Well I am thoroughly confused by arfa brane's stance. He starts by asserting that "colour space" is an Hilbert space, which by definition has a metric (distance and angles are defined), then he tells us it is an affine space, which by definition has no metric, now he is asserting that, after all, you do have fixed distances between "colour vectors" (i.e. a metric) arfa, what is it to be? Do you even understand the links you are quoting?
I think I do understand those links, yes, thanks for the concern. I think it might be possible to put the apparent confusion to rest, but by working backwards from the Euclidean space, specifically the CIE diagram I posted. Please Register or Log in to view the hidden image! But let's include the space occupied by the three different types of cone cell, in the average human retina. It represents a choice (made by evolutionary pressure) of three distinct wavelengths. I can think of an analogy with sound, if we had three types of sound receptor each with a fixed "pure" frequency response (a resonance peak), we could embed human (response to) acoustics in a vector space with three dimensions. The CIE diagram is tuned to the three different retinal responses. That there are three types of cone cell is well-understood; how they transform light signals into neural signals isn't well-understood. But we still have a response model. So the affine color space becomes Euclidean because there are well-defined distances between the response curves of the three cone cell-types. Without color pigments and cone cells that respond to (resonate at) particular frequencies, there is still a color space; this is the affine space referred to by that quote from a recognised journal.
The straight line along the bottom of the CIE diagram is a 1-dimensional subspace of the spectral cone (not to be confused with a retinal cell) at a known, constant intensity, hence the diagram is a 2d subspace of the cone over n distinct colors. In affine geometry, two points are related by a function called affine composition; if the points are (a,b), then a pair of real numbers can be chosen, say z,w, such that z + w = 1, and the relation za "plus" wb holds. Along the lower line between the endpoints of the spectral line in the CIE diagram, this relation holds when z + w = 1 is taken to mean 1 is a constant intensity, and when the cone isn't embedded in the xy system. So this lower line of purples, when you don't have Euclidean distances or angles (you have a pair of endpoints and affine composition) is an affine subspace. Shouldn't be hard to prove that every pair of points is an affine subspace.
That's every pair of points on the spectral line, whose 'coordinates' are wavelengths of visible light. When this line is just the real line and the wavelengths are just numbers, the line is straight. But the real line is an affine space if you forget or omit an origin. Suppose in a graph that the vertices are connected by edges which are real lines. The edges need not be straight lines or have a definite length, they only connect vertices (points). So graphs define this notion of an affine distance, since even when edges are straight lines, the distance between points need not be a metric distance. In the CIE chromaticity diagram, there is a straight line between the two limits of the spectrum, which are violet and red, this line is called the line of purples because it's the set of all affine combinations of violet and red. However the endpoints cannot be affine combinations of two or more other points (colors), so they are the affine basis of the line of purples. Since every point on the spectral line is an element of the affine basis, every pair of points on this locus is an affine subspace.
To wavelengths. The subject of the thread is not wavelengths, but color. No, it isn't. It is a 2d subspace of the spectral cone over a subdivided range of wavelengths. There are n subdivisions, and the range of each is defined in nanometers - a standard measure of wavelength. I already posted a link to an optical illusion featuring exactly that. I have referred you to it several times now. No "standard gamut" of wavelengths is relevant. The CIE diagram is of wavelengths - likewise irrelevant to this question. There is no such region of the CIE diagram, which proves its irrelevance to this particular matter - it cannot, for example, describe or produce the colors visible in that optical illusion. Once again: try posting without reference to wavelengths, as an exercise. Wavelengths are not colors. Proving once again that wavelengths are not colors, which do not have definable fixed distances between them. That is a settled matter now, ok? There is no need to post a single word more about the distances between wavelengths. We can move on. Start possibility: There are no single fixed distances between colors. There is no single fixed order of colors. Why would that surprise me? I am especially unsurprised by your observation that a wavelength is meaningful if there is only one per color in some kind of experimental setup. Clearly one could identify the color with the wavelength and vice versa, for example, and use the two labels interchangeably within that radically simplified experimental setup. Producing a color space from that is not going to be simple or easy, the space produced is unlikely (probably) to be unique. It will not have a defined inner product, for example.
Nice post, iceaura, as always. But you missed this pearl.... Distance, it seems, in this fantasy world, is not a metric i.e. something that can be measured. Moreover it is not at all clear what the meaning is of a "straight line" in affine geometry
Distance along the real line is a fantasy? It isn't clear why the real line is straight even though every point is ordered by value so there are only two directions? This thread is one enormous waste of everyone's time. Here we are arguing about what amounts to whether humans are capable of understanding what color is, or what wavelengths of visible light are, so they can construct a color display . . . I wonder if humans will ever solve this obviously very difficult problem? When do you think it might happen? ps. iceaura appears troubled by the existence of actual papers which define colors (or color response) as either an affine space, or as a Hilbert space. Obviously the authors of such papers haven't considered the problems he keeps reminding everyone here about. Those engineers should just admit they have no real idea what color is. That is what I'm going to do right now. Please ignore everything I've posted, and have a nice life. Oh yeah, and you should all give up trying to understand anything, anything at all. It's a waste of time.
No, it's a 2d subspace with a constant intensity for each distinct color combination. The n subdivisions of the spectral line are entirely arbitrary choices, since n is in fact infinite (the Hilbert space of colors is infinite). Colors have wavelengths, so I really can't understand, at all, why you keep saying colors are not wavelengths. Colors can be combined (try it someday), so wavelengths are also things that can be combined.
The CIE diagram has colors in it, I can see them. It is not a diagram of wavelengths, the wavelengths of the spectral colors are located on that boundary--the spectral line--notice how this line isn't straight (??), and the distances between wavelengths is not constant (what the fuck?). What's wrong with these people?
So what? Does the CIE diagram have colors in it that can be combined so they do produce your optical illusion? If not, why not?
There are fixed distances between the responses to light, of the three types of cone cell in human color vision. Simply put, this is because the different cell types respond to different wavelengths of visible light--i.e. light at particular frequencies. If the responses were at three different frequencies (resp. wavelengths) than the ones humans evolved, the distances would be different for humans. If you build a device that responds to three different frequencies of light, you get to choose which frequencies (wow, huh?).
No, you can use the wavelength of monochromatic light as a reference, a sort-of coordinate. This is not controversial or "radically simplified", it's engineering. No color space is unique, you goddam dick. But you aren't likely to be looking for a job as a display engineer anytime soon, right? I mean, you can't even explain how the CIE diagram appears to have colors in it, can you? Maybe it's an optical illusion?
No. It has wavelengths in it, that can be combined to produce the optical illusion I linked - including the several different colors from one single combination of wavelengths that all humans with normal vision see in that "illusion". So the wavelength distances to that combination will not match the various color distances, and the color distances will vary depending on where in the picture the one single combination of wavelengths is located. Wavelength space is a vector space. It supports a distance function. It is a Hilbert space. We agreed on that pages ago. It is irrelevant. Wavelengths are not colors. So you can't define a distance function over the color space. So it isn't a Hilbert space. Until you learn to quit posting about wavelengths, this is going to go nowhere. So?
Ok. So what you're saying is Wikipedia gets it wrong, with So you can define a distance function over any color space; once you have at least two fixed wavelengths you automatically have a distance between them. Given a single fixed wavelength, how does it output more than one color? Oh right, a wavelength isn't a color (thanks for the helpful hint), but a color with a single wavelength does have a wavelength, right? Why is there a problem with referring to the wavelength of a spectral color (which is a color with a single wavelength)? Why shouldn't engineers do this, what should lecturers say to students about your claim that colors aren't wavelengths? What should they do? How do you tell the difference between two colors with different single wavelengths? According to you, nobody can do this. According to you, it's all an illusion or something. There can't be a three dimensional Euclidean color space, so engineers should stop with all the confusion (according to you). Or perhaps I still have no idea what the hell you think you're talking about. I get that a lot, here. Look, you're probably looking at a color display as you read this. How did humans (with all those pesky color illusions) manage to make it? Was it trial and error (that would be quite an expensive R&D option)?
1) Once again you have overlooked the central issue and reversed the direction of implication. I am tired of repeating this - could you please write it down and post it for me every time you post that muddle? -> several different colors can be produced by one and the same spectral "color" combination (wavelength decomposition and recombination). The optical "illusion" link I posted demonstrates that. -> The distances between colors do not correspond to the distances between points on the spectrum even, much less whatever way you imagine distances between non-spectral colors are measured. The wavelengths are not even in the same order as the colors in many common circumstances. The intransivity of the colors red, yellow, and violet demonstrate that. Again: You don't have two fixed colors, you have two fixed wavelengths - each of which can represent several different colors depending on circumstances, each of which can be produced by many different combinations of wavelengths (plus black and white). Meanwhile: Even for the chosen wavelength combinations you need at least three, plus a way of making black and white. Furthermore: You like Wiki? Ponder: https://en.wikipedia.org/wiki/Visible_spectrum Approximations will of course not fulfill the requirements of a vector space or Hilbert space - so no vector space or Hilbert space involving the wavelength basis engineers use for color screen design and lens processing and so forth is possible for the colors of the perceived world. ? Humans tell the difference between colors by looking at them. I never said humans were unable to perceive differences between colors. All I said was that color space is not a Hilbert space (or even a vector space). I posted the reasons (lacks additivity, does not support an "inner product" or "distance" function, etc). And nothing you post about wavelengths is going to make any difference to that claim or its relevance here because (drum roll) The thread is about colors. Wavelengths are not colors.
Please Register or Log in to view the hidden image! When you look at the CIE diagram above, what are the two colors in the xy system (that you personally, can see)? Why are they at right angles? What do the numbers along the x and y axes of the diagram mean? Why is the spectral line embedded this way, in a Euclidean space, where angles and distances are defined? What's the point? What's the relevance of color displays not being able to reproduce the complete spectrum of perceivable colors? Or for that matter, paint manufacturers?
Notice how the electronics industry likes to use standards. Color spaces are non-unique, so it's nice to have one or two that everyone agrees on, like the one the display you're looking at uses, probably sRGB. I can look at a flat screen display up close, well enough to see the individual LEDs; I can tell when only one of the three LEDs is lit and the other two are not, so I can literally see what the industry would have me believe is a pure spectral color, or primary color. Moreover, the theory behind how it works would have me believe that this color (one of blue, green, or red) is a kind of coordinate, a fixed point in an abstract, quite esoteric, color space. As the following figure of LED output shows, primary colors aren't precisely fixed. Please Register or Log in to view the hidden image! This sRGB color space is a subset of the one in the diagram, it looks like a triangle (triangulation ??) in the Euclidean embedding. In this embedding two primary colors which correspond to the green and red LEDs in my screen, are used like coordinates for the entire CIE colorspace. If you only used the red x coordinate you would recover the straight-line spectrum and see the range of colors across it; because there's a y direction (green), the spectral line gets deformed by the mapping from colors to . . . colors. But one takeaway is that if my display is using the sRGB standard, then it can't actually display the CIE colorspace faithfully. Oh well. That would have to be one of those consumer caveats I guess. Engineers do what they can.
From Wiki: "The spectrum does not contain all the colors that the human eyes and brain can distinguish... purple variations like magenta, for example, are absent because they can only be made from a mix of multiple wavelengths." I had always wondered why magenta appears to be the hue that is most obviously missing from the visible spectrum of sunlight. I also wondered why magenta only became common after RGB color displays became common. Please Register or Log in to view the hidden image! Notice that the fully saturated hues, in order from left to right, are as follows: red, orange, yellow, chartreuse, green, spring green, cyan, azure, blue, violet, magenta, rose, and red again. The visible spectrum of sunlight does not have red at both ends, so it ends around violet. Magenta and rose only arise when red & blue are combined, and red & blue are not actually next to each other on the visible spectrum.
