0 6 aspen leaf petunia 0 5 0 4 0 3 0 2 0 1 0 350 400 450 500 550 600 650 700 Figure 1 Reflectance function of an aspen leaf and a petunia Clark et al 2003 750 Color Vision The L M S cone responses to color C are the inner products of the spectral distribution of color C and the three cone spectral sensitivity functions c C Z S C d cm C Z Sm C d cs C Z Ss C d Metamers Two spectral distributions C and C0 such that Z Z Z S C d Z S C0 d Sm C d Z Sm C0 d Ss C d Z Ss C0 d will be perceived to be the same color by a human observer Such colors are termed metamers 3 L M S 2 5 2 1 5 1 0 5 0 0 5 350 400 450 500 550 600 650 700 750 Figure 2 Cone spectral sensitivity functions S Sm and Ss Stiles and Burch 1955 Color Vision contd C S s S m C s S l C m Figure 3 The L M S cone responses to color C C l Linearity If the M cone response to a light source L1 is cm L1 Z Sm L1 d and to a lightsource L2 is cm L2 Z Sm L2 d then by linearity the M cone response to a mixture of two light sources v1L1 v2L2 is cm v1L1 v2L2 v1 Z Sm L1 d v2 Z Sm L2 d L b L g V b L r V g V r S s C s S m C m S l C l Figure 4 The L M S cone respones to color C can be matched by a weighted sum of three linearly independent source distributions Color Matching The color C can be matched by a mixture of three light sources with linearly independent spectral distributions Lr Lg and Lb by solving the following linear system for the appropriate color mixtures vr vg and vb c m r m g m b cm vr mmr vg mmg vb mmb cs msr msg msb where mi j Si L j d Written slightly differently m r m g m b vr c mmr mmg mmb vg cm msr msg msb vb cs R we see that v M 1c Pure Sources Things are simpler when pure sources are used Lr r Lg g Lb b The CIE1 standard primary sources are r 700 nm g 546 1 nm b 435 8 nm The color matching equations become m r m g m b vr c mmr mmg mmb vg cm msr msg msb vb cs where mi j Si j d Si j R 1 Commission Internationale de L Eclairage L b L g V b L r V g V r S s C s S m C m S l C l Figure 5 The L M S cone respones to color C can be matched by a weighted sum of three pure source distributions Metamers contd Consider three functions Sx Sy and Sz related to the cone spectral sensitivity functions S Sm and Ss as follows a x a y a z S Sx Sy amx amy amz Sm Ss Sz asx asy asz We will show that C and C0 will be metamers with respect to S Sm and Ss if they are metamers with respect to Sx Sy and Sz Metamers contd If C and C0 are metamers with respect to Sx Sy and S2 then Z Z Z Sx C d Z Sx C 0 d Sy C d Z Sy C 0 d Sz C d Z Sz C 0 d It follows that a j j x y z j x y z Z S j C d Z S j C 0 d a j Metamers contd Rearranging things a bit we see that Z a j S j C d j x y z Z a j S j C 0 d j x y z Now because S a xSx a ySy a zSz it follows that Z S C d Z S C 0 d Similar arguments apply to Sm and Ss Consequently C and C0 are metamers with respect to S Sm and Ss Color Matching Functions It follows that it is possible to do color matching with any three functions related to the actual cone spectral sensitivity functions by a linear transformation Suitable functions were first deduced empirically by Wright in 1929 and adopted by the CIE as a standard in 1931 Color Matching Function Deduction Given six pure sources at wavelengths y 6 the values of three color matching functions Sx Sy and Sz can be deduced at these six wavelengths by repeatedly performing a simple color matching task Randomly choose v1 Ci v2 Ci and v3 Ci Adjust v4 Ci0 v5 Ci0 and v6 Ci0 until Ci0 has the same appearance as Ci Color Matching Function Deduction contd This yields three linear equations and eighteen unknowns Sx 1 Sx 2 Sx 3 v1 Ci Sy 1 Sy 2 Sy 3 v2 Ci Sz 1 Sz 2 Sz 3 v3 Ci 0 Sx 4 Sx 5 Sx 6 v4 Ci Sy 4 Sy 5 Sy 6 v5 Ci0 Sz 4 Sz 5 Sz 6 v6 Ci0 Repeating the color matching task six times yields a system of eighteen equations and eighteen unkowns which can be solved by Gaussian elimination 1 8 X Y Z 1 6 1 4 1 2 1 0 8 0 6 0 4 0 2 0 350 400 450 500 550 600 650 700 Figure 6 CIE 1931 color matching functions Sx Sy and Sz Wright 1929 750 Tristimulus Values The tristimulus values for color C are the inner products of the spectral distribution of color C and the three CIE 1931 color matching functions X C Z Sx C d Y C Z Sy C d Z C Z Sz C d The tristimulus values for white are X W Y W Z W 1 Chromaticities The chromaticities for color C are computed by normalizing the tristimulus values by their sum X C x C X C Y C Z C Y C y C X C Y C Z C Z C z C X C Y C Z C Since the chromaticities for any color C always sum to one x C y C z C 1 only two of the three chromaticities e g x and y are independent Plotting the set of visible colors in x and y coordinates results in the chromaticity diagram Hue Saturation and Intensity Another useful coordinate system is HSI where HSI stand for hue saturation and intensity Hue is an angular quantity which correlates closely with wavelength For example as hue varies between 0 and 360 the perceived colors move through the visible spectrum i e red orange yellow green blue indigo violet red Saturation represents the purity of a color i e the absence of white For example red is more saturated than pink Intensity is a …