How to convert Graphics3D object into an Image3D object?

Question

How does one convert a Graphics3D object into an Image3D object? E.g., start with Plot3D[x^2 - y^2, {x, -1, 1}, {y, -1, 1}].

Answer 1

If you already have a Graphics3D object, then you can recreate an Image3D object by stacking slices of your graphics along an axis. Here's an example. We start with your object:

obj = Plot3D[x^2 - y^2, {x, -1, 1}, {y, -1, 1}]

Using the following rudimentary "slice" function, we can generate slices of the function at a given value of $x$:

slice[obj_, x_, dx_] := Show[obj, ViewPoint -> {∞, 0, 0}, 
    PlotRange -> {{x, x + dx}, All, All}, Axes -> False, Boxed -> False]

slice[obj, 0, 0.01]

Now generate such slices for all $x$, rasterize and grab the ImageData and stack the frames:

frames = Table[ImageData@Thinning@ColorNegate@ColorConvert[#, "Grayscale"] &@
    Rasterize@slice[obj, x, 0.05], {x, -1, 1, 0.01}];

Image3D[frames]

As you can see, the reconstruction is not perfect, and this arises from having to artificially sample the Graphics3D object by manipulating the plot ranges. Depending on how quickly the function changes within the chosen dx, the reconstruction could get worse/better. Note that you also need to choose the sampling such that the aspect ratio is maintained (I have only eyeballed it).

A much better reconstruction can be obtained either by generating frames using Plot (you probably can't avoid the Moiré patterns):

frames2 = 
  Table[ImageData@Thinning@ColorNegate@ColorConvert[#, "Grayscale"] &@
    Rasterize@
     Plot[x^2 - y^2, {x, -1, 1}, PlotRange -> {-1.5, 1.5}, 
      Axes -> False, Frame -> False], {y, -1, 1, 0.01}];

Image3D[frames2]

or by directly obtaining the samples as Kuba showed.

Answer 2

You could create a region using DiscretizeGraphics and find points within a certain distance of the surface using RegionDistance

g = Normal @ Plot3D[x^2 - y^2, {x, -1, 1}, {y, -1, 1}];

f = RegionDistance @ DiscretizeGraphics @ g;

data = Array[f[{##}] &, {60, 60, 60}, {-1.1, 1.1}];

Image3D[Clip[data, {0.05, 0.05}, {1, 0}]]

Answer 3

Here's something more fun than practical.

We can simulate an MRI / CT scanner by reconstructing from projected images.

g = AnatomyPlot3D[Entity["AnatomicalStructure", "LeftFemur"], PlotTheme -> "XRay"]

Note that it's important to have some sort of transparency in the objects being 'scanned'. This will better simulate an x-ray.

Normally a CT will only perform a half rotation, but here we will combine 2 CTs by taking a full rotation. This will give a higher quality result. Here are the simulated x-rays:

projectGraphic[g_, α_] := Show[g, ViewPoint -> {Cos[α], Sin[α], 0}, 
  ViewProjection -> "Orthographic", SphericalRegion -> True, ViewAngle -> 1.6]

fcnt = 64;
rsz = 180;

projections = Monitor[
  Table[
    Rasterize[projectGraphic[g, α], RasterSize -> rsz, ColorSpace -> "Grayscale"], 
    {α, 0, 2π - π/fcnt, π/fcnt}
  ], 
  ProgressIndicator[α, {0, 2π}]
];

ListAnimate[projections]

Now we can create the slices:

radons1 = Image3DSlices[ImageRotate[Image3D[projections[[1 ;; fcnt]]], {π/2, {0, -1, 0}}], All, 2];
slices1 = InverseRadon /@ radons1;

radons2 = Image3DSlices[ImageRotate[Image3D[projections[[fcnt+1 ;; -1]]], {π/2, {0, -1, 0}}], All, 2];
slices2 = InverseRadon /@ radons2;

Reconstruct the orignal object by combining both CT scans:

recon = ImageAdjust @ ImageMultiply[
  Image3D[slices1], 
  ImageRotate[Image3D[slices2], π]
];

Image3D[RidgeFilter[recon], BoxRatios -> {1, 1, 1.7}] // ImageAdjust

Answer 4

As of version 11.2 there's RegionImage.

g = Plot3D[x^2 - y^2, {x, -1, 1}, {y, -1, 1}];

RegionImage[DiscretizeGraphics[g]]