Color Measurement and Reproduction

1. Color Measurement and Reproduction Eric Dubois

2. 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?

3. 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

4. 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

5. 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)

6. 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

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

8. 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

9. 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.

10. 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.

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

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

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

14. XYZ frequency sweep X Y Z

15. Specification of a set of primaries • 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

16. Specification of a set of primaries (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. A-1

17. Specification of a set of primaries (3) • 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.

19. Specification of a set of primaries (4) • 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.

20. It follows that A A-T

21. 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

22. 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.

23. 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.

24. 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.

25. 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.

26. 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) lmax l0

27. 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 [C1]=a[C] then C1L=aCL, and if[C]=[C1]+[C2], then CL=C1L+C2L.

28. Luminance • If [C]=C1[P1]+C2[P2]+C3[P3] then it follows thatCL=C1P1L+C2P2L+C3P3L • The luminances of the primaries, CiL are called luminosity coefficients • Note that if [W]= [P1]+[P2]+[P3] , then WL=P1L+P2L+P3L • Typically, everything is normalized such that WL=1

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

30. 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]=C1[P1]+C2[P2]+C3[P3] , we want to choose g such that g[C] lies on this plane. In other words,we want gC1+gC2+gC3=1 and thus

31. Chromaticity • The tristimulus values of the resulting g[C] 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

32. CIE 1931 RGB chromaticity diagram 510 spectrum locus 560 reference white 490 610 470 800 360

33. CIE 1931 XYZ chromaticity diagram Spectrum locus line of purples

34. CIE 1931 XYZ chromaticity diagram

35. The CIE XYZ primaries

36. 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

37. Conversion between tristimulus values and luminance/chromaticity for XYZ space • The luminosity coefficients areXL=0, YL=1, ZL=0 • This leads to

38. Additive reproduction of colors • Let [P1], [P2], [P3] be a set of three primaries. • Let [A], [B], [C] be three physical colors. • Let [Q]=a1[A] + a2[B] +a3[C] 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]

39. Additive reproduction of colors

40. 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 [R]+[G]+[B] = [D65]

41. ITU-R Rec. 709 Primaries

42. ITU-R Rec. 709 Primaries • Calculations for reference white

43. ITU-R Rec. 709 Primaries Luminosity coefficients of primaries Using etc.

44. 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

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

46. ITU-R Rec. 709 Primaries: color matching functions

47. Perceptual non-uniformity of color space Macadam’s ellipses

48. Uniform Chromaticity Scale (UCS) 1976

49. Macadam’s Ellipses in 1960 UCS

50. 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.