Shape theory via affine transformation: Some generalisations

TL;DR

This paper extends affine shape theory to various algebraic fields, deriving general densities applicable to multiple distributions, and demonstrates its application in analyzing brain MRI scans of different patient groups.

Contribution

It generalizes affine shape distributions across real, complex, quaternion, and octonion fields, unifying previous results and enabling new applications in medical imaging.

Findings

Derived complex normal affine density for MRI analysis.

Unified shape distribution framework for multiple algebraic fields.

Applied to distinguish brain scans of normal and schizophrenic patients.

Abstract

This work sets the statistical affine shape theory in the context of real normed division algebras. The general densities apply for every field: real, complex, quaternion, octonion, and for any noncentral and non-isotropic elliptical distribution; then the separated published works about real and complex shape distributions can be obtained as corollaries by a suitable selection of the field parameter and univariate integrals involving the generator elliptical function. As a particular case, the complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients.