3-Principles of Surface Reflection
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Transcript of 3-Principles of Surface Reflection
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Basic Principles of Surface Reflectance
Thanks to Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan
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Radiometry and Image Formation
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Image Intensities
Image intensities = f( normal, surface reflectance, illumination )
Note: Image intensity understanding is an under-constrained problem!
source sensor
surface
element
normal Need to considerlight propagation in
a cone
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Differential Solid Angle and Spherical Polar Coordinates
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Radiometric Concepts
(1) Solid Angle :22
cos'
R
dA
R
dAd i
( steradian )
What is the solid angle subtended by a hemisphere?
d (solid angle subtended by )dA
R'dA
dA
(foreshortened area)
(surface area)
i
(2) Radiant Intensity of Source :
Light Flux (power) emitted per unit solid angle
d
dJ
( watts / steradian )
(3) Surface Irradiance :dA
dE
( watts / m2 )
Light Flux (power) incident per unit surface area.
Does not depend on where the light is coming from!
source
2
(4) Surface Radiance (tricky) :
d
ddA
dL
r)cos(
(watts / m steradian )
dA
r
Flux emitted per unit foreshortened areaper unit solid angle.
Ldepends on direction
Surface can radiate into whole hemisphere.
Ldepends on reflectance properties of surface.
r
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The Fundamental Assumption in Vision
Surface Camera
No Change in
Radiance
Lighting
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Radiance Properties
Radiance is constant as it propagates along ray
Derived from conservation of flux
Fundamental in Light Transport.
1 21 1 1 2 2 2d L d dA L d dA d
2 2
1 2 2 1d dA r d dA r
1 2
1 1 2 22dA dAd dA d dA
r
1 2L L
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Scene
Radiance LLens
Image
Irradiance E
CameraElectronics
Scene
ImageIrradiance E
Relationship between Scene and Image Brightness
MeasuredPixel Values, I
Non-linear Mapping!
Linear Mapping!
Before light hits the image plane:
After light hits the image plane:
Can we go from measured pixel value, I, to scene radiance, L?
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Relation Between Image Irradiance E and Scene Radiance L
f z
surface patchimage plane
id
sd
Ld
idA
s
dA
image patch
si dd 22)cos/(
cos
)cos/(
cos
z
dA
f
dA si 2
cos
cos
f
z
dA
dA
i
s
Solid angles of the double cone (orange and green):
(1)
2
2
)cos/(
cos
4
z
dd L
Solid angle subtended by lens:
(2)
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Relation Between Image Irradiance E and Scene Radiance L
f z
surface patchimage plane
id
sd
Ld
idA
sdA
image patch
Flux received by lens from = Flux projected onto images
dAi
dA
iLs dAEddAL )cos( (3)
From (1), (2), and (3):4
2
cos4
f
dLE
Image irradiance is proportional to Scene Radiance!
Small field of view Effects of 4thpower of cosine are small.
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Relation between Pixel Values I and Image Irradiance E
The camera response function relates image irradiance at the image plane
to the measured pixel intensity values.
Camera
Electronics
Image
Irradiance E
Measured
Pixel Values, I
IEg :
(Grossberg and Nayar)
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Radiometric Calibration
Important preprocessing step for many vision and graphics algorithms such as
photometric stereo, invariants, de-weathering, inverse rendering, image based rendering, etc.
EIg :1
Use a color chart with precisely known reflectances.
Irradiance = const * Reflectance
PixelValues
3.1%9.0%19.8%36.2%59.1%90%
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.
0
255
0 1
g
?
?1g
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The Problem of Dynamic Range
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The Problem of Dynamic Range
Dynamic Range: Range of brightness values measurable with a camera
(Hood 1986)
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!!
Todays Cameras: Limited Dynamic Range
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Images taken with a fish-eye lens of the sky show the wide range of brightnesses.
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).
(Debevec)
(Mitsunaga)
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Computer Vision: Building Machines that See
Lighting
Scene
Camera
Computer
Physical Models
Scene Interpretation
We need to understand the Geometric and Radiometric relations
between the scene and its image.
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Computer Graphics: Rendering things that Look Real
Lighting
Scene
Camera
Computer
Physical Models
Scene Generation
We need to understand the Geometric and Radiometric relations
between the scene and its image.
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Basic Principles of Surface Reflection
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Surface Appearance
Image intensities = f( normal, surface reflectance, illumination )
Surface Reflection depends on both the viewing and illumination direction.
source sensor
surface
element
normal
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BRDF: Bidirectional Reflectance Distribution Function
x
y
z
source
viewing
direction
surfaceelement
normal
incidentdirection
),( ii ),( rr
),( iisurfaceE
),(rr
surfaceL
Irradiance at Surface in direction ),( ii
Radiance of Surface in direction ),( rr
BRDF :
),(
),(),;,(
ii
surface
rr
surface
rriiE
Lf
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Important Properties of BRDFs
x
y
z
source
viewing
direction
surface
element
normal
incident
direction
),( ii ),( rr
BRDF is only a function of 3 variables :
),;,(),;,( iirrrrii ff
Rotational Symmetry (Isotropy):
Appearance does not change when surface is rotated about the normal.
),,( ririf
Helmholtz Reciprocity: (follows from 2ndLaw of Thermodynamics)
Appearance does not change when source and viewing directions are swapped.
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Differential Solid Angle and Spherical Polar Coordinates
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Derivation of the Scene Radiance EquationImportant!
),;,(),(),( rriiiisurface
rr
surface
fEL
iirriiii
src
rr
surface 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
rr
surface dfLL
2/
0
sincos),;,(),(),( iiiirriiiisrc
rr
surface ddfLL
Convert from solid angle to theta-phi representation:
M h i f S f R fl ti
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Mechanisms of Surface Reflection
source
surfacereflection
surface
incident
direction
body
reflection
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
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Mechanisms of Surface Reflection
Body Reflection:
Diffuse ReflectionMatte Appearance
Non-Homogeneous Medium
Clay, paper, etc
Surface Reflection:
Specular ReflectionGlossy Appearance
Highlights
Dominant for Metals
Many materials exhibit both Reflections:
Diff R fl ti d L b ti BRDF
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Diffuse Reflection and Lambertian BRDF
viewing
direction
surface
element
normalincident
direction
i
n
v
s
drriif ),;,( 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 did .cos
source intensity
source intensity I
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Diffuse Reflection and Lambertian BRDF
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White-out Conditions from an Overcast Sky
CANT perceive the shape of the snow covered terrain!
CAN perceive shape in regions
lit by the street lamp!!
WHY?
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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
rr
surface ddfLL
sky
ii
src LL ),(
1
),;,( rriif
sky
rr
surface LL ),(
Specular Reflection and Mirror BRDF
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Specular Reflection and Mirror BRDF
source intensity I
viewing
directionsurface
element
normal
incident
direction 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
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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)
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Phong Model: An Empirical Approximation
An illustration of the angular falloff of highlights:
Very commonly used in Computer Graphics
shinyn
sIL )(cos
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Phong Examples
These spheres illustrate the Phong model as lighting
directionand nshinyare varied:
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Components of Surface Reflection
A Si l R fl ti M d l Di h ti R fl ti
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A Simple Reflection Model - Dichromatic Reflection
Observed Image Color = a x Body Color + b x Specular Reflection Color
R
G
B
Klinker-Shafer-Kanade 1988
Color of Source
(Specular reflection)
Color of Surface
(Diffuse/Body Reflection)
Does not specify any specific model for
Diffuse/specular reflection
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Dror, Adelson, Wilsky
Specular Reflection and Mirror BRDF - RECALL
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Specular Reflection and Mirror BRDF - RECALL
source intensity I
viewing
directionsurface
element
normal
incident
direction 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 : )()(vivis
IL
v r
)()(),;,( vivisvviif
specular albedo
G S f
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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 smooththey show micro-surface geometry (roughness).
Example Models : Phong model
Torrance Sparrow model
Glossy Surfaces
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Blurred Highlights and Surface Roughness
Roughness
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Phong Model: An Empirical Approximation
How to model the angular falloff of highlights:
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.
shinyn
s ERIL ).(
-SR
E
HNN
shinyn
s HNIL ).(
NSNSR ).(2 2/)( SEH
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Phong Examples
These spheres illustrate the Phong model as lighting directionand
nshinyare varied:
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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