Color Theory Equations

  • #1
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Hey guys I figured out something new about color theory in additive color mixing.

So, the primary colors in additive color mixing are R, G, and B (Red, Green, and Blue).

The secondary colors are C, M, Y (Cyan, Magenta, Yellow).

Using basic knowledge we know that

R + G = Y
R + B = M
G + B = C

The secondary colors can also be added to make up primary colors:

M + Y = R
C + Y = G
M + C = B

We know that from looking at Adobe Photoshop's color wheel that these are the opposite colors:

B = -Y
G = -M
R = -C

In further detail:

B = -(R + G) = - Y
G = -(R + B) = - M
R = -(G + B) = - C

So essentially Blue is the inversion of Red and Green. Green is the inversion of Red and Blue. And Red is the inversion of Green and Blue combined.

This can be further proved in the following:

R + G = Y
R + B = M
M + Y = R
M + Y + B = M
B = - Y

This is all I have in terms of simple equations on color.

The rest is analytical.

luminosity.jpg

This is the graph of luminosity of different colors. We notice G is the brightest of primary colors, R is the second brightest, and B is the least bright primary color. We notice there is a bump in brightness at C, M, and Y, which are formed by combining two primary colors. This probably is because combining two colors triggers two cones in the human eye that causes the luminosity to shoot up and make the bump on the graph.

I believe there is a way to define color as the change in luminosity (slope) and the overall luminosity.

I know I am missing a lot of things in this, so can anyone who's knowledgeable about light and physics help me make equations that can define color scientifically and mathematically?

Many thanks in advance.
 
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Answers and Replies

  • #2
tech99
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I should mention that colour mixing does not produce the colour, only the perception of the colour. It is a psycho-perceptual effect. So that for the case of your luminosity diagram, a true Cyan, for instance, does not consist of two colours.
 
  • #3
scottdave
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There have been many YouTube videos about how colors are perceived by our eyes, as well as generated by computer monitors (and 4-color printing). Vsauce and PhysicsGirl are 2 channels which come to mind for that.

So I was wondering about your luminosity measurements. Did this come from an actual device which measures it? You talk about the cones being activated in a person's eye, which leads me to think about a person perceiving one color to be more luminous than another.
 
  • #4
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The measurements come from Adobe Photoshop.

Perhaps Cyan consists of two colors (blue and green) and one color (cyan) at the same time?
 
  • #5
Drakkith
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Perhaps Cyan consists of two colors (blue and green) and one color (cyan) at the same time?

That makes no sense. Cyan can't consist of two primary colors and itself. You'd get a recursive definition.
 
  • #6
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The thing that concerns me the most is how the colors have different luminosity and when mixed the luminosity dont add up. Is there an equation that explains this?
 
  • #7
Drakkith
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The thing that concerns me the most is how the colors have different luminosity

Are you referring to the "brightness" of each of the RGB color channels? There are several different ways of defining luminosity. For example, in video there is the luma, and there are a half-dozen or more different ways when talking about the "lightness" of a color. Here's another page on lightness.

Is there an equation that explains this?

That depends on the exact color theory, model, or format that you're talking about. There are equations in some of the links I just posted. Have a look at those.
 
  • #8
sophiecentaur
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The measurements come from Adobe Photoshop.

Perhaps Cyan consists of two colors (blue and green) and one color (cyan) at the same time?
The eye is not a spectrometer (how many times have I mentioned that fact?) It does the best it can with just three analysis curves. It works with Perceived Colours and not wavelengths.
You can produce Cyan 'colour' with Blue and Green Phosphors and it is also possible to manufacture a Cyan phosphor which would be centred around 490nm. Neither is more correct than the other.
I don't know of an additive system that used the secondaries for phosphors. Three phosphors are sufficient to do a good job of synthesising matching colours by addition. Subtractive mixing is another matter and it's only with (by now ancient) film technology and low quality colour printing that only three primaries are used CMY dyes and pigments give a reasonable result but even humble home printers use six or more inks. High quality printing uses spot colours which are aimed at a particular critical colour - as in advertising logos.
 
  • #9
tech99
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As a matter of interest, Professor Land demonstrated that a perception of any colour can be be obtained using just two primaries, these being colours located at opposite sides of the colour triangle.
 
  • #10
sophiecentaur
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As a matter of interest, Professor Land demonstrated that a perception of any colour can be be obtained using just two primaries, these being colours located at opposite sides of the colour triangle.
It would depend on which colour triangle your talking of.
That's not surprising if you are referring to the appropriate triangle. Two appropriate primaries can produce any colour match that lies on a line between them on the CIE chart. But that's not a very useful fact. More usefully, three primaries can produce any colour in the triangle with the primaries at the vertices. Colour TV relies on that.
 
  • #11
tech99
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I agree with what you are saying, but Edwin Land observed accidentally that certain pairs of colours would stimulate full colour perception, including those colours not lying on their axis.
http://www.greatreality.com/Color2Color.htm
 
  • #12
sophiecentaur
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I agree with what you are saying, but Edwin Land observed accidentally that certain pairs of colours would stimulate full colour perception, including those colours not lying on their axis.
http://www.greatreality.com/Color2Color.htm
I had heard of that already and Land did extensive tests, involving the 'Mondrian' patterns to justify the theory. His 'Retinex' theory was developed in depth. It's true that the blue/violet part of the spectrum contributes very little energy to our vision and that there are many clues about colour in most scenes that allow some 'colour blind' people to identify colours (though perhaps not so good at actual colour matching). It's a set of three ill conditioned equations that the brain has to solve if one channel is missing.
There is strong evidence of a tristimulus colour vision system in humans. A standard (additive) colour reproduction system is based on three sets of curves - the camera analysis with three filters, the display with three phosphors and the eye with three more analysis filters. The camera analysis will give RG and B channel signals, which give the Chrominance values and the Luminance. Bearing in mind how good the brain is at 'filling in' for missing information, I guess it's not too surprising that other clues can make up for lack of a B component in certain scenes. I guess you could say that TV Engineers are lucky that the brain makes such a good job of making sense of an RGB generated image. The Maths that is involved in a three colour reproduction system works very well and the two colour system clearly doesn't work as well - or that could have been what we use today.
 
  • #13
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Guys I looked up more equations on color theory with subtractive color mixing. I figured some out too.

(C = Cyan, M = Magenta, Y = Yellow)

(R = Red, G = Green, B = Blue)

Additive Color Mixing:
R + B = M

R + G = Y

B + G = C

M = -G

Y = -B

C = -R

B - G = R

B - R = G

G - B = R

G - R = B

R - B = G

R - G = B

Subtractive Color Mixing:
C + M = B because they absorb R + G = Y

C + Y = G because they absorb R + B = M

M + Y = R because they absorb G + B = C



C + B (Sky Blue) = -O (Orange) because R + Y = O

Y + G (Lime Green) = -P (Purple) because B + M = P

M + R (Rose Red) = -T (Teal) because B + C = T

O + P = -T

O + T = -P

P + T = -O


------

Question still persists why in subtractive color mixing Blue (Cobalt Blue) + Yellow produces green rather than grey. Also other opposite colors combined produce brown or purple in subtractive color mixing. Why is that? Can anyone figure it out?
 
  • #14
Drakkith
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Question still persists why in subtractive color mixing Blue (Cobalt Blue) + Yellow produces green rather than grey.

As far as I can tell it should be black. It sounds like whatever pigments were mixed weren't mixed in the right amount.

Also other opposite colors combined produce brown or purple in subtractive color mixing. Why is that? Can anyone figure it out?

Mixed in equal proportions, opposite colors produce black. When mixed in non-equal proportions, the color will depend greatly on the specific amount of each pigment.
 
  • #15
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As far as I can tell it should be black. It sounds like whatever pigments were mixed weren't mixed in the right amount.



Mixed in equal proportions, opposite colors produce black. When mixed in non-equal proportions, the color will depend greatly on the specific amount of each pigment.
I wish that would happen but it doesn't. I am a painter with extensive experience in mixing colors. No matter how much you try to balance it, Yellow + Cobalt Blue = Green, not Black/Grey/White. If you add any less Yellow, it becomes distinctly Blue; if you add any less Blue, it becomes distinctly Yellow. And if you add them in just the right amount, it becomes clearly Green.

Same result happens with pretty much every other color. Orange + "Sky Blue" (between Cyan and Blue) = some sort of dark Brown color. Other colors result in some purplish brownish color which is the midpoint.

If you deviate in any way from this mid point, the color starts becoming distinctly one of the two colors that you try to mix to get to the neutral color. Add any more or less to the Sky Blue or Orange? It becomes Sky Blue.... or it becomes Orange, not a brown color that is close to the neutral color that you want. In subtractive color mixing it's impossible to get a pure neutral grey or black or white by mixing 2 colors (pigments).

In order to get neutral grey, you need to add Red or Magenta to the Yellow and Cobalt Blue. Because in subtractive color mixing, Red is opposite of Green, and Blue + Yellow = Green.

The opposites in subtractive color mixing are:

Red = - Green
(Sky) Blue = -Orange
Yellow = -Purple

So the question is... the equations do not support this kind of result. Why is the practical outcome of color mixing different from the theoretical equations that govern color? Something else is going on here that we do not fully understand.
 
  • #16
Drakkith
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I wish that would happen but it doesn't. I am a painter with extensive experience in mixing colors. No matter how much you try to balance it, Yellow + Cobalt Blue = Green, not Black/Grey/White.

According to wikipedia, the RGB values for cobalt blue are (0, 71, 171), meaning that it isn't pure blue. Assuming your paint pigments are at least somewhat accurate to this RGB value, then mixing yellow and cobalt blue results in the yellow pigment absorbing the blue and reflecting the green, turning the whole mixture green.

Same result happens with pretty much every other color. Orange + "Sky Blue" (between Cyan and Blue) = some sort of dark Brown color. Other colors result in some purplish brownish color which is the midpoint.

I'd guess that you aren't working with pure pigments/colors, so you probably won't ever get black by mixing them together. They'll always be reflecting some portion(s) of the spectrum more than others unless you get lucky and have a very specific combination of pigments.

So the question is... the equations do not support this kind of result. Why is the practical outcome of color mixing different from the theoretical equations that govern color? Something else is going on here that we do not fully understand.

I'd recommend not generalizing this to "we". If you do something and it doesn't make sense, chances are that we already know why.
 
  • #17
scottdave
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Red Green and Blue are used for light sources (like your monitor). Any ink will absorb the color which is the opposite, but works by a light source shining and reflecting off the paper, as well. Look at any printed piece with a strong magnifying glass, and you will see individual dots (Cyan Magenta Yellow Black) next to each other in patterns, not usually "mixed". There are some percentage differences between batches of inks, making it difficult to always get gray from equal portions of Cyan, Magenta, and Yellow. Black screening is used to make gray.

Different methods of representing colors have varying gamuts. You may want to read this article. https://en.wikipedia.org/wiki/Gamut
 
  • #18
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According to wikipedia, the RGB values for cobalt blue are (0, 71, 171), meaning that it isn't pure blue. Assuming your paint pigments are at least somewhat accurate to this RGB value, then mixing yellow and cobalt blue results in the yellow pigment absorbing the blue and reflecting the green, turning the whole mixture green.



I'd guess that you aren't working with pure pigments/colors, so you probably won't ever get black by mixing them together. They'll always be reflecting some portion(s) of the spectrum more than others unless you get lucky and have a very specific combination of pigments.



I'd recommend not generalizing this to "we". If you do something and it doesn't make sense, chances are that we already know why.

Well, I use pure Cobalt Blue, not a hue or mixture. I know a lot about pigments, how they behave, and I've used lots and lots of different pigments.

Cobalt Blue and Ultramarine Blue are the closest to the Blue on the computer monitor. The result would be the same if you used an imaginary pigment that was pure Blue straight from the computer monitor. With pigments, in subtractive color mixing, Blue + Yellow = muddy/reddish yellowish Green, not neutral grey or black.

You should try it yourself, buy a small tube of pure Cobalt Blue from Golden Artists Colors (one of best acrylic brands out there) and Cadmium Yellow, and mix them together. You definitely will not get a neutral grey or black, as you would imagine.

I sense a bit of hostility or aggressive language from you, which baffles me. I am purely doing this out of curiosity and exploration.

Red Green and Blue are used for light sources (like your monitor). Any ink will absorb the color which is the opposite, but works by a light source shining and reflecting off the paper, as well. Look at any printed piece with a strong magnifying glass, and you will see individual dots (Cyan Magenta Yellow Black) next to each other in patterns, not usually "mixed". There are some percentage differences between batches of inks, making it difficult to always get gray from equal portions of Cyan, Magenta, and Yellow. Black screening is used to make gray.

Different methods of representing colors have varying gamuts. You may want to read this article. https://en.wikipedia.org/wiki/Gamut

I appreciate this bit of information that the CMYK are not truly mixed as with pigments in paints but rather separately and tightly put together with printer, so it will behave different from mixing of pigments in paints.
 
  • #19
sophiecentaur
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I'd guess that you aren't working with pure pigments/colors,
Obtaining a good black requires total subtraction of all light passing through a combination of filters and is more or less impossible to achieve by using thick and thicker paint / ink pigment; there is always some reflection from the surface. Interference filters (as opposed to pigments) can have very 'rectangular' pass band characteristics and also produce very deep stop band levels and I would imaging that the 'black' they can achieve will be pretty good.
Well, I use pure Cobalt Blue, not a hue or mixture. I know a lot about pigments, how they behave, and I've used lots and lots of different pigments.
I am not sure what you mean by "pure cobalt blue". Any pigment that's worth using will have a wide bandwidth in order to let lots of light through / reflect a lot of light. If you use a pigment that will only let through / reflect a narrow spectral range, the image will be too dark to see. Cobalt Blue is far from monochromatic light.
By mixing only three primary pigments you cannot get good matches over the whole gamut. This is why artists and printing companies use specially produced 'Spot Colours', many of which which are bright and familiar. Even a humble ink jet printer uses grey and black to get low luminance colours, rather than just piling on the CM or Y pigment.
Why is the practical outcome of color mixing different from the theoretical equations that govern color?
The simple answer to that question is that the equations you are using are not adequate; they are over-simplified to give people a start at understanding colour on a very basic quantitative level.
The approach that's used by artists is much more based on experience and lore. It can give stunning results but I don't know of any artist who would rely on purely calculated quantities of pigments to produce a wanted colour.
Additive mixing is far easier to get to grips with and, as we all know, produces pretty damn good colour TV pictures - far far better (higher fidelity) than even the very best produced film ever could.
 
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  • #20
sophiecentaur
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I appreciate this bit of information that the CMYK are not truly mixed as with pigments in paints but rather separately and tightly put together with printer, so it will behave different from mixing of pigments in paints.
Yes. This is a sort of mixture of additive and subtractive mixing when the coloured dots are spatially separate from each other. Obtaining good dark colours must be even more dependent on the Black and Grey dots.
 
  • #21
Drakkith
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I sense a bit of hostility or aggressive language from you, which baffles me. I am purely doing this out of curiosity and exploration.

That's certainly not my intention. Unfortunately, when conversing through text, we don't have all the verbal and non-verbal cues with which to judge a person's attitude and whatnot, making it easy to develop a false impression of the other person. So it's important to both try not to assume you know what the other person's attitude is, and to be extra clear when typing up a message so as to avoid ambiguity and to foster a good conversation. I've obviously failed at the latter, so you have my apologies.
 
  • #22
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Look at any printed piece with a strong magnifying glass, and you will see individual dots (Cyan Magenta Yellow Black) next to each other in patterns, not usually "mixed".
This baffles me. I was taught that printing processes used the color substraction method. I was also taught that the physical explanation of how substraction had to do with how the light behaves upon encountering different colored crystals suspended in a transparent medium, which robs the light of some colors. If the individual dots do not mix, how is it different from the RGB phosphors in a monitor. They are both light coming from a tiny area! Can it be that the printing of colors is not based on substraction after all?
 
  • #23
pinball1970
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This baffles me. I was taught that printing processes used the color substraction method. I was also taught that the physical explanation of how substraction had to do with how the light behaves upon encountering different colored crystals suspended in a transparent medium, which robs the light of some colors. If the individual dots do not mix, how is it different from the RGB phosphors in a monitor. They are both light coming from a tiny area! Can it be that the printing of colors is not based on substraction after all?
In terms of printing (or dyeing) The colour is reflectance (visible spectrum) from the surface that is coloured stimulating the cones in your retina.

If you had a piece of grey cloth for instance with a good enough microscope (in theory- not possible) and could see the individual dye molecules on the fabric you would most probably see Yellow Red and Blue

All sat at the individual bonding sites, same with pigments (CMYK) but larger molecules that are “sat” ( binded not bonded)

The reflectance from each molecule will stimulate your retina and your brain with “interpret” that data as one colour since you cannot resolve the induvial areas as they are too small.

Scale up to a photo graph or print and look at orange for instance, there is no one “orange” colour usually, it is a combination of yellow and red dots next to each other, your eye cannot resolve the dots it interprets both reflectance sets together.

I am not as au fait with monitors but I assume light from the monitor does the same thing in terms of your retina, only this is not reflected wavelengths it is emitted light.

I assume this is a similar principle, each pixel has its colour weighting in RGB adding and subtracting to produce the required Hue/brightness/Depth
 
  • #24
sophiecentaur
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Can it be that the printing of colors is not based on substraction after all?
Perhaps Cyan consists of two colors (blue and green) and one color (cyan) at the same time?
You have two common misconceptions here. Colour is not a 'thing' that can be measured. All you can do is to work out a system of mixing other coloured lights that match (in your brain) the sensation that's produced by another colour.

The tristimulus theory gives us a heuristic method for reproducing colours 'accurately enough'. There are dozens of reasonably good web pages that will give a fairly good way in to colour vision theory and colour reproduction. This one is an example.

The three output levels from our colour sensors are processed together by the brain and, by an iterative method, several decades ago, the equations were developed and the coefficients were chosen to represent a representative sample of individuals' colour vision.
This baffles me. I was taught that printing processes used the color substraction method. I was also taught that the physical explanation of how substraction had to do with how the light behaves upon encountering different colored crystals suspended in a transparent medium, which robs the light of some colors. If the individual dots do not mix, how is it different from the RGB phosphors in a monitor. They are both light coming from a tiny area! Can it be that the printing of colors is not based on substraction after all?
This is a very good point and goes to demonstrate just what a fudge, printing colours turns out to be!! If you are seeing individual dots of different colours then, as you say, there's got to be some additive mixing going on. But are those dots the same colours as the inks in the printer? The problem with simple subtractive mixing is that you need to subtract a lot of light if you want to produce a saturated Green, say by mixing Yellow and Blue. In colour film, you have the problem of 'bright' saturated colours tending to be lower luminance than corresponds to the scene. Colour TV will mix two full-on primary R and G and get a Yellow with all the light that's available. I think colour printers deal with that by mixing coloured dots (made by mixing inks) with different colours. That's additive and subtractive mixing at the same time. Posh printers have multiple inks so that they can start off with primaries, other than the simple CMY inks.

You must remember how poor the old colour prints you used to get from your processed films. Pretty sometimes but not accurate reproduction of the colours.
 
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