Converting 3D graphics primitives to 3D image data

Question

My goal is to produce a 3D dataset, from a scene of simple transformed shapes. The idea is that this data serves as a ground truth for some tomographic reconstruction problems. (For this, it is especially nice that with Mathematica, Ellipsoid, Cuboid and other 3D primitives remain algebraic up to the point of discretization.)

Ideally, I would like to partition the 3D space into voxels (small cuboids), and compute per voxel how much of it is filled by a shape. This does not have to be very precise, I'm happy already if the value is $1$ if it contains something and $0$ otherwise.

However, I do not manage to get the shapes into a discrete dataset of values. So far, I tried something simple:

B = Ball[{10,10,10}, 2];
reg = TransformedRegion[B, ShearingTransform[Pi/5, {1, 0, 0}, {0, 0, 1}]]
DiscretizeGraphics[Graphics3D[reg]];

which produces a triangular mesh (instead of a uniform) for an ellipsoid (sheared ball).

For 2D problems I managed to get grid data from converting the Graphics to grayscale, and subsequently to ImageData. In 3D that doesn't work.

It seems like a relatively simple thing to do, but after a few hours of bugging the documentation, I still did not find a solution. All suggestions are much appreciated.

Answer 1

As I noted in the comments, a judicious combination of RegionImage[] and ImageMesh[] will yield suitable voxelizations of regions. Applied to the OP's example:

reg = TransformedRegion[Ball[{10, 10, 10}, 2], 
                        ShearingTransform[π/5, {1, 0, 0}, {0, 0, 1}]];

ImageMesh[RegionImage[DiscretizeRegion[reg]]]