This file is raw output from pdftotext and may not be ideal for distribution. If you are a maintainer for Hackipedia, please sit down when you have time and clean this text version up. Source PDF: /mnt/fw-js/docs/Color space/pdf/CIE Color Space.pdf Like all conversions the text below should be fully readable as UTF-8 unicode text. --------------------------------------------------------------- Gernot Hoffmann CIE Color Space Contents 1. CIE Chromaticity Diagram 2 2. Color Perception by Eye and Brain 3 3. RGB Color-Matching Functions 4 4. XYZ Coordinates 5 5. XYZ Primaries 6 6. XYZ Color-Matching Functions 7 7. Chromaticity Values 8 8. Color Space Visualization 9 9. Color Temperature and White Points 10 10. CIE RGB Gamut in xyY 11 11. Color Space Calculations 12 12. Matrices 17 13. sRGB 23 14. Barycentric Coordinates 24 15. Optimal Primaries 25 16. References 27 A. Appendix Color Matching 29 1 1. CIE Chromaticity Diagram (1931) The threedimensional color space CIE XYZ is the basis for all color management systems. This color space contains all perceivable colors - the human gamut. Many of them cannot be shown on monitors or printed. The twodimensional CIE chromaticity diagram xyY (below) shows a special projection of the threedimensional CIE color space XYZ. Some interpretations are possible in xyY, others require the threedimensional space XYZ or the related threedimensional space CIELab. 1.0 y sRGB uses ITU-R BT.709 primaries 0.9 Red Green Blue White x 0.64 0.30 0.15 0.3127 520 525 515 y 0.33 0.60 0.06 0.3290 0.8 530 535 AdobeRGB(98) uses Red and Blue 510 540 like sRGB and Green like NTSC 545 0.7 550 CIE-RGB are the primaries for color 555 matching tests: 700 /546.1/435.8nm 505 560 0.6 565 570 500 575 0.5 NTSC CIE sRGB 580 585 590 495 0.4 595 600 605 610 0.3 620 490 635 700 Wavelengths in nm 0.2 485 480 Purple line 0.1 475 470 460 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1.0 2 2. Color Perception by Eye and Brain The retina contains two groups of sensors, the rods and the cones. In each eye are about 100 millions of rods responsible for the luminance. About 6 millions of cones measure color. The sensors are already ’wired’ in the retina - only 1 million nerve fibres carry the information to the brain.The perception of colors by cones requires an absolute luminance of at least some cd/m2 (candela per squaremeter). A monitor delivers about 100 cd/m2 for white and 1 cd/m2 for black. Three types of cones (together with the rods) form a tristimulus measuring system. Spectral information is lost and only three color informations are left. We may call these colors blue, green and red but the red sensor is in fact an orange sensor. The optical system is not color corrected. It would be impossible to focus simultaneously for three different wavelengths. The overlapping sensitivities of the green and the red sensor may indicate that the focussing happens mainly in the overlapping range whereas blue is generally out of focus. This sounds strange, but the gap for image parts on the blind spot is corrected as well - another example for the surprising features of eye and brain. These diagrams show two of several models for the cone sensitivities. These and similar functions cannot be measured directly - they are mathematical interpretations of color matching experiments. The sensitivity between 700nm and 800nm is very low, therefore all the diagrams are drawn for the range 380nm to 700nm. 10.0 2.0 9.0 1.8 8.0 1.6 7.0 1.4 6.0 1.2 _ _ _ _ _ _ p3 p2 p1 p3 p2 p1 5.0 1.0 4.0 0.8 3.0 0.6 2.0 0.4 1.0 0.2 0.0 0.0 380 420 460 500 540 580 620 λ 660 nm 700 380 420 460 500 540 580 620 λ 660 nm 700 Cone sensitivities [3] Cone sensitivities [1] 3 3. RGB Color-Matching The color matching experiment was invented by Her- mann Graßmann (1809 - 1877) about 1853. Three lamps with spectral distributions R,G,B and weight factors R,G,B = 0..100 generate the color impression C = R R + G G + B B. The three lamps must have linearly independent spectra, without any other special specification. A fourth lamp generates the color impression D . View Can we match the color impressions C and D by adjusting R,G,B ? In many cases we can: BlueGreen = 7R + 33G + 39B In other cases we have to move one of the three lamps to the left side and match indirectly: Vibrant BlueGreen + 38R = 42 G + 91B Vibrant BlueGreen = - 38R + 42 G + 91B This is the introduction of ’negative’ colors. The equal Color D Color C sign means ’matched by’. It is generally possible to match a color by three weight factors, but one or even Color matching experiment two can be negative (only one for CIE-RGB) . Data for the example are shown in Appendix A. R,G,B +4.5907 The CIE Standard Primaries (1931) are narrow band light sources (monochromats, line spectra or delta functions) R (700 nm),G (546.1nm) and B (435.8 nm). They replace the red, green and blue lamps in the +1.0000 drawing above. In fact these sources were actually not used - all results were calculated for these prima- +0.0601 ries after tests with other sources. 300 435.8 546.1 700.0 800 CIE Standard Primaries 0.4 The normalized weight factors are called CIE Color- Matching Functions r (λ) , g(λ) , b(λ) . The diagram shows for example the three values for 0.3 matching a spectral pure color (monochromat) with _ _ _ wavelength λ=540nm. This requires a negative value b g r 0.2 for red. RGB colors for a spectrum P(λ) are calculated by these integrals in the range from 380nm to 700nm or 0.1 800nm: R = k ∫ P(λ) r (λ) dλ 0.0 G = k ∫ P(λ) g(λ) dλ -0.1 B = k ∫ P(λ) b(λ) dλ 380 420 460 500 540 580 620 λ 660 nm 700 RGB Color-matching functions 4 4. XYZ Coordinates In order to avoid negative RGB numbers the CIE consortium had introduced a new coordi- nate system XYZ. The RGB system is essentially defined by three non-orthogonal Z base vectors in XYZ. 0.20000 B 0.01063 0.99000   The bottom image explains the sitution for 2D coordinates R,G and X,Y a little simplified. The shaded area shows the hu- man gamut. A plane divides the space in two half spaces. 0.31000 G 0.81240 The new coordinates X,Y are 0.01000   chosen so that the gamut is entirely accessible for positive values. This can be generalized for the 0.49000 R 0.17697 3D space. 0.00000   In the upper image the axes XYZ are drawn orthogonally, in the lower image the axes RGB. X RGB base vectors and color cube in XYZ The coordinates of the base vectors in XYZ (coordinates of the primaries as shown above) for any RGB system are found as columns of the matrix Cxr in chapter 11. G Y Plane R X 2D visualization for RG and XY 5 5. XYZ Primaries The coordinate systems XYZ and RGB are related R, G, B +4.5907 to each other by linear equations. X = C xr R X =+0.49000 R + 0.31000 G + 0.20000 B Y =+0.17697 R + 0.81240 G + 0.01063 B 0 (1) +1.0000 Z =+0.00000 R + 0.01000 G + 0.99000 B +0.0601 R = Crx X 300 435.8 546.1 700.0 800 R =+2.36461 X − 0.89654 Y − 0.46807 Z G =−0.51517 X + 1.42641Y + 0.08876 Z (2) CIE primaries R,G,B B =+0.00520 X − 0.01441Y + 1.00920 Z +2.36461 Another view is possible by introducing synthetical X or ’imaginary’ primaries X,Y,Z. The Standard Primaries R,G,B are monochromatic +0.00031 stimuli. Mathematically they are single delta func- tions with well defined areas. In the diagram the height represents the contribu- tion to the luminance. -2.36499 The ratios are 1.0 :4. 5907: 0.0601. The spectra X ,Y, Z are calculated by the application +6.54822 of the matrix operation (2) and the scale factors. An example: X=1, Y=0, Z=0 : X =+2.36461⋅ 1.0000 R Y −0.51517 ⋅ 4.5907G +0.00520 ⋅ 0.0601 B X =+2.36461R − 2.36499 G + 0.00031B The primaries X,Y, Z are sums of delta functions. -0.00087 X and Z do not contribute to the luminance. This is -0.89654 a special trick in the CIE system. The integrals are zero, here represented by the sum of the heights. The luminance is defined by Y only. Z In color matching experiments negative values or 0.40747 weight factors R, G, B are allowed. 0.06065 Some matchable colors cannot be generated by the Standard Primaries. Other light sources are neces- -0.46807 sary, especially spectral pure sources (mono- chromats). Synthetical primaries X,Y, Z 6 6. XYZ Color-Matching Functions The new color-matching functions x(λ) , y(λ) , z(λ) have non-negative values, as expected. They are calculated from r (λ) , g(λ) , b(λ) by using the matrix Cxr in chapter 5. The functions x(λ) , y(λ) , z(λ) can be understood as 2.0 weight factors. For a spectral pure color C with a 1.8 fixed wavelength λ read in the diagram the three 1.6 values. Then the color can be mixed by the three 1.4 Standard Primaries: 1.2 C = x(λ) X + y(λ) Y + z(λ) Z _ z _ y _ x 1.0 Generally we write 0.8 C = XX +YY+ZZ 0.6 and a given spectral color distribution P(λ) delivers 0.4 the three coordinates XYZ by these integrals in the 0.2 range from 380nm to 700nm or 800nm: 0.0 X = k ∫ P(λ) x(λ) dλ 380 420 460 500 540 580 620 λ 660 nm 700 Y = k ∫ P(λ) y(λ) dλ XYZ Color-matching functions Z = k ∫ P(λ) z(λ) dλ Mostly, the arbitrary factor k is chosen for a normalized value Y=1 or Y=100. Matrix operations are always normalized for R,G,B,Y= 0 to 1. This diagram shows already the human gamut in XYZ. It is an irregularly shaped cone.The intersection with the blue-ish colored plane in the corner will deliver the chromaticity diagram. X Y Human gamut in XYZ 7 7. Chromaticity Values The chromaticity values x,y,z depend only on the z hue or dominant wavelength and the saturation. View Arbitrary They are independend of the luminance: 1 color XYZ X x = X+Y+Z Y y y = X+Y+Z 1 Z z = X+Y+Z Obviously we have x + y + z = 1. All the values are on the triangle plane, projected by a line through the arbitrary color XYZ and the origin, if we draw XYZ and xyz in one diagram. x This is a planar projection. The center of projection 1 is in the origin. Projection and chromaticity plane The vertical projection onto the xy-plane is the chromaticity diagram xyY (view direction). To reconstruct a color triple XYZ from the chromaticity values xy we need an additional information, the luminance Y. z =1− x − y x X = Y y z Z = Y y All visible (matchable) colors which differ only by Z Rendering primaries luminance map to the same point in the chromati- 445 535 city diagram. This is sometimes called ’horseshoe 606 diagram’ (page 2). Halfaxis length 1.0 The right image shows a 3D view of the color- matching functions, connected by rays with the Y origin. The contour is here called ‘locus of unit mono- chromats’ [18]. For spectral colors this is the same as XYZ. Then the contour is mapped onto the plane as above. The spectral loci for blue and for red end nearly in the origin: colors with short and long wavelengths appear rather dark, they are almost invisible for a reasonably limited power. The chromaticity diagram conceals this important fact. The purple line can be considered as a fake. Real purples are inside the horseshoe contour. X 8 8. Color Space Visualization These images are computer graphics. Accurate transformations and a few applications of image processing.The contour of the horseshoe is mapped to XYZ for luminances Y = 0..1 . The purple plane is shown transparent. All colors were selected for readabilty. The colors are not correct, this is anyway impossible. More important is here the geometry. The gamut volume is confined by the color surface (pure spectral colors), the purple plane and the plane Y = 1. The regions with small values Y appear extremely distorted - near to a singularity. For blue very high values Z are necessary to match a color with specified luminance Y = 1. 1 1 2 2 X Y Z X Y 9 9. Color Temperature and White Points The graphic shows the color temperature for the Planck radiator from 2000K to 10000K, the directions of correlated color temperatures and the white points for daylight D50 and D65. Uncalibrated monitors have about 9300K which is here simply called D93. Data by [3]. EPS graphic available here [15]. T/K x y Dir y/x 2000 0.52669 0.41331 1.33101 2105 0.51541 0.41465 1.39021 2222 0.50338 0.41525 1.45962 2353 0.49059 0.41498 1.54240 2500 0.47701 0.41368 1.64291 2677 0.463 0.41121 1.76811 % error in table [3], estimated values 2857 0.446 0.40742 1.92863 3077 0.43156 0.40216 2.14300 3333 0.41502 0.39535 2.44455 3636 0.39792 0.38690 2.90309 4000 0.38045 0.37676 3.68730 4444 0.36276 0.36496 5.34398 5000 0.34510 0.35162 11.17883 5714 0.32775 0.33690 -39.34888 6667 0.31101 0.32116 -6.18336 8000 0.29518 0.30477 -3.08425 10000 0.28063 0.28828 -1.93507 1.0 y 0.9 520 525 515 530 0.8 535 510 540 545 0.7 550 505 555 560 0.6 565 570 500 575 0.5 580 585 590 495 0.4 595 2222 2353 2105 2500 2000 2677 2857 3077 D50 3333 600 3636 4000 D65 605 4444 610 5000 D93 5714 0.3 620 6667 490 635 8000 10000 700 0.2 485 480 0.1 475 470 460 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1.0 10 10. CIE RGB Gamut in xyY The gamut of any RGB system is mostly visualized by a triangle in xyY. For different luminances Y= const. we get the intersection of a vertical plane and the RGB cube (chapter 4). The intersection delivers a triangle, a quadriliteral, a pentagon or a hexagon. These polygons are projected onto the xy-plane The chromaticity diagram below shows the actual gamut for different luminances Y. Low luminances seem to produce a large gamut. But that is a fake - a result of the perspective projection from XYZ to xyY. The gamut appears similarly in all RGB systems. A color outside the triangle (which is defined by the primaries) is always out-of-gamut. A color inside the triangle is not necessarily in- gamut. 1.0 y 0.9 520 525 515 530 0.8 535 510 540 545 0.7 550 505 555 560 0.6 565 570 500 575 0.5 580 585 590 495 0.4 595 Y = 0.05 .. 0.95 600 0.85 605 0.75 0.95 610 0.3 0.65 620 490 635 0.55 700 0.45 0.2 485 0.35 0.25 480 0.15 0.1 475 470 0.05 460 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1.0 11 11.1 Color Space Calculations / General In this chapter we derive the relations between CIE xyY, CIE XYZ and any arbitrary RGB space. It is essential to understand the principle of RGB basis vectors in the XYZ coordinate system. This was shown on previous pages. Given are the coordinates for the primaries in CIE xyY and for the white point: xr ,yr , xg ,yg ,xb ,yb ,xw ,yw . CIE xyY is the horseshoe diagram. Furtheron we need the luminance V. We want to derive the relation between any color set r,g,b and the coordinates X,Y,Z . (1) r = (r, g, b)T Color values in RGB (2) X = (X, Y, Z)T Color values in XYZ (3) x = (x, y, z)T Color values in xyY (4) L = X+Y+Z Scaling value (5) x = X /L y = Y /L z = Z /L (6) z = 1- x - y (7) X = Lx V is the luminance of the stimulus, according to the luminous efficiency function V(λ) in [3 ]. We should not call this immediately Y because Y is mostly normalized for 1 or 100. (8) X = V x / y Y = V Z = Vz/y Basis vectors for the primaries and white point in XYZ: (9) R = L x r = L (x r , y r , z r ) T G = L xg = L (x g , y g , z g ) T B = L xb = L (x b , y b , z b ) T (10) W = L w = L (x w , y w , z w ) T Set of scale factors for the white point correction: (11) u = (u, v, w)T 12 11.2 Color Space Calculations / General For the white point correction, the basis vectors R,G,B are scaled by u,v,w. This does not change their coordinates in xyY .The mapping from XYZ to xyY is a central planar projection. (12) X = L (x, y, z)T = r u R + g v G + b w B For the white point we have r = g = b = 1. (13) W = L (x w , y w , z w ) T = L u (x r , y r , z r ) T + L v (x g , y g , z g ) T + L w (x b , y b , z b ) T This can be re-arranged, L cancels on both sides.: Èx w ˘ Èxr xg xb ˘ È u ˘ Èu˘ (14) Í y w ˙ = Í yr yg yb ˙ Í v ˙ = P Í v ˙ Í ˙ Ízw ˙ Î ˚ Î zr zg zb ˚ Í w ˙ Î ˚ Íw˙ Î ˚ It is not necessary to invert the whole matrix numerically. We can simplify the calculation by adding the first two rows to the third row and find so immediately Eq.(15), which is anyway clear: (15) w = 1- u - v Èx w ˘ Èxr x g xb ˘ È u ˘ (16) Í y w ˙ = Í yr y g yb ˙ Í v ˙ Í ˙ Í ˙ Í1 - u - v ˙ Î ˚ Î ˚Î ˚ (17) x w = (x r - x b ) u + (x g - x b ) v + x b yw = (yr - y b ) u + (y g - y b ) v + y b These linear equations are solved by Cramer’s rule. (18) D = (x r - x b ) ( y g - y b ) - ( y r - y b ) (x g - x b ) U = (x w - x b ) ( y g - y b ) - ( y w - y b ) (x g - x b ) V = (x r - x b ) ( y w - y b ) - ( y r - y b ) (x w - x b ) (19) u = U/D v = V /D w = 1- u - v In the next step we assume that u,v,w are already calculated and we use the general color transformation Eq.(12) and furtheron Eq.(8). We get the matrices Cxr and Crx . È X˘ Èu x r / y w v xg / yw w xb / yw ˘ È r ˘ = V Íu y r / y w (20) Í Y ˙ v yg / y w w y b / y w ˙ Í g˙ Í Z˙ Í ˙ Î ˚ Î uzr / y w v zg / y w w z b / y w ˚ Íb ˙ Î ˚ (21 X = V C xr r - (22) r = (1/ V) C xr1 X = (1/ V) Crx X 13 11.3 Color Space Calculations / General For better readability we show the last two equations again, but now with V = 1, as in most publications. (23) X = C xr r - (24) r = C xr1 X = Crx X Now we can easily derive the relation between two different RGB spaces, e.g. working spaces and image source spaces. (25) X = C xr1 r1 (26) X = C xr 2 r2 - (27) r2 = C xr12 C xr1 r1 (28) r2 = C 21 r1 An example shows the conversion of Rec.709/D65 to D50 and D93. The resulting matrix C21 is diagonal, because the source and destination primaries are the same. The explanation as above is valid for the representation of the same physical color in two different RGB systems. For the simulation of D50 or D93 effects in the same D65 RGB system one has to apply the inverse matrix. Rec.709 Rec.709 xr= 0.6400 yr= 0.3300 zr= 0.0300 xr= 0.6400 yr= 0.3300 zr= 0.0300 xg= 0.3000 yg= 0.6000 zg= 0.1000 xg= 0.3000 yg= 0.6000 zg= 0.1000 xb= 0.1500 yb= 0.0600 zb= 0.7900 xb= 0.1500 yb= 0.0600 zb= 0.7900 D65 D65 xw= 0.3127 yw= 0.3290 zw= 0.3583 xw= 0.3127 yw= 0.3290 zw= 0.3583 D50 D93 xw= 0.3457 yw= 0.3585 zw= 0.2958 xw= 0.2857 yw= 0.2941 zw= 0.4202 Matrix Cxr: X=Cxr*R65 Matrix Cxr: X=Cxr*R65 0.4124 0.3576 0.1805 0.4124 0.3576 0.1805 0.2126 0.7152 0.0722 0.2126 0.7152 0.0722 0.0193 0.1192 0.9505 0.0193 0.1192 0.9505 Matrix Crx: R65=Crx*X Matrix Crx: R65=Crx*X 3.2410 -1.5374 -0.4986 3.2410 -1.5374 -0.4986 -0.9692 1.8760 0.0416 -0.9692 1.8760 0.0416 0.0556 -0.2040 1.0570 0.0556 -0.2040 1.0570 Matrix Dxr: X=Dxr*R50 Matrix Dxr: X=Dxr*R93 0.4852 0.3489 0.1303 0.3706 0.3554 0.2455 0.2502 0.6977 0.0521 0.1911 0.7107 0.0982 0.0227 0.1163 0.6861 0.0174 0.1185 1.2929 Matrix Drx: R50=Drx*X Matrix Drx: R93=Drx*X 2.7548 -1.3068 -0.4238 3.6066 -1.7108 -0.5549 -0.9935 1.9229 0.0426 -0.9753 1.8877 0.0418 0.0771 -0.2826 1.4644 0.0409 -0.1500 0.7771 Matrix Err: R50=Err*R65=Drx*Cxr*R65 Matrix Err: R93=Err*R65=Drx*Cxr*R65 0.8500 0.0000 0.0000 1.1128 0.0000 0.0000 0.0000 1.0250 0.0000 0.0000 1.0063 0.0000 0.0000 0.0000 1.3855 0.0000 0.0000 0.7352 Matrix Frr: R65=Frr*R50=Crx*Dxr*R50 Matrix Frr: R65=Frr*R93=Crx*Dxr*R93 1.1765 0.0000 0.0000 0.8986 0.0000 0.0000 0.0000 0.9756 0.0000 0.0000 0.9938 0.0000 0.0000 0.0000 0.7218 0.0000 0.0000 1.3602 14 11.4 Color Space Calculations / Simplified Now we clean up the mathematics. Eq.(14) delivers: (29) u = P -1 w Eq.(12) and Eq.(20) can be written using the diagonal matrix D with elements u/yw etc.: (30) X = PDr (31) X = C xr r Together with Eq.(29) we find this simple formula for the matrix Cxr: Èu / y w 0 0 ˘ (32) C xr = PÍ 0 v / yw 0 ˙ Í 0 Î 0 w / yw ˙ ˚ The examples in chapter 12 were written by Pascal. Here is a new example in MatLab. Calculation of the matrices for sRGB: % G.Hoffmann % January 14 / 2005 % Matrix Cxr and Crx for sRGB xr=0.6400; yr=0.3300; zr=1-xr-yr; xg=0.3000; yg=0.6000; zg=1-xg-yg; xb=0.1500; yb=0.0600; zb=1-xb-yb; xw=0.3127; yw=0.3290; zw=1-xw-yw; W=[xw; yw; zw]; P=[xr xg xb; yr yg yb; zr zg zb]; u=inv(P)*W % D=[u(1) 0 0; % 0 u(2) 0; % 0 0 u(3)]/yw D=diag(u/yw) Cxr=P*D Crx=inv(Cxr) % Result: % Cxr 0.4124 0.3576 0.1805 % 0.2126 0.7152 0.0722 % 0.0193 0.1192 0.9505 % Crx 3.2410 -1.5374 -0.4986 % -0.9692 1.8760 0.0416 % 0.0556 -0.2040 1.0570 15 11.5 Color Space Calculations / Application The task: red, green and blue lasers generate monochromatic light at wavelengths 671nm, 532nm and 473nm. The powers are to be adjusted so that the three lasers together deliver white light D65. Calculate the matrices, the radiant power ratios and the photometric ratios. In order to test the algorithms we are doing the same for CIE primaries and Equal Energy White, just as if the lasers had these primaries. The results are known in advance, based on standard text books. Thanks to Gerhard Fuernkranz for important clarifications. CIE primaries and white point E Laser primaries and white point D65 % G.Hoffmann % G.Hoffmann % January 19 / 2005 % January 19 / 2005 % Calculations for CIE primaries % Calculations for Laser primaries % x-bar,y-bar,z-bar interpolated % x-bar,y-bar,z-bar interpolated % 700.0 546.1 435.8 nm % 671 532 473 nm xbr=0.011359; xbg=0.375540; xbb=0.333181; xbr=0.0819; xbg=0.1891; xbb=0.1627; ybr=0.004102; ybg=0.984430; ybb=0.017769; ybr=0.0300; ybg=0.8850; ybb=0.1034; zbr=0.000000; zbg=0.012207; zbb=1.649716; zbr=0.0000; zbg=0.0369; zbb=1.1388; % Equal Energy WP % D65 Xw=1; Yw=1; Zw=1; Xw=0.9504; Yw=1.0000; Zw=1.0890; %Chromaticity coordinates %Chromaticity coordinates D=xbr+ybr+zbr; xr=xbr/D; yr=ybr/D; zr=zbr/D; D=xbr+ybr+zbr; xr=xbr/D; yr=ybr/D; zr=zbr/D; D=xbg+ybg+zbg; xg=xbg/D; yg=ybg/D; zg=zbg/D; D=xbg+ybg+zbg; xg=xbg/D; yg=ybg/D; zg=zbg/D; D=xbb+ybb+zbb; xb=xbb/D; yb=ybb/D; zb=zbb/D; D=xbb+ybb+zbb; xb=xbb/D; yb=ybb/D; zb=zbb/D; D=Xw +Yw+ Zw; xw=Xw/D; yw=Yw/D; zw=Zw/D; D=Xw +Yw+ Zw; xw=Xw/D; yw=Yw/D; zw=Zw/D; w=[xw; yw; zw]; w=[xw; yw; zw]; P=[xr xg xb; P=[xr xg xb; yr yg yb; yr yg yb; zr zg zb]; zr zg zb]; u=inv(P)*w u=inv(P)*w D=diag(u/yw) D=diag(u/yw) Cxr=P*D Cxr=P*D % 0.4902 0.3099 0.1999 % 0.6571 0.1416 0.1516 % 0.1770 0.8123 0.0107 % 0.2407 0.6629 0.0964 % 0.0000 0.0101 0.9899 % 0 0.0276 1.0614 Crx=inv(Cxr) Crx=inv(Cxr) % 2.3635 -0.8958 -0.4677 % 1.6476 -0.3435 -0.2042 % -0.5151 1.4265 0.0887 % -0.6005 1.6394 -0.0631 % 0.0052 -0.0145 1.0093 % 0.0156 -0.0427 0.9438 % Radiant power ratios % Radiant power ratios Xbar=[xbr xbg xbb; Xbar=[xbr xbg xbb; ybr ybg ybb; ybr ybg ybb; zbr zbg zbb]; zbr zbg zbb]; W=[Xw; Yw; Zw]; W=[Xw; Yw; Zw]; R=inv(Xbar)*W R=inv(Xbar)*W R=R/R(3) R=R/R(1) % 71.9166 1.3751 1.0000 % 1.0000 0.0934 0.1162 % 72.0962 1.3791 1.0000 Wyszecki & Stiles R=R/R(2) % 10.7111 1.0000 1.2442 % Luminous efficiency ratios R=R/R(3) L=[R(1)*ybr; R(2)*ybg; R(3)*ybb] % 8.6088 0.8037 1.0000 L=L/L(1) % 1.0000 4.5889 0.0602 % Luminous efficiency ratios % 1.0000 4.5907 0.0601 Wyszecki & Stiles L=[R(1)*ybr; R(2)*ybg; R(3)*ybb]; L=L/L(1) % 1.0000 2.7542 0.4004 L=L/L(2) % 0.3631 1.0000 0.1454 L=L/L(3) % 2.4977 6.8791 1.0000 16 12.1 Matrices / CIE + E CIE Primaries and white point E [3]. Page 5 shows the same results. Data are in the Pascal source code. Program CiCalcCi; { Calculations RGB—CIE } { G.Hoffmann February 01, 2002 } Uses Crt,Dos,Zgraph00; Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text; Var Cxr,Crx: ANN; Begin ClrScr; { CIE Primaries } xr:=0.73467; yr:=0.26533; zr:=1-xr-yr; xg:=0.27376; yg:=0.71741; zg:=1-xg-yg; xb:=0.16658; yb:=0.00886; zb:=1-xb-yb; { CIE White Point } xw:=1/3; yw:=1/3; zw:=1-xw-yw; { White Point Correction } D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb); U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb); V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb); u:=U/D; v:=V/D; w:=1-u-v; { Matrix Cxr } Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw; Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw; Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw; { Matrix Crx } HoInvers (3,Cxr,Crx,D,flag); Assign (prn,’C:\CiMalcCi.txt’); ReWrite(prn); Writeln (prn,’ Matrix Cxr’); Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4); Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4); Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4); Writeln (prn,’ Matrix Crx’); Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4); Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4); Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4); Close(prn); Readln; End. Matrix Cxr X 0.4900 0.3100 0.2000 Y 0.1770 0.8124 0.0106 X = Cxr R Z -0.0000 0.0100 0.9900 Matrix Crx R 2.3647 -0.8966 -0.4681 G -0.5152 1.4264 0.0887 R = CrxX B 0.0052 -0.0144 1.0092 17 12.2 Matrices / 709 + D65 / sRGB ITU-R BT.709 Primaries and white point D65 [9]. Valid for sRGB. Data are in the Pascal source code. Program CiCalc65; { Calculations RGB—CIE } { G.Hoffmann February 01, 2002 } Uses Crt,Dos,Zgraph00; Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text; Var Cxr,Crx: ANN; Begin ClrScr; { Rec 709 Primaries } xr:=0.6400; yr:=0.3300; zr:=1-xr-yr; xg:=0.3000; yg:=0.6000; zg:=1-xg-yg; xb:=0.1500; yb:=0.0600; zb:=1-xb-yb; { D65 White Point } xw:=0.3127; yw:=0.3290; zw:=1-xw-yw; { White Point Correction } D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb); U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb); V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb); u:=U/D; v:=V/D; w:=1-u-v; { Matrix Cxr } Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw; Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw; Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw; { Matrix Crx } HoInvers (3,Cxr,Crx,D,flag); Assign (prn,’C:\CiMalc65.txt’); ReWrite(prn); Writeln (prn,’ Matrix Cxr’); Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4); Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4); Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4); Writeln (prn,’ Matrix Crx’); Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4); Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4); Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4); Close(prn); Readln; End. Matrix Cxr X 0.4124 0.3576 0.1805 Y 0.2126 0.7152 0.0722 X = Cxr R Z 0.0193 0.1192 0.9505 Matrix Crx R 3.2410 -1.5374 -0.4986 G -0.9692 1.8760 0.0416 R = CrxX B 0.0556 -0.2040 1.0570 18 12.3 Matrices / AdobeRGB + D65 AdobeRGB(98), D65. Data are in the Pascal source code. Program CiCalc98; { Calculations RGB—AdobeRGB98 } { G.Hoffmann März 28, 2004 } Uses Crt,Dos,Zgraph00; Var r,g,b,x,y,z,u,v,w,d : Double; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Double; prn,cie : Text; Var Cxr,Crx: ANN; Begin ClrScr; { AdobeRGB(98) } xr:=0.6400; yr:=0.3300; zr:=1-xr-yr; xg:=0.2100; yg:=0.7100; zg:=1-xg-yg; xb:=0.1500; yb:=0.0600; zb:=1-xb-yb; { D65 White Point } xw:=0.3127; yw:=0.3290; zw:=1-xw-yw; { White Point Correction } D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb); U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb); V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb); u:=U/D; v:=V/D; w:=1-u-v; { Matrix Cxr } Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw; Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw; Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw; { Matrix Crx } HoInvers (3,Cxr,Crx,D,flag); Assign (prn,’C:\CiMalc98.txt’); ReWrite(prn); Writeln (prn,’ Matrix Cxr’); Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4); Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4); Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4); Writeln (prn,’’); Writeln (prn,’ Matrix Crx’); Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4); Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4); Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4); Writeln (prn,’dummy’); Readln; End. Matrix Cxr X 0.5767 0.1856 0.1882 Y 0.2973 0.6274 0.0753 X = Cxr R Z 0.0270 0.0707 0.9913 Matrix Crx R 2.0416 -0.5650 -0.3447 G -0.9692 1.8760 0.0416 R = CrxX B 0.0134 -0.1184 1.0152 19 12.4 Matrices / NTSC + C NTSC Primaries and white point C [4], also used as YIQ Model. Data are in the Pascal source code. Program CiCalcNT; { Calculations RGB—NTSC } { G.Hoffmann April 01, 2002 } Uses Crt,Dos,Zgraph00; Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text; Var Cxr,Crx: ANN; Begin ClrScr; { NTSC Primaries } xr:=0.6700; yr:=0.3300; zr:=1-xr-yr; xg:=0.2100; yg:=0.7100; zg:=1-xg-yg; xb:=0.1400; yb:=0.0800; zb:=1-xb-yb; { NTSC White Point } xw:=0.3100; yw:=0.3160; zw:=1-xw-yw; { White Point Correction } D:=(xr-xb)*(yg-yb)-(yr-yb)*(xg-xb); U:=(xw-xb)*(yg-yb)-(yw-yb)*(xg-xb); V:=(xr-xb)*(yw-yb)-(yr-yb)*(xw-xb); u:=U/D; v:=V/D; w:=1-u-v; { Matrix Cxr } Cxr[1,1]:=u*xr/yw; Cxr[1,2]:=v*xg/yw; Cxr[1,3]:=w*xb/yw; Cxr[2,1]:=u*yr/yw; Cxr[2,2]:=v*yg/yw; Cxr[2,3]:=w*yb/yw; Cxr[3,1]:=u*zr/yw; Cxr[3,2]:=v*zg/yw; Cxr[3,3]:=w*zb/yw; { Matrix Crx } HoInvers (3,Cxr,Crx,D,flag); Assign (prn,’C:\CiMalcNT.txt’); ReWrite(prn); Writeln (prn,’ Matrix Cxr’); Writeln (prn,Cxr[1,1]:12:4, Cxr[1,2]:12:4, Cxr[1,3]:12:4); Writeln (prn,Cxr[2,1]:12:4, Cxr[2,2]:12:4, Cxr[2,3]:12:4); Writeln (prn,Cxr[3,1]:12:4, Cxr[3,2]:12:4, Cxr[3,3]:12:4); Writeln (prn,’’); Writeln (prn,’ Matrix Crx’); Writeln (prn,Crx[1,1]:12:4, Crx[1,2]:12:4, Crx[1,3]:12:4); Writeln (prn,Crx[2,1]:12:4, Crx[2,2]:12:4, Crx[2,3]:12:4); Writeln (prn,Crx[3,1]:12:4, Crx[3,2]:12:4, Crx[3,3]:12:4); Close(prn); Readln; End. Matrix Cxr X 0.6070 0.1734 0.2006 Y 0.2990 0.5864 0.1146 X = Cxr R Z -0.0000 0.0661 1.1175 Matrix Crx R 1.9097 -0.5324 -0.2882 G -0.9850 1.9998 -0.0283 R = CrxX B 0.0582 -0.1182 0.8966 20 12.5 Matrices / NTSC + C + YIQ NTSC Primaries and white point C [4], YIQ Conversion. Data are in the Pascal source code. Program CiCalcYI; { Calculations RGB—NTSC YIQ } { G.Hoffmann April 01, 2002 } Uses Crt,Dos,Zgraph00; Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text; Var Cyr,Cry: ANN; Begin ClrScr; { NTSC Primaries } xr:=0.6700; yr:=0.3300; zr:=1-xr-yr; xg:=0.2100; yg:=0.7100; zg:=1-xg-yg; xb:=0.1400; yb:=0.0800; zb:=1-xb-yb; { NTSC White Point } xw:=0.3100; yw:=0.3160; zw:=1-xw-yw; { Matrix Cyr, Sequence Y I Q } Cyr[1,1]:= 0.299; Cyr[1,2]:= 0.587; Cyr[1,3]:= 0.114; Cyr[2,1]:= 0.596; Cyr[2,2]:=-0.275; Cyr[2,3]:=-0.321; Cyr[3,1]:= 0.212; Cyr[3,2]:=-0.528; Cyr[3,3]:= 0.311; { Matrix Cry } HoInvers (3,Cyr,Cry,D,flag); Assign (prn,’C:\CiMalcYI.txt’); ReWrite(prn); Writeln (prn,’ Matrix Cyr’); Writeln (prn,Cyr[1,1]:12:4, Cyr[1,2]:12:4, Cyr[1,3]:12:4); Writeln (prn,Cyr[2,1]:12:4, Cyr[2,2]:12:4, Cyr[2,3]:12:4); Writeln (prn,Cyr[3,1]:12:4, Cyr[3,2]:12:4, Cyr[3,3]:12:4); Writeln (prn,’’); Writeln (prn,’ Matrix Cry’); Writeln (prn,Cry[1,1]:12:4, Cry[1,2]:12:4, Cry[1,3]:12:4); Writeln (prn,Cry[2,1]:12:4, Cry[2,2]:12:4, Cry[2,3]:12:4); Writeln (prn,Cry[3,1]:12:4, Cry[3,2]:12:4, Cry[3,3]:12:4); Close(prn); Readln; End. Matrix Cyr Y 0.2990 0.5870 0.1140 I 0.5960 -0.2750 -0.3210 Y = Cyr R Q 0.2120 -0.5280 0.3110 Matrix Cry R 1.0031 0.9548 0.6179 G 0.9968 -0.2707 -0.6448 R = CryY B 1.0085 -1.1105 1.6996 21 12.6 Matrices / NTSC + C + YCbCr NTSC Primaries and white point C [4], YCbCr Conversion. Data are in the Pascal source code. Program CiCalcYC; { Calculations RGB—NTSC YCbCr } { G.Hoffmann April 03, 2002 } Uses Crt,Dos,Zgraph00; Var r,g,b,x,y,z,u,v,w,d : Extended; i,j,k,flag : Integer; xr,yr,zr,xg,yg,zg,xb,yb,zb,xw,yw,zw : Extended; prn,cie : Text; Var Cyr,Cry: ANN; Begin ClrScr; { NTSC Primaries } xr:=0.6700; yr:=0.3300; zr:=1-xr-yr; xg:=0.2100; yg:=0.7100; zg:=1-xg-yg; xb:=0.1400; yb:=0.0800; zb:=1-xb-yb; { NTSC White Point } xw:=0.3100; yw:=0.3160; zw:=1-xw-yw; { Matrix Cxr, Sequence Y Cb Cr } Cyr[1,1]:= 0.2990; Cyr[1,2]:= 0.5870; Cyr[1,3]:= 0.1140; Cyr[2,1]:=-0.1687; Cyr[2,2]:=-0.3313; Cyr[2,3]:=+0.5000; Cyr[3,1]:= 0.5000; Cyr[3,2]:=-0.4187; Cyr[3,3]:=-0.0813; { Matrix Cry } HoInvers (3,Cyr,Cry,D,flag); Assign (prn,’C:\CiMalcYC.txt’); ReWrite(prn); Writeln (prn,’ Matrix Cyr’); Writeln (prn,Cyr[1,1]:12:4, Cyr[1,2]:12:4, Cyr[1,3]:12:4); Writeln (prn,Cyr[2,1]:12:4, Cyr[2,2]:12:4, Cyr[2,3]:12:4); Writeln (prn,Cyr[3,1]:12:4, Cyr[3,2]:12:4, Cyr[3,3]:12:4); Writeln (prn,’’); Writeln (prn,’ Matrix Cry’); Writeln (prn,Cry[1,1]:12:4, Cry[1,2]:12:4, Cry[1,3]:12:4); Writeln (prn,Cry[2,1]:12:4, Cry[2,2]:12:4, Cry[2,3]:12:4); Writeln (prn,Cry[3,1]:12:4, Cry[3,2]:12:4, Cry[3,3]:12:4); Close(prn); Readln; End. Matrix Cyr Y 0.2990 0.5870 0.1140 Note Cb -0.1687 -0.3313 0.5000 This is a linear conversion, as used for JPEG Cr 0.5000 -0.4187 -0.0813 In TV systems the conversion is different Matrix Cry R 1.0000 0.0000 1.4020 Note G 1.0000 -0.3441 -0.7141 Rounded for structural zeros B 1.0000 1.7722 0.0000 22 13. sRGB sRGB is a standard color space, defined by companies, mainly Hewlett-Packard and Micro- soft [9], [12 ]. The transformation of RGB image data to CIE XYZ requires primarily a Gamma correction, which compensates an expected inverse Gamma correction, compared to linear light data, here for normalized values C = R,G,B = 0...1: If C ≤ 0.03928 Then C = C/12.92 Else C = ( (0.055+C)/1.055 ) 2.4 The formula in the document [12 ] is misleading because a bracket was forgotten. 1 Black C = C 2.2 Red sRGB, as above Green ten times the difference 0 0 1 The conversion for D65 RGB to D65 XYZ uses the matrix on page 14, ITU-R BT.709 Prima- ries. D65 XYZ means XYZ without changing the illuminant. [] [ X Y Z = D65 0.4124 0.3576 0.1805 0.2126 0.7152 0.0722 0.0193 0.1192 0.9505 ][ ] R G B D65 The conversion for D65 RGB to D50 XYZ applies additionally (by multiplication) the Bradford correction, which takes the adaptation of the eyes into account. This correction is an improved alternative to the Von Kries corrrection [1]. Monitors are assumed D65, but for printed paper the standard illuminant is D50. Therefore this transformation is recommended if the data are used for printing: [] [ X Y Z = D50 0.4361 0.3851 0.1431 0.2225 0.7169 0.0606 0.0139 0.0971 0.7141 ][ ] R G B D65 23 14.1 Barycentric Coordinates / Concept The corners R,G,B of a triangular gamut, e.g. for a monitor, are described in CIE xyY by three vectors r,g,b which have two components x,y each. A color C is described either by c with two values cx,cy or by three values R,G,B. These are the barycentric coordinates of C . All points inside and on the triangle are reachable by 0 ≤ R,G,B ≤1. Points outside have at least one negative coordinate. The corners R,G,B have barycentric coordinates (1,0,0), (0,1,0) and (0,0,1). G bg rg rg = R RG rb = R RB Line R= 0 Line B = 0 gr = G GR gb = G GB bg = B BG gb C gr br = B BR B R Underline means length of .. rb br Line G = 0 (1) c = R r + G g + B b (2) 1 = R + G + B Substitute R in(1) by (2): (3) R = 1 - G - B (4) G(g - r) + B (b - r) = c - r (4) consists of two linear equations for G,B, which can be solved by Cramer’s rule. R is calculated by (3). (g - r) and (b - r) are the edge vectors from R to G and R to B. The edge vectors are used in (4) as a vector base. o Any point inside the triangle is reached by G + B < 1, which leads to R + G + B = 1. G r+g = (R+G) B B’ B’ b = B B B’ b g r+ C B R 24 14.2 Barycentric Coordinates / Wrong So far the barycentric coordinates remind much to the explanations in [3], chapter 3.2.2. It should be possible to find the relative values R,G,B for a given point c=(cx,cy) by measuring the proportions R=rg/ RG , G=gr/ GR with RG = GR, then B=1-R-G. Unfortunately this interpretation is wrong. The drawing shows the D65 white point and the measurable values R=0.219, G=0.385 and B=0.396 instead of the correct values R=1/3, G=1/3, B=1/3. The base vectors R,G,B in CIE XYZ (chapter 4 for CIE primaries) do not have the same lengths. In [3] the mathematics were explained for unit vectors. So far it is not clear, how the geometrically interpretation for barycentric coordinates could be applied to the actual task. The diagram below shows additionally seven sectors. ’RGB’ means, all values are positive (inside the triangle). ’rGB’ means R<0, G>0, B>0 and so on. Negative values are not prohibited by the definition of coordinates. They just do not appear in technical RGB system. Of course they are essential for the color matching theory. 1.0 y 0.9 520 525 515 530 0.8 535 510 540 545 0.7 550 505 rGb 555 560 0.6 G RGb 565 rg 570 500 575 0.5 580 585 590 495 0.4 595 rGB RGB gr 600 605 610 Rgb 0.3 490 D65 R 620 635 700 0.2 485 RgB 480 0.1 475 470 B rgB 460 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1.0 25 15. Optimal Primaries James A. Worthey had shown in recent publications [18] how to find optimal primaries. This approach is based on ’Amplitude not left out’ . Which primaries should be used if the power is limited for each light source ? The resulting wavelengths are shown by the corners of the triangle below: 445, 536, 604 nm. At least, the wavelengths should be near to these values. For a real system (besides tests in a laboratory) pure spectral colors cannot be used. The corners have to be shifted on a radius towards the white point (which is here indicated by the circle for D65). The optimal red at 604nm is hardly a good candidate for technical systems - it is more a kind of orange instead of vibrant red. Additional illustrations for J.Worthey’s concepts are in [19]. Everything PostScript vector graphics. 1.0 y 0.9 520 525 515 530 0.8 535 510 540 545 0.7 550 505 555 560 0.6 Worthey 565 570 500 575 0.5 sRGB 580 585 590 495 0.4 595 600 605 610 0.3 620 490 635 700 Wavelengths in nm 0.2 485 480 Purple line 0.1 475 470 460 0.0 380 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1.0 26 16.1 References [1] R.W.G.Hunt Measuring Colour Fountain Press, England, 1998 [2] E.J.Giorgianni + Th.E.Madden Digital Color Management Addison-Wesley, Reading Massachusetts ,..., 1998 [3] G.Wyszecki + W.S.Stiles Color Science John Wiley & Sons, New York ,..., 1982 [4] J.D.Foley + A.van Dam+ St.K.Feiner + J.F.Hughes Computer Graphics Addison-Wesley, Reading Massachusetts,...,1993 [5] C.H.Chen + L.F.Pau + P.S.P.Wang Handbook of Pttern recognition and Computer Vision World Scientific, Singapore, ..., 1995 [6] J.J.Marchesi Handbuch der Fotografie Vol. 1 - 3 Verlag Fotografie, Schaffhausen, 1993 [7] T.Autiokari Accurate Image Processing http://www.aim-dtp.net 2001 [8] Ch.Poynton Frequently asked questions about Gamma http://www.inforamp.net/~poynton/ 1997 [9] M.Stokes + M.Anderson + S.Chandrasekar + R.Motta A Standard Default Color Space for the Internet - sRGB http://www.w3.org/graphics/color/srgb.html 1996 [10] G.Hoffmann Corrections for Perceptually Optimized Grayscales http://www.fho-emden.de/~hoffmann/optigray06102001.pdf 2001 [11] G.Hoffmann Hardware Monitor Calibration http://www.fho-emden.de/~hoffmann/caltutor270900.pdf 2001 [12] M.Nielsen + M.Stokes The Creation of the sRGB ICC Profile http://www.srgb.com/c55.pdf Year unknown, after 1998 [13] G.Hoffmann CieLab Color Space http://www.fho-emden.de/~hoffmann/cielab03022003.pdf [14] Everything about Color and Computers http://www.efg2.com 27 16.2 References [15] CIE Chromaticity Diagram, EPS Graphic http://www.fho-emden.de /~hoffmann/ciesuper.txt [16] Color-Matching Functions RGB, EPS Graphic http://www.fho-emden.de /~hoffmann/matchrgb.txt [17] Color-Matching Functions XYZ, EPS Graphic http://www.fho-emden.de /~hoffmann/matchxyz.txt [18] James A. Worthey Color Matching with Amplitude Not Left Out http://users.starpower.net/jworthey/FinalScotts2004Aug25.pdf [19] G.Hoffmann Locus of Unit Monochromats http://www.fho-emden.de/~hoffmann/jimcolor12062004.pdf This document http://www.fho-emden.de/~hoffmann/ciexyz29082000.pdf Gernot Hoffmann December 07 / 2006 Website Load Browser / Click here 28 A. Appendix Color Matching The calculation shows colors as defined by equal distances in CIELab. The corresponding values are drawn in the chromaticity diagram. BlueGreen (4) can be matched by positive weights RGB for CIE primaries. Vibrant BlueGreen (2) requires negative R. BlueGreen (1) is out of human gamut. RGB values are here normalized for 0...100. ColorCalc Med.White: Eq.Energy Primaries: CIE Intent: AbsCol G.Hoffmann Ref.White: D50 Trc: 1.0 Dec.06 / 2006 Input: Lab Bradford: No Set: 13 X 0.090936 0.124317 0.165004 0.213721 0.271192 0.338140 0.415287 0.503358 0.603075 Y 0.281233 0.281233 0.281233 0.281233 0.281233 0.281233 0.281233 0.281233 0.281233 Z 1.271889 0.902067 0.611932 0.391815 0.232047 0.122959 0.054882 0.018146 0.001827 x 0.055312 0.095071 0.155933 0.241011 0.345700 0.455510 0.552683 0.627052 0.680568 y 0.171060 0.215073 0.265774 0.317144 0.358500 0.378851 0.374278 0.350343 0.317371 z 0.773627 0.689856 0.578293 0.441845 0.295800 0.165639 0.073039 0.022605 0.002062 L* 60.0000 60.0000 60.0000 60.0000 60.0000 60.0000 60.0000 60.0000 60.0000 a* - 1 00.0000 - 7 5.0000 - 5 0.0000 - 2 5.0000 0.0000 25.0000 50.0000 75.0000 100.0000 b* - 1 00.0000 - 7 5.0000 - 5 0.0000 - 2 5.0000 0.0000 25.0000 50.0000 75.0000 100.0000 R - 6 3.2553 - 3 8.0477 - 1 4.8426 6.9838 28.0552 48.9952 70.4276 92.9761 117.3231 G 46.7170 41.7155 37.0448 32.5818 28.2034 23.7865 19.2082 14.3453 9.0636 B 127.9629 90.6690 61.4185 39.2362 23.1472 12.1761 5.3480 1.6875 0.0930 R’ 0.0000 0.0000 0.0000 6.9838 28.0552 48.9952 70.4276 92.9761 100.0000 G’ 46.7170 41.7155 37.0448 32.5818 28.2034 23.7865 19.2082 14.3453 9.0636 B’ 100.0000 90.6690 61.4185 39.2362 23.1472 12.1761 5.3480 1.6875 0.0930 CCT none none none 12527 K 5001 K 2504 K none none none RGB out-gam out-gam out-gam in-gam in-gam in-gam in-gam in-gam out-gam 1.0 b* y 100 9 0.9 8 520 525 7 0.8 515 530 535 6 510 540 100 a* 545 5 0.7 550 555 4 505 560 3 0.6 565 2 570 500 575 1 0.5 580 585 590 495 0.4 D50 595 2222 2353 2105 2500 2000 2677 2857 3077 3333 600 D65 3636 4000 6 7 605 4444 8 610 D93 5000 5714 0.3 620 6667 490 4 9 635 8000 10000 700 Trc Matrix Crx 3 25000 0.2 485 2.364998 -0.896709 -0.468149 2 -0.515142 1.426371 0.088744 0.005202 -0.014403 1.008898 1 480 0.1 Matrix Cxr 475 470 0.489921 0.310016 0.200063 460 0.176937 0.812422 0.010641 380 0.000000 0.009999 0.990301 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1.0 29