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Basic Principles of Surface Reflectance
Lecture #3
Thanks to Shree Nayar, Ravi Ramamoorthi, Pat Hanrahan
Methods Relying on Surface Reflectance
Shape from Shading
Photometric Stereo Reflection Separation
Texture Modeling
Computer Vision: Building Machines that See
Lighting
Scene
Camera
Computer
Physical Models
We need to understand the relation between the lighting, surface reflectance and medium and the image of the scene.
Surface Appearance
Image intensities = f ( normal, surface reflectance, illumination )
Surface Reflection depends on both the viewing and illumination direction.
source sensor
surfaceelement
normal
BRDF: Bidirectional Reflectance Distribution Function
x
y
z
source
viewingdirection
surfaceelement
normal
incidentdirection
),( ii ),( rr
),( iisurfaceE
),( rrsurfaceL
Irradiance at Surface in direction ),( ii Radiance of Surface in direction ),( rr
BRDF :),(
),(),;,(
iisurface
rrsurface
rrii E
Lf
Important Properties of BRDFs
x
y
z
source
viewingdirection
surfaceelement
normal
incidentdirection
),( ii ),( rr
BRDF is only a function of 3 variables :
),;,(),;,( iirrrrii ff
• Rotational Symmetry:
BRDF does not change when surface is rotated about the normal.
),,( ririf
• Helmholtz Reciprocity: (follows from 2nd Law of Thermodynamics)
BRDF does not change when source and viewing directions are swapped.
Derivation of the Scene Radiance Equation
),( iisrcL
),( rrsurfaceL
),;,(),(),( rriiiisurface
rrsurface fEL
From the definition of BRDF:
Derivation of the Scene Radiance Equation – Important!
),;,(),(),( rriiiisurface
rrsurface fEL
iirriiiisrc
rrsurface dfLL cos),;,(),(),(
From the definition of BRDF:
Write Surface Irradiance in terms of Source Radiance:
Integrate over entire hemisphere of possible source directions:
2
cos),;,(),(),( iirriiiisrc
rrsurface dfLL
2/
0
sincos),;,(),(),( iiiirriiiisrc
rrsurface ddfLL
Convert from solid angle to theta-phi representation:
Differential Solid Angle and Spherical Polar Coordinates
Mechanisms of Reflection
source
surfacereflection
surface
incidentdirection
bodyreflection
Body Reflection:
Diffuse ReflectionMatte AppearanceNon-Homogeneous MediumClay, paper, etc
Surface Reflection:
Specular ReflectionGlossy AppearanceHighlightsDominant for Metals
Image Intensity = Body Reflection + Surface Reflection
Mechanisms of Surface Reflection
Body Reflection:
Diffuse ReflectionMatte AppearanceNon-Homogeneous MediumClay, paper, etc
Surface Reflection:
Specular ReflectionGlossy AppearanceHighlightsDominant for Metals
Many materials exhibit both Reflections:
Diffuse Reflection and Lambertian BRDF
viewingdirection
surfaceelement
normalincidentdirection
in
v
s
d
rriif ),;,(• Lambertian BRDF is simply a constant :albedo
• Surface appears equally bright from ALL directions! (independent of )
• Surface Radiance :
v
• Commonly used in Vision and Graphics!
snIIL di
d .cos
source intensity
source intensity I
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
• Assume Lambertian Surface with Albedo = 1 (no absorption)
• Assume Sky radiance is constant
• Substituting in above Equation:
Radiance of any patch is the same as Sky radiance !! (white-out condition)
2/
0
sincos),;,(),(),( iiiirriiiisrc
rrsurface ddfLL
skyii
src LL ),(
1
),;,( rriif
skyrr
surface LL ),(
Specular Reflection and Mirror BRDF
source intensity I
viewingdirectionsurface
element
normal
incidentdirection n
v
s
rspecular/mirror direction
),( ii ),( vv
),( rr
• Mirror BRDF is simply a double-delta function :
• Very smooth surface.
• All incident light energy reflected in a SINGLE direction. (only when = )
• Surface Radiance : )()( vivisIL
v r
)()(),;,( vivisvviif specular albedo
Specular Reflections in Nature
Compare sizes of objects and their reflections!
The reflections when seen from a lower viewpoint are always longer than when viewedfrom a higher view point.
It's surprising how long the reflections are when viewed sitting on the river bank.
Specular Reflections in Nature
The reflections of bright objects have better perceived contrast.
Intensity of reflected light is a fraction of the direct light – [Fresnel term (derivation in a later class)]
Papers to Read
http://grail.cs.washington.edu/projects/sam/
Shape and Materials by Example: A Photometric Stereo Approach
http://www.eecs.harvard.edu/~zickler/helmholtz.html
Helmholtz Stereopsis
http://www.eecs.harvard.edu/~zickler/dichromaticediting.html
http://www.eecs.harvard.edu/~zickler/projects/colorsubspaces.html
Color Subspaces as Photometric Invariants
Specularity Removal and Dichromatic Editing
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