This function computes an ONB and correspoinding curvature coefficients, belonging to the parallel transportet orthonormal frame and diagonalizes the curvature tensor along . The set of vectors in a tangent space added as last dimension of the array and consists of vectors.
The following derivation of the orthonormal frame follows [1]. Let . We denote its eigenvalues by and an orthonormal basis of eigenvectors by . Then we can write
For ease of notation we introduce the indexing system
Then the matrices
form an orthonormal basis of $T_x {\mathcal P}(r)$.
The corresponding coefficients characterizing curvature (see Jacobi fields ) are
Matlab Documentation
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% [Xi,k] = TpMONB(x,y) Compute an ONB in TpM and curvature
% coefficients corresponding to the transported frame along g(.,x,y)
%
% INPUT
% p : base point( sets)
% OPTIONAL:
% q : directional indicator( sets) for the first vector(s).
% OUTPUT
% W : orthonormal bases ( n x n x SetDim x Dimension )
% k : (optional) curvature coefficients (Dimension x SetDim)
% 
% MVIRT R. Bergmann, 20171203
See also
 The symmetric positive definite matrices of size $n\times n$
 The parallel transport of a vector along a geodesic
References

Bačák, M, Bergmann, R, Steidl, G and Weinmann, A (2016). A second order nonsmooth variational model for restoring manifoldvalued images. SIAM Journal on Scientific Computing. 38 A567–A597
We introduce a new nonsmooth variational model for the restoration of manifoldvalued data which includes second order differences in the regularization term. While such models were successfully applied for realvalued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the \(n\)sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.@article{BBSW16, author = {Ba{\v{c}}{\'a}k, Miroslav and Bergmann, Ronny and Steidl, Gabriele and Weinmann, Andreas}, title = {A second order nonsmooth variational model for restoring manifoldvalued images}, journal = {SIAM Journal on Scientific Computing}, year = {2016}, volume = {38}, number = {1}, pages = {A567A597}, doi = {10.1137/15M101988X}, eprint = {1506.02409}, eprinttype = {arXiv} }