Color and Radiometry
Digital Image Synthesisg g y Yung-Yu Chuang
with slides by Svetlana Lazebnik, Pat Hanrahan and Matt Pharr
Radiometry
• Radiometry: study of the propagation of
electromagnetic radiation in an environment electromagnetic radiation in an environment
• Four key quantities: flux, intensity, irradiance d di
and radiance
• These radiometric quantities are described by their spectral power distribution (SPD)
• Human visible light ranges from 370nm to 730nmg g
1 1 02
1 04
1 06
1 08
1 01 0
1 01 2
1 01 4
1 01 6
1 01 8
1 02 0
1 02 2
1 02 4
1 02 6
Cosm ic Ra y s Ga m m a
Ra y s X -Ra y s
Ultra - V iolet Infra -
Red Ra d io
H ea t Pow er
1 01 6
1 01 4
1 01 2
1 01 0 1 08
1 06
1 04
1 02
1 1 0-2
1 0-4
1 0-6
1 0-8 Ra y s Ra y s
V iolet Red
W a velength (N M )
7 0 0 6 0 0 5 0 0 4 0 0
IR R G B UV
Electromagnetic spectrum
Why do we see light at these wavelengths?
Human Luminance Sensitivity Function
Why do we see light at these wavelengths?
Because that’s where the sun radiates electromagnetic energy
Basic radiometry
• pbrt is based on radiative transfer: study of the transfer of radiant energy based on radiometric transfer of radiant energy based on radiometric principles and operates at the geometric optics level (light interacts with objects much larger level (light interacts with objects much larger than the light’s wavelength)
It i b d th ti l d l H
• It is based on the particle model. Hence,
diffraction and interference can’t be easily t d f
accounted for.
Basic assumptions about light behavior
• Linearity: the combined effect of two inputs is equal to the sum of effects
equal to the sum of effects
• Energy conservation: scattering event can’t produce more energy than they started with produce more energy than they started with
• Steady state: light is assumed to have reached equilibrium so its radiance distribution isn’t equilibrium, so its radiance distribution isn t changing over time.
• No polarization: we only care the frequency of
• No polarization: we only care the frequency of light but not other properties (such as phases)
• No fluorescence or phosphorescence:
• No fluorescence or phosphorescence:
behavior of light at a wavelength or time
doesn’t affect the behavior of light at other g wavelengths or time
Fluorescent materials
Color
Interaction of light and surfaces
• Reflected color is the
l f i i f
result of interaction of
light source spectrum with surface reflectance
surface reflectance
• Spectral radiometry
– All definitions and units are now All definitions and units are now
“per unit wavelength”
– All terms are now “spectral”
Why reflecting different colors
high
heat/ light
heat/
chemical
g
low low
Light with specific wavelengths b b d
Fluorescent are absorbed.
Primary colors
Primary colors for
addition (light sources) Primary colors for
subtraction (reflection) ( g ) subtraction (reflection)
Heat generates light
• Vibration of atoms or electrons due to heat
generates electromagnetic radiation as well If generates electromagnetic radiation as well. If its wavelength is within visible light (>1000K), it generates color as well
it generates color as well.
• Color only depends on temperature, but not t f th bj t
property of the object.
• Human body radiates IR light under room temperature.
• 2400-2900K: color temperature of incandescent p light bulb
Spectral power distribution
fl li h (日光燈)
400nm (bluish)
650nm (red) 550nm
(green)
fluorescent light (日光燈)
Spectral power distribution
lemmon skin
400nm (bluish)
650nm (red) 550nm
(green)
lemmon skin
Color
• Need a compact, efficient and accurate way to represent functions like these
represent functions like these
• Find proper basis functions to map the infinite- di i l f ll ibl SPD f ti
dimensional space of all possible SPD functions to a low-dimensional space of coefficients
• For example, B(λ)=1 is a trivial but bad approximation
• Fortunately, according to tristimulus theory, all visible SPDs can be accurately represented y p with three values.
The Eye
Slide by Steve Seitz
Density of rods and cones
cone moleculespigment rod
molecules
Rods and cones are non-uniformly distributed on the retina
– Rods responsible for intensity, cones responsible for color
– Fovea - Small region (1 or 2°) at the center of the visual field containing the highest Fovea Small region (1 or 2 ) at the center of the visual field containing the highest density of cones (and no rods).
– Less visual acuity in the periphery—many rods wired to the same neuron
Slide by Steve Seitz
Human Photoreceptors
Color perception
M L Power
S
Wavelength
Rods and cones act as filters on the spectrum
– To get the output of a filter, multiply its response g p , p y p
curve by the spectrum, integrate over all wavelengths
• Each cone yields one number
• Q: How can we represent an entire spectrum with 3 numbers?
• A: We can’t! Most of the information is lost.
As a result two different spectra may appear indistinguishable – As a result, two different spectra may appear indistinguishable
» such spectra are known as metamers
Slide by Steve Seitz
Metamers
different spectrum same perception different spectrum, same perception
t t (鎢絲) b lb l i i i
tungsten (鎢絲) bulb television monitor
Color matching experiment
p1 = 645.2 nm p2 = 525.3 nm
444 4
Foundations of Vision, by Brian Wandell, Sinauer Assoc., 1995
p1 p2 p3 p1 p2 p3
p3 = 444.4 nm
Color matching experiment
Color matching experiment
• To avoid negative parameters
Spectrum
• In core/spectrum.*
T t ti (d f lt)
• Two representations: RGBSpectrum (default) and SampledSpectrum
• The selection is done at compile time with a typedef in core/pbrt.h
typedef RGBSpectrum Spectrum;
• Both stores a fixed number of samples at a Both stores a fixed number of samples at a fixed set of wavelengths.
CoefficientSpectrum
template <int nSamples>
class CoefficientSpectrum { class CoefficientSpectrum {
+=, +, -, /, *, *= (CoefficientSpectrum)
== != (CoefficientSpectrum)
==, != (CoefficientSpectrum) IsBlack, Clamp
* * / / (float)
*, *=, /, /= (float) protected:
fl t [ S l ] float c[nSamples];
}
Sqrt, Pow, Exp
SampledSpectrum
• Represents a SPD with uniformly spaced samples between a starting and an ending samples between a starting and an ending wavelength (400 to 700 nm for HVS). The
number of samples 30 is generally more than number of samples, 30, is generally more than enough.
static const int sampledLambdaStart = 400;
static const int sampledLambdaEnd = 700;
static const int nSpectralSamples = 30;
SampledSpectrum
class SampledSpectrum : public
CoefficientSpectrum<nSpectralSamples> { CoefficientSpectrum<nSpectralSamples> {
… } }
It is possible to convert SPD with irregular spaced It is possible to convert SPD with irregular spaced samples and more or fewer samples into a
SampledSpectrum For example sampled BRDF SampledSpectrum. For example, sampled BRDF.
SampledSpectrum
static SampledSpectrum FromSampled(
float *lambda, float *v, int n) { float lambda, float v, int n) {
<Sort samples if unordered>
SampledSpectrum r;p p ;
for (int i = 0; i<nSpectralSamples; ++i) {
lambda0=Lerp(i/float(nSpectralSamples), sampledLambdaStart, sampledLambdaEnd);
lambda1=Lerp((i+1)/float(nSpectralSamples), sampledLambdaStart, sampledLambdaEnd);
r.c[i]=AverageSpectrumSamples(lambda, v, n, lambda0, lambda1);
}
return r;
}
AverageSpectrumSamples
Human visual system
• Tristimulus theory: all visible SPDs S can be accurately represented for human observers accurately represented for human observers with three values, xλ, yλ and zλ.
Th b i th t l t hi X(λ)
• The basis are the spectral matching curves, X(λ), Y(λ) and Z(λ) determined by CIE (國際照明委員
會) 會).
S
X
d
x ( ) ( )
Y d Sy ( ) ( ) ) ( )
(
Z d Sz ( ) ( )
XYZ basis
pbrt has discrete versions (sampled every 1nm) of these bases in core/color.cpp
360 830
XYZ color
• Good for representing visible SPD to human observer but not good for spectral
observer, but not good for spectral computation.
• A product of two SPD’s XYZ values is likely
• A product of two SPD s XYZ values is likely
different from the XYZ values of the SPD which is the product of the two original SPDs.p g
• It is frequent to convert our samples into XYZ
• In In Init()Init(), we initialize the following, we initialize the following
static SampledSpectrum X, Y, Z;
static float yint; X.c[i][ ] stores the sum of X function within the ith wavelength interval using AverageSpectrumSamples yint stores the
sum of Y.c[i]
XYZ color
void ToXYZ(float xyz[3]) const { xyz[0] = xyz[1] = xyz[2] = 0.;y [ ] y [ ] y [ ]
for (int i = 0; i < nSpectralSamples; ++i) {
xyz[0] += X c[i] * c[i];
xyz[0] += X.c[i] * c[i];
xyz[1] += Y.c[i] * c[i];
xyz[2] += Z.c[i] * c[i];
xyz[2] + Z.c[i] c[i];
}
xyz[0] /= yint;
xyz[0] /= yint;
xyz[0] /= yint;
} }
RGB color
SPD for LCD displays SPD for LED displays
RGB color
SPDs when (0.6, 0.3, 0.2) is displayed on LED and LCD displays We need to know display characteristics to display the color We need to know display characteristics to display the color described by RGB values correctly.
Conversions
(R,G,B) (R,G,B) XYZToRGB
057311 1
204043 0
055648 0
041556 .
0 875992 .
1 969256 .
0
498535 .
0 537150 .
1 240479 .
device dependent 3
Here, we use the f HDTV
FromRGB
xλ, yλ, zλ
S X d
x ( ) ( )
0.055648 0.204043 1.057311
one for HDTV
A heuristic
d Z
S
d Y
S y
d X
S x
) ( ) (
) ( ) (
) ( ) (
ToXYZ
process which satisfies some
criteria z S()Z()d
criteria
spectrum (eg. SampledSpectrum)
RGBSpectrum
• Note that RGB representation is ill-defined.
Same RGB values display different SPDs on Same RGB values display different SPDs on different displays. To use RGB to display a specific SPD we need to know display
specific SPD, we need to know display
characteristics first. But, it is convenient, computation and storage efficient
computation and storage efficient.
class RGBSpectrum : public CoefficientSpectrum<3> {
i C ffi i tS t 3
using CoefficientSpectrum<3>::c;
… }
Radiometry
Photometry
L i
( ) ( )
Y V L d
Luminance
( ) ( )
Y V L d
Basic quantities
Fl (W)
non-directional Flux: power, (W)
Irradiance: flux density per area, (W/m2)
Intensity: flux density per solid angle directional Intensity: flux density per solid angle
Radiance: flux density per solid angle per area
Flux (Φ)
• Radiant flux, power
T l f i h h
• Total amount of energy passing through a surface per unit of time (J/s,W)
Irradiance (E)
• Area density of flux (W/m2)
dA E d
E 2
E
E cos
Lambert’s law Inverse square law
4 r 2 A A
Angles and solid angles
• Angle rrl
circle has 2 radians
• Solid angle A
• Solid angle 2
R
The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere.
sphere has 4 steradians
Intensity (I)
• Flux density per solid angle
I i d ib h di i dl di ib i
I d
• Intensity describes the directional distribution of light
( ) d I ( )
d
Radiance (L)
• Flux density per unit area per solid angle
d
dA d
L d
• Most frequently used,
remains constant along ray remains constant along ray.
• All other quantities can b d i d f di
be derived from radiance
Calculate irradiance from radiance )
, ( x L
d
d
x
dA
L x
d
dA x d
E( ) ( , )cos Light meter
dA
Irradiance Environment Maps
R N
( , )
L E ( , )
Radiance
Environment Map
Irradiance
Environment Map Environment Map Environment Map
Differential solid angles
Goal: find out the relationship between d and dθ d
d
between d and dθ, d Why? In the integral, d
d
r
2
) (
S
d f
y g
dω is uniformly divided. To convert the integral to
f (,)ddWe have to find the relationship We have to find the relationship between d and uniformly
divided dθ and d.
Differential solid angles
• Can we find the surface area of a unit sphere b ?
2 by ?
02 0 dd
02 0 dd Differential solid angles
Goal: find out the relationship between d and dθ d
d
between d and dθ, d
By definition, we know that
dA
sin r
d
d
r 2
r d dA
2
( )( sin )
sin
dA r d r d
r d d
2 sin
d dA2 sin d d
d d d
r
d d cos
Differential solid angles
We can prove that
d 4d sin
r S2
d
d
r
Differential solid angles
We can prove that
d 4d sin
r S2
dd
d
r
sin
2
2
d d
S
sin
sin
0 2
0
0 0
d d
d d
cos 2
1
1
0 0
d
4
1
Isotropic point source
If the total flux of the light source is Φ
light source is Φ,
what is the intensity?
I d
2
4
S
I
4 I
I 4
Warn’s spotlight
If the total flux is Φ, what is the intensity?
I ( ) cos
S
Warn’s spotlight
If the total flux is Φ, what is the intensity?
I ( ) c cos
S
2
otherwise ) 0
(
2
1 1
2
cos
S d cos d 2 c cos
S d cos
0 0
0
c
S d d
S d
1 S
c 2 0
1 1
S c y 2
1 S