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 gitrepository using
1
git clone git@github.com:kellertuer/MVIRT.git
Please refer to the follow up project Manopt.jl in Julia.
Examples are InSAR imaging or when working with phase data, i.e. on the circle , or Diffusion Tensor Imaging (DTI), where every data items are an symmetric positive definite matrices, ..
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

Bergmann, R (2017). MVIRT, A toolbox for manifoldvalued image restoration. IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 17–20, 2017
In many real life application measured data takes its values on Riemannian manifolds. For the special case of the Euclidean space this setting includes the classical grayscale and color images. Like these classical images, manifoldvalued data might suffer from measurement errors in form of noise or missing data. In this paper we present the manifoldvalued image restoration toolbox (MVIRT) that provides implementations of classical image processing tasks. Based on recent developments in variational methods for manifoldvalued image processing methods, like total variation regularization, the toolbox provides easy access to work with these algorithms. The toolbox is implemented in Matlab, open source, and easily extendible, e.g. with own manifolds, noise models or further algorithms. This paper introduces the main mathematical methods as well as numerical examples.@inproceedings{Ber17, author = {Bergmann, Ronny}, title = { {MVIRT}, A toolbox for manifoldvalued image restoration}, year = {2017}, booktitle = {IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 1720, 2017}, doi = {10.1109/ICIP.2017.8296271} }
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 NLMSSE 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 SSNUnit for bringing up the need of regularizing phasevalued 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++ matrixvector classes.
 The overwiew of InSAR Interferometry for providing phase valued data of the Mount Vesuvius.
References

Bergmann, R and Weinmann, A (2016). A second order TVtype approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Journal of Mathematical Imaging and Vision. 55 401–27
In this paper we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space valued data. These kind of data appear when dealing with nonlinear color spaces such as HSV, and they can be obtained by changing the space domain of, e.g., an optical flow field to polar coordinates. For such nonlinear data spaces, we develop algorithms for the solution of the corresponding second order total variation (TV) type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and we apply the algorithms to concrete problems.@article{BW16, author = {Bergmann, Ronny and Weinmann, Andreas}, title = {A second order {TV}type approach for inpainting and denoising higher dimensional combined cyclic and vector space data}, journal = {Journal of Mathematical Imaging and Vision}, year = {2016}, volume = {55}, number = {3}, pages = {401427}, doi = {10.1007/s1085101506273}, eprint = {1501.02684}, eprinttype = {arXiv} }

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} }

Bergmann, R, Persch, J and Steidl, G (2016). A parallel Douglas–Rachford algorithm for minimizing ROFlike functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences. 9 901–37
We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation like regularizing term. To solve the convex minimization problem, we extend the DouglasRachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the DouglasRachford algorithm are reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for KrasnoselskiMann iterations imply the convergence of the DouglasRachford algorithm. Unfortunately, this general results does not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions namely, several distancelike functions and an indicator functions of a special convex sets. We prove that the reflections of certain distancelike functions on Hadamard manifolds are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the DouglasRachford algorithm converges here.
Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or halfquadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.@article{BPS16, author = {Bergmann, Ronny and Persch, Johannes and Steidl, Gabriele}, title = {A parallel {D}ouglas–{R}achford algorithm for minimizing {ROF}like functionals on images with values in symmetric {H}adamard manifolds}, journal = {SIAM Journal on Imaging Sciences}, year = {2016}, volume = {9}, number = {3}, pages = {901937}, doi = {10.1137/15M1052858}, eprint = {1512.02814}, eprinttype = {arXiv} } 
Bergmann, R, Chan, R H, Hielscher, R, Persch, J and Steidl, G (2016). Restoration of manifoldvalued images by halfquadratic minimization. Inverse Problems and Imaging. 10 281–304
The paper addresses the generalization of the halfquadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the halfquadratic minimization method using the notation of the \(c\)transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group \(\operatorname{SO}(3)\), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with \(\operatorname{SO}(3)\)valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.@article{BCHPS16, author = {Bergmann, Ronny and Chan, Raymond H. and Hielscher, Ralf and Persch, Johannes and Steidl, Gabriele}, title = {Restoration of manifoldvalued images by halfquadratic minimization}, journal = {Inverse Problems and Imaging}, year = {2016}, volume = {10}, number = {2}, pages = {281304}, doi = {10.3934/ipi.2016001}, eprint = {1505.07029}, eprinttype = {arXiv} }

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
In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, groundbased astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the onedimensional sphere \(\mathbb S^1\). Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edgepreserving image restoration, such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the socalled staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for \(\mathbb S^1\)valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances, the absolute cyclic second order differences have only values in \([0,\pi]\). We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as realworld data demonstrate the advantageous performance of our algorithm.@article{BLSW14, author = {Bergmann, Ronny and Laus, Friederike and Steidl, Gabriele and Weinmann, Andreas}, title = {Second order differences of cyclic data and applications in variational denoising}, journal = {SIAM Journal on Imaging Sciences}, year = {2014}, volume = {7}, number = {4}, pages = {2916–2953}, doi = {10.1137/140969993}, eprint = {1405.5349}, eprinttype = {arXiv} }

Bergmann, R (2017). MVIRT, A toolbox for manifoldvalued image restoration. IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 17–20, 2017
In many real life application measured data takes its values on Riemannian manifolds. For the special case of the Euclidean space this setting includes the classical grayscale and color images. Like these classical images, manifoldvalued data might suffer from measurement errors in form of noise or missing data. In this paper we present the manifoldvalued image restoration toolbox (MVIRT) that provides implementations of classical image processing tasks. Based on recent developments in variational methods for manifoldvalued image processing methods, like total variation regularization, the toolbox provides easy access to work with these algorithms. The toolbox is implemented in Matlab, open source, and easily extendible, e.g. with own manifolds, noise models or further algorithms. This paper introduces the main mathematical methods as well as numerical examples.@inproceedings{Ber17, author = {Bergmann, Ronny}, title = { {MVIRT}, A toolbox for manifoldvalued image restoration}, year = {2017}, booktitle = {IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 1720, 2017}, doi = {10.1109/ICIP.2017.8296271} }

Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Infimal Convolution Coupling of First and Second Order Differences on ManifoldValued Images. Scale Space and Variational Methods in Computer Vision: 6th International Conference, SSVM 2017, Kolding, Denmark, June 48, 2017, Proceedings. Springer International Publishing, Cham. 447–59
Recently infimal convolution type functions were used in regularization terms of variational models for restoring and decomposing images. This is the first attempt to generalize the infimal convolution of first and second order differences to manifoldvalued images. We propose both an extrinsic and an intrinsic approach. Our focus is on the second one since the summands arising in the infimal convolution lie on the manifold themselves and not in the higher dimensional embedding space. We demonstrate by numerical examples that the approach works well on the circle, the 2sphere, the rotation group, and the manifold of positive definite matrices with the affine invariant metric.@inproceedings{BFPS17b, author = {Bergmann, Ronny and Fitschen, Jan Henrik and Persch, Johannes and Steidl, Gabriele}, editor = {Lauze, Fran{\c{c}}ois and Dong, Yiqiu and Dahl, Anders Bjorholm}, title = {Infimal Convolution Coupling of First and Second Order Differences on ManifoldValued Images}, booktitle = {Scale Space and Variational Methods in Computer Vision: 6th International Conference, SSVM 2017, Kolding, Denmark, June 48, 2017, Proceedings}, year = {2017}, publisher = {Springer International Publishing}, address = {Cham}, pages = {447459}, isbn = {9783319587714}, doi = {10.1007/9783319587714_36} }

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
Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and realworld data demonstrate the performance of our algorithm.@inproceedings{BW15, author = {Bergmann, Ronny and Weinmann, Andreas}, title = {Inpainting of cyclic data using first and second order differences}, year = {2015}, pages = {155–168}, editor = {Tai, X.C. and Bae, E. and Chan, T. F. and Leung, S. Y. and Lysaker, M.}, publisher = {Springer}, booktitle = {Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong}, doi = {10.1007/9783319146126_12}, eprint = {1410.1998}, eprinttype = {arXiv} }