appearance and reflectance - cs.cmu.edu
TRANSCRIPT
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Appearance and reflectance
16-385 Computer VisionSpring 2018, Lecture 13http://www.cs.cmu.edu/~16385/
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• Apologies for cancelling last Wednesday’s lecture.
• Homework 3 has been posted and is due on March 9th.- Any questions about the homework?- How many of you have looked at/started/finished homework 3?
• Office hours for Yannis’ this week: Wednesday 3-5 pm.
• Results from poll for adjusting Yannis’ regular office hours: They stay the same.
• Many talks this week: 1. Judy Hoffman, "Adaptive Adversarial Learning for a Diverse Visual World," Monday
March 5th, 10:00 AM, NSH 3305.
2. Manolis Savva, "Human-centric Understanding of 3D Environments," Wednesday March 7, 2:00 PM, NSH 3305.
3. David Fouhey, "Recovering a Functional and Three Dimensional Understanding of Images," Thursday March 8, 4:00 PM, NSH 3305.
• How many of you went to Pulkit Agrawal’s talk last week?
Course announcements
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• Appearance phenomena.
• Measuring light and radiometry.
• Reflectance and BRDF.
Overview of today’s lecture
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Slide credits
Most of these slides were adapted from:
• Srinivasa Narasimhan (16-385, Spring 2014).
• Todd Zickler (Harvard University).
• Steven Gortler (Harvard University).
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1. Image processing.
2. Geometry-based vision.
3. Physics-based vision.
4. Learning-based vision.
5. Dealing with motion.
Course overview
We are starting this part now
Lectures 7 – 12
See also 16-822: Geometry-based Methods in Vision
Lectures 1 – 7
See also 18-793: Image and Video Processing
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Appearance
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Appearance
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“Physics-based” computer vision
(a.k.a “inverse optics”)
reflectance
illumination
shape
shape , reflectance, illumination
Our challenge: Invent computational representations of
shape, lighting, and reflectance that are efficient: simple
enough to make inference tractable, yet general enough to
capture the world’s most important phenomena
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Example application: Photometric Stereo
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Why study the physics (optics) of the
world?
Lets see some pictures!
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Light and Shadows
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Reflections
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Refractions
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Interreflections
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Scattering
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More Complex Appearances
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Measuring light and
radiometry
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Solid angle
๏ The solid angle subtended by a small surface patch with respect to point O
is the area of its central projection onto the unit sphere about O
Depends on:
๏ orientation of patch
๏ distance of patch
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Solid angle
๏ The solid angle subtended by a small surface patch with respect to point O
is the area of its central projection onto the unit sphere about O
Depends on:
๏ orientation of patch
๏ distance of patch
One can show:
Units: steradians [sr]
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Solid angle
๏ The solid angle subtended by a small surface patch with respect to point O
is the area of its central projection onto the unit sphere about O
Depends on:
๏ orientation of patch
๏ distance of patch
One can show:“surface foreshortening”
Units: steradians [sr]
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Solid angle
๏ To calculate solid angle subtended by a surface S relative to O you must
add up (integrate) contributions from all tiny patches (nasty integral)
One can show:“surface foreshortening”
Units: steradians [sr]
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Question
๏ Suppose surface S is a hemisphere centered at O. What is the solid angle
it subtends?
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Question
๏ Suppose surface S is a hemisphere centered at O. What is the solid angle
it subtends?
๏ Answer: 2\pi (area of sphere is 4\pi*r^2; area of unit sphere is 4\pi; half of
that is 2\pi)
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Quantifying light: flux, irradiance, and radiance
๏ Imagine a sensor that counts photons passing through planar patch X in
directions within angular wedge W
๏ It measures radiant flux [watts = joules/sec]
๏ Measurement depends on sensor area |X|
radiant flux
* shown in 2D for clarity; imagine three dimensions
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Quantifying light: flux, irradiance, and radiance
๏ Irradiance:
A measure of incoming light that is independent of sensor area |X|
๏ Units: watts per square meter [W/m2]
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Quantifying light: flux, irradiance, and radiance
๏ Irradiance:
A measure of incoming light that is independent of sensor area |X|
๏ Units: watts per square meter [W/m2]
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Quantifying light: flux, irradiance, and radiance
๏ Irradiance:
A measure of incoming light that is independent of sensor area |X|
๏ Units: watts per square meter [W/m2]
๏ Depends on sensor direction normal.
๏We keep track of the normal because a planar sensor with
distinct orientation would converge to a different limit
๏ In the literature, notations n and W are often omitted, and
values are implied by context
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Quantifying light: flux, irradiance, and radiance
๏ Radiance:
A measure of incoming light that is independent of sensor area |X|,
orientation n, and wedge size (solid angle) |W|
๏ Units: watts per steradian per square meter [W/(m2∙sr)]
๏Has correct units, but still depends on sensor orientation
๏ To correct this, convert to measurement that would have
been made if sensor was perpendicular to direction ω
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Quantifying light: flux, irradiance, and radiance
๏ Radiance:
A measure of incoming light that is independent of sensor area |X|,
orientation n, and wedge size (solid angle) |W|
๏ Units: watts per steradian per square meter [W/(m2∙sr)]
๏ Has correct units, but still depends on sensor orientation
๏ To correct this, convert to measurement that would have been
made if sensor was perpendicular to direction ω
“foreshortened area”
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Quantifying light: flux, irradiance, and radiance
๏ Radiance:
A measure of incoming light that is independent of sensor area |X|,
orientation n, and wedge size (solid angle) |W|
๏ Units: watts per steradian per square meter [W/(m2∙sr)]
๏ Has correct units, but still depends on sensor orientation
๏ To correct this, convert to measurement that would have been
made if sensor was perpendicular to direction ω
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Quantifying light: flux, irradiance, and radiance
๏ Radiance:
A measure of incoming light that is independent of sensor area |X|,
orientation n, and wedge size (solid angle) |W|
๏ Units: watts per steradian per square meter [W/(m2∙sr)]
๏ Has correct units, but still depends on sensor orientation
๏ To correct this, convert to measurement that would have been
made if sensor was perpendicular to direction ω
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๏ Attractive properties of radiance:
• Allows computing the radiant flux measured by any finite sensor
Quantifying light: flux, irradiance, and radiance
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๏ Attractive properties of radiance:
• Allows computing the radiant flux measured by any finite sensor
Quantifying light: flux, irradiance, and radiance
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๏ Attractive properties of radiance:
• Allows computing the radiant flux measured by any finite sensor
• Constant along a ray in free space
Quantifying light: flux, irradiance, and radiance
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The Fundamental Assumption in Vision
Surface Camera
No Change in
Surface Radiance
Lighting
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๏ Attractive properties of radiance:
• Allows computing the radiant flux measured by any finite sensor
• Constant along a ray in free space
• A camera measures radiance (after a one-time radiometric calibration;
more on this later). So RAW pixel values are proportional to radiance.
Quantifying light: flux, irradiance, and radiance
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Question
๏ Most light sources, like a heated metal sheet, follow Lambert’s Law
radiant intensity [W/sr]
“Lambertian
area source”
๏ What is the radiance of an infinitesimal patch [W/sr・m2]?
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Question
๏ Most light sources, like a heated metal sheet, follow Lambert’s Law
radiant intensity [W/sr]
“Lambertian
area source”
Answer: (independent of direction)
๏ What is the radiance of an infinitesimal patch [W/sr・m2]?
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Question
๏ Most light sources, like a heated metal sheet, follow Lambert’s Law
radiant intensity [W/sr]
“Lambertian
area source”
๏ What is the radiance of an infinitesimal patch [W/sr・m2]?
Answer: (independent of direction)
“Looks equally bright when viewed from any direction”
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Appearance
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“Physics-based” computer vision
(a.k.a “inverse optics”)
reflectance
illumination
shape
shape , reflectance, illumination
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Reflectance and BRDF
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Reflectance
๏ Ratio of outgoing energy to incoming energy at a single point
๏ Want to define a ratio such that it:
• converges as we use smaller and smaller incoming and outgoing
wedges
• does not depend on the size of the wedges (i.e. is intrinsic to the
material)
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Reflectance
๏ Ratio of outgoing energy to incoming energy at a single point
๏ Want to define a ratio such that it:
• converges as we use smaller and smaller incoming and outgoing
wedges
• does not depend on the size of the wedges (i.e. is intrinsic to the
material)
๏ Notations x and n often implied by context and omitted;
directions \omega are expressed in local coordinate system
defined by normal n (and some chosen tangent vector)
๏ Units: sr-1
๏ Called Bidirectional Reflectance Distribution Function (BRDF)
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BRDF: Bidirectional Reflectance Distribution Function
x
y
z
source
viewing
direction
surface
element
normal
incident
direction
),(ii
),(rr
),(ii
surfaceE
),(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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Reflectance: BRDF
๏ Units: sr-1
๏ Real-valued function defined on the double-hemisphere
๏ Allows computing output radiance (and thus pixel value) for any configuration
of lights and viewpoint
reflectance equation
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Important Properties of BRDFs
x
y
z
source
viewing
direction
surface
element
normal
incident
direction
),(ii
),(rr
• Conservation of Energy:
Why smaller
than or equal?
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Important Properties of BRDFs
x
y
z
source
viewing
direction
surface
element
normal
incident
direction
),(ii
),(rr
),;,(),;,(iirrrrii
ff
• Helmholtz Reciprocity: (follows from 2nd Law of Thermodynamics)
BRDF does not change when source and viewing directions are swapped.
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Property: “Helmholtz reciprocity”
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Important Properties of BRDFs
x
y
z
source
viewing
direction
surface
element
normal
incident
direction
),(ii
),(rr
Can be written as a function of 3 variables :
• Rotational Symmetry (Isotropy):
BRDF does not change when surface is rotated about the normal.
),,(riri
f
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Common assumption: Isotropy
Bi-directional Reflectance Distribution Function (BRDF)
4D → 3D
[Matusik et al., 2003]
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Simplification: Bivariate
Bi-directional Reflectance Distribution Function (BRDF)
4D → 3D → 2D
“half-angle”
“diffe
rence
angle
”
Original 3D Reduced 2D
[Romeiro et al., 2008]
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Reflectance: BRDF
๏ Units: sr-1
๏ Real-valued function defined on the double-hemisphere
๏ Allows computing output radiance (and thus pixel value) for any configuration
of lights and viewpoint
reflectance equation
Why is there a cosine in the reflectance equation?
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Derivation of the Scene Radiance Equation
),(ii
srcL
),(rr
surfaceL
),;,(),(),(rriiii
surface
rr
surfacefEL
From the definition of BRDF:
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Derivation of the Scene Radiance Equation
),;,(),(),(rriiii
surface
rr
surfacefEL
iirriiii
src
rr
surfacedfLL cos),;,(),(),(
From the definition of BRDF:
Write Surface Irradiance in terms of Source Radiance:
Integrate over entire hemisphere of possible source directions:
2
cos),;,(),(),(iirriiii
src
rr
surfacedfLL
2/
0
sincos),;,(),(),(iiiirriiii
src
rr
surfaceddfLL
Convert from solid angle to theta-phi representation:
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BRDF
Bi-directional Reflectance Distribution Function (BRDF)
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Mechanisms of Reflectionsource
surface
reflection
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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Example Surfaces
Body Reflection:
Diffuse Reflection
Matte Appearance
Non-Homogeneous Medium
Clay, paper, etc
Surface Reflection:
Specular Reflection
Glossy Appearance
Highlights
Dominant for Metals
Many materials exhibit
both Reflections:
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BRDF
Bi-directional Reflectance Distribution Function (BRDF)
Lambertian (diffuse) BRDF: energy equally distributed in all directions
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Diffuse Reflection and Lambertian BRDF
viewing
directionsurface
element
normalincident
direction
i
n
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!
snIILd
i
d.cos
source intensity
source intensity I
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BRDF
Bi-directional Reflectance Distribution Function (BRDF)
Specular BRDF: all energy concentrated in mirror direction
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Specular Reflection and Mirror BRDFsource 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 :
• Valid for very smooth surfaces.
• All incident light energy reflected in a SINGLE direction (only when = ).
• Surface Radiance : )()(vivis
IL
v r
)()(),;,(vivisvvii
f
specular albedo
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BRDF
Bi-directional Reflectance Distribution Function (BRDF)
Glossy BRDF: more energy concentrated in mirror direction
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Trick for dielectrics (non-metals)
• BRDF is a sum of a Lambertian diffuse component and non-Lambertian
specular components
• The two components differ in terms of color and polarization, and under
certain conditions, this can be exploited to separate them.
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Trick for dielectrics (non-metals)
• BRDF is a sum of a Lambertian diffuse component and non-Lambertian
specular components
• The two components differ in terms of color and polarization, and under
certain conditions, this can be exploited to separate them.
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Trick for dielectrics (non-metals)
• BRDF is a sum of a Lambertian diffuse component and non-Lambertian
specular components
• The two components differ in terms of color and polarization, and under
certain conditions, this can be exploited to separate them.
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Trick for dielectrics (non-metals)
• BRDF is a sum of a Lambertian diffuse component and non-Lambertian
specular components
• The two components differ in terms of color and polarization, and under
certain conditions, this can be exploited to separate them.
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Trick for dielectrics (non-metals)
Often called the dichromatic BRDF:
• Diffuse term varies with wavelength,
constant with polarization
• Specular term constant with
wavelength, varies with polarization
• BRDF is a sum of a Lambertian diffuse component and non-Lambertian
specular components
• The two components differ in terms of color and polarization, and under
certain conditions, this can be exploited to separate them.
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DIFFUSE
= +
SPECULAR
Trick for dielectrics (non-metals)
• In this example, the two components were separated using linear polarizing
filters on the camera and light source.
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DIFFUSE
= +
SPECULAR
Trick for dielectrics (non-metals)
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Diffuse and Specular Reflection
diffuse specular diffuse+specular
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Tabulated 4D BRDFs (hard to measure)
[Ngan et al., 2005]
Gonioreflectometer
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Low-parameter (non-linear) BRDF models
Blinn:
Lafortune:
Ward:
Phong:
๏ A small number of parameters define the (2D,3D, or 4D) function
๏ Except for Lambertian, the BRDF is non-linear in these parameters
๏ Examples:
Lambertian:
Where do these constants come from?
α is called the albedo
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Recent progress: “Active appearance capture”
Reciprocity[ICCV 2001; ECCV 2002; CVPR 2003; CVPR 2006; Sen et al., 2005;
Hawkins et al., 2005, SIGGRAPH 2010]
Isotropy[Lu and Little 1999; CVPR 2007; Alldrin and Kriegman 2007; CVPR 2008;
CVPR 2009]
Separability[Sato & Ikeuchi, 1994; Schluns and Wittig, 1993; ...; Barsky and Petrou, 2001;
CVPR 2005; CVPR 2006; IJCV 2008; ...]
Spatial regularity[Lensch et al., 2001; Hertzmann & Seitz, 2003; EGSR 2005; PAMI 2006;
Lawrence et al., 2006; Weistroffer et al., 2007; CVPR 2008; Garg et al. 2009;...]
Tangent-plane symmetry[SIGGRAPH Asia 2008]
...
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Reflectance ModelsReflection: An Electromagnetic Phenomenon
h
T
Two approaches to derive Reflectance Models:
– Physical Optics (Wave Optics)
– Geometrical Optics (Ray Optics)
Geometrical models are approximations to physical models
But they are easier to use!
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Reflectance that Require Wave Optics
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References
Basic reading:• Szeliski, Section 2.2.• Gortler, Chapter 21.