# Point relaxation inside of Chromaticity Plot 3D

I'm trying out the new ChromaticityPlot3D function in Mathematica and noticed I can extract the mesh of the output graph using //InputForm.

I'd like to find the most even/relaxed distribution of points inside a subset luminosity of a ChromacityPlot3D in the CMYK (or other device) gamut projected in LAB color space, since it's perceptually uniform. My ultimate goal is to find the x most different print colors inside a luminosity range.

Here's the subset mesh I'm trying to relax points inside:

ChromaticityPlot3D["CMYK", "LAB", PlotRange -> {{.65, .82}, {-1, 1}, {-1, 1}}]

Successive application VoronoiMesh seems like the best way to solve this. I see another question applying this function iteratively, but I'm having trouble applying VoronoiMesh to the InputForm of ChromacityPlot3D.

• Well, VoronoiMesh[] doesn't (yet) work for 3D, so things are going to be a bit more complicated. – J. M. will be back soon Feb 1 '17 at 19:14
• CIELab is not perceptually uniform. – R Hall Feb 14 '17 at 14:17
• Isn't that the whole reason the decadly Lab revisions exist, to be more and more perceptually uniform? – Lucent Feb 16 '17 at 16:35
• @Lucent CIELAb was created "to be" perceptually uniform, but alas it was found that in blues it is not. See this link. – R Hall Feb 16 '17 at 17:01

I admit this is not the most elegant solution, but here is a function that I wrote which finds a solution of relaxed points in a colourspace (i.e. a constrained 3D vector space).

The function takes an initial point, a ChromaticityPlot3D object, the number of desired points, and the min and max constraints in all three dimensions (including the luminosity). It populates the space with random points and the essentially treats the points as identically charged particles confined within the specific region. So the points iteratively push each other around until they are as spread out as they can be.The function then returns the coordinates of all the points in colourspace.

I do have a clause at the end of the main while loop which helps shake things up (in case the solution gets stuck in a local maximum). It's a Monte Carlo method, so it's designed to be fast(ish), so it won't necessarily give the optimal answer. If you don't like the first answer it gives, change the random seed and run it again.

 RelaxedPoints[givenpoint_, chrmaticityplot3d_, npts_, lims_] :=
Module[{iters = 1000, set, inc, region, points = {}, pnt, dm,
vectors, magnitude, move, olddm = Table[Infinity, npts, npts],
n = 0, m = 2, res, err, tol = 0.075, mean, ids, plt},
set = Flatten[InputForm[chrmaticityplot3d][[1, 1, 1, ;; , 1]], 1];
set = Reap[Do[
inc =
set[[i, 1]] > lims[[1, 1]] && set[[i, 1]] < lims[[1, 2]] &&
set[[i, 2]] > lims[[2, 1]] && set[[i, 2]] < lims[[2, 2]] &&
set[[i, 3]] > lims[[3, 1]] && set[[i, 3]] < lims[[3, 2]];
If[ inc, Sow[set[[i]]]];
, {i, 1, Length[set]}]][[2, 1]];
region = ConvexHullMesh[set];

pnt = givenpoint;
If[! RegionMember[region, givenpoint],
pnt = RegionNearest[region, givenpoint]];
AppendTo[points, pnt];

Do[
pnt = RandomReal[{-1, 1}, 3];
If[! RegionMember[region, pnt], pnt = RegionNearest[region, pnt]];
AppendTo[points, pnt];
, {i, 2, npts}];

dm = DistanceMatrix[points];
err = Total[Abs[dm - olddm], 2];
While[err > tol && n < iters,
vectors =
Table[Normalize[points[[i]] - points[[j]]], {i, 1, npts}, {j, 1,
npts}];
magnitude = (1.0/(dm^2 + IdentityMatrix[npts] + 0.0001))/500.;

move = magnitude*vectors;
ids = Drop[Take[Ordering[#], Round[Log2[npts]*2]], 1] & /@ dm;

Do[
points[[i]] = points[[i]] + Total[move[[i, ids[[i]]]]];
If[! RegionMember[region, points[[i]]],
points[[i]] = RegionNearest[region, points[[i]]]]
, {i, 2, npts}];

If[n == 100*Fibonacci[m],
ids = Ordering[Flatten[Drop[Take[Sort[#], 2], 1] & /@ dm]];
If[ids[] == 1,
points[[ids[]]] = Mean[points];,
points[[ids[]]] = Mean[points];
];
m = m + 1;
];

olddm = dm;
n = n + 1;
dm = DistanceMatrix[points];
err = Total[Abs[dm - olddm], 2];
];
Return[points];
]


Here is an example:

chplt = ChromaticityPlot3D["RGB", "LAB", FillingStyle -> Opacity[0.5]];
pt = {0.7, 0.1, 0.1};
regionlims = {{0.5, 0.7}, {-10, 10}, {-10, 10}};
pts = RelaxedPoints[pt, chplt, 9, regionlims];
Show[{chplt,
Graphics3D[{PointSize[0.07], LABColor[#[], #[], #[]],
Point[#]} & /@ pts], Graphics3D[Point[#] & /@ pts]}] Techniques that I used in to solve this problem include extracting the data set from the Plot object (using InputForm) and creating a Region object to work with (using ConvexHullMesh). Finally, it's not fool-proof so if you use if for something it's not designed for it might break.