Scaling-rotation distance and interpolation of symmetric positive-definite matrices. (English) Zbl 1321.15020
Summary: We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed at characterizing 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 nonunique 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 for evaluating 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.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 53C20 | Global Riemannian geometry, including pinching |
| 53C22 | Geodesics in global differential geometry |
Keywords:
symmetric positive-definite matrices; eigendecomposition; Riemannian distance; geodesics; diffusion tensors; eigenvalue; eigenvector; trace; determinantReferences:
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