# Relaxed Optimisation for Tensor Principal Component Analysis and Applications to Recognition, Compression and Retrieval of Volumetric Shapes

## Abstract

The mathematical and computational backgrounds of pattern recognition are the geometries in Hilbert space used for functional analysis and the applied linear algebra used for numerical analysis, respectively. Organs, cells and microstructures in cells dealt with in biomedical image analysis are volumetric data. We are required to process and analyse these data as volumetric data without embedding into higher-dimensional vector spaces from the viewpoint of object-oriented data analysis . Therefore, sampled values of volumetric data are expressed as three-way array data. The aim of the paper is to develop relaxed closed forms for tensor principal component analysis (PCA) for the recognition , classification , compression and retrieval of volumetric data. Tensor PCA derives the tensor Karhunen-Loève transform, which compresses volumetric data, such as organs, cells in organs and microstructures in cells, preserving both the geometric and statistical properties of objects and spatial textures in the space.

## Notes

### Acknowledgements

This research was supported by the “Multidisciplinary Computational Anatomy and Its Application to Highly Intelligent Diagnosis and Therapy” project funded by a Grant-in-Aid for Scientific Research on Innovative Areas from MEXT, Japan, and by Grants-in-Aid for Scientific Research funded by the Japan Society for the Promotion of Science.

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