small.
Relation between Pixel Values I and Image Irradiance E
Image
Irradiance E
Camera
Electronics
Measured
Pixel Values, I
The camera response function relates image irradiance at the image plane
to the measured pixel intensity values.
g:E I
(Grossberg and Nayar)
Radiometric Calibration
•Important preprocessing step for many vision and graphics algorithms such as
photometric stereo, invariants, de-weathering, inverse rendering, image based rendering, etc.
1
g :I E
•Use a color chart with precisely known reflectances.
255
Pixel Values
g 1
?
g
0
0
90% 59.1% 36.2% 19.8% 9.0% 3.1%
?
1
Irradiance = const * Reflectance
• Use more camera exposures to fill up the curve.
• Method assumes constant lighting on all patches and works best when source is
far away (example sunlight).
• Unique inverse exists because g is monotonic and smooth for all cameras.
The Problem of Dynamic Range
The Problem of Dynamic Range
• Dynamic Range: Range of brightness values measurable with a camera
(Hood 1986)
• Today’s Cameras: Limited Dynamic Range
High Exposure Image
Low Exposure Image
• We need 5-10 million values to store all brightnesses around us.
• But, typical 8-bit cameras provide only 256 values!!
High Dynamic Range Imaging
• Capture a lot of images with different exposure settings.
• Apply radiometric calibration to each camera.
• 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
We need to understand the Geometric and Radiometric relations
between the scene and its image.
Computer Graphics: Rendering things that Look Real
Lighting
Camera
Physical Models
Computer
Scene
Scene Generation
We need to understand the Geometric and Radiometric relations
between the scene and its image.
Basic Principles of Surface Reflection
Surface Appearance
sensor
source
normal
surface
element
Image intensities = f ( normal, surface reflectance, illumination )
Surface Reflection depends on both the viewing and illumination direction.
BRDF: Bidirectional Reflectance Distribution Function
source
z
incident
direction

(i , i )
y
viewing
direction
(r , r )
normal

surface
element
x
E surface( i , i ) Irradiance at Surface in direction (i , i )
Lsurface( r , r )
BRDF :
Radiance of Surface in direction
f ( i , i ;  r , r ) 
(r ,r )
Lsurface( r , r )
E surface( i , i )
Important Properties of BRDFs
source
z
incident
direction

(i , i )
viewing
direction
(r , r )
normal
y

surface
element
x
• Rotational Symmetry (Isotropy):
Appearance does not change when surface is rotated about the normal.
BRDF is only a function of 3 variables :
f (i ,  r , i  r )
• Helmholtz Reciprocity: (follows from 2nd Law of Thermodynamics)
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:
surface
L
( r , r )  L ( i , i ) f ( i , i ;  r , r ) cos  i di
src
Integrate over entire hemisphere of possible source directions:
src
L
 (i ,i ) f (i ,i ;  r ,r ) cosi di
Lsurface( r , r ) 
2
Convert from solid angle to theta-phi representation:
Lsurface( r , r ) 
  /2


src
L
 (i ,i ) f (i ,i ; r ,r ) cosi sin i di di
0
Mechanisms of Surface Reflection
source
incident
direction
surface
reflection
body
reflection
surface
Body Reflection:
Diffuse Reflection
Matte Appearance
Non-Homogeneous Medium
Clay, paper, etc
Surface Reflection:
Specular Reflection
Glossy Appearance
Highlights
Dominant for Metals
Image Intensity = Body Reflection + Surface Reflection
Mechanisms of Surface Reflection
Body Reflection:
Diffuse Reflection
Matte Appearance
Non-Homogeneous Medium
Clay, paper, etc
Many materials exhibit both Reflections:
Surface Reflection:
Specular Reflection
Glossy Appearance
Highlights
Dominant for Metals
Diffuse Reflection and Lambertian BRDF
source intensity I
incident
direction
s
normal
n
i
viewing
direction
v
surface
element
• Surface appears equally bright from ALL directions! (independent of
• Lambertian BRDF is simply a constant :
• Surface Radiance :
d
L
I cos  i

f ( i , i ;  r , r ) 

• Commonly used in Vision and Graphics!
d
I n.s

v)
d

source intensity
albedo
Diffuse Reflection and Lambertian BRDF
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?
Diffuse Reflection from Uniform Sky
Lsurface( r , r ) 
  /2


src
L
 (i ,i ) f (i ,i ; r ,r ) cosi sin i di di
0
• Assume Lambertian Surface with Albedo = 1 (no absorption)
f ( i , i ;  r , r ) 
1

• Assume Sky radiance is constant
Lsrc ( i , i )  Lsky
• Substituting in above Equation:
Lsurface( r , r )  Lsky
Radiance of any patch is the same as Sky radiance !! (white-out condition)
Specular Reflection and Mirror BRDF
source intensity I
incident
direction
(i , i )
s
normal
specular/mirror
direction
r (r ,r )
n
viewing
direction
surface
element
v (v , v )
• Very smooth surface.
• All incident light energy reflected in a SINGLE direction. (only when
• Mirror BRDF is simply a double-delta function :
specular albedo
f (i , i ; v , v )   s  (i   v )  (i    v )
• Surface Radiance :
L  I  s  (i   v )  (i    v )
v = r)
BRDFs of Glossy Surfaces
• Delta Function too harsh a BRDF model
(valid only for polished mirrors and metals).
• Many glossy surfaces show broader highlights in addition to specular reflection.
• Example Models : Phong Model (no physical basis, but sort of works (empirical))
Torrance Sparrow model (physically based)
Phong Model: An Empirical Approximation
• An illustration of the angular falloff of highlights:
L  I  s (cos  )
•
nshiny
Very commonly used in Computer Graphics
Phong Examples
• These spheres illustrate the Phong model as lighting
• direction and nshiny are varied:
Components of Surface Reflection
A Simple Reflection Model - Dichromatic Reflection
Observed Image Color = a x Body Color + b x Specular Reflection Color
Klinker-Shafer-Kanade 1988
R
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
incident
direction
(i , i )
s
normal
specular/mirror
direction
r (r ,r )
n
viewing
direction
surface
element
v (v , v )
• Very smooth surface.
• All incident light energy reflected in a SINGLE direction. (only when
• Mirror BRDF is simply a double-delta function :
specular albedo
f (i , i ; v , v )   s  (i   v )  (i    v )
• Surface Radiance :
L  I  s  (i   v )  (i    v )
v
=
r
)
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 Model: An Empirical Approximation
• How to model the angular falloff of highlights:
N
-S
N
R
H
E
L  I  s ( R.E )
R   S  2( N .S ) N
nshiny
Phong Model
L  I  s ( N .H )
H  (E  S ) / 2
nshiny
Blinn-Phong Model
• Sort of works, easy to compute
• But not physically based (no energy conservation and reciprocity).
• Very commonly used in computer graphics.
Phong Examples
• These spheres illustrate the Phong model as lighting direction and
nshiny are varied:
Those Were the Days
“In trying to improve the quality of the synthetic
images, we do not expect to be able to display
the object exactly as it would appear in reality,
with texture, overcast shadows, etc. We hope
only to display an image that approximates the
real object closely enough to provide a certain
degree of realism.”
– Bui Tuong Phong, 1975