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Compression of Voxelized Vector Field Data by Boxes is Hard

Simon Zhang

TL;DR

The paper studies rate-distortion lossy compression for voxelized vector field data and establishes a precise complexity landscape. It shows LossyIndexedCompression is undecidable in general due to the continuous vector space, then defines a decidable subproblem $(k,D)$-RectLossyVVFCompression based on box coverings. The authors prove decompression is polynomial-time, but compression is NP-hard and, for $k,D\ge 2$, APX-hard via a sequence of PTAS reductions from geometric rectangle-cover problems, effectively delineating tractable versus intractable regimes for box-based voxel-field compression. These results illuminate fundamental limits for exact and approximate box-based codings of high-dimensional voxel data and guide the design of practical compression schemes that balance efficiency and accuracy.

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 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.

Compression of Voxelized Vector Field Data by Boxes is Hard

TL;DR

The paper studies rate-distortion lossy compression for voxelized vector field data and establishes a precise complexity landscape. It shows LossyIndexedCompression is undecidable in general due to the continuous vector space, then defines a decidable subproblem -RectLossyVVFCompression based on box coverings. The authors prove decompression is polynomial-time, but compression is NP-hard and, for , APX-hard via a sequence of PTAS reductions from geometric rectangle-cover problems, effectively delineating tractable versus intractable regimes for box-based voxel-field compression. These results illuminate fundamental limits for exact and approximate box-based codings of high-dimensional voxel data and guide the design of practical compression schemes that balance efficiency and accuracy.

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 the -RectLossyVVFCompression problem. We show four main results about the -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 -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 . Assuming , this shows that when there can be no exact polynomial time algorithm nor can there even be a PTAS approximation algorithm for the -RectLossyVVFCompression problem.
Paper Structure (29 sections, 43 theorems, 189 equations, 3 figures, 5 algorithms)

This paper contains 29 sections, 43 theorems, 189 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Voxelized Vector Field Data
  3. Preliminaries
  4. Vector Fields
  5. Clustering in Metric Spaces
  6. Voxelized Vector Field Compression
  7. Voxelized Vector Field Compression as a Box Covering Problem
  8. Outline of Main Results
  9. Decompressing is Easy
  10. A Decompression Algorithm
  11. $(k,D)$-RectLossyVVFCompression is NP-Hard
  12. $(k,D)$-RectLossyVVFCompression is APX-Hard
  13. A Collection of Rectangles on the Plane can be Embedded into $[2N]^2$
  14. A Sweep-Line Algorithm to Cover the Complement
  15. $\textsc{SpecialRect-3SC} \leq_{PTAS} \textsc{VGridSpecialRect-3SC}$
  16. ...and 14 more sections

Key Result

Lemma 1

In the lossy indexed compression problem with Kolmogorov decoder: where $C$ is dependent only on the choice of the prefix universal Turing machines.

Figures (3)

  • Figure 1: An illustration of a $(3,7)$-dimensional Voxelized Vector Field with the kinematic energy function providing the coloring at every voxel.
  • Figure 2: A $0$-$1$ matrix instance of the RectiLinearImgCompression Problem. This problem is NP-Complete.
  • Figure 3: Illustration of a Sweep Line Algorithm with its Top Events and Bottom Events

Theorems & Definitions (149)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Corollary 1
  • proof
  • Example 1
  • ...and 139 more