Color coordinates of the system and chromaticity. color model

The three-dimensional nature of color perception allows it to be displayed in a rectangular coordinate system. Any color can be represented as a vector whose components are the relative weights of red, green and blue, calculated by the formulas

Since these coordinates always add up to one, and each of the coordinates lies in the range from 0 to 1, then all points of space represented in this way will lie in one plane, and only in a triangle cut off from it by the positive octant of the coordinate system (Fig. 2.5 a). It is clear that with such a representation, the entire set of points of this triangle can be described using two coordinates, since the third is expressed in terms of them by means of the relation

Thus, we pass to a two-dimensional representation of the area, i.e. to the projection of the area onto the plane (Fig. 2.5 b).

Rice. 2.5.

With the use of such a transformation in 1931, international standards for the definition and measurement of colors were developed. The basis of the standard was the so-called two-dimensional CIE color chart. Since, as shown by physical experiments, not all possible color shades can be obtained by adding the three primary colors, other parameters obtained on the basis of a study of the standard reactions of the eye to light were chosen as the basic ones. These parameters - - are purely theoretical, since they are built using negative values ​​of the primary color components. The primary color triangle was built to cover the entire spectrum of visible light. In addition, an equal amount of all three hypothetical colors adds up to white. Chromaticity coordinates are built in the same way as in the above formula:

When projecting this triangle onto a plane, a CIE color graph is obtained. But chromaticity coordinates define only the relative amounts of primary colors, not specifying the brightness of the resulting color. The brightness can be set by the coordinate , and can be determined based on the values, according to the formulas

The MCO color chart is shown in fig. 2.6. The area bounded by the curve covers the entire visible spectrum, and the curve itself is called the line of spectral chromaticities. The numbers in the figure indicate the wavelength at the corresponding point. The point corresponding to midday illumination under continuous cloudiness is taken as the reference white color.

The color chart is handy for a number of tasks. For example, it can be used to get an additional color: to do this, you need to draw a ray from a given color through a reference point until it intersects with the other side of the curve (the colors are additional to each other if, when added together in the appropriate proportion, a white color is obtained). To determine the dominant wavelength of any color, a beam is also drawn from the reference point to the intersection with the given color and continues until the intersection with the nearest point of the color line.

Grassmann's laws are used to mix two colors. Let two colors be given on the CIE chart by coordinates and . Then mixing them gives the color. If we introduce the notation , then we get

One way or another, when working with any images (photos, layout of a printed or web page, drawings, etc.), you have to deal with color. Before you become familiar with color management systems, you need to understand the essence of the processes that underlie them. This article will be useful not only for beginners in the field of digital imaging, but also for experienced professionals, as it will help to systematize a lot of accumulated knowledge and help clarify some details.

To begin with, let's try to define the concept that interests us most now - this is color.

Color is electromagnetic radiation that our eyes can perceive and distinguish by wavelength. Yes, but this statement does not explain the existence of purple, which is not in the spectrum.

Color is the ability of the surface of an object to selectively reflect the radiation that falls on it. Yes, but a color photograph in low light is perceived as almost black and white, and in sunlight it is perceived as rich full color.

Color is the spectral composition of visible electromagnetic radiation. Yes, but different (sometimes significantly) in the spectral composition of the radiation can cause a sensation of the same color.

The above definitions first come to mind for most people, however, as you can see, they do not give an exhaustive definition of color and are not accurate.

A fairly complete definition of the concept of "color" will be as follows:

color is a sensation that arises in the human mind when exposed to its organs of vision of electromagnetic radiation in the visible range of the spectrum.

That is, the radiation of a certain spectral composition is only stimulus for our eyes, and color is already feeling, which arises in our minds as a result of the action of such a stimulus. It is necessary to clearly distinguish between the concepts of a color stimulus and the color itself.

Here the question may arise: why not use the measured spectral distribution of radiation to accurately describe the color, if it is precisely this that will cause the sensation of color in our minds? That is, to describe the color by the stimulus that causes it. Firstly, this method will not be convenient, since one stimulus will be given by about 35 values ​​of the spectral transmittance, reflection or emission (ie, the range of 390-740 nm with a step of 10 nm). Secondly, and most importantly, this way of describing color does not take into account the peculiarities of the perception of visible radiation by our visual system. This can be illustrated by the following figure, which shows the spectral reflectances of two objects (black and white plots, respectively):

Try to analyze these two graphs and say what color the surfaces of these two bodies will be perceived and how these colors will differ from each other. The only conclusion that seems to lie on the surface is that these two bodies are most likely of different colors. This conclusion suggests itself because of the significant difference in the curves of the spectral reflection coefficients. However, the stimuli shown on the graph will be perceived by us as absolutely identical colors. These two incentives are called metameric. The phenomenon of metamerism cannot be explained solely by physics or optics, therefore, in order to interpret the data of spectral measurements, it is necessary to know how the human visual system will react to various color stimuli.

To take into account the peculiarities of the perception of color stimuli and solve the issue of color measurements, in 1931 the International Commission on Illumination CIE (Commission Internationale de l "Eclairage) proposed a system that takes into account the perception of color stimuli by the so-called CIE Standard Observer, which characterizes the color perception of an average person with normal vision.

The set of data that defines the CIE Standard Observer was obtained empirically on a certain number of real observers. But how did the researchers manage to measure the sensation of color under the influence of the desired stimuli, if a direct measurement of such a quantity as "feeling" on a person is impossible to make?

Since every science starts with measurements, colorimetry couldn't get by with just subjective data about the color that a person can express (bright, dull, red, pale, bluish, etc.). Computers can also work only with numbers, therefore, the need to measure the perception of color by a person is not only of scientific interest, but also necessary for practical activities.

In the 1920s, independently of each other, scientists Guild and Wright conducted a series of experiments to study human color vision. The experiments were carried out using the device shown schematically in the figure:

The operation of such a device (visual colorimeter) is based on the principle of additive color synthesis, according to which, by adding two or more radiations to each other (for example, on a screen), you can get a feeling of a certain number of colors, while adjusting the brightness of each of these basic radiations. Such basic stimuli are selected based on the need to reproduce as many colors as possible with the least amount of these basic radiations. A standard CIE observer was generated with respect to three spectrally pure stimuli that evoke the sensation of red, green, and blue (R, G, B) at wavelengths of 700, 546.1, and 435.8 nm, respectively.

These three radiations are projected onto the top of the screen, and the radiation from which the sense of color was tried is projected onto the bottom. The participants of the experiment needed to get a feeling of the same color on both parts of the field, while adjusting the brightness of the three main radiations. Quantities (brightness) of the main radiations that cause the feeling of the desired color and are the numerical values ​​(coordinates) of this color. That is, the researchers succeeded measure the sensation of color, by its reproduction and visual evaluation by a person.

However, it turned out that a significant part of the monochromatic radiations cannot be reproduced by this method. In order to bypass this limitation and measure the color coordinates of these stimuli that cannot be reached by this method, one of the main radiations was projected not on the upper, but on the lower part of the screen, thereby "polluting" the studied stimulus projected onto it. The principle of color measurement does not change in this case: it is also necessary to adjust the brightness of the main radiations in order to achieve color equality between the two fields of the device. In this case, the amount of the main radiation projected onto the object under study (the lower part of the field) is taken with a minus sign, that is, a negative color coordinate appears.

Having measured the color coordinates of all spectrally pure radiations of the visible zone of the spectrum, we will obtain a coordinate system of all possible colors. The presence of negative coordinates in this system made it inconvenient to use, since most of the calculations at that time were carried out manually. This was one of the reasons for creating the XYZ system, in which all color coordinates are positive.

The XYZ system is also based on additive mixing of stimuli, however, unlike the RGB system used in the visual colorimeter described above, XYZ uses unrealistic, mathematically described stimuli, which are selected to facilitate calculations. That is, when obtaining the XYZ system, not experiments were used, but mathematical transformations of the data from the experiments of Guild and Wright. XYZ color coordinates do not have negative values, and it is this system that is used to describe the CIE Standard Observer.

XYZ data can be obtained by measurement on colorimeters that have scales directly graduated in XYZ (this is possible despite the unreality of the main XYZ stimuli), or by performing calculations from the data of the spectral distribution of reflection, transmission or emission energy. Having calculated the color coordinates of the above metameric curves in the XYZ system, we will obtain the same color coordinates of these two stimuli. Regardless of the spectral distribution, stimuli that produce the same color sensation will have the same XYZ color coordinates. That is, this system describes how color stimuli will be perceived by our visual system and can be used to numerically describe color.

In practice, most often, the derivative of the XYZ coordinate system is used - xyY, which was obtained by simple recalculation from XYZ:

where x and y- chromaticity coordinates, and Y— luminance coefficient, which remains unchanged (setting the brightness of the color by the Y value was laid down when the XYZ system was created).

Chroma is a two-dimensional quantity that includes the concept of hue and saturation. It is the xy chromaticity diagrams that can most often be seen when graphically displaying color coordinates. This diagram is shown in the following figure:

The black closed curve is the chromaticity coordinates of all spectrally clear and magenta stimuli. Inside it are all other colors, the saturation of which decreases as the white point approaches (for example, for daylight, the white point has xy coordinates of 0.31 and 0.33, respectively).

The xy diagram allows you to visually show the chromaticity of various stimuli, color gamuts of devices and compare them. However, this diagram has one significant drawback: the same distances on the graph do not correspond to the same color difference experienced by our visual system. This unevenness is illustrated by two white segments in the previous figure. The lengths of these segments correspond to the sensation of the same color difference. In other words, the same distance on the graph in one of its zones can be perceived as a clearly noticeable difference in color, while in another zone no difference will be observed.

To overcome this shortcoming, the CIE committee in the 60-70s of the twentieth century developed a series equally contrasting(uniform for perception) graphs and scales in which the unit of the scale always corresponds to the same difference in color perception. The most common among them is the CIE LAB system, or L * a * b * or simply Lab. This system is equally contrasting not only with respect to color, but also with respect to the perception of stimulus brightness, i.e. lightness. The L* value is an equal contrast lightness scale, while a* and b* are uniform color scales. Since this system is three-dimensional, it is customary to call it Lab color space.

The Lab space is obtained by mathematical transformations of the XYZ space, that is, the Lab data can be obtained from the XYZ or xyY data, and vice versa.

An important advantage of the Lab space, which follows from its equal contrast, is the ability to numerically specify the difference in the compared colors. The value of this difference will be the usual geometric distance between the coordinates of these colors, which is denoted as ∆E.

To learn how the XYZ and Lab color coordinate systems are used by modern color management systems, as well as get instructions and tips on how to set them up, you can read this site.

The basis of the modern theory of color is the theory of Helmholtz and Hering about tricolor color sensations. The currently accepted color theory is based on the three laws of color addition established by Grassmann.

In accordance with the first law, any color can be considered as a set of three linearly independent colors, i.e., such three colors, none of which can be obtained by adding the other two.

It follows from the second law that the entire range of colors is continuous, that is, there cannot be a color that is not adjacent to other colors. By continuous changes in radiation, any color can be changed into another.

The third law of color addition states that some color, obtained by adding several components, depends only on their colors and does not depend on their spectral compositions. Based on this law, the same color can be obtained by different combinations of other colors. It is generally accepted nowadays to consider any color as a combination of blue, green and red, which are linearly independent. However, according to the third law of color mixing, there are countless other combinations of three linearly independent colors.

The International Commission on Illumination (CIE) adopted the colors of monochromatic radiation with wavelengths of 700, 546.1 and 435.5 nm as the three primary colors, denoted R, G, B.

If these three primary colors are arranged in space in the form of three vectors emanating from one point, denoting the corresponding unit vectors r, g, b, then any color F, can be expressed as a vector sum:

F=Rr+Gg+Bb

where R, G, B - modules of colors proportional to the number of primary colors in the resulting total color; these modules are called color coordinates.

Color coordinates uniquely characterize the color, i.e. a person does not feel the difference in colors that have the same coordinates. However, equal color coordinates do not mean the same spectral composition. Samples whose color is characterized by different spectra, but having the same color coordinates, are called metameric. The perceived color of a stained sample depends on the source in which it is viewed. Metameric samples that appear the same color in the light of one source are different in the light of another.

The system adopted for expressing color measurement data is X, Y, Z. In this system, the three primary colors are taken to be colors that do not really exist, but are linearly related to the colors R, G and AT.
Color in the system XYZ is expressed as a vector sum:

F=xx+Yy + Zz

Unlike the system RGAT all real colors in the system XYZ have positive coordinates. Luminosity of Primary Colors X and y taken equal to zero, so the brightness of the color F can be characterized by only one color coordinate Y,

The specific coordinates of spectrally pure colors of different wavelengths (specific color coordinates) are shown in fig.

The ratio of a color coordinate to the sum of all three coordinates is called color coordinate. The chromaticity coordinates corresponding to the color coordinates are denoted X, y,z

x=X/(X+Y+Z) etc.

It's obvious that:

X+ y +z=1

It is also obvious that the chromaticity coordinates remain unchanged when proportionally increasing or decreasing all color coordinates. Thus, the chromaticity coordinates uniquely characterize only color, but do not take into account brightness colors. The fact that the sum of all chromaticity coordinates is equal to one allows using only two coordinates to characterize the chromaticity, which, in turn, makes it possible to graphically represent the chromaticity in Cartesian coordinates.

Graphic representation of chromaticity in coordinates X, y is called a color chart (Fig.).

Points corresponding to spectrally pure colors are plotted on the color chart. They are located on an open curve. White corresponds to point C with chromaticity coordinates X = 0.3101 and y = 0.3163. The ends of the curve are contracted by a segment on which the purple tones that are absent in the spectrum are located. The wavelength of the magenta tone is indicated by a number with a stroke and is equal to the wavelength of the complementary color, i.e., the color located at the point at the intersection of the straight line passing through the point of the given magenta color and the point FROM, with a curve of spectrally pure colors. On the segments connecting the white point with the points on the periphery of the diagram, there are colors of the same color tone.

Color tone (dominant wavelength) - this is the wavelength corresponding to the maximum in the reflectance spectrum of the sample (or the transmission spectrum of a transparent sample), or the wavelength of monochromatic radiation that must be added to white in order to obtain a given color.

Color purity (saturation) of any color is defined as the ratio of the brightness of the monochromatic component to the sum of the brightness of the monochromatic and white components. Brightness - this is a value that characterizes the amount of light reflected from the sample. As already noted, the brightness in the three-color system is taken as the value of the color coordinates Y.

If we take some color on the color chart and denote it with a dot a, then its total brightness will be equal to Ya, and the brightness of the monochromatic component, proportional to the relative distance of the color from the white point, will be expressed by the ratio: Yll2/(l1+l2).

Thus, color can be characterized in three ways, using in any case three quantities to characterize it:

1) color coordinates X, Y, Z,

2) chromaticity coordinates X and at in conjunction with the Y color coordinate;

3) color tone l, color purity R and brightness Y.

Whiteness measurement.
One of the main indicators of white pigments and fillers is their whiteness. whiteness called the degree of approximation of color to ideal white. Ideally white is a surface that diffusely reflects all the light incident on it in the entire visible region of the spectrum. However, another preferred white sample can be taken as a reference.

There are quite a few different spectrophotometric and colorimetric methods for evaluating whiteness. Most often, the values ​​of color differences between the measured sample and the accepted standard are used to evaluate the whiteness of white pigments. Whiteness W in this case is calculated by the formula:

DE - complete color difference.

Rice. 6.10. Scheme for calculating color coordinates according to the general method Rice. 6.11. Scheme for calculating color coordinates by the method of chosen ordinates Rice. 6.12. Threshold ellipse: C - base color; A, B, C, D, E, F, G, H, I - colors that differ from the base one by one threshold Rice. 6.13. Threshold McAdam ellipses Rice. 6.14. Equal Contrast UV Color Plot Rice. 6.15. Lab color body and ab chart Rice. 6.16. Color discrimination thresholds in subtractive synthesis: 1 - unsaturated colors; 2- saturated; 3 - commemorative

To reproduce color, it is necessary to know the characteristics of both the reproduced object and the result obtained (for example, a color original and its reproduction). In this case, to assess the quality, one cannot do without color measurements, without a strict description of the color. The study of color measurement is called colorimetry or color metrology.

Color theory uses basically two ways of describing color - using colorimetric systems and specification systems. In this section, only the principles of constructing colorimetric systems will be considered.

One way to determine the color is based on measuring it according to the principle of synthesis. In devices - colorimeters (they are discussed in more detail in subsection 8.1), where this principle is implemented, with the help of three main ones, a color is synthesized that is identical to the measured one.

The two faces of the prism form a photometric field. The measured radiation C is directed to one half of the field, and the main R, G, B are directed to the other half. By adjusting the amounts of the main ones, the colors of both halves of the field can be equalized. Knowing the characteristics of light-absorbing devices (diaphragms, wedges), one can find the quantities of the main ones, and from them - the coordinates of the measured color. Having defined the Color Coordinates, it is easy to reproduce the color itself.

Sometimes, instead of color coordinates, the psychophysical characteristics of color are determined: the dominant wavelength, color purity and brightness. Their definition is based on the fact that the spectrum contains all colors except magenta. Therefore, for any light beam, it is possible to select a spectral color that is identical to the one measured by the color tone. On fig. 6.1 shows a measurement scheme according to this principle.

In this case, the standard is monochromatic radiation M, isolated from the spectrum. Since the measured and monochromatic beams can differ in saturation, white radiation B is also directed to the edge of the prism along with the monochromatic one. Knowing the wavelength of monochromatic radiation M, its amount and the amount of white necessary to obtain a color identical to C, the psychophysical characteristics of the measured colors.

The wavelength of monochromatic radiation, identical to the measured color, is called dominant wavelength(defined "> Color saturation C is characterized by colorimetric purity of color. It determines the proportion of that monochromatic radiation, which, when mixed with white, provides visual identity with the considered radiation (color), is calculated by the formula

formula" src="http://hi-edu.ru/e-books/xbook322/files/Blam- brightness of monochromatic radiation; transition" href="part-005.htm#i1304"> section 5.1.7).

The choice of primary colors, limited only by the condition of linear independence between them, allows you to have an unlimited number of colorimetric systems.

One of these systems is the main physiological system of the GLC. In this system, the color coordinates K, 3 and C are the levels of excitation of the three receivers of the eye in single values ​​of the CCD - color components. The peculiarity of the physiological system lies in the fact that, unlike all other systems (including those that will be considered further), in it any color is not only expressed by the sum of the three main ones, but is also determined by the level and ratio of the reactions of the three color-perceiving receptors of the eye (see Fig. 4.7). In this regard, this system is of particular importance where there is a need to analyze the reactions of color-sensing receptors, color adaptation, etc.

The main difficulty in constructing this system lies in the impossibility of accurately measuring the spectral sensitivity of each of the three color-sensing receptors.

The first RGB colorimetric system was proposed and adopted in 1931 by the International Commission on Illumination (CIE), in the literature the CIE abbreviation from the French Commision Internationale de I "Eclairage is often used instead of CIE. The choice of primary colors of this system was carried out based on the following requirements.

1. Selected main ones should be easy to reproduce.

2. Each of the selected main ones should excite, if possible, only one group of color-sensing receptors.

Taking into account the year of the development of the first colorimetric system, it should be noted that at that time the most reproducible radiation was considered to be gas-light lamps, from which monochromatic radiation was easily separated with the help of light filters. In this regard, CIE radiations were chosen as the main ones:

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For the transition to energy quantities, not luminance coefficients, but luminance units are taken as units of quantities of basic RGB: the formula "src="http://hi-edu.ru/e-books/xbook322/files/6-1-2.gif" border="0" align="absmiddle" alt="(!LANG:

Given that the brightness is proportional to the light fluxes, we can assume that with the ratio of light fluxes, the formula "src="http://hi-edu.ru/e-books/xbook322/files/219-3.gif" align="absmiddle" alt="(!LANG:in lumens:

formula" src="http://hi-edu.ru/e-books/xbook322/files/Fo-lambda.gif" border="0" align="absmiddle" alt="(!LANG:.gif" border="0" align="absmiddle" alt="(!LANG:

Whereas def. "> R + GG + BB . (6.1.4)

To go to the chromaticity equation, the color module m is found - the sum of the color coordinates (m = R + G + B) and then each of the terms of equation (6.1.4) is divided by the module:

where r, g, b are chromaticity coordinates.

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Taking into account (6.1.2), passing from luminance units to luminance coefficients, we obtain

formula" src="http://hi-edu.ru/e-books/xbook322/files/6-1-7.gif" border="0" align="absmiddle" alt="(!LANG:

The sum in brackets expresses the brightness of a single color..gif" border="0" align="absmiddle" alt="(!LANG:

Replacing the sum in brackets of the expression (6.1..gif" border="0" align="absmiddle" alt="(!LANG:

The determination of the psychophysical characteristics of the dominant wavelength and color purity in CIERGB is carried out according to the chromaticity diagram rg obtained using addition curves.

Addition curves formula" src="http://hi-edu.ru/e-books/xbook322/files/rgb.gif" border="0" align="absmiddle" alt="(!LANG:.gif" border="0" align="absmiddle" alt="(!LANG:) (Fig. 6.2). Therefore, the values ​​of the ordinates of the addition curves are called specific, i.e. per unit of power.

In CIERGB, the ordinates of the addition curves (specific coordinates) were established empirically. Experimentally, specific coordinates were found by selecting a mixture of basic RGB radiations to spectral radiations of arbitrary power and then dividing their coordinates by power:

formula" src="http://hi-edu.ru/e-books/xbook322/files/r-lam-.gif" border="0" align="absmiddle" alt="(!LANG:has a negative value in a certain area. This suggests that in order to obtain color equality, one of the primary colors must be mixed with the investigated spectral.

Using addition curves selection "> Fig. 6.3

). On the line thus obtained (in the figure it is shown by a dotted line) there are single magenta colors of maximum saturation. There are no purple colors in the spectrum. They are obtained artificially by mixing red and purple colors in various quantities. The area bounded by the locus and the dashed line is called the area of ​​real colors. Outside this area, colors are more saturated than real.

As can be seen from fig. 6.3, the color triangle rОg is entirely located inside the region, limited by the locus. All colors within the triangle have positive chromaticity coordinates. For colors outside the triangle, one of the chromaticity coordinates has a negative value. This is due to the presence of a region of negative values ​​\u200b\u200bof the addition curve selection "\u003e Fig. 6.4). The locus is obscured by a line of magenta colors.

This rg color chart is characterized by the following colorimetric properties.

1. White point B has coordinates (0.33; 0.33).

2. Color saturation increases from white point to locus.

3. On the straight line connecting the white point with the locus lie the colors of a constant hue.

4. Locus is the border of the most saturated (spectral) colors.

The method for finding color characteristics - the dominant wavelength and color purity - is discussed in subsection. 7.1.5.2.

To conclude this section, two remarks should be made regarding the CIERGB system.

1. The CIERGB system discussed above is a colorimetric system. However, in current terminology, "RGB system" is sometimes understood to mean a color description system that is not a standard colorimetric system. This is most often encountered in prepress processes when processing color image information. The colors, so-called in this case, the "RGB system" depend on the specific device, such as a monitor or scanner. They cannot be characterized by a constant, specific wavelength. For example, it is known that the color in the wavelength range from 620 nm to 700 nm is red, and any radiation of arbitrary power in this interval can be called "R". The same applies to "G" and "B". Different monitors can reproduce the same color in different ways, since each of them has its own personal characteristics (color temperature, phosphors, etc.). But these characteristics are not constant and can change over time, as well as from device to device. Therefore, the device-dependent colors of the "RGB system" have nothing to do with the RGB colorimetric system adopted in 1931.

2. The RGB colorimetric system is currently practically not used. It should be viewed as a guide to better understand the general principles of color metrology based on actual primary colors. Therefore, it is given attention in this textbook.

It should be noted that CIERGB served as the basis for most of the colorimetric systems developed later. Therefore, the shortcomings that were the basis of this colorimetric system were later transferred to others.

Simultaneously with the RGB colorimetric system, another one was adopted. Colors more saturated than spectral colors were chosen as the main ones in it. Due to the fact that there are no such colors in nature, they were designated by the symbols XYZ, and the colorimetric system itself was called CIEXYZ. The development of this colorimetric system was prompted by a number of reasons associated with some inconvenience when working with the CIERGB system.

One of the disadvantages of the CIERGB system is the presence of negative coordinates for a number of real colors, which makes it difficult to calculate color characteristics from spectral curves. Another significant drawback of the CIERGB system is the need to determine all three color components in order to determine the quantitative characteristic of color - brightness.

In this regard, the following provisions were taken as the basis for the construction of the XYZ colorimetric system:

1) all real colors must have only positive coordinates;

2) brightness should be determined by one color coordinate;

3) the coordinates of the white color of the equal-energy source (for the equal-energy source, see Subsection 7.1.8) must have coordinates 0.33; 0.33.

Through mathematical transformations, taking into account the above requirements, it was possible to make the transition from real CIERGB colors to unreal (supersaturated) CIEXYZ.

In accordance with the second condition for constructing the XYZ colorimetric system, the colors X and Z have brightness coefficients equal to zero, the formula "src="http://hi-edu.ru/e-books/xbook322/files/Ly.gif" " align="absmiddle" alt="(!LANG:= 1). In this case, the formula for calculating brightness B is greatly simplified:

formula" src="http://hi-edu.ru/e-books/xbook322/files/6-1-12.gif" border="0" align="absmiddle" alt="(!LANG:

In general, the color equation in CIEXYZ is written as follows:

C \u003d XX + YY + ZZ.

The transition to the chromaticity equation in CIEXYZ is carried out through m in the same way as in the CIERGB system (see formula 6.1.5):

formula" src="http://hi-edu.ru/e-books/xbook322/files/xyz.gif" border="0" align="absmiddle" alt="(!LANG:.

As mentioned earlier, when developing the XYZ colorimetric system, the condition was set that real colors should not have negative coordinates..gif" border="0" align="absmiddle" alt="(!LANG:do not have negative values ​​(Fig. 6.5). They are determined by formulas (6.1.13) and have the same meaning as the ordinates of the curves in the CIERGB system:

formula" src="http://hi-edu.ru/e-books/xbook322/files/y-lam-.gif" border="0" align="absmiddle" alt="(!LANG:coincides in shape and position with the relative luminous efficiency curve..gif" border="0" align="absmiddle" alt="(!LANG:.gif" border="0" align="absmiddle" alt="(!LANG:is explained by the conditions for converting CIERGB to CIEXYZ. The areas bounded by each curve and coordinate axis are the same.

During the use of the CIEXYZ system, it was found that the values ​​of the specific color coordinates of the selection "> Fig. 6.6

(Xyz addition curves 1931 and the formula" src="http://hi-edu.ru/e-books/xbook322/files/225-1.gif" border="0" align="absmiddle" alt="(! lang:in 1964 were recommended by the CIE as additional color reagents. The selection system "> Fig. 6.7 does not fundamentally differ in the rg chromaticity diagram. Its properties are the same, the only difference is that the locus is located inside the unit chromaticity triangle. The white point corresponds to the coordinates of the equal-energy source E (0.33; 0.33).

The color graph xy is used to find the qualitative characteristics of the color of the dominant wavelength selection "> Fig. 6.8 (Definition formula" src="http://hi-edu.ru/e-books/xbook322/files/226.gif" border=" 0" align="absmiddle" alt="(!LANG:. Connect point E with point C and extend the line to the intersection with the locus..gif" border="0" align="absmiddle" alt="(!LANG:). This means that the color C is green (green has an interval in the spectrum from 510 to 565 nm).

Determining the characteristics of the chromaticity of magenta colors has its own peculiarity. They are not in the spectrum, and therefore, the points expressing the chromaticities of purple colors with a certain wavelength are also absent on the locus (on the xy color chart, the ends of the locus characterizing red and purple colors are connected to each other by a line of purple colors).

Taking the point P near this line, which characterizes the purple color (see Fig. 6.8), we express its color tone. To do this, as in the previous example, we connect point E with point P and extend until the intersection with the locus..gif" border="0" align="absmiddle" alt="(!LANG:, in the opposite direction to the intersection with the locus..gif" border="0" align="absmiddle" alt="(!LANG:expresses a color complementary to the color P.

In the considered examples, the colors lying on the lines are the formula" src="http://hi-edu.ru/e-books/xbook322/files/6-1-14.gif" border="0" align="absmiddle" alt ="(!LANG:

where the formula is" src="http://hi-edu.ru/e-books/xbook322/files/227.gif" border="0" align="absmiddle" alt="(!LANG:.gif" border="0" align="absmiddle" alt="(!LANG:- coordinates of the light source (in our case E). The one of the formulas is used, the numerator of which is the transition "href="part-007.htm#i1751"> Sec. 7.1.8), with respect to which constructions are carried out.

Taking into account the various requirements put forward by the practice of color reproduction, several colorimetric systems have been created. In each of them, the main ones were chosen on certain conditions.

As a rule, the transition from one color coordinate system to another was carried out using recalculation. This is how the recalculation from the real colors of the CIERGB system to the unreal CIEXYZ was carried out. Since the coordinates of unreal (more saturated than spectral) colors cannot be determined experimentally, the recalculation method is, in essence, the only one. It follows from Grassmann's law that there should be a linear relationship between the coordinates of any colors expressed in different systems. In this regard, the transformation of colorimetric systems is based on the solution of linear equations.

To move from one colorimetric system to another, it is necessary to measure the main old system in the coordinates of the new system. Let's look at this with an example.

Let the color be expressed by the equation in the basic RGB system:

C \u003d RR + GG + BB. (6.1.15)

Defines the coordinates of this color, but in the base XYZ system:

C \u003d XX + YY + ZZ.

For such a transition, it is necessary to measure the coordinates of the old main CIERGBs in the new CIEXYZ.

Let the following result be obtained (similar to (5.1.1)), showing the principles of transition from one system to another:

formula" src="http://hi-edu.ru/e-books/xbook322/files/6-1-17.gif" border="0" align="absmiddle" alt="(!LANG:

This formula shows the relationship between the coordinates of the old and new systems:

radiation" and "source". The term "source" refers to a physical object that gives one or another radiation (for example, the sun, etc.). The term "radiation" refers to a certain spectral distribution of energy falling on the object. the spectral distribution need not be derived from a single source.

In 1931, the CIE established a number of standard emissions and sources. Their brief description is given below.

Standard Radiation A characterized by the same distribution of radiation in the visible part of the spectrum as a completely black body at T \u003d 2856 K. This is the average color temperature of an incandescent lamp.

Standard radiation B reproduces the distribution of energy in the spectrum of direct sunlight with a correlated color temperature T = 4874 K.

Standard radiation C reproduces the radiation of the daytime sky covered with clouds with a correlated color temperature T = 6774 K.

As later studies have shown, daylight radiation is not always accurately reproduced by B and C radiation. In addition, it became necessary to more fully take into account the ultraviolet range of the daylight spectrum, especially when evaluating the color characteristics of luminescent objects. In this regard, the CIE in 1963 determined the spectral distribution of the various phases of daylight in the range of 300-830 nm and recommended several new D radiations. D65 radiation - with a correlated color temperature of 6504 K. It is now accepted by the CIE as a standard. Since the use of only D65 radiation met the necessary requirements, D50, D55 and D75 radiations were proposed to the CIE. O50 and D55, respectively with a correlated color temperature of 5000 K and 5500 K, are for those applications where a yellowish daylight phase is required, and D75 for a bluer daylight phase.

Studies have shown that the chromaticity of daylight radiation does not coincide with the chromaticity of a black body, and it is possible to characterize daytime radiation by the temperature of a black body only in a certain approximation. Therefore, the color temperature of daylight radiation is called the correlated color temperature.

Standard CIE sources (A, B, C,....gif" border="0" align="absmiddle" alt="(!LANG:with T \u003d 6504 K. In addition to them, the CIE established emission sources "\u003e Fig. 6.9 - curves of the relative spectral distribution of energy in the emission spectrum of a number of sources recommended by the CIE.

Chromaticity coordinates of standard CIE sources

Source	Chromaticity x	Chromaticity y	Color temperature, K
BUT	0,4476	0,4074	2856
AT	0,3484	0,3516	4874 (4800)
FROM	0,3101	0,3162	6774 (6500)
D55	0,3324	0,3475	5503
D65	0,3127	0,3290	6504
D75	0,2990	0,3150	7504

Which arise when working with images, and many other topics, for example, on the topic of image processing, one way or another affect the issues of color and color reproduction. But, unfortunately, most of these articles describe the concept of color and the features of its reproduction very superficially, or they draw hasty conclusions or even errors. The number of articles and questions on specialized forums about the practical aspects of accurate color reproduction, as well as many incorrect attempts to answer these questions even by experienced specialists, suggests that problems when working with color arise quite often, and to find reasoned and clear answers to them difficult.

Insufficient or erroneous knowledge of most IT specialists regarding color reproduction, in my opinion, is explained by the fact that very little time is spent on studying color theory, since its basics are deceptively simple: since there are three types of cones on the retina, mixing certain three colors can without problems to get the whole rainbow of colors, which is confirmed by the RGB or CMYK controls in some program. For most, this seems enough, and their craving for knowledge in this area ends. But, the processes of obtaining, creating and reproducing images prepare you for many nuances and possible problems that an understanding of color theory, as well as the processes on which it is based, will help to solve. This topic is intended to fill the knowledge gap in the field of color science, and will be useful to most designers, photographers, programmers, and, I hope, other IT specialists.

Try to answer the following questions:

• why can't physics define the concept of color?
• Which of the seven basic SI units is based on the properties of the human visual system?
• What color tone is not in the spectrum?
• How was it possible to measure the perception of color by a person 90 years ago?
• where are colors used that have no brightness?

If at least one question you did not find the answer, I recommend looking under the cat, where you can find answers to all these questions.

Definition of the concept of color. His dimension

We all know that science cannot do without measurements and units of measurement, and the science of color is no exception. Therefore, we will first try to define the concept of color, and based on this definition, we will try to find ways to measure it.

No one will be surprised to hear that colors are perceived by us with the help of our eyes, which for this purpose capture the light of the world around us. Light is electromagnetic radiation in the 390-740 nm wavelength range (visible to the eye), so let's try to find the key to how to measure color in the properties of these rays, assuming that color is the features of the light that has entered our eyes. This does not contradict our thoughts in any way: it is the light that enters the eyes that makes a person perceive color.

Physics knows and can easily be measured such parameters of light as power and its spectral composition (that is, the distribution of power over wavelengths - the spectrum). By measuring the spectrum of reflected light, for example, from blue and red surfaces, we will see that we are on the right track: the power distribution plots will differ significantly, which confirms our assumption that color is a property of visible radiation, since these surfaces are of different colors. The first difficulty that lies in wait for us is the need to record at least 35 numerical values ​​of the spectrum (visible wavelength range 390-740 nm with a step of 10 nm) to describe one color. Before we even start thinking about ways to solve this minor problem, we find that the spectra of some samples that are identical in color behave strangely (red and green graph):

We see that the spectra differ significantly, despite the unmistakably identical color of the samples (in this case, gray; such two radiations are called metameric). The formation of the color perception of these samples is influenced only by the light that is reflected from them (we will skip the influence of the background color, the level of adaptation of the eye to lighting and other minor factors here), because its spectral distribution is all that physical measurements of our samples can give us. In this case, two significantly different spectrum distributions define the same color.

Let us give a second example of the problem of the spectral description of color. We know that the rays of each section of the visible spectrum are colored for us in a certain color: from blue in the region of 400 nm, through blue, green, yellow, orange to red with a wavelength of 650 nm and above. Yellow is somewhere in the region of 560-585 nm. But we can choose such a mixture of red and green radiation, which will be perceived as yellow despite the complete absence of any radiation in the "yellow" range of 560-585 nm.

It turns out that no physical parameters can explain the identity of the color in the first situation and the presence of yellow coloring of the rays in the second situation. Strange situation? Where did we go wrong?

When conducting an experiment with measuring spectra, we assumed that color is a property of radiation, but our results refute this, because there were different rays of light beyond the spectrum, which are perceived as the same color. If our assumption were correct, each noticeable change in the spectrum curve would cause a perceived change in color, which is not observed. Since we are now looking for ways to measure color, and we have seen that the measurement of spectra cannot be called color measurement, we need to look for other ways in which this will be feasible.

In fact, in the first case, two experiments were carried out: one using a spectrometer, which resulted in two graphs, and the other, a visual comparison of samples by a person. The first way measures spectral composition light, and the second matches Feel in the mind of man. In view of the fact that the first method does not suit us, let's try to use a person to measure color, assuming that color is the sensation that a person experiences when light is applied to his eyes. But how to measure a person's feelings, understanding the complexity and uncertainty of this concept? Do not offer electrodes to the brain or an encephalogram, because even now such methods do not provide the necessary accuracy for such a subtle concept as color. Moreover, this problem was successfully solved back in the 20s of the twentieth century without the availability of most of the current technologies.

Brightness

The first problem for the solution of which it became necessary to numerically express the visual sensations of a person was the task of measuring the brightness of light sources. Measurement of the radiation power of lamps (namely, the radiation power, in joules, or watts, and not the consumed electrical power) did not answer this question, because, firstly, a person does not see radiation with wavelengths less than 380 and more than 780 nm, and therefore any radiation outside this range does not affect the brightness of the source. Secondly, as we have already seen with spectra, the perception of color (and brightness) is a more complex process than simply fixing the characteristics of the light that enters our eyes: human vision is more sensitive to some zones of the spectrum, and less to others. For example, green radiation is much brighter than blue radiation of identical power. Obviously, to solve the problem of numerical expression of the brightness of light sources, it is necessary to quantify the sensitivity of the human visual system for all individual wavelengths of the spectrum, which can then be used to calculate the contribution of each wavelength of the source to its total brightness. Like the problem raised above with the measurement of color, this one also boils down to the need to measure the sensation of brightness by a person.

It was possible to measure the sensation of brightness from the radiations of each wavelength by visually comparing the brightness of the radiations with known powers by a person. This is quite simple: by controlling the radiation intensity, you need to equalize the brightness of two monochromatic (spectral as narrow as possible) streams, while measuring their powers. For example, in order to equalize the brightness of monochromatic radiation with a wavelength of 555 nm with a power of one watt, two-watt radiation with a wavelength of 512 nm should be used. That is, our visual system is twice as sensitive to the first radiation. In practice, for high accuracy of the results, a more complex experiment was carried out, but this does not change the essence of what was said (the process is described in detail in the original scientific work of 1923). The result of a series of such experiments for the entire visible range is the spectral luminous efficiency curve (you can also find the name “visibility curve”):

Wavelengths are plotted along the X axis, and the relative sensitivity of the human visual system to the corresponding wavelength is plotted along the Y axis.

Having a device with the same spectral sensitivity, you can easily determine the brightness of the desired light emissions on it. It is under such a curve that the sensitivity of various photometers, luxmeters and other devices is carefully adjusted, in the work of which it is important to determine the brightness perceived by a person. But the sensitivity of such devices is always only an approximation to the human spectral luminous efficiency curve, and for more accurate measurements of brightness, the spectral distribution of the light source of interest is used.

The spectral distribution is obtained by dividing the radiation into narrow spectral zones and measuring the power of each of them separately. We can consider the brightness of our source as the sum of the brightness of all these spectral zones, and for this we determine the brightness of each of them (the formula for those who are not interested in reading my explanations on the fingers): we multiply the measured power by the sensitivity of our visual system corresponding to this wavelength ( the y and x axes of the previous graph, respectively). Summing up the luminances of all zones of the spectrum obtained in this way, we will obtain the brightness of our primary radiation in photometric units, which give an accurate idea of ​​the perceived brightness of certain objects. One of the photometric units is included in the Basic SI units - candela, which is determined through the spectral light efficiency curve, that is, based on the properties of the human visual system. The relative sensitivity curve of the human visual system was adopted as an international standard in 1924 by the International Commission on Illumination (abbreviation CIE can be found in Soviet literature), or CIE - Commission Internationale de l "Éclairage.

CIE RGB system

But, the spectral luminous efficiency curve gives us an idea only about the brightness of light radiation, and we can name its other characteristics, for example, saturation and hue, which cannot be expressed with its help. According to the way brightness is measured, we now know that only a person can “measure” color directly (do not forget that color is a sensation) or some model of his reaction, such as a curve of spectral luminous efficiency, which allows you to numerically express sensations of brightness. Suppose that in order to measure color, it is necessary to create experimentally with the help of a person, by analogy with the luminous efficiency curve, a certain system that will display the color response of the visual system to all possible options for the spectral distribution of light.

One property of light rays has long been known (in fact, this is a feature of our visual system): if you mix two different-colored radiations, you can get a color that will be completely different from the original. For example, by directing green and red light of certain powers onto a white sheet of paper at one point, you can get a pure yellow spot without impurities of green or red hues. By adding a third radiation, and blue is better suited to the existing two (because there is no way to get it with a mixture of red and green), we get a system that will allow us to get many colors.

If we visually equalize some test radiation in such a device, we will get three indicators: the intensity of the red, green and blue emitters, respectively (like the voltage applied to the lamps, for example). That is, with the help of our device (called a visual colorimeter), which reproduces color, and our visual system, we managed to obtain numerical values ​​for the color of a certain radiation, which is what we were striving for. These three meanings are often called color coordinates, because it is convenient to represent them as coordinates of three-dimensional space.

Similar experiments were successfully carried out in the 1920s independently by scientists John Guild and David Wright. Wright used monochromatic radiations of red, green and blue colors with wavelengths of 650, 530 and 460 nm, respectively, as the main radiations, while Guild used more complex (non-monochromatic) radiations. Despite significant differences in the equipment used and the fact that the data were averaged over only 17 observers with normal vision (10 for Wright and 7 for Guild), the final results of both researchers were very close to each other, which indicates a high accuracy of measurements. carried out by scientists. Schematically, the measurement procedure is shown in the figure:

A mixture of radiation from three sources is projected onto the upper part of the screen, and the studied radiation is projected onto the lower part, and the participant in the experiment sees them simultaneously through a hole in the curtain. The researcher sets the task for the participant to equalize the color between the fields of the device, and at the same time directs the studied radiation to the lower field. The participant adjusts the power of the three radiations until he succeeds, and the researcher records the intensity of the three sources.

In a number of cases, it is not possible to equalize certain monochromatic radiations in such an experiment: the test field at any position of the three radiation regulators remains more saturated than the mixture used. But, due to the fact that the purpose of the experiment is to obtain color coordinates, and not reproduce it, the researchers went for a trick: they mixed one main radiation of the device not with the other two, but directed it to the lower part of the screen, that is, they mixed it with test radiation :

Further, the equalization is carried out as usual, but the amount of the radiation that is mixed with the studied one will be considered negative. Here we can draw an analogy with changing the sign when transferring a number to another part of the usual equation: since visual equality is established between the two parts of the colorimeter screen, its upper part can be considered as one part of the equation, and the lower part as the other.

Both researchers made visual measurements of all individual monochromatic emissions in the visible spectrum. Studying the properties of the visible spectrum in this way, scientists assumed that their results could be used to describe any other radiation. Scientists operated with the powers of three independent radiations and the result of a series of such experiments are three curves, and not one, as was done when creating the luminous efficiency curve.

To create a convenient and versatile color specification system, the CIE committee averaged the measurement data of Gild and Wright, recalculating their data for the trio of basic radiations with wavelengths of 700, 546.1 and 435.8 nm (red, green and blue, red, green, blue - RGB). Knowing the ratio of the brightness of the main radiations of such an average system, which are needed to reproduce white color (respectively 1: 4.5907: 0.0601 for red, green and blue rays, which was established experimentally with subsequent recalculation) and using the spectral efficiency curve, the CIE members calculated the curves of the specific color coordinates , which show the required number of three main radiations of this system for the equation of any monochromatic radiation with a power of one watt:

Wavelengths are plotted along the X-axis, and the required amounts of the three radiations necessary to reproduce the color caused by the corresponding wavelength are plotted along the Y-axis. The negative sections of the graphs correspond to those monochromatic emissions that cannot be reproduced by the three main emissions used in the system, and for their specification it is necessary to resort to the trick described above when adjusting.

To build such a system, we can choose any other three radiations (remembering that none of them should be reproduced by a mixture of the other two), which will give us other specific curves. The main radiations selected in the CIE RGB system reproduce a large number of spectrum radiations, and its specific curves are obtained with high accuracy and standardized.

Curves of specific color coordinates eliminate the need to use a cumbersome visual colorimeter, with its slow visual adjustment method to obtain color coordinates using a human, and allow them to be calculated only from the spectral distribution of radiation, which are quite quickly and easily obtained using a spectrometer. Such a method is possible because any radiation can be represented as a mixture of monochromatic rays, the powers of which correspond to the intensity of the corresponding zone of the spectrum of this radiation.

Now let's check our two samples, before which physics has given up, showing different spectra for one-color objects, using the curves of specific coordinates formula spectral distribution, but three curves are used here). The result will be three numbers, R, G and B, which are the color coordinates in the CIE RGB system, that is, the number of three radiations of this system, the mixture of which is identical in color to the measured one. We will get three identical RGB values ​​for our two samples, which corresponds to our identical color sensation and confirms our assumption that color is a sensation and can only be measured with the participation of our visual system, or its model in the form of three curves of the CIE RGB system or some other another, whose specific coordinates are known (another such system based on other primary colors, we will consider in detail a little later). Using the CIE RGB colorimeter to measure the light reflected from the samples directly, that is, visually equalizing the color of the mixture of the three radiations of the system with the color of each sample, we will get the same three RGB coordinates.

It should be noted that in colorimetric systems it is customary to normalize the amounts of basic radiation so that R=G=B=1 corresponds to the white color adopted in the system. For the CIE RGB system, this white color is the color of a hypothetical equal-energy source that radiates evenly at all wavelengths of the visible spectrum. Without such a normalization, the system turns out to be inconvenient, because the brightness of the blue source is very small - 4.5907:0.0601 against green, and on the graphs most of the colors would "stick" to the blue axis of the diagram. Having introduced such a normalization (respectively 1:4.5907:0.0601 for the red, green and blue rays of the system), we will pass from photometric to colorimetric units, which will make such a system more convenient.

It should be noted that the CIE RGB system is not based on any theory of color vision, and the curves of the specific color coordinates are not the spectral sensitivity of the three types of human retinal cones, as they are often erroneously interpreted. Such a system easily dispenses with data on the properties of retinal cone pigments and without any data on the most complex processes of visual information processing in our brain. This speaks of the exceptional ingenuity and foresight of the scientists who created such a system despite the negligible information about the properties of the human visual apparatus at that time. Moreover, the CIE RGB system underlies the science of color with virtually no changes so far, despite the enormous progress of science over the past time.

It should also be noted that even though the monitor also uses three emitters for color reproduction, like the CIE RGB system, the monitor's three color component (RGB) values ​​will not strictly specify the color, because different monitors reproduce color differently with a fairly large spread. , and besides, the main emissions of monitors are quite different from the main emissions of the CIE RGB system. That is, you should not take the monitor's RGB values ​​as some kind of color definition absolute.

For a better understanding, it should be noted that when we say "radiation/source/wavelength/lamp is green" we really mean that "radiation/source/wavelength/lamp makes you feel Green colour". Visible radiation is only stimulus for our visual system, and color is the result of the perception of this stimulus and color properties should not be attributed to electromagnetic waves. For example, as in the example above, no waves from the yellow range of the spectrum appear when red and green monochromatic rays are mixed, but we perceive their mixture as yellow.

Unreal colors. CIE XYZ system

In 1931, at Trinity College, Cambridge University (Great Britain), at a regular meeting of the CIE, a system based on Gild and Wright's data was adopted as an international standard. Also, a group of scientists, led by the American Deane B. Judd, in order not to wait for the next committee meeting, which will take place no earlier than a year later, proposed another color specification system, the final data of which were calculated only the night before the meeting. The proposed system turned out to be so convenient and successful that it was accepted by the committee without any serious discussion.

To understand on the basis of what such a system was created, the color must be represented as a vector, because the addition of two or more colors obeys the same rules as the addition of vectors (this emerges from Grassmann's laws). For example, the result of mixing red and green radiation can be represented as the addition of two vectors with lengths that are proportional to the brightness of these radiations:

The brightness of the mixture will be equal to the length of the vector obtained by addition, and the color will depend on the ratio of the brightness of the radiations used. The more the ratio is in favor of one of the primary colors, the more the resulting radiation will be closer in color to this radiation:

Let's try to graphically depict the color mixing in the colorimeter used to create the CIE RGB colorimeter in a similar way. As you remember, it uses three radiations of red, green and blue. No color of this triple can be obtained by the sum of the other two, therefore, it will be necessary to represent all possible mixtures of these radiations in three-dimensional space, which does not prevent us from using the vector properties of color addition in this case:

It is not always convenient to draw three-dimensional diagrams, so a simplified graph is often used, which is a projection of all the necessary colors onto a single plane (highlighted in blue) of a three-dimensional diagram:

The result of such a projection of the color vector will be a point on the diagram, the axes of which will be the sides of the triangle, which are set by the points of the primary colors of the CIE RGB system:

Such a point will have coordinates in the system of this triangle in the form of a distance from any two of its sides (the third coordinate is superfluous, since any point in a triangle can be determined by two distances from the vertices or sides). The coordinates in such a triangle are called chromaticity coordinates, and they determine such color parameters as hue (blue, cyan, green, etc.) and saturation (gray, pale, saturated, etc.). Due to the fact that we have moved from a three-dimensional to a flat diagram, it does not allow showing the third color parameter - brightness, but for many cases, determining only the color value will be enough.

In order not to get confused, we separately highlight that the coordinates colors- this is the position of the end of the color vector in the three-dimensional system, and they are denoted by capital letters (RGB, XYZ, for example), and the coordinates chromaticity- this is the position of the color point on the flat color diagram, and they are denoted by lowercase letters (rg, xy) and two of them are enough.

The use of a coordinate system in which there is no right angle between the axes is not always inconvenient, therefore, in colorimetry, such a system of three vectors is more often used, the unit plane of which forms a right triangle. Its two sides near the right angle are used as the axes of the chromaticity diagram:

Let us now place all possible chromaticities on such a diagram, the limit of which will be a line of spectrally pure radiations with a line of magenta chromaticities, often called a locus, which limits the region of real colors on the diagram (red line):

The line of magenta chromaticities lies between the chromaticities of the radiations of the extreme blue and red ends of the spectrum. We cannot associate purple colors with any zone of the spectrum, as it can be done with any other color, because the sensation of purple arises from the simultaneous action of blue and red rays on our visual system, and not just one.

A significant part of the locus (in the zone of 380-546 nm) goes beyond the triangle bounded by the chromaticities of the main radiations, that is, it has negative chromaticity coordinates, because this part of the spectral radiations could not be equalized on the CIE colorimeter. This corresponds to the curves of the specific color coordinates, in which the same part of the spectrum has negative coordinates (in the range of 380-440 nm, these are small values ​​invisible on the graph).

The presence of negative color and chromaticity coordinates made colorimetric calculations a difficult task: in the 1920s and 1930s, most calculations were carried out using a slide rule, and the amount of calculations in colorimetric works is rather big.

The previous diagram shows us that all positive coordinates have only colors that lie within the triangle, which form the chromaticities of the basic radiations used in this system. If the locus lay in the middle of the triangle, all colors would have positive coordinates, which would greatly simplify the calculations. But it is completely impossible to find such three points on the locus that could include it in itself, due to its convex shape. Later it was found that the reason for this form of the locus lies in the peculiarities of the spectral sensitivity of the three types of cones in our eye, which overlap with each other and any radiation excites cones that are responsible for another zone of the spectrum, which lowers the level of color saturation.

But what if we go beyond the locus and use colors that cannot be reproduced and seen, but whose coordinates can be easily used in equations along with the coordinates of real colors? Since we have already moved from experiments to calculations, nothing prevents us from using such unrealistic colors, because all the properties of color mixing are preserved! We can use any three colors whose triangle can include the locus of real colors, and we can easily draw many such triplets of unreal primary colors (it will be advisable to build such a triangle as tightly as possible around the locus, so there will be less unnecessary areas on the diagram):

With such freedom in choosing the dots of the new primary colors, the scientists decided to extract some useful possibilities from this for the new tricolor system. For example, the ability to determine the photometric brightness directly using the created system without additional calculations or measurements (in the CIE RGB system, the brightness must be calculated), that is, to combine it somehow with the photometric standard of 1924.

To justify the choice of a trio of new colors (remember that they exist only in calculations), which were eventually chosen by scientists for this, let's return to our three-dimensional color coordinate chart. For clarity and ease of understanding, we will use the usual rectangular coordinate system. Let's place a plane on it, on which all colors will have the same photometric brightness. As you remember, the unit brightness of red, green and blue basic radiations in the CIE RGB system are related as 1: 4.5907: 0.0601, and in order to go back to photometric units, they must be taken in the ratio of 1/1 to 1/4.59 to 1/0, 0601, that is, 1:0.22:17 which will give us a plane of colors with the same photometric brightness in the CIE RGB colorimetric system (the point of intersection of the plane with the B-axis is outside the figure at position 17):

All colors whose coordinates are on this plane will have the same photometric brightness. If we draw a parallel plane twice as low as the previous one (0.5:0.11:8.5), we get the position of the colors with half the brightness:

Similarly, below you can draw a new parallel plane that intersects the origin, on which all colors with zero brightness will be placed, and even lower you can draw even planes of negative brightness. This may seem absurd, but remember that we are working with a mathematical representation of a three-color system, where all this is possible in the equations, which we will use.

Let's go back to the flat diagram rg by projecting a plane of zero brightness onto it. The projection will be a line of zero brightness - alychna, which crosses the origin:

There are chromaticities on the alychne that do not have brightness, and if you use the color placed on it in a color equation (not real, with mixing light fluxes, but in equations where such colors are possible), it will not affect the brightness of the resulting mixture. If we place two colors of the three-color system on the alychne, then the brightness of the entire mixture will be determined by only one remaining color.

Let me remind you that we are looking for color coordinates of such three hypothetical colors that can equalize the colors of all real radiations without using negative values ​​(the triangle must include the entire locus) and at the same time, the new system will include the photometric brightness standard directly. By placing two colors on the alychne (named X and Z) and a third above the locus (Y), we solve both problems:

The locus of real colors is completely in a triangle, which is limited by three selected colors, and the brightness has completely passed to one of the three components of the system - Y. Depending on the normalization of the values ​​​​and the nature of the measurements, the Y coordinate can express the brightness directly in candelas per m 2, a percentage of the maximum brightness of some system (display, for example), the percentage of transmission (transparent samples, slides for example) or the brightness relative to some standard (when measuring reflective samples).

Transforming the resulting triangle into a rectangular one, we get the xy chromaticity diagram familiar to many:

It must be remembered that the xy diagram is a projection of the system with the main XYZ points onto a unit plane, similarly so is the rg diagram and the RGB system. This diagram allows you to conveniently illustrate the chromaticities of various emissions, for example, the color gamuts of various devices. The diagram has one useful property: the chromaticity coordinates of the mixture of two radiations will be located strictly on the line that connects the points of these two radiations on the diagram. Therefore, the color gamut of the monitor, for example, in such a diagram will be a triangle.

The xy chart also has one drawback to keep in mind: equal bars in different areas of the chart do not mean the same perceived difference in color. This is illustrated by the two white lines in the previous figure. The lengths of these segments correspond to the sensation of the same color difference, but the segments differ in length by a factor of three.

Let us calculate the curves of the specific color coordinates of the resulting system, which show the required number of three primary colors XYZ for the equation of any monochromatic radiation with a power of one watt:

We see that there are no negative sections in the curves (which was observed in the RGB system), which was one of the goals of creating the XYZ system. Also, the y curve (y with a dash on top) completely coincides with the curve of the spectral light efficiency of human vision (it was discussed above when explaining the definition of the brightness of light emissions), so the Y value determines the color brightness directly - it is calculated in an identical way as the photometric brightness from that same curve. This is achieved by placing the other two colors of the system on the plane of zero brightness. Therefore, the 1931 colorimetric standard incorporates the 1924 photometric standard, eliminating unnecessary calculations or measurements.

These three curves define the Standard Colorimetric Observer, the standard that is used in the colorimetric interpretation of spectral measurements and has underpinned all color science almost unchanged to this day. Although the XYZ visual colorimeter cannot physically exist, its properties allow highly accurate color measurements and it helps many industries to predictably reproduce and communicate color information. All further advances in color science are based on the XYZ system, for example, the familiar CIE L * a * b * system and the like, as well as the latest CIECAM systems that use modern color profiling programs.

Results

1. Accurate work with color requires its measurement, which is as necessary as measuring length or weight.
2. Measuring the perceived brightness (one of the attributes of the visual sensation) of light emissions is impossible without taking into account the features of our visual system, which have been successfully studied and included in all photometric quantities (candela, lumen, lux) in the form of a curve of its spectral sensitivity.
3. A simple measurement of the spectrum of the light under investigation does not in itself give an answer to the question of its color, because it is easy to find different spectra that are perceived as one color. Different quantities that express the same parameter (color, in our case) indicate the failure of such a method of determination.
4. Color is the result of the perception of light (color stimulus) in our minds, and not a physical property of this radiation, so this sensation needs to be measured in some way. But direct measurement of human sensations is not possible (or was not possible at the time of the creation of the colorimetric systems described here).
5. This problem was circumvented by visual (with the participation of a person) equalizing the color of the studied radiation by mixing three radiations, the amounts of which in the mixture will be the desired numerical expression of the color. One of the systems of such three radiations is CIE RGB.
6. Having experimentally equalized with the help of such a system all monochromatic radiations separately, they obtain (after some calculations) the specific coordinates of this system, which show the required quantities of its radiations for the color equation of any monochromatic radiation with a power of one watt.
7. Knowing the specific coordinates, it is possible to calculate the color coordinates of the studied radiation from its spectral composition without visual color equalization by a person.
8. The CIE XYZ system was created by mathematical transformations of the CIE RGB system and is based on the same principles - any color can be precisely specified by the number of three radiations, the mixture of which is perceived by a person as identical in color. The main difference of the XYZ system is that the color of its main "emissions" exists only in colorimetric equations, and it is physically impossible to obtain them.
9. The main reason for creating the XYZ system is to facilitate calculations. The color and chromaticity coordinates of all possible light emissions will be positive. Also, the Y color coordinate expresses the photometric brightness of the stimulus directly.

Conclusion

The areas closest to IT specialists, which are based on the principles and systems described in this article, are image processing and their reproduction in various ways: from photography to web design and printing. Color management systems work directly with colorimetric systems and color measurement results, allowing predictable color reproduction in a variety of ways. But this topic is already beyond the scope of this article, because the fundamental aspects of color theory, and not color reproduction, are affected here.

This topic does not pretend to give exhaustive and complete information about the topic raised, but is only a “picture to attract attention” for IT specialists, many of whom are simply obliged to understand the basics of color science. To facilitate understanding, much here is simplified or stated in passing, therefore, I give a list of sources that will be of interest to those who want to get acquainted with color theory in more detail (all books can be found on the net):
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