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Sampled values of volumetric data are expressed as three-way array data, which is expressed as tensors. We are required to process and analyse volumetric data without embedding into higher-dimensional vector space from the viewpoints of object oriented data analysis. Multi-way forms of volumetric data require quantitative methods for the discrimination of multi-way forms. Therefore, we define a distance metric for subspaces of multi-way data arrays using transportation between the Stiefel manifolds constructed by subspaces of multi-way data arrays.
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