$ \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}} $

The symmetric positive definite $\SPD{n}$ matrices of size $n\times n$

The manifold consists of all matrices such that for all vectors the inequality

holds. 1

In this class we use the affine metric as for example described in [1], but there is also the alternative way of using the Log-Euclidean metric [2].

For vizualization we can use the following approach. Since is symmetric, all eigenvalues are real valued; since is positive definite, all eigenvalues are (strictly) positive. This yields a vizualization for the cases and :

For we take the eigenvectors of scaled to unit length and denot the eigenvalues with . Assume further that the eigenvalues are sorten, i.e. . We can now interprete the eigenvectors as axis of an ellipse: as major axis and minor axis . An example is the signal from [3].

Visualization of a P(2) signal
Visualizazion of $\SPD{2}$-valued signal as ellipses.

Note that the mid points of the ellipses are placed on a regular grid. This grid has no fixed size and should be set to the same value for images that are showing comparable data.

For one can do the similar approach with all three eigenvalues and their eigenvectors to visualize an ellipsoid. Furthermore this can be used for image, or even 3D/volumetric, data. The following is an atrificial data set from [4]

Visualization of P(3) image
Visualizazion of an image of $\SPD{3}$ valued pixel as ellispids.

Here, the grid is also arbitrarily chosen, see last note. This construction does not yet yield a color. We empoloy the geometric anisotropy index as presented in [5].

Footnotes

  1. While this page usually sets vectors in small letters and bold, matrices in capital letters and bold, we refrain from this notation to emphasize the matrices here being points on the manifold. This way we keep pages consistent, that are about general manifold theory 

Functions

The orthonormal basis in a tangent space
1
% [Xi,k] = TpMONB(x,y) Compute an ONB in TpM and curvature

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.

more
Add (Rician or Gaussian white) noise
1
% addNoise(x,sigma) add (Rician or Gaussian) noise to data x

Adds Rician (standard) or Gaussian (set 'Distribution' to 'Gaussian') with mean zero (using the tangent space) to data.

The inner product on symmetric positive definite matrices
1
% d = dist(x,y) compute the distance between x,y from P(n).

The distance on the symmetric positive definite matrices is given by

where denotes the matrix logarithm and is the Frobenius norm of the matrix.

The inner product on symmetric positive definite matrices
1
% dot(x,xi,nu) inner product of two tangent vectors in TxP(m)

The affine invariant metric on the symmetric positive definite matrices is defined for and as

where is the trace of a matrix.

The exponential map on symmetric positive definite matrices
1
% y = exp(x,xi) exponential map at x from P(n) towards xi in TxP(n)

The exponential map on the sphere is given for , by the formula

where denotes the matrix exponential.

The logarithmic map on symmetric positive definite matrices
1
% xi = log(x,y) logarithmic map at the point(s) x of points(s) y

The logarithmic map on symmetric positive definite matrices is given by

where where denotes the matrix logarithm.

The parallel transport on symmetric positive definite matrices
1
% eta = parallelTransport(x,y,xi) parallel transport xi along g(.,x,y)

This function parallel transports a vector along the unique geodesic . The transport is given by

See also

References

  1. Sra, S and Hosseini, R (2015). Conic Geometric Optimization on the Manifold of Positive Definite Matrices. SIAM Journal on Optimization. 25 713–39
  2. Arsigny, V, Fillard, P, Pennec, X and Ayache, N (2005). Fast and simple calculus on tensors in the Log-Euclidean framework. International Conference on Medical Image Computing and Computer-Assisted Intervention. 3749 115–22
  3. Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Priors with coupled first and second order differences for manifold-valued image processing
  4. Bergmann, R, Persch, J and Steidl, G (2016). A parallel Douglas–Rachford algorithm for minimizing ROF-like functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences. 9 901–37
  5. Moakher, M and Batchelor, P G (2006). Symmetric Positive-Definite Matrices: From Geometry to Applications and Visualization. Visualization and Processing of Tensor Fields. Springer Berlin Heidelberg, Berlin, Heidelberg. 285–98