Compression of Voxelized Vector Field Data by Boxes is Hard
Simon Zhang

TL;DR
This paper investigates the complexity of lossy compression of voxelized vector field data, proving that while decompression is efficient, the compression process is computationally hard, with NP-hardness and APX-hardness results.
Contribution
The paper introduces the $(k,D)$-RectLossyVVFCompression problem, proving its decidability and polynomial-time decompression, but also establishing NP-hardness and APX-hardness for the compression stage.
Findings
Decompression for the problem is polynomial time tractable.
Lossy compression of voxelized vector fields is undecidable in general.
Exact and approximate compression algorithms are computationally hard to develop.
Abstract
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…
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Taxonomy
TopicsDigital Image Processing Techniques · Computational Geometry and Mesh Generation · Topological and Geometric Data Analysis
