Compression of Voxelized Vector Field Data by Boxes is Hard

Voxelized vector field data consists of a vector field over a high dimensional lattice. The lattice consists of integer coordinates called voxels. The voxelized vector field assigns a vector at each voxel. This data type encompasses images, tensors, and voxel data. Assume there is a nice energy function on the vector field. We consider the problem of lossy compression of voxelized vector field data in Shannon's rate-distortion framework. This means the data is compressed and then decompressed up to an error bound on the energy distortion at each voxel. Our first result is that under general conditions, lossy compression of voxelized vector fields is undecidable to compute. This is caused by having an infinite number of Euclidean vectors. We formulate this problem instead in terms of clustering the finite number of indices of a voxelized vector field by boxes. We call this problem the $(k,D)$-RectLossyVVFCompression problem. We show four main results about the $(k,D)$-RectLossyVVFCompression problem. The first is that it is decidable. The second is that decompression for this problem is polynomial time tractable. This means that the only obstruction to a tractable solution of the $(k,D)$-RectLossyVVFCompression problem lies in the compression stage. This is shown by the two hardness results about the compression stage. We show that the compression stage is NP-Hard to compute exactly and that it is even APX-Hard to approximate for $k,D\geq 2$. Assuming $P\neq NP$, this shows that when $k,D \geq 2$ there can be no exact polynomial time algorithm nor can there even be a PTAS approximation algorithm for the $(k,D)$-RectLossyVVFCompression problem.