Colorimetry - Trichromatic Theory of Color Vision, Color Matching Functions, Cone fundamentals, CIE
values opponent tristimulus cmfs
SABINE SÜSSTRUNK, Ph.D.
Ecole Polytechnique Fédérale de Lausanne
Trichromatic Theory of Color Vision
The trichromatic theory of color vision, also referred to as the Young-Helmholtz three-component theory [You70, vH62], assumes that the signals generated in the three cone types ( LMS ), which are independent and have different spectral sensitivities ( L for long wavelength sensitivity, M for medium wavelength sensitivity, and S for short wavelength sensitivity), are transmitted directly to the brain where “color sensations” are experienced that correlate in a simple and direct way to the three cone signals. This theory has been found to hold in a series of color-matching experiments. The experimental laws of color matching assume that for a given observation condition, test color stimulus C t (?) can be matched completely by an additive mixture of three fixed primary stimuli C r (?), C g (?), C b (?) with adjustable radiant power:
R, G , and B are the relative intensities of C r (?), C g (?), C b (?), respectively, and are called the tristimulus values of C t (?). Any set of primaries can be used, as long as none of the primaries can be color matched with a mixture of the other two.
The results of color matches obey certain linearity laws, as first formulated by Grassman in 1853 [WS82]. If C 1 (?), C 2 (?), C 3 (?), and C 4 (?) are color stimuli and the symbol = has the meaning of “visual match,” then:
- Symmetry Law: if C 1 (?) = C 2 (?), then C 2 (?) = C 1 (?)
- Transitivity Law: if C 1 (?) = C 2 (?) and C 2 (?) = C 3 (?), then C 1 (?) = C 3 (?)
- Proportionality Law: if C 1 (?) = C 2 (?), then a C 2 (?) = a C 2 (?), where a is a positive factor that increases or
reduces the radiant power of the color stimulus while its relative SPD remains the same. - Additivity Law: if C 1 (?) = C 2 (?) and C 3 (?) = C 4 (?), then ( C 1 (?) + C 3 (?)) = ( C 2 (?) + C 4 (?))
These generalized laws of trichromacy ignore the dependence of color matches on the observational conditions, such as different radiant power, viewing eccentricity, stimulus surround, and adaptation to previous stimuli. To control viewing conditions, color matching experiments are therefore usually done with a visual colorimeter . A visual colorimeter is a device with a partitioned viewing area, where one half displays the reference color stimulus and the other half the mixture of the three primaries that can be adjusted by the observer to match the reference (see Figure 28).
Color Matching Functions
In 1931, the CIE (Commission Internationale de l’Eclairage) standardized a set of Color Matching Functions (CMFs) based on color-matching experiments by Wright and Guild [WS82] using a colorimeter with a 2-inch bipartite field. Assuming that additivity holds and the luminous efficiency function V (?) of the HVS is a linear combination of the CMFs, they established a set of r, ?, b¯ color matching functions with “real” red C r (?) (? = 700 nm), green C g (?) (? = 546.1 nm), and blue C b (?) (? = 435.8 nm), monochromatic primaries based on the chromaticity coordinates of their experimental primaries. These r, ?, b¯ , CMFs [CIE86] illustrate the relative amount R, G, B of primaries C r , C g , and C b needed to additively mix a monochromatic light source at a given wavelength (see Figure 29).
The CIE additionally standardized a set of x¯, ?, z¯ color matching functions, based on imaginary X, Y, Z primaries, which are a linear combination of the color matching functions derived from the original primaries. The transform was designed so that the x¯, ?, z¯ CMFs do not contain any negative values, primarily to design physical measuring devices, and that the ? color matching function corresponds to V (?) [WS82, Hun98]:
x¯ = 0.49 r (?) + 0.31 ? (?) + b¯ (?)
? = 0.17697 r (?) + 0.81240 ? (?) + 0.01063 b¯ (?)
z¯ = 0.0 r (?) + 0.01 ? (?) + 0.099 b¯ (?)
The x¯, ?, z¯ color matching functions are also called the CIE 1931 standard observer and are used for colorimetric calculations (see section 2) when the size of the stimulus does not extend 4° of visual angle. These CMFs are still an international standard today, even though Judd [Jud51] and later Vos [Vos78] proposed a modification based on a corrected luminous efficiency function, called V M (?). The original V M (?) of 1924 used in the derivation of the CIE 1931 CMFs underestimates the sensitivities at wavelength below 460 nm. Today, the color vision research community almost exclusively uses the Judd-Vos modified 2° CMFs [SS01], while the color science and color imaging communities still use the original CIE 1931 2° CMFs (see Figure 30).
In 1964, the x¯ 10 , ? 10 , z¯ 10 color matching functions for the CIE 1964 supplementary standard observer were developed by Judd [WS82], based on experimental investigations by Stiles and Burch [SB59] and Speranskaya [Spe59] with stimuli sizes of 10°.
Cone fundamentals
The color-matching functions described above are not cone sensitivities, or absorption spectra of the cone pigments. They are based on color-matching experiments, and their shape is determined by the choice of the primaries. However, if we assume that the basic principle of the trichromatic theory of color vision is correct, then cone responses also behave additively, and the cone sensitivities (also called cone fundamentals ) are a linear combination of color-matching functions (divided by intra-ocular medium absorption spectra). While this assumption is somewhat questionable when considering the complexity of the HVS, there are advantages when modeling visual processing.
Physical measurements, such as reflectances and spectral power distributions of light sources, can easily be linearly transformed into cone responses. Using color-matching data and experimental data of color-deficient observers, several sets of cone fundamentals that are linear combinations of either 2° or 10° CMFs were published (see [SS01] for an overview and [SS03] for data). Vos and Walraven [VW71] and Smith and Pokorny [SP75] base their cone sensitivities on the modified 2° X, Y, Z color-matching functions. Stockman and Sharpe [SS00] base their 2° and 10° cone fundamentals on the Stiles and Burch 10° CMFs. Figure 31 illustrates the different 2° cone fundamentals.
As can be seen in Figure 31, the L and M cone sensitivities are very correlated—their spectral distributions overlap significantly. Additionally, they have a broad base. From the point of view of quantum efficiency, broadband sensors are able to capture more quanta and are thus more sensitive to radiation overall. However, from the point of view of coding efficiency, having two nearly identical sensors is inefficient as they both carry similar information [DB91]. As discussed below, the human visual system has found a way to de-correlate these sensor responses by its ability to encode the difference of the signals instead of the absolute responses.
Opponent color modulations
The theory of opponent colors is commonly attributed to Hering [Her78], although Goethe (1832) [vG91] previously discussed the concept. They both observed that certain colors are never perceived together, i.e. their names do not mix. We never see bluish-yellows or reddish-greens, where as bluish-greens (turquoise) and yellowish-reds (orange) are very common mixture descriptions. Hering also observed that there is a distinct pattern to the color of after-images. For example, if one looks at a unique red patch for a certain amount of time, and then switches to look at a homogeneous white area, one will perceive a green patch in the white area. Hering hypothesized that this antagonism between colors occurred in the retina, and that there are two major opponent classes of processing: spectrally opponent processes (red vs. green and yellow vs. blue) and spectrally non-opponent processes (black vs. white).
Experimental psychophysical support of Hering’s theory was first given by Jameson and Hurvich [JH55, HJ57]. They conducted a set of hue-cancellation experiments, where observers used monochromatic opponent light to “cancel” any hue that was not perceived as unique: red was canceled with green, blue was canceled with yellow, and vice versa. Repeating the experiment for all spectral lights and using the amount of opponent light needed as an indicator, they established opponent color curves over the visible spectrum.
Subsequent physiological experiments ([DSKK58, DKL84, LH88, Hub95]) corroborated the presence of an opponent encoding mechanism in the human visual system. Additional psychophysical experiments [CKL75, WW82, SH88, PW93] have shown that such a representation correlates much better with experimental color discrimination and color appearance data than the additive theory of color vision. Figure 32 illustrates the opponent color responses Poirson and Wandell [PW93] derived from a color appearance experiment involving spatial patterns.
Luminance and opponent color sensitivities are considered to be orthogonal, and are generally modeled as a de-correlating transformation of cone fundamentals [EMG01]. Using the image formation model, this assumption allows us to derive opponent color representations from physical measurements. The red-green opponent channel is usually a function of L – M , the blue-yellow channel of (L +M) – S , and luminance of L +M . However, the color opponent responses are not directly related to quantum catches of the cones, due to the neural interactions in the retina. Opposition works in cone contrast, i.e. the relative cone responses compared to the environment.
Contrast can either be taken into account with a Webertype contrast function, where the difference of stimulus and environmental (background) stimulus is normalized by the environmental stimulus. These contrast signals are then linearly transformed to opponent signals. In color science and computer vision, contrast is usually modeled by a logarithmic or power function. Color responses are normalized by the color response of the environment (a white surface or the illuminant) and then non-linearly encoded to account for lightness perception before being transformed to opponent signals.
CIE colorimetry
Colorimetry is the part of color science that deals with the measurement of physically defined color stimuli and their numerical representation. The CIE [CIE78, CIE86, CIE95, ISO98] has published several standards and recommendations pertaining to colorimetry that are summarized below. However, note that the basic principles of colorimetry remain the same, whether standardized or modified color matching functions are used.
A color response can be characterized by its relative tristimulus values X, Y, Z according to the image formation models, using physical measurements E (?) and S (?) of illuminant and surface reflectance, respectively:
Sensors are either the color-matching functions of the CIE 1931 standard observers (2°) or the CIE 1964 supplementary standard observer (10°), dependent on the stimuli size. The CIE X, Y, Z tristimulus values follow the trichromatic color matching laws described in section 1. For example, two stimuli with equal specifications will look the same when viewed by observers with normal color vision under identical observation conditions, i.e. they color match.
K is a normalization factor that is calculated as follows:
so that Y = 100 for a perfect diffuser and S (?) = 1 for a perfect reflector. E (?) is the spectral power distribution of the illuminant.
A few more comments on Y :
- Y represents the luminance of a color.
- Y L is the absolute luminance in cd/m 2 if K = 1 and E (?) is not normalized in eq. 2.
- If Y = 100 for a perfect reflector or diffuser, then Y color = reflectance (transmittance) factor in percentage.
- Most calculations are based on relative values: Y = 100 for the whitest point (= white point).
- X and Z do not correspond to any perceptual attributes.
For the purpose of emphasizing relative magnitudes of the tristimulus values, which are related to color attributes, the X, Y, Z tristimulus values are often normalized by dividing by the sum of their components:
The x, y, z chromaticity values therefore represent the percentage of X, Y, Z of a particular color response. x, y chromaticity values are often used to graphically represent tristimulus values. For example, the x, y chromaticity coordinates of the CIE 1931 standard observer spectral X, Y, Z tristimulus values graphically indicate the two-dimensional gamut of the human visual system—the color coordinates that are visually achievable. In the case of CIE 1931 or CIE 1964 r, ?, b¯ and x¯, ?, z¯ CMFs, the color gamut boundary is called the spectral locus . The gamut of any RGB color encoding whose transformation from RGB to XYZ is known can, of course, also be plotted, as well as the gamut of any device colors (see Figure 33).
When modeling visual or imaging systems and evaluating psychophysical experiments, it is often more useful to measure or predict the difference of color responses rather than their actual or relative values. The X, Y, Z and x, y, z color representations are not perceptually uniform , i.e., equal Euclidean distances do not equate to equal perceptual color differences [Mac43, Mac44].
In 1976, the CIE therefore published additional, more perceptually uniform representations to facilitate the interpretation of color differences: CIE u’,v’ chromaticity diagram, CIE L*, u*, v * (CIELUV), and CIE L*, a*, b * (CIELAB) [CIE78].
The CIE u’,v’ chromaticity values are derived from X, Y, Z and x, y as follows [Hun98]:
In the CIE u’,v’ chromaticity diagram, perceptually equal color differences result in (almost) equal Euclidean distances. In other words, the u’,v’ chromaticity diagram is a better visual indicator of gamut differences of two devices (see Figure 34).
Both CIELAB and CIELUV [WS82, CIE86, Hun98] are opponent color spaces, where L * represents the lightness of a color response, a * or u * its red-greenness, and b * or v * its yellow-blueness. The CIELUV system is commonly used for lighting and display, whereas the CIELAB system is more often used for reflecting stimuli, although the CIE did not specify any preferred usage [Rob90].
The transformation from X, Y, Z tristimulus values to CIELUV values is as follows [Hun98]:
CIE colorimetry
where X n , Y n , Z n are the tristimulus values of the nominally white object-color stimulus, usually the illuminant normalized to Y n = 100, and u’ n , u’ n its u’, v’ values.
The transformation from X, Y, Z tristimulus values to CIELAB values is as follows [Hun98]:
where X n , Y n , Z n are the tristimulus values of the nominally white object-color stimulus, usually the illuminant normalized to Y n = 100. If = 0.008856, then in eq. 7 are replaced with 7.787 F + 16/116 where F is , respectively.
Color differences ? E are then expressed as the Euclidean distance between the CIELAB coordinates:
where ? H is a measure of hue difference:
? H = v(? a ) 2 + (? b ) 2 – (? C ) 2
The hue angle h and chroma C are defined as:
C = v>a 2 + b 2
In 1994, the CIE introduced a modified color difference formula, CIE ? E 94 [CIE95], which correlates better with visual perception of small color differences. It decreases the weights given to differences in ? C and ? H with increasing C . Equation (9) is modified as follows:
where S L = 1, S C = 1 + 0.045 v C 1 C 2 , S H = 1 + 0.015 v C 1 C 2 and k L = k C = k H = 1. C 1 and C 2 refer to the chroma of the two color responses under consideration.
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