Is a 2d diagram of a color gamut such as this one.....



basically an overhead view of an RGB cube with the white corner in the center?

No, it is not. A 2D projection of a cube would not give you the smooth shape of your first diagram.

The term given to the object depicted in the first diagram is the spectral locus.

See http://en.wikipedia.org/wiki/CIE_1931_color_space

Wikipedia also says this about Color Gamut Diagrams here




"A typical CRT gamut:
The grayed-out horseshoe shape is the entire range of possible chromaticities. The colored triangle is the gamut available to a typical computer monitor; it does not cover the entire space. The corners of the triangle are the primary colors for this gamut; in the case of a CRT, they depend on the colors of the phosphors of the monitor. At each point, the brightest possible RGB color of that chromaticity is displayed, resulting in the bright Mach band stripes corresponding to the edges of the RGB color cube."

Correct, the spectral locus is the shape and range of perceivable colors for an average viewer.

This is not the same thing as a RGB cube, which is a mathematical abstraction of an additive color space with three primaries.

As indicated above in the diagram you posted, if the primaries are visible (i.e., the vertices of the triangles lie within the horseshoe shape), then there will be physical colors that cannot be reproduced in that RGB space. This is clear, because no matter where you place the three vertices of the triangle within the horseshoe, there will be colors that lie outside the triangle.

Jeremy,
If you want to see a visual description of how 2D color space relates to 3D color space, download ColorThink Pro. Even the free demo version allows you to choose between Lab space, Yxy, and Luv spaces. Then, clicking on the 2D graph or 3D graph buttons will automatically animate the change between the two. It's a great way to get your head around these concepts.

https://www.chromix.com/colorthink/download

Both of these pictures seem to be showing the same thing: A 2d representation of a 3d RGB color space. The edges of the RGB cube even seem to show up in the color gamut diagram. Are they not doing the same thing: Mixing Red, Green, And blue together to show every possible color?

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They are related, but definitely not the same. The spectral locus shows, approximately, the range of colors visible to humans. The cube shows the set of colors that can be generated by three primaries. These are not the same. Pick any set of real primaries. There are colors that are visible to us, that cannot be produced by those primaries, no matter how you add them up.

Furthermore, these shapes are quite different, which should be a straightforward giveaway. It's like comparing a square and a circle. They have different shapes, represent different things, and have different properties.

I didn't know that. Wouldn't that make the spectral locus pretty incomplete when viewed on an RGB monitor or printed with a printer that only uses CMYK?

That is correct. There is no display nor printer that can reproduce such saturated red, green and blue colors as depicted in the CIE (x,y)-chromaticity diagram (AKA spectral locus).

For the RGB cube and 2D x,y-chromaticity diagram, there is a way to transform the cube into spectral locus representation, but the opposite is not true. RGB values can be transformed into CIE-XYZ values using a linear transformation. XYZ values then can be further transformed into x,y coordinates which are the basis of the spectral locus diagram. However, along the way, the Y (luminance) component is lost and only 2D graph projected. Not knowing the luminance values, one cannot get back to XYZ, thus precluding calculation of RGB values. As Eric Chan pointed out, the RGB gamut (and cube) shown in your previous post would not match the x,y-diagram. However if you had a hypothetical RGB color space (gamut) that has primaries of the spectral locus, then there would be a mathematical translation going in direction RGB->XYZ>x,y. Prolem is that while RGB space is linear, x,y-chromaticity diagram is not. That means that same changes in the RGB cube would not correspond to perceived changes in the x,y diagram.

Some more details and pictures are at this webpage. For transformation of sRGB space to XYZ (->y,y), see equation VII-inv.

So the XYZ is a color space made in 1931 by CIE that encloses the visible gamut with three primaries that are actually outside the human spectral locus gamut, meaning that it will never be outdated and there will never be a color outside of it. And this makes it the standard reference space for viewing other color spaces.

Is that right?

The digital cinema initiative has incorporated that thinking into canonizing XYZ space. Otherwise, actually that "out of gamut" concept is more applicable when you are ready to display, and at that time you need to consider the gamut of colors that the display can accommodate. Any color, including out of gamut, and those outside of spectral locus may be expressed in any standard RGB model where the primaries are inside the spectral locus -- only that some of the values for the intensities of these primaries will be negative, and/or in normalized values greater than +1. And, when it comes to display, or for whatever reason, these negative or greater than +1 values may be dropped or mapped into the target gamut under consideration. Even in the XYZ space, since the primaries are physically non-realizable, when it comes to display then some sort of gamut conversion stage has to be employed to map the colors outside the gamut of the display device into colors that it can display, therefore some reduction happens even during XYZ->display gamut.

Think about it: the spectral locus is outside the CIE RGB primaries, i.e., the triangle formed by the location of these RGB primaries. So that means negative numbers for at least one of the CIE RGB intensities. But, CIE still measured those RGB numbers and then converted them to XYZ, since XYZ is always positive for the spectral locus. (CIE did that by adding some of the RGB primaries to the test color to make the overall system positive.)

I agree with your summary. XYZ system is quite nonintuitive, so other color systems were created based on it (Lab, xyY, RGB). Think of the XYZ color space as of computational rather than visual system. And yes, it is the underlying system of color science, digital photography, imaging ,..

How is this

converted into this?

The following tools offer even more transformations and mappings: Couleur.org (free download, install).

Your second graph (the wiggly one) is actually the monochromatic stimuli of constant radiance for all wavelengths shown in the first graph. When this wiggly curve is projected on the plane x+y+z=1, you get that "standard" spectral locus diagram shown in the figure in the first message (post # 1) of this thread, that I copy below (the figure below is a further projection on the z-axis to show just x-y diagram, instead of an x-y-z diagram):



I.e., from the origin draw a straight line connecting any point on the wiggly curve and extend it to the plane x+y+z=1. Where this line meets the plane draw a dot; do that for all points on the wiggly curve and you have now a bunch of dots on the plane. Connect the dots on the plane and you get your standard spectral locus diagram shown in post #1.

I'm sorry but this stuff is really getting to be over my head.

Do you think someone here can give me an easy to understand, very basic idea on how these three diagrams are converted into each other?


This

Becomes This

Which Becomes This

Trying again. Pick any single color (monochromatic) from your first diagram and note its radiant power (used later). Then measure the amount of X, Y, and Z primaries, which are calibrated using a certain process, that you need to match that color, and this gives you 3 numbers. Plot the 3 numbers and you get a point on the wiggly curve in your second diagram. Now divide each of the X,Y,Z by their sum (X+Y+Z), and you get another set of 3 numbers. Plot them again and that gives you a point on spectral locus on the third diagram. Now go back to the first diagram and pick another color at the same radiant power as the first color picked and repeat the process. Keep on doing until you have done for all/enough colors in the first diagram. Now you have a whole bunch of points on the second and third diagrams and you just join them together to get the curves shown.

How do I find a color's radiant power?

Firstly, your second curve doesn't seem fully okay to me. I have a slight problem trying to reconcile it with one of the primaries. But that is fine, as the process I outlined will result in a wiggly nature of the curve and after that "beauty pass" of dividing by X+Y+Z your third diagram will emerge.

As far as measuring the radiant power you will need certain specialized hardware for that and even for setting up an experiment for matching the colors.

I think I pretty much understand everything except for this:

"Now divide each of the X,Y,Z by their sum (X+Y+Z), and you get another set of 3 numbers. Plot them again and that gives you a point on spectral locus on the third diagram."

What exactly are the second set of 3 numbers? Do you ignore the Z number in order to plot this set into the 2d spectral locus?

Yep, sorry, you are right, to get the 3rd diagram you ignore the Z number and get the 2D spectral locus. If you don't want to ignore the Z number, then that is fine and you would get a 3D plot of spectral locus instead of 2D. The 3D spectral locus shape will be similar to 2D and won't look like the wiggly plot of the 2nd diagram.

Do the X, Y and Z primaries used to form the 3d spectral locus correspond to the 3 cones in the human eye?

No, they are different, but people have tried to find a transformation relating the two responses to varying degree of success.

Then what are the 3 primaries for the spectral locus? (Not the CIE XYZ space)

In theory any 3 primaries can be used as long as they are not in the same direction and all three don't lie in a plane. Recall, we want to define any color with 3 numbers, i.e. any color would be represented by a point in 3D, therefore we just need 3 vectors in a 3D space to represent any point. The primaries are these vectors. We don't want to point them in the same direction or lie in the same plane otherwise we would have effectively a 2D space (a plane), or a line (1D) embedded in a 3D space as we have suppressed one or two degrees of freedom, respectively. So if you want to use a different set of primaries it is just a reorientation of the original set of primaries, i.e., why we multiply primaries by a matrix to transform one set of primaries to another.

In practise it is advisable to have them spaced farther apart so that they cover as many colors that can be physically displayable (i.e., those with positive values). Primaries in the regions of Red, green and blue fulfill that criteria.

And, as I mentioned before, if we want to measure the values for the spectral locus, then for any set of "real" primaries (i.e., those we can display later) it would mean at least one primary has negative value. But that is fine, we are not going to display it at this stage, we are just measuring it. This is how CIE measured these numbers, including negative numbers, using RGB. But CIE did not want to work in negative numbers thinking that people would make mistakes with negative numbers. Therefore a transformation was found to convert RGB in such a way that resulted in all positive numbers, i.e., RGB->XYZ.

I though that because of this picture.



But now this picture is just confusing me.

Please don't let the graph confuse you. I have to ascertain how close are LMS primaries to XYZ, however, typically, the shape of the spectral locus in that "beauty pass" plot (that horse-shoe like spectral locus) is going to look similar for various different primaries that are close by. It would seem to get stretched, elongated, etc., but overall similar shape.

BTW, which software did you use to generate that "wiggly" spectral locus shape?

I found it on wikipedia. I am still having trouble understanding how the "wiggly" 3d spectral locus shape is made. Heres another one.



"Spectral locus in XYZ and xy (demonstration of chromaticity derivation)"

Have a look at www.handprint.com, especially this section. The site goes into quite a lot of detail about chromaticity diagrams, etc.

Cliff

I actually found that site yesterday. It's great.

Here is another diagram that seems to show that the 3d spectral locus dimensions correspond to the 3 human eye cones.

Nevermind, that was a dumb question.

I'm confused about the green and purple lines in this spectrum locus diagram. I'm reading that the purple one is the V luminosity function which is L plus M and that the green line is the contrast between L and M which is L minus M. I don't really get what this means or what the lines are representing.

Does the green line represent the hues that appear brightest to the human eye in photopic vision?

The colors that extend the farthest in the direction of the green line appear more luminous.

The blue line to 525nm is along the crease where the spectral locus seems to bend, marking a division of color perception. Quoting Handprint:



Cliff

Thanks. I think I understand now.

Do the colors that extend the farthest in the direction of the green line appear more luminous than the white point?

Yes, quoting further from Handprint:



I'm not sure, but I think this might be related to the Helmholtz-Kohlrausch effect, where in general, more chromatic colors will appear brighter than a white of the same luminance.

Cliff

Isn't brightness usually measured by the amount if white there mixed into the hue?

I think you're describing purity or chroma; the amount of white mixed with a spectral hue is a measure of purity.

Brightness is an absolute perception of a color's intensity.

Maybe a good book about color will help. "The Reproduction of Colour" by Hunt is an excellent resource, expensive but can probably be found in a local library.

Cliff

Im confused because this quote from handprint:



doesn't seem to match up with this HSV color model. In the HSV model, the value or brightness goes up when the color starts containing more white and less black. So I've always thought of color as a variety of hues that have a brightness determined by the shade of gray from black to white present in the color, and a saturation level determined by the amount of that shade of gray compared to the amount of the actual hue present in the color. Is this view wrong?

You're mixing pieces of models from computer graphics with models of color perception.

The HSV model is a device-dependent RGB model from computer graphics that has little to do with perception-based color models like CIELAB, CIELUV, or color appearance models like CIECAM02. It's not clear to me what the "V" or value in HSV is supposed to correspond to: lightness, brightness, Munsell Value, or (probably) something else.

CIELAB in cylindrical coordinates, which is Lightness, Chroma, and hue, is similar to your HSV model. Note that the white-gray-black axis in CIELAB is Lightness, not Brightness. There is a difference between Lightness and Brightness, (and saturation and Chroma) and you should be careful to differentiate the two while delving into Handprint.

Cliff

Why are there extratraspectral hues mixed by Red and Violet between 620 nm and 445 nm?

Those are the purple hues that we can see, but don't exist anywhere as a monochromatic color on the spectrum/spectral locus. Those hues can only be produced by mixing red and violet.

So is the reason they're different than other mixed colors because they make up an edge that is past monochromatic colors in the cone excitation space? Or is this completely wrong?

I thought I was beginning to understand the spectral locus as the monochromatic wavelengths curving around in a 3d space but the extraspectral colors are confusing me because they look like they're outside the spectral locus.

Pretty much. Take magenta as an example of an extraspectral color. There is no single pure wavelength of light that looks magenta - there is no magenta on the spectral locus. You can only get that color by mixing at least two different wavelengths from opposite ends of the spectrum, such as red and blue. The extraspectral colors are different because you can't match them with a single pure spectral color (or pure spectral color plus an amount of white light), like you can with the "non-extraspectral" ones.

So all the other colors that are not monochromatic are mixtures of monochramitic colors and grayscale colors, except for the extraspectral colors?

Yes, that is the concept of "Dominant Wavelength" .

So when you take extraspectral colors into consideration, this doesn't show every color we can see right?



But this does?