How to modify tick's range in Image3D

Question

Suppose we have an array of color data

colorArray = ParallelTable[With[{v = RandomReal[]},
Append[List @@ ColorData["LightTemperatureMap"][v], If[v >= .8, .5, 0]]
], {z, 0, .1, .005}, {y, 0, 1, .1}, {x, 0, 10, 2}];

And we use Image3D to visualize them

is = 350;
Image3D[
colorArray, "Real", ColorSpace -> "RGB",
ViewPoint -> {1.3, -2.4, 2}, SphericalRegion -> True,
Boxed -> True, BoxRatios -> {1, 1, 1},
Axes -> True, AxesLabel -> {"X", "Y", "Z"},
ImageSize -> is, ImageResolution -> 3*is]

The problem is that the ticks are the relative position of each RGBColor element in the array, not the actual range of x, y, and z ({z, 0, .1, .005}, {y, 0, 1, .1}, {x, 0, 10, 2}). My question is how can we change the ticks to show the actual range of x, y, and z? A direct implement seems impossible because in colorArray no data about the original x, y, and z values are stored (actually we cannot store them in colorArray due to the requirement of Image3D).

Answer 1

Edit Thanks to matheorem for simplifying this a good deal.

Image3D is relatively new and doesn't have all the options that other Graphics3D objects have, but we can work with it because it can be used with Show like any other 3D graphics object. The strategy here is to only give those options to Image3D which are unique to it, while giving all other options to Show.

Image3D produces a plot where the data ranges for the three dimensions are equal to the dimensions of the input array, similar to the 2D functions ArrayPlot and MatrixPlot. However, unlike those functions we can't supply the option DataRange to Image3D. So we need to manually specify a scaling function for the tickmarks. That is, when you give the Ticks manually, you can specify the position and the label for each tick mark, i.e. in a list like $\{\{x_1, label_1\},\{x_2, label_2\},...\}$. This is tedious to do by hand, so we can do it functionally.

There are two ways I can think of to do this. The first is by using the CustomTicks package and the second is by sending a function to Ticks.

I like to use the CustomTicks package, available here: http://library.wolfram.com/infocenter/Demos/5599/ It makes it really easy to rescale the tickmark values. Just download the CustomTicks.m file and place it somewhere in your $Path.

And this isn't really clear from this example, because the data is random, but in order for the plot ranges to match up it is necessary to reverse the first two levels of the 3D array.

Needs["CustomTicks`"]

colorArray = 
  ParallelTable[
   With[{v = RandomReal[]}, 
    Append[List @@ ColorData["LightTemperatureMap"][v], 
     If[v >= .8, .5, 0]]], {z, 0, .1, .005}, {y, 0, 1, .1}, {x, 0, 10,
     2}];
{xmax, ymax, zmax} = {10, 1, .1};
{zdim, ydim, xdim} = Dimensions[colorArray][[;; 3]];


Show[Image3D[Reverse[colorArray,{1,2}], "Real", ColorSpace -> "RGB"],
 ViewPoint -> {1.3, -2.4, 2},
 SphericalRegion -> True,
 Boxed -> True,
 BoxRatios -> {1, 1, 1},
 Axes -> True,
 AxesLabel -> {"X", "Y", "Z"},
 ImageSize -> 350,
 ImageResolution -> 750,
 Ticks -> {
   LinTicks[0, 10, TickPostTransformation -> (# xdim/xmax &)], 
   LinTicks[0, 1, TickPostTransformation -> (# ydim/ymax &)], 
   LinTicks[0, .1, TickPostTransformation -> (# zdim/zmax &)]}]

A somewhat less elegant solution that doesn't rely on an external package is

Show[Image3D[Reverse[colorArray,{1,2}], "Real", ColorSpace -> "RGB"],
 ViewPoint -> {1.3, -2.4, 2},
 SphericalRegion -> True,
 Boxed -> True,
 BoxRatios -> {1, 1, 1},
 Axes -> True,
 AxesLabel -> {"X", "Y", "Z"},
 ImageSize -> 350,
 ImageResolution -> 750,
 Ticks -> {
   Function[{min, max}, {# xdim/xmax, NumberForm[N@#, {3, 2}] } & /@ 
     FindDivisions[Round[{min, max}] xmax/xdim, 5]], 
   Function[{min, max}, {# ydim/ymax, NumberForm[N@#, {3, 2}] } & /@ 
     FindDivisions[Round[{min, max}] ymax/ydim, 5]], 
   Function[{min, max}, {# zdim/zmax, NumberForm[N@#, {3, 2}] } & /@ 
     FindDivisions[Round[{min, max}] zmax/zdim, 5]]
   }]