### 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 0****2**

**1 0****4**

**1 0****6**

**1 0****8**

**1 0****1 0**

**1 0****1 2**

**1 0****1 4**

**1 0****1 6**

**1 0****1 8**

**1 0****2 0**

**1 0****2 2**

**1 0****2 4**

**1 0****2 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 0****1 6**

**1 0****1 4**

**1 0****1 2**

**1 0****1 0**
**1 0****8**

**1 0****6**

**1 0****4**

**1 0****2**

**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 _{molecules}^{pigment}
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**

p_{1} = 645.2 nm
p_{2} = 525.3 nm

444 4

Foundations of Vision, by Brian Wandell, Sinauer Assoc., 1995

p_{1 } p_{2 } p_{3}
p_{1 } p_{2 } p_{3}

p_{3} = 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, x*_{λ}*, y*_{λ}*and z*_{λ}.

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*

*S*

*y* ( ) ( )
)
(
)

(

###

###

###

###

###

*Z*

*d*

*S*

*z* ( ) ( )

**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**

*x*_{λ}*, y*_{λ}*, z*_{λ}

*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/m^{2})

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/m^{2})

*dA*
*E* *d*

*E* 2

*E*

*E* cos

Lambert’s law Inverse square law

*4 r* 2 *A* *A*

**Angles and solid angles**

• Angle ^{} ^{} _{r}_{r}^{l}

** 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 ^{d}^{}l 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}^{(}

^{}

^{,}

^{}

^{)}

^{d}^{}

^{d}^{}

We 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 ?

###

0^{2}

^{ }0

*d*

^{}

*d*

^{}

###

0^{2}

^{ }0

*d*

^{}

*d*

^{}

**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* *dA*_{2} sin *d d*

*d* *d d*

*r*

*d*
*d cos*

**Differential solid angles**

We can prove that ^{} ^{}

###

^{d}^{}

^{}

^{4}

^{}

*d*
sin

*r* ^{S}^{2}

*d*

*d*

*r*

**Differential solid angles**

We can prove that ^{} ^{}

###

^{d}^{}

^{}

^{4}

^{}

*d*
sin

*r* ^{S}^{2}

###

^{d}*d*

*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}

^{I}

^{S}^{}

**Warn’s spotlight**

If the total flux is Φ, what is the intensity?

### _{I} _{(} _{} _{)} _{} _{} ^{} ^{c} ^{cos}

_{I}

^{c}

^{S}^{} ^{} ^{}

^{}

^{2}

###

*otherwise* ) 0

### (

### 2

1 1

2

###

^{}

### ^{cos}

^{S}^{} ^{d} ^{cos} ^{} ^{d} ^{} ^{2} ^{c} ^{cos}

^{d}

^{d}

^{S}^{} ^{d} ^{cos} ^{}

^{d}

0 0

0

###

###

### ^{c}

^{c}

^{S}^{} ^{d} ^{} ^{d} ^{} ^{}

^{d}

^{d}

^{S}^{} ^{d} ^{}

^{d}

### 1 S

### c 2 0

### 1 1

### S c y 2

1 S

###

###

###

###

^{}