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12/4/2013 1 Newton’s Color Experiments 1671 White light has seven constituent components: red, orange, yellow , green, blue, indigo and violet. Dispersed light can be recombined to form white light. Magenta and purple can be obtained by combining only portions of the spectrum. Newton’s Conclusions
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Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Page 1: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

1

Newton’s Color Experiments 1671

• White light has seven constituent components: red, orange, yellow, green, blue,indigo and violet.

• Dispersed light can be recombined to form white light.

• Magenta and purple can be obtained by combining only portions of the spectrum. 

Newton’s Conclusions

Page 2: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

2

Color and Color Vision

The perceived color of an object depends onfour factors:1. Spectrum of the illumination source2. Spectral Reflectance of the object3. Spectral response of the photoreceptors

(including bleaching)4. Interactions between photoreceptors

Light Sources

400 700nm

SpectralOutput

400 700nm

SpectralOutput

400 700nm

SpectralOutput

White Monochromatic Colored

Page 3: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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3

Pigments

Red Green Blue

400 700nm

SpectralReflectance

400 700nm

SpectralReflectance

400 700nm

SpectralReflectance

Subtractive ColorsPigments (e.g. paints and inks) absorb different portions of the spectrum

400 700nm

SpectralOutput

400 700nm

SpectralOutput

400 700nm

SpectralOutput

400 700nm

400 700nm

400 700nm

Multiply the spectrum of the light source by the spectral reflectivityof the object to find the distribution entering the eye.

Page 4: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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4

Blackbody Radiation

For lower temperatures,blackbodies appear red.As they heat up, theshift through the spectrumtowards blue.

Our sun looks like a6500K blackbody.

Incandescent lights are poorefficiency blackbodiesradiators.

Gas‐Discharge & Fluorescent Lamps

A low pressure gas or vapor isencased in a glass tube. Electricalconnections are made at the ends ofthe tube. Electrical discharge excitesthe atoms and they emit in a seriesof spectral lines. We can use individual lines for illumination (e.g. sodium vapor) or ultraviolet lines to stimulate phosphors.

Page 5: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

5

CIE Standard Illuminants

Illuminant A - Tungsten lamp looking like a blackbody of 2856 KIlluminant B - (discontinued) Noon sunlight.Illuminant C - (discontinued) Noon sunlight.

Illuminant D55 - (Occasionally used) 5500 K blackbodyIlluminant D65 - 6500 K blackbody, looks like average sunlight and

replaces Illuminants B and C.Illuminant D75 - (Occasionally used) 7500 K blackbody

Illuminants A and D65

0

50

100

150

200

250

300

300 500 700

Wavelength (nm)

Sp

ectr

al R

adia

nce

Illuminant A

Illuminant D65

Page 6: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

6

Additive ColorsSelf-luminous Sources (e.g. lamps and CRT phosphors) emit differentspectrums which combine to give a single apparent source.

400 700nm

SpectralOutput

400 700nm

SpectralOutput

400 700nm

SpectralOutput

Add the spectrums of the different light sources to get the spectrum of the apparent sourceentering the eye.

Color Models

• Attempt to put all visible colors in a ordered system.

• Mathematics based, art based and perceptually based systems.

Page 7: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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7

RGB Color Model

A red, green and blue primary are mixed in different proportionsto give a color

(1,0,0) is red(0,1,0) is green(0,0,1) is blue

24 bit color on computer monitors devote 8 bits (256 values) to eachprimary color (i.e. red can take on values (0…255) / 255)

(1,1,1) is white(0,0,0) is black

HSB (HSV) Color Model

Hue – color is represented by angleSaturation – amount of white representedby radial positionBrightness (Value) – intensity is representedby the vertical dimension

Page 8: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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8

HLS Color Model

Hue – color is represented by angleLightness – intensity is representedby the vertical dimensionSaturation – amount of white representedby radial position

CIE

• 90 year old commission on color

• Recognized as the standards body for illumination & color. 

• Has defined standard illuminants and human response curves.

Page 9: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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9

Luminosity FunctionsThe spectral responses of the eye are called the luminosity functions. TheThe V() curve (photpic response) is for cone vision and the V’() curve (scoptopic response) is for rod vision. V() was adopted as a standard by the CIE in 1924. There are some errors for <500 nm, that remain. TheV’() curve was adopted in 1951 and assumes an observer younger than30 years old.

00.10.20.30.40.50.60.70.80.9

1

380 480 580 680 780

Wvaelength (nm)

Rel

ativ

e S

pec

tral

Sen

siti

vity

Photopic

Scotopic

V()V’()

Purkinje Shift

Note how the brightness of reds and blues change with decreasingillumination. This is due to the sensitivity of the eye shifting from photopic to scoptic

Page 10: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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10

Human Color Models

RG

B

C

Observer adjusts the Luminances of R, G and Blights until they match C

C r’(R + g’(G + b’(B

R = 700.0 nmG = 546.1 nmB = 435.8 nm

1931 CIE Color Matching Functions

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

380 430 480 530 580 630 680 730 780

Wavelength (nm)

Ch

rom

ati

city

Co

ord

ina

tes

r'()g'()

b'()

435.8 546.1 700.0

Page 11: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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11

Color Matching Functions

R

G

B

C

What does a negative value of the ColorMatching Function mean? Bring one light to the other side of the field. Observer now adjusts, for example, the Luminances of G and B lights until they match C+ R

C + r’(R g’(G + b’(B

R = 700.0 nmG = 546.1 nmB = 435.8 nm

1931 CIE Color Matching Functions

The CIE defined threetheoretical primaries x’, y’ and z’ such that the color matching functions are everywhere positive and the “green” matching function is the same as the photopic response of the eye. -0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

380 430 480 530 580 630 680 730 780

Wavelength (nm)

Ch

rom

ati

city

Co

ord

ina

tes

x'()y'()

z'()

Page 12: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

12

Conversion x’y’z’ to r’g’b’

'z

'y

'x

17860.000255.000092.0

01571.025243.009117.0

08283.015860.041846.0

'b

'g

'r

The x’, y’, z’ are more convenient from a book-keeping standpointand the relative luminance is easy to determine since it is related to y’.

The consequence of this conversion is that the spectral distribution ofthe corresponding primaries now have negative values. This meansthey are a purely theoretical source and can not be made.

Tristimulus Values X, Y, Z

d)('z)(PZ

d)('y)(PY

d)('x)(PX

0

0

0

The tristimulus values are coordinatesin a three dimensional color space. Theyare obtained by projecting the spectraldistribution of the object of interest P()onto the color matching functions.

Page 13: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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13

Chromaticity Coordinates x, y, z

yx1ZYX

Zz

ZYX

Yy

ZYX

Xx

The chromaticity coordinatesare used to normalize out thebrightness of the object. This way,the color and the brightness can bespearated. The coordinate z is notindependent of x and y, so this isa two dimensional space.

Chromaticity Coordinates

Light SourceSpectralDistribution

Object SpectralReflectance orTransmittance

Spectral Distributionof Light entering theeye.

x =

Calculate TristimulusValues X, Y, Z

Project ontox’, y’, z’

CalculateChromaticityCoordinates

Normalize

Plot (x, y)

Page 14: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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14

Example ‐ Spectrally Pure Colors

Suppose P() = (-o)

)('zd)('z)(Z

)('yd)('y)(Y

)('xd)('x)(X

o

0

o

o

0

o

o

0

o

yx1z

)('z)('y)('x

)('yy

)('z)('y)('x

)('xx

ooo

o

ooo

o

Plotting x vs. y for spectrally pure colors gives the boundary of color vision

CIE Chromaticity Chart

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

x chromaticty coordinate

y ch

rom

atic

ity

coo

rdin

ate

Page 15: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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15

Example ‐White Light

Suppose P() =

1d)('zZ

1d)('yY

1d)('xX

0

0

0

33.0yx1z

33.0111

1y

33.0111

1x

CIE Chromaticity Chart

Page 16: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

16

Example ‐MacBeth Color Checker

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

380 430 480 530 580 630 680 730 780

Wavelength (nm)

Sp

ectr

al R

efle

ctan

ce

Blue Sky

Example ‐ Blue Sky Patch

0

20

40

60

80

100

120

140

380 480 580 680 780

Wavelength (nm)

Rel

ativ

e R

adia

nce

Illuminant C

After Reflection

The Macbeth colorchecker assumes thatIlluminant C is usedfor illumination.

Multiply the spectraldistribution of IlluminantC by the spectral reflectance of the color patch to get the light entering the eye.

Page 17: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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17

Example Blue Sky Patch

253.)9.3796.1921.188/(6.192y

247.)9.3796.1921.188/(1.188x

9.379)('z)(PZ

6.192)('y)(PY

1.188)('x)(PX

Example ‐MacBeth Color Checker

Dark Skin

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

380 430 480 530 580 630 680 730 780

Wavelength (nm)

Sp

ectr

al R

efle

ctan

ce

Page 18: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

18

Example ‐ Dark Skin Patch

Dark skin is much darker than the blue sky. The chromaticity coordinates,however, remove the luminance factor and onlylook at color.

0

20

40

60

80

100

120

140

380 480 580 680 780

Wavelength (nm)

Rel

ativ

e R

adia

nce

Illuminant C

Dark Skin

Blue Sky

Example Dark Skin

35.0)9.3796.1921.188/(6.192y

41.0)9.3796.1921.188/(1.188x

2.66)('z)(PZ

8.98)('y)(PY

6.113)('x)(PX

Page 19: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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19

CIE Chromaticity Diagram

Think of this as a distortedversion of the HSB color model.

W = White Point (0.33,0.33)D = Dominant Wavelength (hue)

W

D

AWD

WApurity excitationp C

BC = Complimentary Color

WC

WBpurity excitationp

MacAdam EllipsesJust noticeable differences for two similar colors is nonlinear onthe CIE diagram. Would like acolor space that these ellipses become circles. (ellipses are 3xlarger than actuality)

Page 20: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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20

1976 CIELUV

Television and Video

3y12x2

y9

Z3Y15X

Y9'v

3y12x2

x4

Z3Y15X

X4'u

v'v*L13*v

u'u*L13*u

008856.0Y

Yfor

Y

Y292.903*L

008856.0Y

Yfor 16

Y

Y116*L

n

n

nn

n

3/1

n

L* is related to the lightness andis nonlinear to account for thenonlinear response of the visualsystem to luminance. The u’s and v’s distort the CIE diagram to make the MacAdam ellipses more round.

Yn, un and vn are for white

1976 CIELUV

222 *v*u*LE

DifferenceColor CIELUV

*

*

uv

***uv

u

vtanh

vuC

1

22

180

Polar Coordinates

Page 21: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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21

1976 CIELAB

Plastic, Textile & Paint L* is related to the lightness andis nonlinear to account for thenonlinear response of the visualsystem to luminance. The a’s andb’s distort the CIE diagram to make the MacAdam ellipses moreround.

Xn, Yn and Zn are for white

0.008856sfor 16/1167.787sf(s)

0.008856sfor sf(s) where

Z

Zf

Y

Yf200*b

Y

Yf

X

Xf500*a

16Y

Yf116*L

3/1

nn

nn

n

1976 CIELAB

*

*

ab

***ab

a

btanh

baC

1

22

180

Polar Coordinates

222 ****ab baLE

Color Difference

Page 22: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

12/4/2013

22

Color Difference Formulas

• DE = 1 is approximate threshold for Just Noticeable Difference.

• CIELAB used primarily today instead of CIELUV

• CIELAB DE > 5, but some variation for smaller differences.

• CIEDE2000 performs better for small color differences.

CIEDE2000

Westland – Computational Colour Science using Matlab

Page 23: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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23

Comparison of 10 Samples

CIELAB Space

Page 24: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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24

Color Difference Comparison

Human Cone Sensitivities

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

380 430 480 530 580 630 680 730 780

Wavelength (nm)

Re

lati

ve S

en

siti

vity

LS M

Only during the 1990swere researcher able todistinguish the conesensitivities

Page 25: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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25

LMS Color Space

Project P onto LMS color matchingfunctions. The (l,m) is analogous to(x,y) chromaticity coordinates.

Color Blindness

• Protanopia ‐ No L Cones

– 1% men, rare in women

• Deutranopia ‐ No M Cones

– 1% men, 0.01% women

• Tritanopia ‐ No S Cones

– rare

Page 26: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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26

Color Blindness

Color Blindness

Normal Protonopia Deuteranopia

No L Cones No M Cones

Page 27: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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27

Color Blindness Examples

ProtanopiaDeuteranopiaNormal

Color Space Conversions

709

709

709

B

G

R

950227.0119193.0019334.0

072169.0715160.0212671.0

180423.0357580.0412453.0

Z

Y

X

This is a conversion from an RGB color model to XYZ trichromaticcoordinates. The 709 refers to a standard set of phosphors used inmost displays. RGB colors are assumed to range from 0..1. However,in the computer they usually range from 0..255. Simply divide the computer value by 255 to normalize.

Page 28: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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28

Color Space Conversions ‐ Example

0

0

1

950227.0119193.0019334.0

072169.0715160.0212671.0

180423.0357580.0412453.0

Z

Y

X

X = 0.412453Y = 0.212671Z = 0.019334

x = 0.412453/(0.412453+0.212671+0.019334)x = 0.64y = 0.212671/(0.412453+0.212671+0.019334)y = 0.33

Example (1, 0, 0)

Page 29: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Color Space Conversions

L+M Channels

0

0.2

0.4

0.6

0.8

1

1.2

380 480 580 680 780

Wavelength (nm)

Re

lati

ve S

en

siti

vity

0.68273L+0.35235M

Photopic Response

Page 30: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Opponent Processes

Opponent Colors

Page 31: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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31

Camera White Balance

Photofocus.com

Von Kries Transform

1

1

1

12

12

12

2

2

2

00

00

00

S

M

L

SS

MM

LL

S

M

L

WW

WW

WW

Sometimes called “wrong” von Kries transform when done on XYZ instead of LMS.

Page 32: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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32

Chromatic Adaptation Transform

• Von Kries is a little too simple due to interaction between the channels.

• CATs try to match real data

1

1

1

12

12

121

2

2

2

00

00

00

Z

Y

X

M

ZZ

YY

XX

M

Z

Y

X

CAT

WW

WW

WW

CAT

Questions on Trichromatic Theory

• What does bluish‐yellow look like?

• What does greenish‐red look like?

• Why do colorblind people either lose red‐green or yellow‐blue colors in pairs?

• Red, green and blue appear to be pure colors (i.e. not a mixture of other colors).  Why does yellow also appear as a pure color?

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33

Opponent Process

Opponent Process Test

How much yellowis needed to cancelblue tint?

How much blueis needed to cancelyellow tint?

How much greenis needed to cancelred tint?

How much redis needed to cancelgreen tint?

Page 34: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Opponent Process

Opponent Process

Page 35: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Opponent Process

Opponent Colors

Page 36: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Receptive Fields

The lighter photoreceptors are the ones that activate in response to light while the others are the photoreceptors that activate in the absence of light. As you can see, some of the dark photoreceptors encircle the lighter ones and vice versa. In reality, however, these two types of photoreceptors look the same.

Receptive Fields

Center-ON

Center-OFF

Page 37: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Hermann’s Grid

Hermann’s Grid

Page 38: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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38

Mach Bands

Mach Bands

Page 39: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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39

Mach Bands

Lightness Contrast

Page 40: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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40

Lightness Contrast

Color Contrast

Page 41: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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41

Complex Color Shifts

Rubik’s Cube??

Page 42: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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Rubic’s Cube

Blue Dot

Page 43: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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43

Blue Dot

Color Mach Bands

Page 44: Newton’s Color 1671...Newton’s Color Experiments 1671 • White light has seven constituent components: red, orange, yellow, green,blue, indigo and violet. • Dispersed light

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44

Color Contrast Example

R=200G=200B=200

R=249G=245B=24

R=249G=106B=71

Color Contrast

Rnorm = R/255

Scale XYZ to make Yw = 100

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CIECAM02 Output

• Brightness is the subjective appearance of how bright an object appears given its surroundings and how it is illuminated. 

• Lightness is the subjective appearance of how light a color appears to be. 

• Colorfulness is the degree of difference between a color and grey. 

• Chroma is the colorfulness relative to the brightness of another color that appears white under similar viewing conditions. This allows for the fact that a surface of a given chroma displays increasing colorfulness as the level of illumination increases. 

• Saturation is the colorfulness of a color relative to its own brightness. 

• Hue is the degree to which a stimulus can be described as similar to or different from stimuli that are described as red, green, blue, and yellow.

Source: Wikipedia

Lightness Contrast

Sample Color

White

BackgroundLuminance (Yb)

CIECAM02

Chroma & Hue unchanged,Brightness increased

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CIECAM02 Example

Hyperspectral Imaging

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Singular Value Decomposition (SVD)

A matrix Amxn with m rows and n columns can be decomposed into

A = USVT

where UTU = I, VTV = I (i.e. orthogonal) and S is a diagonal matrix.

If  Rank(A) = p, then Umxp, Vnxp and Spxp

OK, but what does this mean is English?

SVD by Example

Keele Data on Reflectance of Natural Objectsm = 404 rows of different objectsn = 31 columns, wavelengths 400‐700 nm in 10 nm stepsRank(A) = 31 means at least 31 independent rows

A404x31=

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SVD by Example

UTU = I means dot product of two different columns of U equalszero. VTV = I means dot product of two different columns of V (rowsof VT) equals zero.

A404x31 U404x31 S31x31 VT31x31=

Basis Functions

V31x31=

Columns of V are basis functions that can be used to representthe original Reflectance curves.

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Basis Functions

First column handles most of the variance, then the second columnetc.

Singular Values

S31x31=

The square of diagonal elements of S describe the varianceaccounted for by each of the basis functions.

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SVD Approximation

The original matrix can be approximated by taking the first dcolumns of U, reducing S to a d x d matrix and using the first drows of VT.

A404x31 U404xd Sdxd VTdx31~

SVD ReconstructionThree Basis Functions

Five Basis Functions