# Murdoch redux. Colorimetry as Linear Algebra. Math of additive mixing. Approaching color mathematically. RGB colors add as vectors

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1 Murdoch redux Colorimetry as Linear Algebra CS 465 Lecture 23 RGB colors add as vectors so do primary spectra in additive display (CRT, LCD, etc.) Chromaticity: color ratios (r = R/(R+G+B), etc.) color without regard for overall brightness Color matching and metamers any spectrum can be matched by combining 3 primaries metamers: different spectra that look the same CIE colorimetry X, Y, and Z: standardized hypothetical primaries x!, y!, z!: color matching functions for X, Y, Z 2005 Steve Marschner Steve Marschner 2 Approaching color mathematically Three distinct ideas relating color values to stimuli Primaries and additive color: R, G, and B tell how much you turn up three primary spectra Sensitivities and color detection: R, G, and B are the outputs of detectors with three sensitivity functions Color matching functions and metamers: R, G, and B are the amounts of three primaries required to match a given spectrum Math of additive mixing Simply add contributions of primaries per wavelength key property: all wavelengths change by the same scale factor [sources unknown] 2005 Steve Marschner Steve Marschner 4

2 A simple light detector A simple light detector Produces a number when photons land on it value depends strictly on the number of photons detected each photon has a probability of being detected that depends on the wavelength (can t distinguish signals caused by different wavelengths) This model works for many detectors: based on semiconductors (such as in a digital camera) based on visual photopigments (such as in human eyes) 2005 Steve Marschner Steve Marschner 6 Light detection math Light detection in the eye Same math carries over to power distributions spectrum entering the detector is s(!) detector has its spectral sensitivity or spectral response, r(!) Recall there are three types of cones call them S, M, L for short, medium, long wavelengths eye therefore detects three values from a spectrum, corresponding to three response functions: measured signal detector s sensitivity input spectrum [Michael Murdoch Kodak] 2005 Steve Marschner Steve Marschner 8

3 Spectra as vectors Additive synthesis and detection correspond to basic linear algebra concepts for concreteness, think of spectra as having a finite number of little bands continuous spectrum s(!) becomes discrete spectrum s[i] Color operations as vector algebra Additive display (synthesis): linear combination of spectra: is like linear combination of vectors, or matrix multiplication: 2005 Steve Marschner Steve Marschner 10 Color operations as vector algebra Color detection (analysis): linear measurement of spectra: Color operations as vector algebra Color detection (analysis): three-band linear measurement of spectra corresponds to three dot products, or a matrix multiplication: is like a dot product of vectors: 2005 Steve Marschner Steve Marschner 12

4 Pseudo-geometric interpretation A dot product is a projection We are projecting a high dimensional vector (a spectrum) onto three vectors differences that are perpendicular to all 3 vectors are not detectable For intuition, we can imagine a 3D analog 3D stands in for high-d vectors 2D stands in for 3D Then vision is just projection onto a plane Pseudo-geometric interpretation The information available to the visual system about a spectrum is three values this amounts to a loss of information analogous to projection on a plane Two spectra that produce the same response are metamers 2005 Steve Marschner Steve Marschner 14 Color reproduction CRT display primaries Have a spectrum s; want to match on RGB monitor match means it looks the same any spectrum that projects to the same point in the visual color space is a good reproduction Must find a spectrum that the monitor can produce that is a metamer of s Emission (watts/m 2 ) [cs417 Greenberg] wavelength (nm) R, G, B? 2005 Steve Marschner 15 Curves determined by phosphor emission properties 2005 Steve Marschner 16

5 LCD display primaries Color reproduction Say we have a spectrum s we want to match on an RGB monitor match means it looks the same any spectrum that projects to the same point in the visual color space is a good reproduction So, we want to find a spectrum s that the monitor can produce that matches s that is, we want to display a metamer of s on the screen Curves determined by (fluorescent) backlight and filters 2005 Steve Marschner Steve Marschner 18 Color reproduction We want to compute the combination of r, g, b that will project to the same visual response as s. Color reproduction as linear algebra What color do we see when we look at the display? Feed C to display Display produces s Eye looks at s and produces V 2005 Steve Marschner Steve Marschner 20

6 Color reproduction as linear algebra Goal of reproduction: visual response to s and s a is the same: Substituting in the expression for s, color matching matrix for RGB Color matching functions Used like response functions, but give primary weights e.g. R,G,B color matching functions, dotted with a spectrum, tell how much of a particular R, G, and B are required to match the spectrum Just derived them for a particular display also can measure directly in fact, from visual experiments we can only get color matching functions, not S, M, and L Recall previous discussion: CIE XYZ system standard hypothetical primaries defined only via color matching functions 2005 Steve Marschner Steve Marschner 22 Color matching in practice Color matching in practice In practice, we have color matching functions, not the S, M, and L sensitivities but any color matching functions are just as good as SML for matching colors any colors with the same X, Y, Z values have the same S, M, L values (they have to, because the colors match!) so in practice color matching is done thus: you can compute the point s using any basis for the human visual subspace (you are just matching the response to s and s ) and the results are the same as with M SML because any color matching matrices span the same space 2005 Steve Marschner Steve Marschner 24

7 Basic colorimetric concepts Luminance the overall magnitude of the the visual response to a spectrum (independent of its color) corresponds to the everyday concept brightness determined by product of SPD with the luminous efficiency function V! that describes the eye s overall ability to detect light at each wavelength e.g. lamps are optimized to improve their luminous efficiency (tungsten vs. fluorescent vs. sodium vapor) [Stone 2003] Luminance, mathematically Y just has another response curve (like S, M, and L) r Y is really called V! V! is a linear combination of S, M, and L Has to be, since it s derived from cone outputs 2005 Steve Marschner Steve Marschner 26 Color spaces Standard color spaces Need three numbers to specify a color but what three numbers? a color space is an answer to this question Common example: monitor RGB define colors by what R, G, B signals will produce them on your monitor (in math, s = RR + GG + BB for some spectra R, G, B) device dependent (depends on gamma, phosphors, gains, ) therefore if I choose RGB by looking at my monitor and send it to you, you may not see the same color also leaves out some colors (limited gamut), e.g. vivid yellow Standardized RGB (srgb) makes a particular monitor RGB standard other color devices simulate that monitor by calibration srgb is usable as an interchange space; widely adopted today gamut is still limited 2005 Steve Marschner Steve Marschner 28

8 A universal color space: XYZ Standardized by CIE (Commission Internationale de l Eclairage, the standards organization for color science) Based on three imaginary primaries X, Y, and Z (in math, s = XX + YY + ZZ) imaginary = only realizable by spectra that are negative at some wavelengths key properties any stimulus can be matched with positive X, Y, and Z separates out luminance: X, Z have zero luminance, so Y tells you the luminance by itself Perceptually organized color spaces Artists often refer to colors as tints, shades, and tones of pure pigments tint: mixture with white shade: mixture with black tones: mixture with black and white gray: no color at all (aka. neutral) This seems intuitive white grays black tints shades tints and shades are inherently related to the pure color same color but lighter, darker, paler, etc. pure color [after FvDFH] 2005 Steve Marschner Steve Marschner 30 Perceptual dimensions of color Hue the kind of color, regardless of attributes colorimetric correlate: dominant wavelength artist s correlate: the chosen pigment color Saturation the colorfulness colorimetric correlate: purity artist s correlate: fraction of paint from the colored tube Lightness (or value) the overall amount of light colorimetric correlate: luminance artist s correlate: tints are lighter, shades are darker Perceptual dimensions: chromaticity In x, y, Y (or another luminance/chromaticity space), Y corresponds to lightness hue and saturation are then like polar coordinates for chromaticity (starting at white, which way did you go and how far?) [source unknown] 2005 Steve Marschner Steve Marschner 32

9 Perceptual dimensions of color There s good evidence ( opponent color theory ) for a neurological basis for these dimensions the brain seems to encode color early on using three axes: white black, red green, yellow!blue the white black axis is lightness; the others determine hue and saturation one piece of evidence: you can have a light green, a dark green, a yellow-green, or a blue-green, but you can t have a reddish green (just doesn t make sense) thus red is the opponent to green another piece of evidence: afterimages (recall flag illusion) RGB as a 3D space A cube: (demo of RGB color picker) 2005 Steve Marschner Steve Marschner 34 Perceptual organization for RGB: HSV Uses hue (an angle, 0 to 360), saturation (0 to 1), and value (0 to 1) as the three coordinates for a color the brightest available RGB colors are those with one of R,G,B equal to 1 (top surface) each horizontal slice is the surface of a sub-cube of the RGB cube [FvDFH] Perceptually uniform spaces Two major spaces standardized by CIE designed so that equal differences in coordinates produce equally visible differences in color LUV: earlier, simpler space; L*, u*, v* LAB: more complex but more uniform: L*, a*, b* both separate luminance from chromaticity including a gamma-like nonlinear component is important (demo of HSV color pickers) 2005 Steve Marschner Steve Marschner 36

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