Color Presentation

Alexander Liss

07/18/01

Human eye has four different sensors, and one would expect a theory describing human perception of light (color) with four independent parameters. However, we use theories with three (RGB) or even two parameters.

Following is a consistent four-parameter theory of color, including color transformation by a substance, when light passes it through, as in a case of glass, or it is reflected by it, as in a case of a surface.

Physics of Light

We break a range of wavelengths of visible light [Z_{0},Z_{1}] with intermediate points

z_{0}=Z_{0}, z_{1}, ..., z_{n}=Z_{1}

and analyze separately components of the light, which wavelengths belong to the segment [z_{i-1},z_{i}]. For each such component, we determine the intensity of light s_{i}. (We arrive to a familiar characteristic of spectrum of light, when we make segments [z_{i-1},z_{i}] small and their number large. We will not make this step, because for practical measurements we will operate with this finite division of the segment [Z_{0},Z_{1}] any way.) This discrete spectrum defines light with some precision.

We define now characteristics of a substance in relation to light. The substance can reflect light, or light can pass through it, in any case, light is transformed by the substance.

For each component j of original light, we look at its transformation by the substance. It has a full spectrum, and we take its component i.

When we take "standard" original light - its component j is equal to a unit of measurement of light intensity, we get a standard matrix. This matrix is a characteristic of substance.

Physiology of Sight

Following are a few facts from physiology.

1. Human eye has four types of receptors of light. If we draw a graph of sensitivity to the light of particular wavelength for each type of sensors, we get for curves.

One curve is uniform - this receptor (rods, achromatic) reacts on the presence of light.

Others (related to cones, chromatic sensors) are non-uniform. They have one apex - each type in its area of wavelengths.

2. For each component of the light (as we had defined above), the reaction of a sensor to the intensity of light is roughly proportional to a logarithm of intensity of this component.

3. A reaction on a sum of components is a sum of reactions. This corresponds well with the pattern of reaction of our other sensors.

4. Reactions of the sensors are based on a light induced reaction (photoreaction) of a special sensor's substance (rhodopsin). Characteristics of the reaction in the current moment depend on a degree of illumination of the sensor in a previous moment. Hence, sensors have some "memory".

One can observe such "memory" by viewing a long time some object of a definite color and switching viewing to a white surface. One observes a faint colored image of the object (the color is different than one observed originally).

5. Our eyes create two-dimensional snapshot, albeit uneven - the central part of this image is much richer (more sensors are involved), than peripheral. The perpetual automatic scanning feeds our brain with the series of such partially overlapped snapshots. Consecutive snapshots interfere, because of the sensors' "memory".

6. There is a threshold of sensitivity, which is different for different sensors. Chromatic sensors do not deliver useful signals to brain, when light is dim.

Perception

Following are facts related to interpretation of images of light in our minds.

1. Our mind operates with four types of signals from eyes. This is the base of our color-images.

2. We perceive colored surfaces and space, but we can define an abstraction of the color in one point. This abstraction we associate with a uniform color of some (small) area.

3. Our perception of the same colored surface changes as the level of illumination changes; this effect is felt especially, when a level of illumination reaches some of thresholds of sensitivity, as in dusk.

4. Our perception of a color-image depends on the surroundings; for example, the same color image can look darker on a bright background.

5. We have a strong perception of relations between different colors: some colors form families of related colors (red, rose, etc.) and some color combinations we perceive as harmonious.

Dual System of Concepts

There is a system of concepts, with which physics operates, when it describes light, and there is a different system of concepts, which we have to use when we describe color-images. Thus, we have presentation of light with the spectrum on one side, and RGB presentation of the color on the other.

One system is traditionally used in physics and another is used when color-images are described, for example in business or art presentations.

Physical Presentation of Light and Mental Color-Image

A sensitivity of the eye sensor's substance to light is measured by a degree of substance reaction to light of given intensity.

If this sensitivity is r_{i} for a light component i, and s_{i} is an intensity of component i, then a cumulative reaction of the substance is a sum of s_{i}*r_{i} by all light components:

s_{1}*r_{1}+ ...+s_{n}*r_{n}.

There is a general pattern, how a body (including brain) reacts on stimuli. Reaction on light follows this pattern.

A relation between a physical stimulus and corresponding perception follows a rule: a small change in the level of stimulus leads to a proportional small change in a level of perception; it is inversely proportional to a current level of stimulus. When a change of the level of perception *y* is *dy*, and a change in the level of stimulus *x* is *dx* and coefficient of proportionality is *a*, we can write this rule in a following formula

*dy=a*dx/x.*

Hence

*y=a*ln(x)+c,*

where

*ln()* - natural logarithm, and

*c* - constant.

The constant *c* in this formula, we can find from a following fact. When the level of stimulus is below *x0* there is no reaction (there is a threshold of sensitivity). Hence:

*0=a*ln(x0)+c, or*

c=-a*ln(x0).

It means:

*y = a*ln(x/x0), when x ³
x0,*

y = 0, when x < x0

When we know a level y, a corresponding level *x* is:

*x = x0*exp(y/a), y ³
0.*

We apply this reasoning to an eye sensor. In this case, the stimulus is the reaction of the eye sensor's substance:

x = s_{1}*r_{1}+ ...+s_{n}*r_{n}.

Values *a,* *x0 *and r_{i} depend on the sensor. Hence, we need to know values a_{j}, x0_{j}, and

r_{1j}, ..., r_{nj}

for each sensor j=1,2,3,4.

As soon we have this information, we can compute reactions of eye sensors to the light of any spectrum.

Color Presentations

We can present color of the light with spectrum

s1, ..., sn

as a vector (x_{1}, ..., x_{4}), where

x_{j} = s_{1}*r_{1j}+ ...+s_{n}*r_{nj}

or as a vector (X_{1}, ..., X_{4}), where

X_{j}=x_{j}/x0_{j}

and, if we introduce normalized coefficients

R_{ij} = r_{ij}/x0_{j},

then

X_{j} = s_{1}*R_{1j}+ ...+s_{n}*R_{nj},

or we can present color as a vector (y_{1},...,y_{4}), where

y_{j} = a_{j}*ln(x_{j}/x0_{j}), when x_{j} ³
x0_{j},

y_{j} = 0, when x_{j} < x0_{j}.

or

y_{j} = a_{j}*ln(X_{j}), when X_{j} ³
1,

y_{j} = 0, when X_{j} < 1.

All these presentations are equivalent. We call them *x-form*, *X-form* and *y-form* correspondingly.

Classes of Related Colors

There are classes of colors, which we perceive as closely related. It is possible, that these classes emerge from an observation of the same object with different illumination.

One of the sensors - we give it number four, is achromatic, it is present differently in our classification, than three other sensors.

All colors with the same combination of (x_{1},x_{2},x_{3}) regardless of the value x_{4} are perceived as closely related. Hence a combination (x_{1},x_{2},x_{3}) we call *tint* of the color, and x_{4} we call *intensity* of the color.

Tints are characterized with three parameters. In this respect they are close to RGB presentation. However, linear transformations in the space of tints have good visual associations, and this is not true in RGB.

Colors with the same ratio x_{1}:x_{2}:x_{3} (and any possible value of x_{4}) form a class of related colors. We call this ratio *hue* of the color.

Hues can be interpreted as classes of Tints. They can be described with two parameters.

Base Intensity Level

There are shiny colors - which intensity is high and dull colors - with low intensity. The boundary between them - *Base Intensity Level* depends on a degree of illumination of the color-image and surrounding area. Our visual system makes an automatic adjustment for the degree of illumination.

Hence, if one cares to deliver a particular impression with a color-image, one has to either control a Base Intensity Level or has to manipulate an image to adjust to an existing Level.

Brightness of Tint and Base Brightness Level

Tints (x_{1},x_{2},x_{3}) of the same hue x_{1}:x_{2}:x_{3 }have different *brightness* - some are dark tints others are bright tints, for example brown and yellow. We can describe brightness with the value

1/3*(x_{1}+x_{2}+x_{3}),

which we call a *level* of tint. This is a technical characteristic, which corresponds to a perception of brightness of color.

A boundary between dark and bright tints is *Base Brightness Level.* It depends on a degree of illumination of the color-image and on a surrounding area.

If we want a more precise characteristic of the brightness of color, we have to define it with y-forms. x-forms allow us only convenient description of the boundary between bright and dark colors.

Visible Colors

There are values of the vector representing color (x_{1},x_{2},x_{3},x_{4}), or values of the vector representing tint (x_{1},x_{2},x_{3}), or even values representing hue (x_{1}:x_{2}:x_{3}), which do not correspond to any light. For example, red sensor reacts always, when any other chromatic sensor reacts; hence, it is impossible to have a zero component, corresponding to the red sensor, with non-zero component, corresponding to another chromatic sensor.

Values, which correspond to some light, form areas in four-dimensional color space, in three-dimensional tint space and in two-dimensional hue space.

Hence, our presentation of color with tint is different from the similar three-dimensional RGB presentation, where any combination of coordinates corresponds to some light.

We need one property of the areas of visible colors. If there is a color with the given hue, then all possible tints with this hue exist.

This is because, if the spectrum of the light with the given hue is s_{1}, ..., s_{n}, then lights with spectrums c*s_{1}, ..., c*s_{n}, with all possible values of c form the class of tints of given hue.

It is important that when eyes get tired, for example from prolonged observation of the same color, the characteristics of sensors change and the areas described above move somewhat. This effect allows creation of "unseen" colors - we observe the color, which we did not see before. Creators of color-images sometimes use this effect.

Mixing Lights and Paints

When we mix paints or lights the effect can be unexpected. However, mixing colors is a basic operation of color manipulation.

When we mix lights, we add-up their spectrums - if one light has a spectrum a_{1}, ..., a_{n}, and the other has a spectrum b_{1}, ..., b_{n}, then their mixture has a spectrum

a_{1}+b_{1}, ..., a_{n}+b_{n}

and we can compute its color. If the color of first light is presented with the vector (x_{1},x_{2},x_{3},x_{4})_{1}, and the second light with the vector (x_{1},x_{2},x_{3},x_{4})_{2}, then the color of their mixture is presented with the vector

(x_{1},x_{2},x_{3},x_{4})_{1} + (x_{1},x_{2},x_{3},x_{4})_{2}

where sign plus means ordinary addition of vectors.

When we mix paints, and we have fixed conditions of illumination, we can compute results of light transformation by paint. If the spectrum of light transformed by first paint is a_{1}, ..., a_{n}, and the spectrum of light transformed by second paint is b_{1}, ..., b_{n}, and we mix these paints in ratio p_{1}:p_{2} (p_{1}+p_{2}=1), then light, transformed by mixed paint has the spectrum

p_{1}*a_{1 }+ p_{2}*b_{1}, ..., p_{1}*a_{n }+ p_{2}*b_{n}

and we can compute its color. If the color of light transformed by first paint is presented with the vector (x_{1},x_{2},x_{3},x_{4})_{1}, and the color of light transformed by second paint - with the vector (x_{1},x_{2},x_{3},x_{4})_{2}, then the color of light transformed by their mixture is presented with the vector

p_{1}*(x_{1},x_{2},x_{3},x_{4})_{1} + p_{2}*(x_{1},x_{2},x_{3},x_{4})_{2}

where sign plus means ordinary addition of vectors and multiplication sign is a multiplication of the vector by scalar.

We have similar formulae in X-forms.

White-black Color

One group of colors is very important for the description of perception of color. This group includes all white and black colors, depending on the color intensity and brightness. We call this group *white-black* color. This group is better defined in y-form.

All colors with

y_{1 }= y_{2 }= y_{3}

(and any possible value of y_{4}) form this special class. In X-form it is

a_{1}*ln(X_{1}) = a_{2}*ln(X_{2}) = a_{3}*ln(X_{3})

When a_{1}, a_{2} and a_{3} are close to each other, we have an approximation of this equation

X_{1 }= X_{2 }= X_{3}

Stain

When we mix paints, we often get surprising results. However, there is one color, mixing of which produces predictable results - white-black color.

This creates one additional classification. All hues, which can be presented as a mixture of paint with given hue and other paint of white-black color, belong to the same *stain*. The set of all possible stains is one-dimensional.

Basis for Color Harmony

We described above how eyes scan the view and how snapshots interfere. As a result of such interference, we can see colors-mixtures, which are not present in the view. Because we want to see the real picture, we reduce speed of scanning to reduce such interference.

Interference during an observation of some color combinations produces only white-black colors, which can be easily filtered out. Such convenience causes a natural attachment to these color combinations. We call them harmonious.

Fast jumping of the eye focus from one color to the other produces effect similar to mixing of the colors (in approximately same ratio). Hence, we have a quantitative definition of the color harmony. The set of colors is harmonious, if the mixture of them in equal ratios is a white-black color.

The mixture of lights reaching the eyes (X_{11},X_{21},X_{31}) and (X_{12},X_{22},X_{32}) in equal proportion is a color with parameters

0.5*(X_{11},X_{21},X_{31}) + 0.5*(X_{12},X_{22},X_{32})

where addition and multiplication are ordinary vector operations. For colors to be perceived as harmonious it has to have equal components in y-form (this gives approximately white-black color):

a_{1}*ln(X_{11} + X_{12}) = a_{2}*ln(X_{21} + X_{22}) = a_{3}*ln(X_{31} + X_{32}).

This formula can be generalized for many colors.

When a_{1}, a_{2} and a_{3} are close to each other, we have an approximation of this equation

X_{11} + X_{12} = X_{21} + X_{22} = X_{31} + X_{32}

Distance between Colors

The concept of distance between different colors is vague. However, we need to define it in the most reasonable way, because we want to manipulate color-images.

For technical color approximation, we define it as a distance between two vectors of y-form presentation of color. Hence, if we have two colors with presentations (y_{1},y_{2},y_{3},y_{4})_{1} and (y_{1},y_{2},y_{3},y_{4})_{2}, then we define the distance between them as a distance between these two vectors.

Effect of Small Variation of Color on Surface and in Space

The particular properties of color-image can be determined by the small random variations of color on the surface or in the space (as in glass). Examples are faded colors (with mixed-in white dots) and dirty colors (with mixed-in black dots).

There are many ways to describe this variation, we present here one of them.

We present the color of surface (or space) as a set of randomly placed small dots of uniform color. Hence, if we have dots of colors d_{1}, ..., d_{m}, we need only define the corresponding probabilities p_{1}, ..., p_{m}, to define this type of the color-image.

Color Transformation by Substance

We can compute transformation of color by substance (paint, color filter, etc.) with the help of special *effect coefficients*, which we define here.

When we want to define the change of color perception induced by substance, we do not need to know the entire transformation of spectrum induced by this substance, we need to know less.

We take light component i of the intensity equal to the unit of measurement. We define the effect coefficient q_{ij} as the degree of reaction of the eye sensor j (its sensitive substance) to this light component i after the transformation by this transforming light substance (paint).

If the light has spectrum

s_{1}, ..., s_{n}

and it is transformed by given substance, then we can compute the color-vector (x_{1},x_{2},x_{3},x_{4}) of a transformed light:

x_{j}=q_{1j}*s_{1} + ... + q_{nj}*s_{n}

We can normalize coefficients q_{ij}

Q_{ij} = q_{ij}/x0_{j}

and compute color-vector (X_{1},X_{2},X_{3},X_{4}):

X_{j}=Q_{1j}*s_{1} + ... + Q_{nj}*s_{n}.

Mixing Paints and Covering Surface or Space

Paint includes small colored particles, which determine its color. Geometric qualities and degree of transparency of these particles determine how they compete with the colored particles of the other paint at mixing and how well they cover the color of the surface or transform the passing light.

This property we describe with the *covering coefficient. *Covering coefficient is a minimum amount of paint, which we need to apply to the unit of surface area to cover the surface completely (or to apply in the unit of volume, to have no holes for passing light).

Covering coefficient depends on the method of application (as placing dots by the printer), and type of the surface (paper, textile, wall, etc.). Usually, these factors are fixed, or we have only a few classes of possible combinations of these factors.

It is a number, which allows calculation of the qualities of light reflected by paint, as a function of the applied amount of paint.

Color-producing Instruments

With our presentation of color, we can have the universal presentation of color-producing instrument, including its gamut. We distinguish two cases. One is Lamp instrument, which produces light (as TV set, for example) and the other is Paint instrument, which produces something transforming light (as printer, for example). Color-producing Instruments have Elements, which actions we combine to produce color, as inks in inkjet printer.

Lamp Instrument

We assume that the Lamp Element does not change the shape of its spectrum; it only changes its intensity (in some boundaries). Hence universal characteristics of the state of Element k are:

intensity of the light c_{k} (variable),

its boundaries [C0_{k},C1_{k}],

and four parameters

(g_{1k},g_{2k},g_{3k},g_{4k})

which are equal to the color-vector (x_{1},x_{2},x_{3},x_{4}) of the light produced by this element, when intensity c_{k}=1.

This Element can produce light with the color-vector

c_{k}*(g_{1k},g_{2k},g_{3k},g_{4k}), for any c_{k} from [C1_{k},C2_{k}].

If we have K Elements, we can produce light with the color-vector

c_{1}*(g_{11},g_{21},g_{31},g_{41}) + ... + c_{K}*(g_{1K},g_{2K},g_{3K},g_{4K})

for any combination of c_{1}, ..., c_{K} as long each c_{k} belongs to its segment [C0_{k},C1_{k}].

This is the gamut of Lamp Instrument in our x-presentation of color. Obviously, if one wants to monitor only tint of the color, one works only with three parameters of color instead of four.

This defines a figure with linear boundaries in the four-dimensional space - a polyhedron. The presence of lower boundaries can make this figure complex with the kind of depression or even a hole in the middle.

Manufacturers of color-producing instruments can compensate this handicap of *unavailable low intensity of the light of the Elements* by adding Elements with lights, which colors are near black-white color.

Paint Instrument

Paint as color-transforming substance has effect coefficients and covering coefficient. Hence, Paint, as an Element k of the Paint Instrument, has effect coefficients

q_{ijk} i=1,...,n, j=1,...,4

and the covering coefficient

u_{k}.

The variable corresponding to the Paint Element is amount of paint. In our universal presentation we use normalized amount of paint c_{k} - the ratio of the amount of paint applied to the unit of surface area (or volume in the case of space) to the covering coefficient u_{k}. Light reflected by the paint is proportional to the normalized amount of paint. The normalized amount of paint we can vary from zero to one.

We apply a few Paints to the surface (or in the space), which has its own ability to transform the light and its own color.

First, we do not apply more than needed to cover the color of surface (or the space), hence the sum of normalized amounts of Paints should not be greater than one.

Second, if we do not cover completely the color of surface (or the space), we have to compute the effect of combination of Paints and the surface.

The color of surface (or the space) we should treat as Paint - an additional artificial Element. The normalized amount of this Paint is what supplements the sum of normalized amounts of real Paints to one.

If we know for sure that we cover the surface (or the space) completely, we do not need to add this artificial Element.

Now, for the given illuminating light with the spectrum s_{1}, ..., s_{n}, we can compute the vector (g_{1k},g_{2k},g_{3k},g_{4k}), where

g_{jk} = x_{jk} = q_{1jk}*s_{1} + ... + q_{njk}*s_{n}

which is the presentation of color of the given Paint.

Normalized amounts of paint, including the artificial Element formed by the color of surface (or the space), are non-negative and not greater than one: 0 £
c_{k} £
1. In addition, we have the constraint

c_{1} + … + c_{K+1} = 1,

which simply states that we took in consideration all light transforming substances and did not use excess of them.

The color of surface (the space) is

c_{1}*(g_{11},g_{21},g_{31},g_{41}) + .... + c_{K+1}*(g_{1K+1},g_{2K+1},g_{3K+1},g_{4K+1}).

This defines a polyhedron in four-dimensional space.

In the case of printer, usually there are four paints - three chromatic and black. If we add a bright white Paint, then the polyhedron can be increased dramatically - this means better color approximation (color matching). This has an additional advantage - we do not need to use the white color of paper, which vary from one type of paper to the other, we can cover the surface completely.

Color Presentation

There is a family of coordinated color presentations. The most complete one presents (uniform) color with four-dimensional vector - color-vector. It has three forms

x-form (x_{1},x_{2},x_{3},x_{4}),

X-form (X_{1},X_{2},X_{3},X_{4}) and

y-form (y_{1},y_{2},y_{3},y_{4}),

which complement each other.

y-form reflects the perception of color. Three components of these vectors are chromatic, and fourth component is achromatic.

The colors of the same tint form one-dimensional subspace in four-dimensional color space of x-forms where achromatic component can vary. Hence, if the limited characteristic of color - tint, is sufficient, then we can operate in three-dimensional space of vectors-tints (first tree components of the color-vector).

The vectors-tints, which are proportional, form a one-dimensional subspace in the tint space. These are tints, which have the same hue and differ in brightness. If it is sufficient to work with hues, we can operate in two-dimensional space of hues. The convenient presentation of the hue space is a triangle which is formed by non-negative values of tint-vectors (x_{1},x_{2},x_{3}), when x_{1} + x_{2} + x_{3} = 1.

Finally, all hues of colors, which can be presented as a mixture of the white-black color and a given color, form a one-dimensional subspace in the hue space - stain. Hence if the minimal characteristic of the color - stain, is sufficient, we can operate in one-dimensional space of stains. The convenient presentation of the stain space is the perimeter of the hue triangle or a circumference.

Spectrum of Light and Color

In many cases, there is a practical need in the computing the presentation of color from the spectrum of light. We divide the segment of wavelengths of visible light [Z_{0},Z_{1}] with intermediate points

z_{0}=Z_{0}, z_{1}, ..., z_{n}=Z_{1}

and define spectrum

s_{1}, ..., s_{n}

where s_{i} is intensity of the part of light, which wavelengths are from the segment [z_{i-1},z_{i}].

We define coefficients r_{ij}, which depend only on the characteristics of human visual system, and

x_{j} = s_{1}*r_{1j}+ ...+s_{n}*r_{nj}

Also we define coefficients a_{j} and z0_{j}, which depend only on the characteristics of human visual system, and

y_{j} = a_{j}*ln(x_{j}/x0_{j}), when x_{j} ³
x0_{j},

y_{j} = 0, when x_{j} < x0_{j}.

Thus, we have the relationships between two forms of presentation of color - x-form and y-form, and the formula of computation of color-vectors from the spectrum of light.

Color with Small Variation

For creators of color-images, there is a way of presentation and generation of a color on a surface or in a space with small variations.

We present the color of a surface (or a space) as a set of randomly placed small dots of uniform color. We have dots of colors d_{1}, ..., d_{m}, (which are presented with color-vectors) and the corresponding probabilities p_{1}, ..., p_{m}. This defines this type of color-image.

To generate this type of color-image we use random number generator to compute placement of dots on the surface or in the space and after we place colored dots accordingly.

Base Intensity Level and Base Brightness Level

A color-image has two important characteristics, which are not encoded in the colors of its points, but in the surroundings of it - Base Intensity Level and Base Brightness Level. These values define the division of colors on shiny or dull and bright or dark. These values have to be the part of description of a color-image. In the process of reproduction of a color-image, they have to be matched to the conditions of the observation of the reproduced color-image.

Effect and Covering Coefficients

We define effect coefficients for the light transforming substances (as paint). We take light, which wavelengths belong to the segment [z_{i-1},z_{i}] of the intensity equal to the unit of measurement. We define the effect coefficients (q_{i1,} q_{i2,} q_{i3,} q_{i4})_{,} as the color-vector (x_{1},x_{2},x_{3},x_{4}) of this light after the transformation by this light-transforming substance.

The color-vector of light with the spectrum

s_{1}, ..., s_{n}

after the transformation by this substance is

x_{j}=q_{1j}*s_{1} + ... + q_{nj}*s_{n}.

We define the *covering coefficient* - the minimum amount of paint we need to apply to the unit of surface area to cover the surface completely (or to apply in the unit of volume, to have no holes for passing light). It depends on the method of application, and type of the surface (or material in space). When we have only a few classes of possible combinations of these factors we define covering coefficient for each class.

These effect coefficients and covering coefficients can be supplied with the substance (with products as paints, printer inks, filters, etc.) in a convenient form (labels, bar codes, digital data, etc.). The presentation of this information can be more compact, if there is a set of standard divisions of the segment of the wavelengths of visible light, at which we can refer.

Based on this information, a customer can compute the proper mix of the substances, which delivers the needed color effect, without experimentation.

Harmonious Colors

Creators of the color-images can form harmonious combinations of colors using following rule: The sum of corresponding tint-vectors in X-form should have equal components in y-form. As an approximation: The sum of corresponding tint-vectors in X-form should have equal components.