We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function $(1-det(X)^beta)/beta$. We give explicit expressions for the geodesics and distance function, under suitable co
nditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As $beta$ tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric associated with the matrix geometric mean; we show that the geodesic and the distance associated with the power potential converge to the weighted matrix geometric mean and the distance associated with the logarithmic potential, respectively.
In this work, we propose a multiscale approach for a nonstandard higher-order PDE based on the $p(cdot)$-Kirchhoff energy. First, we consider a topological gradient approach for a semilinear case in order to detect important object of image. Then, we
consider a fully nonlinear $p(cdot)$-Kirchhoff equation with variables exponent functions that are chosen adaptively based on the map furnished by the topological gradient in order to preserve important features of the image. Then, we consider the split Bregman method for the numerical implementation of our proposed model. We compare our model with other classical variational approaches such that the TVL and biharmonic restoration models. Finally, we present some numerical results to illustrate the effectiveness of our approach.
In this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then gi
ve closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determinant $alpha$-divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case. These diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of Hermitian positive-definite matrices. Numerical computations based on a modified Newton method are presented and commented.
We present a novel approach for the derivation of PDE modeling curvature-driven flows for matrix-valued data. This approach is based on the Riemannian geometry of the manifold of Symmetric Positive Definite Matrices Pos(n).
