Polyadic Decomposition of Voxel Hypercubes in ISR

Abstract

Greg Passmore

Volumetric datasets in intelligence, surveillance, and reconnaissance (ISR) typically extend beyond three-dimensional spatial fields and often include additional measurements such as spectral, polarization, or temporal components. A voxel hypercube is treated as a high-rank tensor in which fixed spatial indices are combined with one or more attribute dimensions. This method provides a mathematical structure for multi-dimensional voxel data and permits the direct use of tensor operations, including compression and polyadic decomposition, on the resulting arrays. The approach begins with a spatial rank-3 tensor representation and incorporates additional axes for parameters such as spectral frequency, polarization state, and time. Although all axes can be expressed uniformly within the tensor, time differs conceptually because it indexes repeated realizations of the complete spatial–attribute block rather than adding new attributes within a single voxel. The tensor model supports CP, Tucker, and related decompositions, allowing spatial factors to be separated from spectral and polarization components. The treatment of the temporal axis is examined both as an additional tensor rank and as a sequence of discrete hypercubes, with consequences for analysis and computation. Representing voxel hypercubes in this tensor form provides a consistent and general framework in which time encodes evolution of the volume, while the remaining axes encode the set of measurements associated with each voxel.

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