Color Basics - ראשי

Color Basics

Lecture 2

Gamma X rays Infrared Radar FM TV AMUltra-violet

10-12

10-8

10-4

104

1 108

electricityACShort-

wave

400nm 500nm 600nm 700nm

Wavelength

Wavelength in meters (m)

Visible light

Wavelength EncodingTrichromatic Color Theory

Color Matching Experiments

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Newton’s Experiment

1665, Cambridge University


Electromagnetic Radiation -Spectrum

Gamma X rays Infrared Radar FM TV AMUltra-violet

10-12

10-8

10-4

104

1 108

electricityACShort-

wave

400 nm 500 nm 600 nm 700 nmWavelength in nanometers (1nm=10-9 m)

Wavelength in meters (m)

Visible light


The Spectral Power Distribution (SPD) of a light is a function f(l) which defines the energy at each wavelength.

Wavelength (λ)

400 500 600 7000

0.5

1

Re

lative

Po

wer

Spectral Power Distribution


Monochromators

Monochromators measure the power or energy at different wavelengths


Examples of Spectral power Distributions

Blue Skylight Tungsten bulb

Red monitor phosphor Monochromatic light

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0.5

1

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0.5

1

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0.5

1

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0.5

1


Superposition of Light SPDs

wavelength

pow

er

pow

er

pow

er

wavelength

wavelength


• Low illumination levels (Scotopic vision).

• Highly sensitive (respond to a single photon).

• 100 million rods in each eye.

• No rods in fovea.

Retinal Photoreceptors

Rods -

Wavelength (nm)

Re

lative

sensiti

vity

400 500 600 7000

0.25

0.5

0.75

1

Rod Spectral Sensitivity


Cones - • High illumination levels (Photopic vision)

• Less sensitive than rods.

• 5 million cones in each eye.

• Only cones in fovea (aprox. 50,000).

• Density decreases with distance from fovea.

• 3 cone types differing in their spectral

sensitivity: L , M, and S cones.

Retinal Photoreceptors

Wavelength (nm)

Re

lative

sensiti

vity

Cone Spectral Sensitivity

400 500 600 7000

0.25

0.5

0.75

1

ML

SM


L and M Cones -Density decreases with distance from fovea.None past 40 deg.

S Cones -None in the fovea (central 25’).Very sparse elsewhere.

Sensitivity to color decreases with distance from fovea in the order: green, red, yellow, blue.


Color Sensitivity test – distance from fovea


Color Deficiency

Trichromats - use 3 sensors

Dichromats - use 2 sensors (8% males, 0.05% females)protanopia - missing red conedeuteranopia - missing green conetritanopia - missing blue cone

Monochromats - use 1 sensor.

protanopia deuteranopia tritanopia

Dichromatic confusions:


Color Deficiency

ProtanopeNormal

Deuteranope Tritanopia


Ishihara Plates (1917).


Reverse Ishihara Plates

Normal Should see Nothing.CVD should see 5

Normal Should see both 2 and 6Deutanopes should see 2 more easilyProtanopes should see 6 more easily

Normal Should see 3.CVD should see 5

Normal Should see 73.CVD should nothing.


Wavelength (λ)

400 500 600 7000

0.5

1

Re

lative

Po

wer

Spectral Power Distribution

High dimensional data

RGB – 3 dimensional


Trichromatic Color Theory

Thomas Young (1773-1829) -A few different retinal receptors operating with different wavelength sensitivities will allow humans to perceivethe number of colors that they do.Suggested 3 receptors.

Helmholtz & Maxwell (1850) -Color matching with 3 primaries.

Trichromatic: “tri”=three “chroma”=colorcolor vision is based on three primaries(i.e., it is 3 dimensional).


Color Matching Experiment

Thomas Young 1802Helmholtz & Maxwell 1850Wright 1929Stiles & Burch 1959Judd & Wyszeki 1975

Metamer - two lights that appear the same visually. They might have different SPDs(spectral power distributions).

400 500 600 7000

400

800

400 500 600 7000

100

200

Wavelength (nm)

Pow

er

The phosphors of the monitor were set to match the tungsten light.

Tungsten light Monitor emission


+ -

+ -

+ -

test match

Three primary lights are set to match a test light.

=~

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1

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Test light Match light



If

and matches

matches

then matches

Color Matching Experimentis Linear

Homogeneity + additivity


Color Matching Experimentis a Linear System

There exists a system matrix thatmaps test SPD to Match intensities.

400 500 600 700

Test Match

LinearSystem

C T = M


Given a set of primaries, one can determine for every spectral wavelength, the intensity of the guns required to match a monochromatic light of that spectral wavelength.

Color Matching Functions

Monochromatic lights

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0.5

1

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0.5

1

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0.5

1

Primary Intensities


These values form the Color Matching Functions associated with the primaries.

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0.2

0.4

0.6

0.8

1

Wavelength (nm)


The intensity of the primaries required to match any spectra can then be determined by inner product of the spectra with the 3 color matching functions.


Wavelength (λ)400 500 600 700

0

0.5

1

Wavelength (λ)400 500 600 700

0

0.5

1

Test Light = sum of monochromatic lights

=


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0.2

0.4

0.6

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1

Wavelength (nm)

ColorMatchingFunctions

TristimulusValues


)(3 λM

Wavelength (λ)

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0.5

1

Tristimulus Values = Inner product of SPD and CMF

Tristimulus Calculation

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0.2

0.4

0.6

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1

dotproduct

λλλλ

dMte ∫= )()( 11

)(λt)(1 λM

)(2 λM

λλλλ

dMte ∫= )()( 22

λλλλ

dMte ∫= )()( 33

Wavelength (λ)


Wavelength (λ)400 500 600 700

0

0.5

1

Rela

tive

Po

we

r

Light

Matrix Representation of the color matching system

Wavelength (λ)400 500 600 700

0

0.5

1

Rela

tive

Po

we

r

Wavelength (λ)

0

0.5

1

Se

nsi

tivity

400 500 600 700

Sensor

s

0.5

Wavelength (λ)

0

1

Se

nsi

tivity

400 500 600 700

I


Wavelength (λ)400 500 600 700

0

0.5

Rela

tive

Po

we

r

s

Wavelength (λ)

0

0.5

Se

nsi

tivity

400 500 600 700

t

Sensor Response Calculation

e = <s,t> = sT t =


M1

t

Tristimulus Calculation

)(3 λM

Wavelength (λ)

400 500 600 7000

0.5

1

400 500 600 7000

0.2

0.4

0.6

0.8

1

dotproduct

)(λt)(1 λM

)(2 λM

Wavelength (λ)

M2

M3

e1

e2

e3

=


+ -

+ -

+ -

test match


Primaries

P1 P2 P3

Primaries

=

SpectraTristimulus

values

e1

e2

e3

t~


Thus there must exist a 3 x nλ system matrix that mapsinput to output:

nλ

Color Matching is Linear

Color Matching is a linear system.i.e.

Color Matching defines a linear mapping from the test spd (nλ x 1 vector) to 3 primary intensities (3 x 1 vector).

M3

M2

M1

=

e1

e2

e3

t

SpectraTristimulus

values

e1

e2

e3

t

e = C t

nλ


[0 0 . . . 1 ]t

[1 0 . . . 0 ]t

Measure intensities of primaries for monochromatic test lights:

e1

en

. . .

.

λ

. . .

.

t2

t1 =

=

e are the columns of the color matching system matrix C.

i

C = [ e1

. . . .

enλ

]

Now, given the color matching functions, we

can calculate the response to any test light t :

e = C t

Calculating the system matrix


The rows of the system matrix are the color matching functions

with respect to the given primaries.


M3

M2

M1

=C

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0.2

0.4

0.6

0.8

1

Wavelength (nm)

ColorMatchingFunctions


M3

M2

M1

=1 0 00 1 00 0 1

MatchingFunctions Primaries

Note an important relationship between Primaries and

their Matching Functions:

P1 P2 P3


400 500 600 7000

20

40

60

80

Wavelength (nm)

Every color can be represented by 3 values.

Space of visible colors is 3 Dimensional.

“tri”=three “chroma”=color


e1

e2

e3


Stiles & Burch (1959) Color matching functions.Primaries are: 444.4 525.3 645.210 deg field..

Given the color matching functions, we can describeany light with 3 values (CIE-RGB):

r(λ)

g(λ)b(λ)

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0

1

2

3

Wavelength (nm)

Prim

ary

Inte

nsity

(85, 38, 10) (21, 45, 72) (65, 54, 73)

Color Representation


Caveat: For some matches e may be negative.i

t = e p + e p + e p1 1 22 3 3

~ e < 01

This does not make sense physically, however mathematically:

t = - |e1| p1 + e2 p2 + e3 p3~

t + |e1| p1 = e2 p2 + e3 p3~then

with all positive coefficients.

Physically this can be interpreted as adding

primary light p to the test : 1

+ -

+ -

+ -

test match

e.g.:


The Color Matching Experiment Predicts Metameric Matches

S and T match perceptually; they are metamers

400 500 600 7000

400

800

400 500 600 7000

100

200

Wavelength (nm)

Pow

er

Tungsten light Monitor emission

M3

M2

M1

t =M3

M2

M1

s


Using different primary lights

1) Primary lights must be visually independent.

2) Uniqueness of the color matching functions:

p p p1 2 3

cmf C

or

This is true for all t.

primaries p p p` ` `1 2 3

C̀

given a test light t: e = C t e = C t``

denote P = [p1 p2 p3 ] P = [p p p ]1 2 3

` ` ` `

Since t is metameric to Pe and to Pe :` `

C t = C P e = C P e` `

e = C P e` `

t = P e t = P e` `we have ~ ~


test match

Primaries

test match

Primaries

test match

PrimariesPrimaries


C = (C P ) C` `

CP’ is a 3 x 3 matrix relating the two sets of color matching functions.

The color matching functions are unique up to a free 3 x 3 linear transformation.

i.e.

For monochromatic t:e are the columns of C,e are the columns of C.

e = C P e` ` for all t.

``

3 x 3 matrix


Linear Transformation of The CMFsPredict the Same Metamers:

S and T are metamers

M3

M2

M1

t =M3

M2

M1

s

M3

M2

M1

t =M3

M2

M1

s LL

The color matching functions are uniqueup to a free linear transformation.


CMF vs Human Photoreceptors

Single Unit Cone Photocurrent Measurements(Schnapf, Baylor et al – 1987).


Current Recordings From a Cone


Cone Spectral Sensitivities

Wavelength (nm)

Re

lative

sensitiv

ity

Cone Spectral Sensitivity

400 500 600 7000

0.25

0.5

0.75

1

ML

SM


There are Three Types of Cone Wavelength Responsivity


Stimuli Causing Equal Cone Signals Match Perceptually

L M S

L M S

Ab

sorp

tion

rat

e

Cone type


Behavioral CMFs are accurately predicted by cone responsivities

B

R

G


The cone responsivities are a linear transformation from the CMFs

SML

=M3

M2

M1L

3x3 matrix

SensorResponsivities

Color MatchingFunctions

= L

3x3 matrix

SensorRespones

TristimulusValues

e1

e2

e3

L

M

S


400 500 600 7000

20

40

60

80

Wavelength (nm)

Every color can be represented by 3 values.

Space of visible colors is 3 Dimensional.

“tri”=three “chroma”=color


Color Representation ?

e1

e2

e3


Color Matching Predicts Matches, Not Appearance

Albers (1975)


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Color Basics - ראשי

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