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