We introduce a new geometric framework for the set of symmetricpositive-definite (SPD) matrices, aimed to characterize deformations of SPDmatrices by individual scaling of eigenvalues and rotation of eigenvectors ofthe SPD matrices. To characterize the deformation, the eigenvalue-eigenvectordecomposition is used to find alternative representations of SPD matrices, andto form a Riemannian manifold so that scaling and rotations of SPD matrices arecaptured by geodesics on this manifold. The problems of non-uniqueeigen-decompositions and eigenvalue multiplicities are addressed by findingminimal-length geodesics, which gives rise to a distance and an interpolationmethod for SPD matrices. Computational procedures to evaluate the minimalscaling--rotation deformations and distances are provided for the most usefulcases of $2 \times 2$ and $3 \times 3$ SPD matrices. In the new geometricframework, minimal scaling--rotation curves interpolate eigenvalues at constantlogarithmic rate, and eigenvectors at constant angular rate. In the context ofdiffusion tensor imaging, this results in better behavior of the trace,determinant and fractional anisotropy of interpolated SPD matrices in typicalcases.
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机译：我们为对称正定（SPD）矩阵集引入了新的几何框架，旨在通过特征值的个体缩放和SPD矩阵的特征向量的旋转来表征SPD矩阵的变形。为了表征变形，特征值-特征向量分解用于查找SPD矩阵的替代表示形式，并形成黎曼流形，以便SPD矩阵的缩放和旋转由该流形上的测地线捕获。通过找到最小长度的测地线，可以解决非唯一特征分解和特征值多重性的问题，这会增加SPD矩阵的距离和内插方法。计算最小比例缩放的计算程序-对于$ 2 x 2 $和3x 3 $ SPD矩阵中最有用的情况，提供了旋转变形和距离。在新的几何框架中，最小比例缩放-旋转曲线以恒定对数速率插入特征值，以恒定角速率插入特征向量。在扩散张量成像的情况下，在典型情况下，这会导致内插SPD矩阵的迹线，行列式和分数各向异性更好的行为。
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