$ \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}} $

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 , 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 [1]

  1. Publication illustration image
    Bergmann, R (2017). MVIRT, A toolbox for manifold-valued image registration. 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. [1][2][3][4][5]

References

  1. Publication illustration image
    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. Publication illustration image
    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. Publication illustration image
    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. Publication illustration image
    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. Publication illustration image
    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][2][3]

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