$ \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}} $

Bernoulli's Lemniscate on the Sphere

The example of Bernoulli’s Lemniscate on the sphere denoises a signal which is sampled from the curve defined as Bernoulli’s Lemniscate in the tangential space of the north pole and then brought onto the manifold by the exponential map. This signal is obstructed by noise and the example denoises this noisy artificial signal using the CPP algorithm for the additive first and second order differences model. The script itself performs a small parameter test. This example follows the first example of Section 5.1 in [1] and the initial data are the black points in the following figure.

Sampling Bernoulli's lemniscate on the sphere \(\mathbb S^2\) (black) and creating noisy data (blue).
Sampling Bernoulli's lemniscate on the sphere \(\mathbb S^2\) (black) and creating noisy data (blue).

A denoising result setting and is shown in the folling figure.

Denoising with \(\alpha=0.45\) and \(\beta=9\) (blue) compared to the original (black).
Denoising with \(\alpha=0.45\) and \(\beta=9\) (blue) compared to the original (black).

See also

References

  1. Bačák, M, Bergmann, R, Steidl, G and Weinmann, A (2016). A second order non-smooth variational model for restoring manifold-valued images. SIAM Journal on Scientific Computing. 38 A567–A597