$ \DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}} $

Jacobi field

For the geodesic to a Jacobi field is a smooth vector field along fulfilling the ODE

where denotes the (partial) covariant derivative, is the curvature tensor and is a geodesic variation, , i.e. for any the function is a geodesic; and we have .

This function evaluates certain special Jacobi fields given a weight function , where refers to the index of the eigenvalue, and

By [1] there exists a geodesic variation for every Jacobi field. The Jacobi field itself is unique given initial conditions for some .

Note that instead of initial conditions, the ODE might also be given with boundary conditions.

With an parallel transported orthonormal basis we can write

which on a symmetric Riemannian manifold simplifies the ODE characterizing a Jacobi field to , where and .

If we choose a basis that diagonalizes the operator then the matrix is a diagonal matrix and the system of ODEs simplifies to $d$ ODEs

where is the eigenvalue corresponding to . They can be interpreted as characterizing curvature. With this formulation they are independent of the length of the geodesic, because we use the unit speed geodesic . Using just introduces a factor of to the eigenvalues . We prefer to introduce the eigenvalues similar to the ONB with respect to unit vectors in the other arguments of the curvature tensor.

Finally, the parameter specifies one of the initial conditions depending on the geodesic variation (i.e. the weights herein) involved; see the corresponding differentials for details.

Note that for fixed and

Matlab Documentation

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% xi = JacobiField(x,y,t,eta) - evaluate a Jacobi field
%    along the geodesic geo(x,y) at point t, where the
%    'weight'-function f(k,t,d) determines the boundary
%    conditions of the field and hence the meaning of eta,
%
% INPUT
%    x   : a point on the manifold Sn (or a vector of points)
%    y   : a point on the manifold Sn (or a vector of points)
%    t   : a value from [0,1] indicating the point geo(x,y,t)
%           (or a vector of values)
%    eta : an initial condition of the Jacobi field, where the
%           following weights determine the type of initial
%           condition.
%
% OPTIONAL
%    'weights' : [@(k,t,d) = (k==0)*t
%       + (k>0)*sin(sqrt(k)*t*d)/sin(sqrt(k)*d)
%       + (k<0)*sinh(sqrt(-k)*d*t)/sinh(sqrt(-k)*d)
%       provides the weight depending on the eigenvalue (k) of
%       the curvature tensor coresponding to the ONB basis
%       vector, the position t along the Jacobi field and d the
%       length of the geodesic.
%       For the standard value, eta is a tangential vector at 0
%       and the second boundary condition is J(1)=0, i.e. the
%       Jacobifield corresponds to D_x\gamma_{xy}(t)[\eta]
%
% ---
% Manifold-valued Image Restoration Toolbox 1.3
% R. Bergmann | MVIRT | 2017-12-01

See also

used in

References

  1. Lee, J M (1997). Riemannian Manifolds. An Introduction to Curvature. Springer, New York