Scaling-Rotation Distance and Interpolation of Symmetric Positive-Definite Matrices

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 scalingrotation deformations and distances are provided for the most useful cases of 2 × 2 and 3 × 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.

Geometric foundations for scaling-rotation statistics on symmetric positive definite matrices: Minimal smooth scaling-rotation curves in low dimensions

“…To provide a geometric framework for these curves, Section 2 is devoted to systematic characterization of fibers and its connection to the stratification of Sym + (p). This allows us to build upon the scaling-rotation framework for SPD matrices proposed in [25], which provided a geometric interpretation for the scaling-rotation curves in [36]. In particular, our characterization of fibers is essential in understanding differential topology and geometry of this framework.…”

“…Our eigenvalue-multiplicity stratification categorizes the ellipsoids associated with the SPD matrices into distinct shapes, which in the case p = 3 are known as spherical, prolate/oblate, and tri-axial (scalene). We believe that the scalingrotation framework studied in this work and in [25,20] will be highly useful in developing new methodologies of smoothing, registration and regression analysis of diffusion tensors.…”

“…This stratification is tied inextricably to our main goal in this paper: the systematic characterization and analysis of minimal smooth scaling-rotation curves in low dimensions. Such curves were defined in [25] as smooth curves whose length minimizes the amount of scaling and rotation needed to transform an SPD matrix into another. The techniques developed in this paper, when applied to the case p = 3, allow us to find new explicit formulas for such curves.…”

“…In previous work [25], we introduced a fourth framework, the "scaling-rotation framework", that is the subject of this paper. In [25,Section 5], we presented evidence of advantages of this framework over the popular log-transformation-based frameworks for tensor interpolations. In Section 1.2 of the present paper, we briefly discuss some other advantages of the scaling-rotation framework in statistical analysis.…”

“…Eigen-decomposition, upon which the scaling-rotation framework is based, determines both a stratification of Sym + (p), defined by eigenvalue multiplicities, and fibers of the "eigencomposition" map SO(p) × Diag + (p) → Sym + (p). This leads to the notion of scaling-rotation distance [Jung et al (2015)], a measure of the minimal amount of scaling and rotation needed to transform an SPD matrix, X, into another, Y, by a smooth curve in Sym + (p). Our main goal in this paper is the systematic characterization and analysis of minimal smooth scalingrotation (MSSR) curves, images in Sym + (p) of minimal-length geodesics connecting two fibers in the "upstairs" space SO(p)×Diag + (p).…”