$ \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 Sphere $\mathbb S^1$

The Sphere can be parametrized by an angle from . While it is also possible to use an embedding into , see the sphere , the amount of data to store is smaller for this special case.

We introduce as the set of congruence class representants of . Then we identify and introduce as the mapping of a real valued number to its congruence class representant with respect to in

If and are antipodal, i.e. , both half circles connecting the points are shortest paths. Otherwise, the shorter connecting arc is the shortes connection and hence the geodesic.

A typical visualization of cyclic data is the HSV colorbar, i.e. using the hue channel.

Colorization of the values on the sphere S1
Colorization of the values on the sphere $\mathbb S^1$.

Functions

The orthonormal basis in a tangent space
1
% [V,k] = TpMONB(p,q) an onb of the tangent space including log_pq

Since the dimension of is 1, there is only one direction (), and this direction is along the geodesic, hence the “curvature” is just zero.

Add (Gaussian white) wrapped noise
1
% fn = addNoise(f,sigma) add wrapped Gaussian noise.

Add noise to the data, which is done by adding Gaussian noise and “wrapping” it to the .

The distance on the sphere
1
% dist(a,b) computes the length of the smaller arc of a,b on S1

The distance on the sphere is defined for any , as the length of a shorter arc connecting them.

where denotes the symmetric modulo operation.

The inner product for cyclic data
1
% S1.dot(x,xi,nu)

The tangent vectors on lie on a line (when looking at the embedded circle), and hence the dot prodcut is the product of the tangent vectors, which are here values.

The exponential map on cyclic data
1
% exp(x,xi) - Exponential map at x with respect to xi in TxS1

The exponential map on the sphere is given by the formula

where denotes the symmetric modulo operation.

The logarithmic map for cyclic data
1
% log(x,y) - Inverse Exponential Map at x to y.

The logarithmic map on the sphere is defined for , not antipodal to , by

where denotes the symmetric modulo operation.

The parallel transport the sphere
1
% W = parallelTransport(x,y,xi) parallel transport a tangential

The parallel transport is the identity, since the only non-Euclidean change is the base point, which might get a hence

See also