Mathematics and Colour. Research Matters. Professor Nick Higham. February 25, Director of Research. School of Mathematics

Transcription

1 Mathematics and Colour Research Matters Professor Nick Higham February 25, 2009 Director of Research School Nick Higham of Mathematics The Director University of Research of Manchester School of Mathematics 1 / 6

2 What is Colour? Human retina has 3 types of cones. Nick Higham Mathematics and Colour 2 / 60

3 What is Colour? Human retina has 3 types of cones. Colour space is 3-dimensional ( trichromatic theory ). Can mathematics help us understand colour? Nick Higham Mathematics and Colour 2 / 60

4 There s Something about Yellow Springer Undergraduate Mathematics Series David F. Griffiths John W. Dold David J. Silvester Essential Partial Differential Equations Analytical and Computational Aspects Nick Higham Mathematics and Colour 3 / 60

5 There s Something about Yellow Springer Undergraduate Mathematics Series David F. Griffiths John W. Dold David J. Silvester Essential Partial Differential Equations Analytical and Computational Aspects Why does yellow appear so bright? Nick Higham Mathematics and Colour 3 / 60

6 There s Something about Yellow Springer Undergraduate Mathematics Series David F. Griffiths John W. Dold David J. Silvester Essential Partial Differential Equations Analytical and Computational Aspects Why does yellow appear so bright? Nick Higham Mathematics and Colour 3 / 60

7 Colour Blindness John Dalton ( ). Described his own c.b. in lecture to M/cr Lit & Phil Soc, He was a deuteranope. Nick Higham Mathematics and Colour 4 / 60

8 Vector Space Model of Colour (1) Model responses of the 3 cones as c i = λmax λ min s i (λ)f (λ)dλ, i = 1: 3, where f = spectral distrib. of light, s i = sensitivity of ith cone, [λ min, λ max ] = wavelengths of visible spectrum. Nick Higham Mathematics and Colour 5 / 60

9 Vector Space Model of Colour (1) Model responses of the 3 cones as c i = λmax λ min s i (λ)f (λ)dλ, i = 1: 3, where f = spectral distrib. of light, s i = sensitivity of ith cone, [λ min, λ max ] = wavelengths of visible spectrum. Discretizing gives c = S T f, c R 3, S R n 3, f R n. For standardized S, c is the tristimulus vector. Nick Higham Mathematics and Colour 5 / 60

10 Vector Space Model of Colour (2) Let columns of P = [p 1 p 2 p 3 ] be colour primaries. }{{} n 3 Assuming S T P is nonsingular, S T f = }{{} S T P (S T P) 1 S T f S T Pa(f ), 3 3 where a(f ) = (S T P) 1 S T f. Colour of any spectrum f can be matched by primaries. Nick Higham Mathematics and Colour 6 / 60

11 Vector Space Model of Colour (2) Let columns of P = [p 1 p 2 p 3 ] be colour primaries. }{{} n 3 Assuming S T P is nonsingular, S T f = }{{} S T P (S T P) 1 S T f S T Pa(f ), 3 3 where a(f ) = (S T P) 1 S T f. Colour of any spectrum f can be matched by primaries. Need a i 0 not all visible spectra can be produced. Compensate a i < 0 by adding a i p i to f, There exist spectra f, g, f g, such that metamers. Both good and bad. S T }{{} 3 n f = S T g: Nick Higham Mathematics and Colour 6 / 60

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13 R Matrix Theory of Cohen Cohen (2001) stresses the importance of R = S(S T S) 1 S T = SS +, the orthogonal projector on range(s). Independent of the choice of primaries used for colour matching (S SZ ). F matrix defined as Q in the factorization S = QL (Q R n 3, Q T Q = I, L R 3 3 lower triangular). Proposes use of tricolor coordinates F T f. Nick Higham Mathematics and Colour 8 / 60

14 A Nonlinear, Imperfect World Limitations on how far the mathematical model can take us. We all see colour slightly differently. Our eyes do not behave linearly. Brain processing of colour is complicated (colour temp, opponent-process theory) and leads to various illusions. Most colours we see are artificially generated: camera, screen, print, paints,... all these devices have limitations. Nick Higham Mathematics and Colour 9 / 60

15 CMYK All printing is done using four colours: cyan, yellow, magenta, and black. C + M + Y = K = black. One redundant coordinate. Why do we need K? Nick Higham Mathematics and Colour 10 / 60

16 CMYK All printing is done using four colours: cyan, yellow, magenta, and black. C + M + Y = K = black. One redundant coordinate. Why do we need K? Printing 3 layers makes the paper very wet. Black as 3 layers requires accurate registration. C + M + Y will not give a true, deep black due to ink imperfections. Coloured ink is more expensive. Nick Higham Mathematics and Colour 10 / 60

17 CMYK All printing is done using four colours: cyan, yellow, magenta, and black. C + M + Y = K = black. One redundant coordinate. Why do we need K? Printing 3 layers makes the paper very wet. Black as 3 layers requires accurate registration. C + M + Y will not give a true, deep black due to ink imperfections. Coloured ink is more expensive. What order to lay down the inks? CMYK or KCMY are standard. Note that C + M + Y M + C + Y. Nick Higham Mathematics and Colour 10 / 60

18 Complex Beauties Calendar [link] Nick Higham Mathematics and Colour 11 / 60

19 CMYK vs RGB CMYK produces a different range of colors than RGB. Cannot produce some of the brilliant blues. Whenever we print a document on a laser printer we view a CMYK representation of the colors. Original Print Scan of print Nick Higham Mathematics and Colour 12 / 60

20 CIE Chromacity Coordinates Projective transformation of 3-dimensional colour space. Nick Higham Mathematics and Colour 13 / 60

21 Projective Transformation Nick Higham Mathematics and Colour 14 / 60

22 Excursion into LAB Space Change from RGB space to CIE L*a*b* (LAB, 1976): L = lightness, A = green magenta, B = blue yellow. Separates luminosity from colour. More perceptually uniform. Nick Higham Mathematics and Colour 15 / 60

23 Transformation XYZ LAB Let X n, Y n, Z n be tristimuli of white stimulus. where f (x) = Range: 0 L 100. L = 116f (Y /Y n ) 16, A = 500 [f (X/X n ) f (Y /Y n )], B = 200 [f (Y /Y n ) f (Z /Z n )]. A = B = 0 no colour. { x 1/3, x , 7.787x , x Euclidean distance used as colour difference metric. Nick Higham Mathematics and Colour 16 / 60

24 Dan Margulis on LAB (2006) Nick Higham Mathematics and Colour 17 / 60

25 Editing in LAB LAB separates luminosity (L) from colour (A,B). Colour noise can be handled by blurring the A, B channels. Much bigger space than srgb with many imaginary colours. Good for boosting contrast, enhancing colours, and sharpening. Nick Higham Mathematics and Colour 18 / 60

26 LAB Example: Original Nick Higham Mathematics and Colour 19 / 60

27 LAB Example: Via LAB Nick Higham Mathematics and Colour 20 / 60

28 LAB Example: Explanation 1 Convert from RGB to LAB. 2 Apply Image to itself in overlay mode: L f (L), A f (A), B f (B), where Apply Image: L 75% old L + 25% new L. 4 Curves adjustment on L channel: Nick Higham Mathematics and Colour 21 / 60

29 Rainbow Colour Maps Nick Higham Mathematics and Colour 22 / 60

30 Rainbow Considered Harmful Not perceptually uniform: colours change at different rates. Confusing: no natural ordering (ROYGBIV). Introduces artefacts: sharp transitions between hues. Loses information in grayscale Nick Higham Mathematics and Colour 23 / 60

31 MATLAB Parula Colour Map (2014) Nick Higham Mathematics and Colour 24 / 60

32 MATLAB Parula Colour Map (2014) Nick Higham Mathematics and Colour 25 / 60

33 Adobe Photoshop Photoshop 1.0 (Mac), Market leader for commercial bitmap/image manipulation. Supports RGB, LAB, CMYK. Excels in non-destructive editing (layers). Adobe Photoshop software includes a counterfeit deterrence system (CDS) that prevents the use of the product to illegally duplicate banknotes. Nick Higham Mathematics and Colour 26 / 60

34 Adobe Photoshop Photoshop 1.0 (Mac), Market leader for commercial bitmap/image manipulation. Supports RGB, LAB, CMYK. Excels in non-destructive editing (layers). Adobe Photoshop software includes a counterfeit deterrence system (CDS) that prevents the use of the product to illegally duplicate banknotes. Nick Higham Mathematics and Colour 26 / 60

35 Adobe Photoshop Photoshop 1.0 (Mac), Market leader for commercial bitmap/image manipulation. Supports RGB, LAB, CMYK. Excels in non-destructive editing (layers). Adobe Photoshop software includes a counterfeit deterrence system (CDS) that prevents the use of the product to illegally duplicate banknotes. Nick Higham Mathematics and Colour 26 / 60

36 JPEG JPEG (1992) stores RGB images in compressed form. It converts from RGB to YC b C r colour space where Y = luminance, C b = blue, C r = red by Y C b = R G. C r B Transformation must be inverted to display a JPEG image. Human vision more sensitive to luminance than colour, so can more heavily compress C b, C r coordinates. Nick Higham Mathematics and Colour 27 / 60

37 Fingerprints FBI Digitized at 500dpi 10Mb. Compression > 10:1 req d. Standardized on wavelet compression (1993). Jpeg: resonance of 8-pixel tiling w/ 500dpi scans, many edges. Wavelets: gradual blurring as compression increased. Nick Higham Mathematics and Colour 28 / 60

38 Adobe DNG XYZtoCamera matrix is n 3, n = dim of camera colour space, usually 3 or 4. Translating Camera Neutral Coordinates to White Balance xy Coordinates 1 Guess an xy value. Use that guess to find the interpolation weighting factor between the color calibration tags. Find the XYZtoCamera matrix as above. 2 Find a new xy value by computing: XYZ = Inverse (XYZtoCamera) * CameraNeutral (If the XYZtoCamera matrix is not square, then use the pseudo inverse.) 3 Convert the resulting XYZ to a new xy value. 4 Iterate until the xy values converge to a solution. Nick Higham Mathematics and Colour 29 / 60

39 Rounding Errors Every editing operation executes p ij = round(f ij (p ij )). Rounding errors can potentially cause deterioration. Controversy over 8-bit vs. 16-bit editing. Controversy over colour space: choice & conversions. Nick Higham Mathematics and Colour 30 / 60

40 Arithmetic on Images: Brightening Simple arithmetic on images (+,,,/) can be very effective! Let R, G, B [0, 1] with black = (0, 0, 0), white = (1, 1, 1). To brighten an image we need to increase the coordinates. Nick Higham Mathematics and Colour 31 / 60

41 Original Nick Higham Mathematics and Colour 32 / 60

42 Simple Brightening Transformation Nick Higham Mathematics and Colour 33 / 60

43 Better Brightening Transformation Map each coordinate x 1 (1 x) 2. Photoshop: Apply Image with Screen Blending Mode Nick Higham Mathematics and Colour 34 / 60

44 Pixel-Dependent Brightening Nick Higham Mathematics and Colour 35 / 60

45 Final Image Nick Higham Mathematics and Colour 36 / 60

46 Change Autumn into Summer Nick Higham Mathematics and Colour 37 / 60

47 Looking at the Numbers Sample colours from photo. Typical RGB values for green tree leaves: (R, G, B) = (110, 103, 53), (50, 55, 12), (135, 125, 81). Nick Higham Mathematics and Colour 38 / 60

48 Looking at the Numbers Sample colours from photo. Typical RGB values for green tree leaves: (R, G, B) = (110, 103, 53), (50, 55, 12), (135, 125, 81). Typical RGB values for yellow tree leaves: (R, G, B) = (250, 193, 73), (152, 88, 90), (194, 112, 18). Nick Higham Mathematics and Colour 38 / 60

49 Looking at the Numbers Sample colours from photo. Typical RGB values for green tree leaves: (R, G, B) = (110, 103, 53), (50, 55, 12), (135, 125, 81). Typical RGB values for yellow tree leaves: (R, G, B) = (250, 193, 73), (152, 88, 90), (194, 112, 18). Solution Make R = G by copying the green coordinates into the red. Nick Higham Mathematics and Colour 38 / 60

50 It s Summer Nick Higham Mathematics and Colour 39 / 60

51 Original Nick Higham Mathematics and Colour 40 / 60

52 With Mask to Protect Sky Nick Higham Mathematics and Colour 41 / 60

53 Repainting University Place Nick Higham Mathematics and Colour 42 / 60

54 RePainted Nick Higham Mathematics and Colour 43 / 60

55 Flip Sign of A Channel UoM turquoise is (L, A, B) (85, 12, 3). Convert to LAB then A A. Now have (L, A, B) (85, 12, 3). Nick Higham Mathematics and Colour 44 / 60

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66 Mean Nick Higham Mathematics and Colour 55 / 60

67 Median Nick Higham Mathematics and Colour 56 / 60

68 Max Nick Higham Mathematics and Colour 57 / 60

69 Min Nick Higham Mathematics and Colour 58 / 60

70 Variance Nick Higham Mathematics and Colour 59 / 60

71 Summary Maths intrinsic to modelling colour, and defining, analyzing and exploiting colour spaces. Can go a long way in manipulating the colour of images with elementary maths. All the maths needed to understand colour is covered in the Manchester honours degree maths programme. Talk, including references, available at digphot_long.pdf Nick Higham Mathematics and Colour 60 / 60

72 Acknowledgements for Graphics Wikipedia: CIE1931_XYZCMF.png commons/b/b0/ciexy1931.png compvis/colourintro/colourintro.htm Nick Higham Mathematics and Colour 1 / 5

73 References I J. B. Cohen. Visual Color and Color Mixture: The Fundamental Color Space. University of Illinois Press, Urbana and Chicago, USA, B. Fraser. Raw capture, linear gamma, and exposure. linear_gamma.pdf. Nick Higham Mathematics and Colour 2 / 5

74 References II N. J. Higham. Color spaces and digital imaging. In N. J. Higham, M. R. Dennis, P. Glendinning, P. A. Martin, F. Santosa, and J. Tanner, editors, The Princeton Companion to Applied Mathematics, pages Princeton University Press, Princeton, NJ, USA, A. R. Hill. How we see colour. In R. McDonald, editor, Colour Physics for Industry, pages Society of Dyers and Colourists, Bradford, England, Nick Higham Mathematics and Colour 3 / 5

75 References III D. M. Hunt, K. S. Dulai, J. K. Bowmaker, and J. D. Mollon. The chemistry of John Dalton s color blindness. Science, 267: , JPEG file interchange format, version D. Margulis. Photoshop LAB Color: The Canyon Conundrum and Other Adventures in the Most Powerful Colorspace. Peachpit Press, Berkeley, CA, USA, D. Margulis. Professional Photoshop. The Classic Guide to Color Correction. Peachpit Press, Berkeley, CA, USA, fifth edition, Nick Higham Mathematics and Colour 4 / 5

76 References IV C. Poynton. A guided tour of color space, G. Sharma and H. J. Trussell. Digital color imaging. IEEE Trans. Image Processing, 6(7): , S. Westland and C. Ripamonti. Computational Colour Science Using MATLAB. Wiley, New York, Nick Higham Mathematics and Colour 5 / 5