Computer Graphics

Inf4/MSc

Computer Graphics

Lecture Notes #12

Colour: physics and light
Computer Graphics
Inf4/MSc

The Elements of Colour

Perceived light of different
wavelengths is in
approximately equal weights
– achromatic.
>80% incident light from
white source reflected from
white object.
<3% from black object.
Narrow bandwidth reflected
– perceived as colour

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The Visible Spectrum

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Measuring Light and
Colour
Physics: Radiometry
The amount of power per wavelength interval
• Termed radiance, we will often use intensity
• Psychophysics
Photometry
The relative brightness of a light source (colour or
black/white) when compared to a standard candle
• Termed luminance
Uniform perceptual scale
• Termed lightness
• Colourimetry

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Colour Matching
Experiment.
Adjust brightness of 3 primaries to “match” colour
C - colour to be matched, RGB - laser sources (R=700 nm,
G=546 nm, B=435 nm)

C R G C R B G
B

C=R+G+B C+R=G+B

Therefore: humans have trichromatic color vision

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Human Colour Vision.

• There are 3 light sensitive pigments in your cones (L,M,S),
each with different spectral response curve.

L = ∫ L (λ ) ⋅ E (λ )

M = ∫ M (λ ) ⋅ E (λ )
S = ∫ S (λ ) ⋅ E (λ )

• Biological basis of colour blindness

– genetic disease. © Pat Hanrahan.

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Colour Matching is Linear!

Grassman’s Laws

1. Scaling the colour and the primaries by the

same factor
preserves the match :
2C=2R+2G+2B
2. To match a colour formed by adding two
colours, add
the primaries for each colour
C1+C2=(R1 +R2)+(G1 +G2 )+(B1 +B2)
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Spectral Matching Curves

Red, Green & Blue primaries.

Match each pure colour

in the visible spectrum
with the 3 primaries, and
record the values of the
three as a function of
wavelength.

Note : We need to specify a negative amount

of one primary to represent all colours.
© Pat Hanrahan.

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Luminance Compare colour source

to a grey source

• Luminance

Y = .30R + .59G + .11B

Colour signal on a B&W tv

(Except for gamma, of course)

• Perceptual measure : Lightness

L* = Y 1/3

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Inf4/MSc

CIE Colour Space

For only positive mixing coefficients, the CIE (Commission
Internationale d’Eclairage) defined 3 new hypothetical light
sources x, y and z (as shown) to replace red, green and blue.

Primary Y intentionally
has same response as
luminance response of
the eye.
The weights X, Y, Z
form the 3D CIE XYZ
space (see next slide).

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Chromaticity Diagram.
CIE Colour Coordinates Often convenient to work
 X  2.77 1.75 1.13   Rλ  in 2D colour space, so 3D
 Y  = 1.00 4.59 0.06 G 
    λ  colour space projected onto
 Z  0.00 0.57 5.59  Bλ  the plane X+Y+Z=1 to
X yield the chromaticity
x= diagram.
X +Y + Z
Normalise by the total
Y The projection is shown
y= amount of light energy.
X +Y + Z opposite and the diagram
Z appears on the next slide.
z=
X +Y + Z

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CIE Chromaticity Diagram

C is “white” and close to x=y=z=1/3

The dominant wavelength of a

colour, eg. B, is where the line
E from C through B meets the
F D spectrum, 580nm for B (tint).
i B A and B can be mixed to
j produce any colour along the
C
line AB here including white.
A True for EF (no white this time).
k
True for ijk (includes white)

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Some device colour “gamuts”

The diagram can be
used to compare the
gamuts of various
devices. Note
particularly that a
C colour printer can’t
reproduce all the
colours of a colour
monitor. Note no
triangle can cover
all of visible space.

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Colour Cube.

R,G,B model is additive, i.e

we add amounts of 3
primaries to get required
colour.

Can visualise RGB space as

cube, grey values occur on
diagonal K to W.

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Intuitive Colour Spaces.

Artist specification of colours Tints
resulting from a pure pigment : White
Saturated → Pure
• Tint – Adding white to a pure Pigment
pigment
Greys Tones
• Shade – Adding black to a pure Shades
pigment.

• Tone – Add both black & white.

Black

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CMYK – subtractive colour model.

R = (1-C) (1-K) W
G = (1-M) (1-K) W
B = (1-Y) (1-K) W

K = G(1-max(R,G,B))
C = 1 - R/(1-K)
M = 1 - G/(1-K)
Y = 1 - B/(1-K)

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Radiometry : Radiance.
Radiometry is the science of light energy measurement

Definition: The radiance (luminance) is the power per unit

area per unit solid angle.

 W 
L( x , w )  2 
 m .sr 
Properties:
1. Fundamental quantity
2. Stays constant along a ray
3. Response of a sensor proportional to radiance

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Radiometry: Irradiance and Radiosity.

Definition: The irradiance (illuminance) is the power per

unit area incident on a surface.

E= ∫ L cosθ dω
2Π
i i i

W 
E( x)  2 
m 

tion: The radiosity (luminosity) is the power per unit area leaving a sur

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Irradiance: Distant Source

E = Es cosθ s
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Irradiance: Point Source

Φ
E= cosθ s
4Π r 2

• Inverse square law fall off

• Still has cosine dependency.

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What does Irradiance look like?

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The Reflection Equation.

• Linear response

2. Bidirectional
reflectance
distribution function
(BRDF) defines
outgoing radiance for a
given incoming
irradiance –
characteristic property
Lr ( x, ω r ) = ∫f
2Π
x ( x, ω i → ω r ) Li ( x,of
ω i surface.
) cosθ i dω i

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Approximating the BRDF.

• All illumination models in graphics are
approximations to the BRDF for surfaces.
• Frequently chosen for their visual effect,
and ease of implementation, rather than on
physical principles.
• BRDF is approximated by reflection
functions.
• Usually a total hack !
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Types of Reflection Functions

• Ambient.
• Ideal Specular
– Mirror
– Reflection Law
• Ideal Diffuse
– Matte
– Lambert’s Law
• Specular
– Glossiness and
Highlights
– Phong and Blinn Models

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Ambient Reflection.
• Simplest illumination model.
• There is assumed to be global ambient
illumination in the scene, Ia
• Amount of ambient light reflected from a
surface defined by ambient reflection
coefficient, ka.
• Ambient term is I = Ia.ka
• No physical basis whatsoever !
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Mirror: Ideal Specular Surface

Calculation of the reflection vector involves
mirroring L about N.
Law of Reflection Both L and N are normalised.

Projection of L onto N is N cosθ

N
By vector subtraction and congruent triangles :
L S S R S = N cosθ − L
So :
N cosθ

θi θ R = 2 N cosθ − L
r

Subsitute N .L for cos θ :

R = 2 N .( N .L ) − L

θr= θi
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Matte: Ideal Diffuse Reflection.

• Dull surfaces such as chalk exhibit diffuse
or Lambertian reflection.
• Reflect light with equal intensity in all
directions.
• For a given surface, brightness depends
only on the angle between the surface
normal and the light source.

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Matte: Ideal Diffuse Reflection.

Ip N 2 effects to consider :
L • The amount of light reaching the
surface.
θ • Beam intercepts an area dA/ cos
θ
• cos θ dependence.

• The amount of light seen by the

θ
viewer.
dA dA • Also cos θ dependence per unit
cosθ surface area
• BUT amount of surface seen by
viewer also has cos θ dependence.

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Matte: Ideal Diffuse Reflection.

Ip N
L
The diffuse lighting equation is :
θ
I = I p k d cosθ
If N and L are both normalized :

θ I = I p k d ( N .L )
dA dA
cosθ

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Matte: Ideal Diffuse Reflection.

• Diffuse coefficient defined for each surface.
• Diffusely lit objects often look harshly lit
– Ambient light often added.
• Poor physical basis for diffuse reflection.
– Internal reflections inside the material etc…

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Specular reflection.
• Can be observed on a shiny surface, e.g nice red
apple lit with white light.
• Observe highlights on surface.
• Highlight appears as the colour of the light, rather
than of the surface.
• Highlight appears in the direction of ideal
reflection. Now view direction important.
• Materials such as waxy apples, shiny plastics have
transparent reflective surface.
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The Phong model.

R Assume specular highlight is at a
N maximum when α = 0 , and falls off
L rapidly with larger values of α
θ θ V • Fall-off depends on cosn α.
α

• n referred as specular exponent.

• For perfect reflector, n is infinite.

I λ = I a k a + I p [k d cosθ + k s cos n α ]

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The Phong model.

H R • An alternative formulation uses
N halfway vector, H
L β
• It’s direction is halfway between
θ θ V viewer and light source.
α
• If the surface normal was oriented at
H, viewer would see brightest highlights.

H = (L +V ) / L +V • Note α ≠ β , both formulations are

approximations.
Specular term is now
( N .H )n

If viewer and light source at infinity, H is constant

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Rough Surface : Microfacet distribution.
Physical justification for Phong model is
that the surface is rough and consists of
microfacets which are perfect specular
reflectors.

Distribution of microfacets determines

specular exponent.

L N
N′ R

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Material Selection.
Ambient 0.52 Ambient 0.39
Diffuse 0.00 Diffuse 0.46
Specular 0.82 Specular 0.82
Shininess 0.10 Shininess 0.75

Light intensity 0.31 Light intensity 0.52

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Summary of Lighting.

• Surface reflection specified by BRDF.

• BRDF approximated by ambient, diffuse
and specular reflection.
• Lambertian reflection.
• Phong Lighting model.

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