# Colorimetry

The colorimetry is the study of the dimension designations of colors . It uses mathematical formulas to numerically represent the visual result of a color observation or a color comparison.

## Subdivision

• The lower colorimetry is based on Graßmann's laws and assesses the equality of colors . This is not about the primary color stimulus , but about the color valence , insofar as the colorimetry is a color valence metric. The additive color mixture uses the lower colorimetry, since three components, according to the sensitivity of the cones , are combined in such a way that the same (here desired) color is achieved.
• The higher colorimetry is based on the application of the lower colorimetry and turns to the color differences . The main goal is to optimally adapt the dimensions of the color valences to the color sensations .

## application

The colorimetry evaluates color measurements so that the visual perception “color” can be fixed as color valence with color measures .

## Color measurement

Spectrum of light-emitting diodes in red, green, blue and white. Color valence corresponds to the name, the light in the spectrum is the color stimulus

Here color is always color valence, the sensation perceived by the eye from a color stimulus. The measurement objective is not the (physical, spectral) color stimulus, but the (effective) color valence . The term color valence measurement is less common but more precise . The measurement is carried out in principle according to the Lambert-Beer law , which is only fulfilled with monochromatic measurement. Therefore (as narrow as possible) intervals of wavelengths are formed and measured.

So far, only an instrumental recording of the color stimulus has been possible , the desired numerical representation of the color valence as a color system therefore requires a mathematical apparatus or suitable material filtering. In other words, the measurement is carried out instrumentally according to the spectral composition of the recorded light, the conversion (mapping) to the three cone absorptions is carried out by calculation. Finding the exact mapping function, the design of the color space , is the current problem of color valence measurement.

### Demarcation

The three human cones inevitably produce three color valences that have to be evaluated. Color measurement must be a "sensory" measurement of three color valences. The determination of other measurement numbers , such as the whiteness of paper, the iodine color number, the degree of bleaching or colorimetry , are not to be understood here in the narrower sense as color measurement. Likewise, color recognition cannot be assigned to color measurement, since the result of color recognition gives a color name or color number - but not a color measure.

### Measurement method

There are various methods for measuring color (color valences).

Equality procedure
In this method, the examination pattern is compared with a series of known standard patterns using a technical device or visually with the eye until the equality is reliably established. The selected (three) basic colors can also be offered proportionally. Technical implementations are the color spinning top or Maxwell's approach. In the first case the temporal resolution of the measuring device (e.g. the eye) is undershot due to a quick change, in the second case a spatial distribution of the basic colors is brought to a (seemingly) common surface by a defocusing and thus perceived by the eye as a uniform color impression. Usually, this method uses the equality judgment of the normally sighted eye, so it is actually subjective. The development of expensive technical devices has been stopped by improved computing technology in favor of the following two methods, which, however, require calculations.
Luminosity method (three-area method)
The color stimulus hits such a receiver, the spectral sensitivity of which corresponds to the primary color spectral values ​​due to the addition of suitable color filters . The measuring element ( photocell , nowadays photodiodes ) then measures a “brightness” that (ideally) corresponds to the stimulus on the cones . The measured value thus determined corresponds to the color valence. Filters based on the standard spectral value curves are most suitable. If the three color filters (or color filter combinations) defined in this way are connected upstream one after the other, the three standard color values ​​result immediately. The measurement requirement is that the Luther condition is met. The measurement accuracy depends on how well the spectral composition of the color filters is adapted. This is the principle behind color sensors that have three photodiodes in one housing with three upstream filters.
Spectral method
Each color valence is the integral over all spectral (monochromatic) color valences. The spectrum (i.e. associated intensities) of the light color or body color to be examined is measured over the wavelength range of visible light . In the case of body colors, the illuminating light must also be included. Thanks to more than a hundred years of device development ( spectrophotometer , spectrometer) with connected computing technology, powerful devices make this method the most widely used today.

### evaluation

For the evaluation of the spectral numerical values ​​from the measuring device, the required color coordinates must be converted. The standard spectral color values ​​defined by the International Commission on Illumination (CIE) have prevailed . These basic numbers are available in tabular form at intervals of one nanometer. ${\ displaystyle {\ overline {x}} {,} \; {\ overline {y}} {,} \; {\ overline {z}}}$

### Weight ordinate method

The basis of this calculation method is that the color valence is calculated from the sum of the spectral color valences (λ) and spans the color space perceived by the eye . The standard color values ​​are usually preferred. These are tabulated by the CIE for the standard valence system. ${\ displaystyle {\ mathfrak {F}}}$${\ displaystyle {\ mathfrak {S}}}$

${\ displaystyle {\ mathfrak {F}} = r * \, {\ mathfrak {R}} + g * \, {\ mathfrak {G}} + b * \, {\ mathfrak {B}}}$according to the laws of transformations in vector space , so too
${\ displaystyle {\ mathfrak {F}} = X * \, {\ mathfrak {X}} + Y * \, {\ mathfrak {Y}} + Z * \, {\ mathfrak {Z}}}$if the introduced standard valence system is used.

For the calculation, this can now be resolved for the respective wavelength-indicated values ​​A λ as A w at the point . For the spectral radiance of the light source A λ so etc. and thus dλ integrates dw here. ${\ displaystyle \ lambda = w}$${\ displaystyle S_ {w}}$

${\ displaystyle {\ mathfrak {F}} = \ int _ {380} ^ {760} \, S_ {w} \, {\ mathfrak {S}} _ {w} \, \ mathrm {d} w = \ int _ {380} ^ {760} \, S_ {w} \, ({\ overline {x}} _ {w} {\ mathfrak {X}} + {\ overline {y}} _ {w} {\ mathfrak {Y}} + {\ overline {z}} _ {w} {\ mathfrak {Z}}) \, \ mathrm {d} w}$

For the practical calculation it is necessary to switch from the integral to the sum dλ → δλ. Depending on the required accuracy, measurement intervals of 10 nm are common in color measurement today, while simple color measurement devices still use 16 values ​​at 20 nm intervals. When using the tables at 5 nm intervals, 5 nm can be selected in accordance with the ISO and DIN standards for higher demands.

${\ displaystyle {\ mathfrak {F}} = \ left (\ sum S_ {w} {\ overline {x}} _ {w} \ right) \, {\ mathfrak {X}} + \ left (\ sum S_ {w} {\ overline {y}} _ {w} \ right) \, {\ mathfrak {Y}} + \ left (\ sum S_ {z} {\ overline {z}} _ {w} \ right) \, {\ mathfrak {Z}}}$

The addition of the product now results in the color value X, correspondingly with the color value and with then . However, it is still standardized, mostly on , i.e. with . ${\ displaystyle S _ {\ lambda} {\ overline {x}} _ {w}}$${\ displaystyle S _ {\ lambda} {\ overline {y}} _ {w}}$${\ displaystyle Y}$${\ displaystyle S _ {\ lambda} {\ overline {z}} _ {w}}$${\ displaystyle Z}$${\ displaystyle Y = 100}$${\ displaystyle k = \ sum S_ {w} {\ overline {y}} _ {w}}$

However, in the case of body colors or reflective colors, the radiation S λ emanating from the light source is changed by the spectral properties of the surface in question. For the color stimulus function φ (λ) that hits the eye, the color in the real sense, this “influenced” spectral radiance must be used. This is either the spectral reflectance curve β (λ) for surface colors or the spectral transmission curve τ (λ) for reflective colors. The spectral color stimulus φ λ is now φ λ = S λ · β λ for the spectral reflectance and accordingly φ λ = S λ · τ λ for the spectral transmittance

And finally the color values ​​are obtained by weighting the ordinate values ​​of the spectrum accordingly

${\ displaystyle X = \ sum _ {380} ^ {760} \ varphi _ {\ lambda} {\ overline {x}}}$
${\ displaystyle Y = \ sum _ {380} ^ {760} \ varphi _ {\ lambda} {\ overline {y}}}$
${\ displaystyle Z = \ sum _ {380} ^ {760} \ varphi _ {\ lambda} {\ overline {z}}}$

### Further evaluation methods

Selection coordinate procedure
With this method, the multiplication is omitted due to the conversion of the integrals. Using a set of tabulated standard values, the spectral measured value is determined at suitable support points. The selected β λ and τ λ are determined here and so only an addition of these numerical values ​​is necessary.
Spectral band method
The other way around, the radiation distribution of the light source can also be summarized and measured in this spectral interval. The color values ​​are correspondingly obtained by measuring the color stimuli in these intervals.

### Colorimeters

Since the 1980s, color measuring devices have mostly been spectrophotometers that automatically register the spectral curve and then carry out the necessary integration of the measured values ​​obtained on the chip used. The measured values ​​can then of course be output in different coordinates (according to the desired color space ) or as a spectral curve. The color differences between the color template and a series of patterns can then also be output through storage . By converting to different (preferably standardized) types of light , the metamerism index can also be calculated from template to sample.

## Models

### Color and brightness

V (lambda) curve : Brightness sensitivity curve for night (left) and daytime vision

The color world made up of three components is only given from a certain brightness , the trichromatic vision with the cones, each of which contains different opsins (this brightness range is shown in the V (lambda) curve ). These three types of cones, the excitations of which supply the color valence of the incident radiation as an inseparable overall effect of the three individual excitations, have different spectral sensitivity curves for the average color-normal-sighted observer . Normalized to the same total areas of the three curves, the standard spectral value functions result . So if each receptor delivers 1/3 of the total excitation , then achromatic (white, gray or black) is perceived. The size of the total excitation ( ) gives the color brightness . ${\ displaystyle B + G + R}$

### The cone excitation space as a color space model

The CIE standard color table represents a spatial illustration of the relative excitations and . The third component is clearly defined by. (The colors shown are for guidance only).${\ displaystyle r}$${\ displaystyle g}$${\ displaystyle b}$${\ displaystyle b = 1-rg}$

Hue is by the relative excitations , , given: etc. As is true: , you only two shares (needs and ) indicate clearly mark a shade. In a - plane only a triangle is possible because there are no negative excitations. The corners of the triangle cannot be reached because there is no color stimulus that only excites one color receptor. The spectrum color train does not close. To close the arc, you need the mixed colors between purple and red, the purple line . The standard color table that is used in DIN 5033 results in the CIE standard valence system . ${\ displaystyle b}$${\ displaystyle g}$${\ displaystyle r}$${\ displaystyle b = B / (B + G + R)}$${\ displaystyle b + g + r = 1}$${\ displaystyle r}$${\ displaystyle g}$${\ displaystyle r}$${\ displaystyle g}$

Different saturations of the colors towards white or black can not be taken into account with a two-dimensional standard color table . For this you need a three-dimensional structure, the color space , like a sphere with a white pole and a black pole and a color circle forms the equator .

If all color tones are to be perceived as being equally far apart, this ball changes to a “strangely” shaped body of color . In the blue the ball gets a belly - it becomes more convex. In the case of purple and red, the ball flattens out and in the case of yellow it has a “knee” that protrudes far - a corner. This subjectively determined body of color of perception coincides with the possible arousal space calculated from the functions of the cone excitations .

### Imaginary colors

The designation imaginary colors stands for nonexistent, unreal "colors" or better unreal color valences , for such there are no color stimuli in the physical world of electromagnetic waves . On the illustration of the CIE standard color table , it is the gray areas, the parts designated there as theoretical colors . In the LMS color space of the pins all perceptible colors (valences) can be described. The measurements in preparation for the CIE system and subsequent microspectrophotometric determinations on the eye could only objectively determine real colors. In principle, however, any primary valences can be used as coordinates in the three-dimensional color space, resulting in an equally large number of conceivable color spaces. Such a (mathematical) space can be larger than it corresponds to a transformation of the cone space . The "outside" and therefore imperceptible color constructs are called imaginary colors , they are ultimately aids in calculating color spaces. In order to achieve such color locations in terms of measurement technology, measurements in the color comparison are not changed on the "actual" light, but (in fact as a subtraction) on the "target" light; the measurement process involves an external color mixture .