Color Measurement and Reproduction

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Color Measurement and Reproduction

• Eric Dubois

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How Can We Specify a Color Numerically?

• What measurements do we need to take of a colored
  light to uniquely specify it?
• How can we reproduce the same color on a display?
• on a printer?

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Color Vector Space

• The appearance of a colored light is determined
  by its power spectral density
• A color is a set of all that appear
  identical to a human viewer, denoted or
  
• The set of colors can be embedded in a
  3-dimensional vector space. A basis for the
  vector space is the set of primaries ,
  ,
• Any color can be expressed

Tristimulus values
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Determination of tristimulus values
Color matching functions
The color matching functions are determined by
subjective experiment ONCE for one set of
primaries P1, P2, P3. For any color
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CIE 1931 Red, Green and Blue primaries
B(l)d(l-435.8) G(l)d(l-546.1) R(l)d(l-700.0)
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Transformation of primaries

• Obtaining the tristimulus values with respect to
  a new set of primaries is a change of basis
  operation.
• is a given set of primaries and
  is a different set.
• We can express the primaries
  in terms of

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Transformation of primaries (2)
For an arbitrary color
From which we can identify
In matrix form
or
A
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Transformation of primaries (3)
The relationship between the sets of primaries
can also be expressed in matrix form
AT
Note that this is a symbolic equation involving
elements of the color vector space C
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Transformation of primaries (4)
Recognizing that color matching functions specify
tristimulus values for each l
Each new color matching function can be viewed as
a linear combination of the three old color
matching functions.
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The CIE XYZ primaries

• In 1931, the CIE defined the XYZ primaries so
  that all the color matching functions are
  positive, and the Y component gives information
  about the brightness (luminance, to be
  discussed).
• These new primaries are not physical primaries.

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The CIE XYZ primaries (2)
If C is an arbitrary color that can be
expressed
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The CIE XYZ primaries (3)
Applying this to each l, we get the XYZ color
matching functions
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The CIE XYZ primaries (4)
The set of physical colors in XYZ space
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XYZ frequency sweep
X
Y
Z
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Specification of a set of primaries

1. Each new primary is expressed in terms of
   existing primaries, usually XYZ, i.e. X, Y,
   Z take the role of the The matrix AT
   is specified.For example, the 1976 CIE Uniform
   Chromaticity Scale (UCS) primaries are given by

It follows that
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Specification of a set of primaries (2)

1. The matrix equation to calculate the tristimulus
   values of an arbitrary color with respect to the
   new primaries as a function of the tristimulus
   values for the XYZ primaries is given, i.e. the
   matrix A-1 is specified. For the same example as
   1.

A-1
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Specification of a set of primaries (3)

1. The spectral density of one member of the
   equivalence class Pi is provided for each i.
   For example, this could be the spectral density
   of the light emitted by each type of phosphor in
   a CRT display. The XYZ tristimulus values of each
   primary can be calculated using the XYZ color
   matching functions.

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(No Transcript)
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Specification of a set of primaries (4)

1. The set of three color-matching functionsare
   provided. However, to be valid color-matching
   functions, each one must be a linear combination
   of

• An example is the spectral sensitivities of the
  L, M and S cones of the human retina.

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It follows that
A
A-T
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Luminance and chromaticity

• Luminance is a measure of relative brightness. If
  two lights have equal luminance, they appear to
  be equally bright to a viewer, independently of
  their chromatic attributes.
• Chromaticity is a measure of the chromatic (hue
  and saturation) attribute of a color,
  independently of its brightness.

different luminance
different chromaticity
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Luminance

• It may be difficult to judge if two very
  different colors, say, a red light and a green
  light, have equal brightness when viewing them
  side by side.

• This judgement is easier if they are viewed in
  alternation one after the other.

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Luminance

• It may be difficult to judge if two very
  different colors, say, a red light and a green
  light, have equal brightness when viewing them
  side by side.

• This judgement is easier if they are viewed in
  alternation one after the other.

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Luminance

• It may be difficult to judge if two very
  different colors, say, a red light and a green
  light, have equal brightness when viewing them
  side by side.

• This judgement is easier if they are viewed in
  alternation one after the other at a high enough
  frequency.

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Luminance

• As the switching frequency increases and passes a
  certain limit, the two colors merge into one,
  which flickers if they have different brightness.
• The intensity of one of the lights can be
  adjusted until the flickering disappears. At this
  point, the two lights have equal perceptual
  brightness.
• This brightness depends on the power density
  spectrum of the light.
• A light with a spectrum concentrated near 550 nm
  appears brighter than a light of equal total
  power with a spectrum concentrated near 700 nm.

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Luminance

• This property is captured by the relative
  luminous efficiency curve V(l).

• The curve tells us that a monochromatic light at
  wavelength l0 with power density spectrum d(l-l0)
  appears equally bright as a monochromatic light
  with power density spectrum V(l0) d(l-lmax),
  where lmax is about 555 nm.

V(l0)
l0
lmax
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Luminance

• Note that V(l) is the same (up to a scale factor)
  as
• Consider an arbitrary light with power spectral
  density C(l). Because of linearity of brightness
  matching, C(l) is a brightness match to
• The quantity where
  Km is a constant is referred to as the luminance
  of C.
• Note that if C1aC then C1LaCL, and
  ifCC1C2, then CLC1LC2L.

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Luminance

• If CC1P1C2P2C3P3 then it follows
  thatCLC1P1LC2P2LC3P3L
• The luminances of the primaries, CiL are called
  luminosity coefficients
• Note that if W P1P2P3 , then
  WLP1LP2LP3L
• Typically, everything is normalized such that WL1

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Luminance scaling
aC
Chromatic attribute does not change along the
line only the brightness
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Chromaticity

• The chromatic attribute of the color is specified
  by identifying the line through the origin
  passing through the color. This can be done by
  locating the intersection of the line with the
  plane
• If CC1P1C2P2C3P3 , we want to choose g
  such that gC lies on this plane. In other
  words,we want gC1gC2gC31 and thus

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Chromaticity

• The tristimulus values of the resulting gC
  lying on the given plane are
• The ci are called chromaticity coefficients
• Only two of them need to be specified, usually c1
  and c2
• A set of colors plotted in the c1c2 plane is
  called a chromaticity diagram

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CIE 1931 RGB chromaticity diagram
510
spectrum locus
560
reference white
490
610
470
800
360
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CIE 1931 XYZ chromaticity diagram
Spectrum locus
line of purples
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CIE 1931 XYZ chromaticity diagram
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The CIE XYZ primaries
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Determination of tristimulus values from
luminance and chromaticities

• Given primaries P1, P2, P3 and their
  luminosity coefficients P1L, P2L, P3L
• the luminance CL and the chromaticities c1 and c2
  of a color C.
• Find the tristimulus values.
• Solution

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Conversion between tristimulus values and
luminance/chromaticity for XYZ space

• The luminosity coefficients areXL0, YL1, ZL0
• This leads to

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Additive reproduction of colors

• Let P1, P2, P3 be a set of three
  primaries.
• Let A, B, C be three physical colors.
• Let Qa1A a2B a3C be an additive
  mixture of A, B and C with non-negative
  coefficients ai 0
• Then
• The chromaticities q1,q2 lie within a triangle in
  the chromaticity diagram whose vertices are the
  chromaticities of A, B and C

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Additive reproduction of colors
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ITU-R Rec. 709 Primaries

• Representative of phosphors of typical modern RGB
  CRT displays
• The reference white is D65, a CIE standard white
  meant to be representative of daylight
• Good model for accurate reproduction of color on
  CRTs we use here it illustrate standard
  computations with color.
• The primaries are specified by their XYZ
  chromaticity coordinates, along with RGB
  D65

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ITU-R Rec. 709 Primaries
Red Green Blue White D65
x 0.640 0.300 0.150 0.3127
y 0.330 0.600 0.060 0.3290
z 0.030 0.100 0.790 0.3582
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ITU-R Rec. 709 Primaries

• Calculations for reference white

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ITU-R Rec. 709 Primaries

• Luminosity coefficients of primaries

Using
etc.
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ITU-R Rec. 709 Primaries

• Tristimulus values of R G B in XYZ space

We now know the chromaticities and luminance of
the RGB primaries, so we can compute the
tristimulus values using etc
AT
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ITU-R Rec. 709 Primaries

• Conversion of tristimulus values

A
A-1
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ITU-R Rec. 709 Primaries color matching functions
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Perceptual non-uniformity of color space
Macadams ellipses
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Uniform Chromaticity Scale (UCS) 1976
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Macadams Ellipses in 1960 UCS
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Nonlinear spaces CIELUV and CIELAB

• These are non-linear spaces, but still described
  by three coordinates. However these coordinates
  do not sum when we add two colors.
• CIELAB is the most widely used one in color FAX
  and color profiles so I only present that one.
• CIELUV is often called Luv
• CIELAB is often called Lab
• They both use the same L.
• These spaces require choice of a reference white.

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CIELAB L component
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CIELAB a and b components
Otherwise the corresponding cube root is replaced
by a linear segment as for L, although such
small values are not normally encountered.
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CIELAB color difference
An approximately uniform measure of difference in
CIELAB space between C1 (C1L,C1a,C1b) and
C2 (C2L,C2a,C2b) is given as follows
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Device Space (CRT display)

• The light output of a CRT display is related to
  the voltage applied approximately by a power law
• intensity voltageg
• A better model is
• intensity (voltage e )2.5
• To compensate, RGB values are gamma corrected
  before appliying them to the display device

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Device space gamma correction

• The new space RGB is more perceptually uniform
  than RGB.
• RGB values are not tristimulus values

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Device space gamma correction
ITU-R Rec. 709 gamma correction
with similar expressions for QG and QB. The
inverse law is
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Device space ITU-R gamma
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Illustration of display gamma (1)
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Illustration of display gamma (2)
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LUMA-Color Difference Space

• This is a device dependent non-linear space that
  starts from gamma-corrected RGB.
• This type of space used in TV, JPEG, MPEG, etc.
• ITU-R Rec. 601

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LUMA- color difference space
In matrix form
For 8-bit integer values between 0 and 255, we
have
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Step pattern with equal luminance steps
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Step pattern with equal luma steps
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Relevant Properties of Human Vision

• In an imaging system, we want to deliver the
  highest image quality in the most economical
  fashion
• What information is important to the visual
  system, and what is not important?
• How do we measure image quality?
• Can we predict the visibility of impairments in
  an image --- like noise, blurring, artifacts,
  etc.
• Ideally, we would like a numerical measure of
  image quality or image distortion that could be
  used in the optimization of an imaging system
• In the absence of any pattern, image color is
  specified by three tristimulus values, or three
  values in a perceptually uniform space like
  CIELUV or CIELAB.
• What happens in the presence of spatial and
  spatiotemporal patterns?