Scaling-rotation distance and interpolation of symmetric positive-definite matrices

TL;DR

This paper presents a novel geometric framework for symmetric positive-definite matrices that models deformations via eigenvalue scaling and eigenvector rotation, enabling improved interpolation and distance measurement.

Contribution

It introduces a new Riemannian manifold-based approach for SPD matrices that addresses eigen-decomposition ambiguities and provides computational methods for 2x2 and 3x3 cases.

Findings

Provides minimal scaling-rotation geodesics for SPD matrices

Enhances interpolation of diffusion tensors in imaging

Improves behavior of trace, determinant, and fractional anisotropy

Abstract

We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed to characterize deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices. To characterize the deformation, the eigenvalue-eigenvector decomposition is used to find alternative representations of SPD matrices, and to form a Riemannian manifold so that scaling and rotations of SPD matrices are captured by geodesics on this manifold. The problems of non-unique eigen-decompositions and eigenvalue multiplicities are addressed by finding minimal-length geodesics, which gives rise to a distance and an interpolation method for SPD matrices. Computational procedures to evaluate the minimal scaling--rotation deformations and distances are provided for the most useful cases of $2 \times 2$ and $3 \times 3$ SPD matrices. In the new geometric framework, minimal scaling--rotation curves interpolate eigenvalues at constant logarithmic rate, and eigenvectors at constant angular rate. In the context of diffusion tensor imaging, this results in better behavior of the trace, determinant and fractional anisotropy of interpolated SPD matrices in typical cases.