Do you know why the gamut of icc profiles (for APN) defined in the XYZ space are larger ones defined in space CIE Lab?
I found that with several profilers without understanding even if the equations give the possibility to move from one space to another.
The difference between the 2 spaces being that CIE Lab is perceptual intent rendering.
The ICC gamut for a camera does not really say much about what real colors the camera actually can capture. A camera is an input device, and you can't see the gamut the same way as for an output device such as a printer or a display.
A ICC Lab LUT will take raw RGB inputs and convert to Lab output, that is 3 values in 3 values out -- a 3D LUT. The LUT needs to cover all combinations of input values (with a limited resolution). As a camera have overlapping color filters it's impossible to get an input signal with say 100% green an 0% red and blue, but that still needs to have some output value in the LUT.
Actually this has been a very hard problem for when designing DCamProf, say 30% of all LUT values are invalid colors and since the calibration makes a model based on real colors and interpolate the rest, the impossible input value combinations get thrown out to crazy positions way outside the human locus, negative lightness and such things. First I just clipped those values, but in practice you then get an "unstable" ICC profile which can lead to strange color effects when you apply an extreme white balance (which may make some of those "impossible" input combinations arise). So I had to fake the impossible values by a special type of interpolation that makes sure that you get sane color output also for insane inputs. It seems like most other profile designers does the same thing in one way or another.
Try pulling the white balance slider to the lowest temperature possible (that will make the whole image blue) and see how the ICC profile behaves. If you have a profile that haven't cared to interpolate outputs the "impossible" inputs, you will probably see colors get clipped to black and possibly get thrown into totally different colors like green, magenta or yellow rather than staying blue.
A LUT can also have XYZ output rather than Lab, but it's not as common so I assume when you refer to XYZ you think about matrix profiles. In this case there's just a simple matrix multiplication that makes the colors, and the matrix has been optimized to produce sane colors in the range it was profiled for, typically a CC24 or similar. It means that for extreme input values you will get crazy output values. The corners of the "gamut triangle" you see in an ICC viewer are 100% green, 100% red and 100% blue, and you can't really reach those points with real inputs, and even if you could the matrix will map those to crazy positions for sure as it has been optimized to make normal colors correct, and that will cause highly saturated colors to get to bad positions.
So in other words, the gamut you see in a camera profile doesn't say anything about the camera's ability to capture colors.
With DCamProf I made a color separation analysis function that can give you some insights into how well the camera can separate colors:
http://www.ludd.ltu.se/~torger/dcamprof.html#ssf_csepIt's common to say that "a camera doesn't have a gamut", and sure it does not have a gamut in the same well-defined way as a printer+paper, but it does have limitations concerning color separation. There is no standardized way to define those limits though, and unfortunately you can't get any of that information from the ICC profile.
Of course the ICC profile won't output values outside it's gamut, so you know what values you can't get, but you cannot see which values that represent those "impossible" combinations and thus are meaningless parts of the gamut.