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 then decompressed up to a bound on the distortion of the energy at each voxel. We formulate this in terms of compressing a single voxelized vector field by a collection of box summary pairs. We call this problem the $(k,D)$-RectLossyVFCompression} problem. We show three main results about this problem. The first is that decompression for this problem is polynomial time tractable. This means that the only obstruction to a tractable solution of the $(k,D)$-RectLossyVFCompression 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 $(k,D)$-RectLossyVFCompression problem.

翻译：体素化矢量场数据包含一个定义在高维格点上的矢量场。该格点由称为体素的整数坐标构成。体素化矢量场为每个体素分配一个矢量。此类数据涵盖图像、张量及体素数据。假设矢量场上存在一个良好的能量函数。我们在香农率失真框架下考虑体素化矢量场数据的有损压缩问题。这意味着数据经过压缩后需解压缩，且每个体素上的能量失真需控制在限定范围内。我们通过使用一组盒摘要对来压缩单个体素化矢量场的形式化该问题，并将其称为$(k,D)$-RectLossyVFCompression问题。我们展示了关于该问题的三个主要结果。首先，该问题的解压缩阶段具有多项式时间可解性。这表明$(k,D)$-RectLossyVFCompression问题可解性的唯一障碍在于压缩阶段。这一点通过关于压缩阶段的两个硬度结果得以证明。我们证明压缩阶段在精确计算上是NP-Hard问题，且当$k,D\geq 2$时，甚至不存在APX-Hard意义上的近似解。在假设$P\neq NP$的前提下，这证明当$k,D \geq 2$时，$(k,D)$-RectLossyVFCompression问题既不存在精确的多项式时间算法，也不存在PTAS近似算法。