
# Manifold-valued Image Restoration Toolbox

In many application measured data appears nonlinear, i.e. restricted in a certain range or equipped with a different distance measure.

This toolbox provides an easy access to image processing tasks for such data, where the values of each measurement are points on a manifold. You can get started by downloading the source code from Github or by cloning the git-repository using

1
git clone git@github.com:kellertuer/MVIRT.git


Examples are InSAR imaging or when working with phase data, i.e. on the circle $\mathbb S^1$, or Diffusion Tensor Imaging (DTI), where every data items are an $n\times n$ symmetric positive definite matrices, $\SPD{n}$..

If you are using the toolbox, it would be nice, if you give us a note. The Toolbox is available under the GPL 3 license, so you can use as long as you stick to the terms of that license. If you use the toolbox within one of your scientific works, please cite [1]

1. Bergmann, R (2017). MVIRT, A toolbox for manifold-valued image restoration. IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 17–20, 2017

and if you use a specific algorithm, the corresponding paper, too.

The tutorials are a good point to start. An overview on manifolds. Many paper examples provide illustrations of the algorithms.

## Acknowledgements

• First and foremost Johannes Persch contributed a lot to this toolbox, especially the algorithms on NL-MSSE are his implementation work.
• The Manopt toolbox by Nicolas Boumal inspired a few function and interface design decisions and is a great toolbox for optimization on manifolds.
• The MFOPT Matlab Library by Jan Lellmann also does TV regularization of manifold valued images, though with a different algorithmical approach.
• The SSN-Unit for bringing up the need of regularizing phase-valued data which started the project.
• UCL Camino Diffusion MRI Toolkit for providing real life diffusion data and allowing us to provide a small part of that data within one of the examples.
• The eigen library for their nice C++ matrix-vector classes.
• The overwiew of InSAR Interferometry for providing phase valued data of the Mount Vesuvius.

# References

1. Bergmann, R and Weinmann, A (2016). A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Journal of Mathematical Imaging and Vision. 55 401–27
2. 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
3. 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
4. Bergmann, R, Chan, R H, Hielscher, R, Persch, J and Steidl, G (2016). Restoration of manifold-valued images by half-quadratic minimization. Inverse Problems and Imaging. 10 281–304
5. Bergmann, R, Laus, F, Steidl, G and Weinmann, A (2014). Second order differences of cyclic data and applications in variational denoising. SIAM Journal on Imaging Sciences. 7 2916–53

1. Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Infimal Convolution Coupling of First and Second Order Differences on Manifold-Valued Images. Scale Space and Variational Methods in Computer Vision: 6th International Conference, SSVM 2017, Kolding, Denmark, June 4-8, 2017, Proceedings. Springer International Publishing, Cham. 447–59
2. Bergmann, R (2017). MVIRT, A toolbox for manifold-valued image restoration. IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 17–20, 2017
3. Bergmann, R and Weinmann, A (2015). Inpainting of cyclic data using first and second order differences. Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong. Springer. 155–68