Color Measurement and Reproduction Eric Dubois

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?

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

Determination of tristimulus values Color matching functions The color matching functions are determined by subjective experiment ONCE for one set of primaries [P 1 ], [P 2 ], [P 3 ]. For any color

CIE 1931 Red, Green and Blue primaries B( )=  ( ) G( )=  ( ) R( )=  ( )

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

Transformation of primaries (2) For an arbitrary color: From which we can identify In matrix form:or A

Transformation of primaries (3) The relationship between the sets of primaries can also be expressed in matrix form: ATAT Note that this is a symbolic equation involving elements of the color vector space C

Transformation of primaries (4) Recognizing that color matching functions specify tristimulus values for each : Each new color matching function can be viewed as a linear combination of the three old color matching functions.

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.

The CIE XYZ primaries (2) If [C] is an arbitrary color that can be expressed:

The CIE XYZ primaries (3) Applying this to each, we get the XYZ color matching functions

The CIE XYZ primaries (4) The set of physical colors in XYZ space

XYZ frequency sweep X Y Z

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 A T is specified. For example, the 1976 CIE Uniform Chromaticity Scale (UCS) primaries are given by It follows that

2. 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. Specification of a set of primaries (2) A -1

3. The spectral density of one member of the equivalence class [P i ] 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. Specification of a set of primaries (3)

4. The set of three color-matching functions are provided. However, to be valid color-matching functions, each one must be a linear combination of Specification of a set of primaries (4) An example is the spectral sensitivities of the L, M and S cones of the human retina.

It follows that A A -T

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

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.

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.

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.

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. Luminance

This property is captured by the relative luminous efficiency curve V(  Luminance The curve tells us that a monochromatic light at wavelength 0 with power density spectrum  ( - 0 ) appears equally bright as a monochromatic light with power density spectrum V( 0 )  ( - max ), where max is about 555 nm. 0 max V0)V0)

Luminance Note that V( ) is the same (up to a scale factor) as Consider an arbitrary light with power spectral density C( ). Because of linearity of brightness matching, [C( )] is a brightness match to The quantity where K m is a constant is referred to as the luminance of [C]. Note that if [C 1 ]=  [C] then C 1L =  C L, and if [C]=[C 1 ]+[C 2 ], then C L =C 1L +C 2L.

Luminance If [C]=C 1 [P 1 ]+C 2 [P 2 ]+C 3 [P 3 ] then it follows that C L =C 1 P 1L +C 2 P 2L+ C 3 P 3L The luminances of the primaries, C iL are called luminosity coefficients Note that if [W]= [P 1 ]+[P 2 ]+[P 3 ], then W L =P 1L +P 2L+ P 3L Typically, everything is normalized such that W L =1

Luminance scaling  [C] Chromatic attribute does not change along the line – only the brightness

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 [C]=C 1 [P 1 ]+C 2 [P 2 ]+C 3 [P 3 ], we want to choose  such that  [C] lies on this plane. In other words, we want  C 1 +  C 2 +  C 3 =1 and thus

Chromaticity The tristimulus values of the resulting  [C] lying on the given plane are The c i are called chromaticity coefficients Only two of them need to be specified, usually c 1 and c 2 A set of colors plotted in the c 1 c 2 plane is called a chromaticity diagram

CIE 1931 RGB chromaticity diagram spectrum locus reference white

CIE 1931 XYZ chromaticity diagram Spectrum locus line of purples

CIE 1931 XYZ chromaticity diagram

The CIE XYZ primaries

Determination of tristimulus values from luminance and chromaticities Given: primaries [P 1 ], [P 2 ], [P 3 ] and their luminosity coefficients P 1L, P 2L, P 3L ; the luminance C L and the chromaticities c 1 and c 2 of a color [C]. Find the tristimulus values. Solution

Conversion between tristimulus values and luminance/chromaticity for XYZ space The luminosity coefficients are X L =0, Y L =1, Z L =0 This leads to

Additive reproduction of colors Let [P 1 ], [P 2 ], [P 3 ] be a set of three primaries. Let [A], [B], [C] be three physical colors. Let [Q]=  1 [A] +  2 [B] +  3 [C] be an additive mixture of [A], [B] and [C] with non-negative coefficients a i ≥ 0 Then The chromaticities q 1,q 2 lie within a triangle in the chromaticity diagram whose vertices are the chromaticities of [A], [B] and [C]

Additive reproduction of colors

ITU-R Rec. 709 Primaries Representative of phosphors of typical modern RGB CRT displays The reference white is D 65, 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 [R]+[G]+[B] = [D 65 ]

ITU-R Rec. 709 Primaries RedGreenBlueWhite D 65 x y z

ITU-R Rec. 709 Primaries Calculations for reference white

ITU-R Rec. 709 Primaries Luminosity coefficients of primaries Usingetc.

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 ATAT

ITU-R Rec. 709 Primaries Conversion of tristimulus values A A -1

ITU-R Rec. 709 Primaries: color matching functions

Perceptual non-uniformity of color space Macadam’s ellipses

Uniform Chromaticity Scale (UCS) 1976

Macadam’s Ellipses in 1960 UCS

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 L*u*v* CIELAB is often called L*a*b* They both use the same L*. These spaces require choice of a reference white.

CIELAB L* component

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.

CIELAB color difference An approximately uniform measure of difference in CIELAB space between [C 1 ] (C 1L*,C 1a*,C 1b* ) and [C 2 ] (C 2L*,C 2a*,C 2b* ) is given as follows

Device Space (CRT display) The light output of a CRT display is related to the voltage applied approximately by a power law intensity = voltage   better model is intensity = (voltage +  ) 2.5  compensate, RGB values are gamma corrected before appliying them to the display device

Device space – gamma correction The new space R’G’B’ is more perceptually uniform than RGB. R’G’B’ values are not tristimulus values

Device space – gamma correction ITU-R Rec. 709 gamma correction with similar expressions for Q’ G and Q’ B. The inverse law is

Device space – ITU-R gamma

Illustration of display gamma (1)

Illustration of display gamma (2)

LUMA-Color Difference Space This is a device dependent non-linear space that starts from gamma-corrected R’G’B’. This type of space used in TV, JPEG, MPEG, etc. ITU-R Rec. 601:

LUMA- color difference space In matrix form: For 8-bit integer values between 0 and 255, we have

Step pattern with equal luminance steps

Step pattern with equal luma steps

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?