WaveletLowRank

by Tommi Heikkilä, LUT University
0000-0001-5505-8136
tommheik

MultiResolution Low-Rank (MRLR) Algorithm for dynamic X-ray tomography.

(And some other low-rank algorithms.)

From left to right: MRLR, LLR, L+S and FTVNNR methods. Example reconstructions using the STEMPO data. Time step t=4 is shown.

From left to right: ground truth, MRLR, LLR and FTVNNR methods. Example reconstructions using simulated data. All 32 times steps are shown (gifs not necessarily in sync).

These codes were used to obtain the numerical results in

[1] T. Heikkilä,
"MultiResolution Low-Rank Regularization of Dynamic Imaging Problems" Scale Space and Variational Methods in Computer Vision. SSVM 2025. Lecture Notes in Computer Science, vol 15667. Springer. (2025)
DOI: 10.1007/978-3-031-92366-1_6
arXiv: 2502.20977

Please cite the following if you use these codes:

@inproceedings{heikkila2025multiresolution,
    title={{MultiResolution Low-Rank Regularization of Dynamic Imaging Problems}},
    author={Heikkilä, Tommi},
    booktitle={International Conference on Scale Space and Variational Methods in Computer Vision},
    pages={70--82},
    year={2025},
    series={LNCS, volume 15667},
    organization={Springer}
}

Contents

• LowRank_comparison_2d_main.m runs all algorithms using either the provided simulated data or the STEMPO dataset which is available in Zenodo: 10.5281/zenodo.8239013. The underlying problem is dynamic X-ray tomography.

  The Low-rank based regularization algorithms are

  • [MRLR] Multiresolution Low-rank: wavelet domain patched low-rank as explained in the article [1].
  • [LLR] Local low-rank: traditional patch based method as explained in the article [1].
  • [L+S] Low-rank + sparse: as explained in [2] and [3].
  • [FTVNNR] Fast Total Variation + global low rank regularization: as explained in [4] and [5].

• simulate_dynamic_data.m is used to generate the simulated data.

• algorithm contains the different reconstruction algorithms and some standalone codes. Standalone codes may be slightly out of date.

• util contains additional utility codes.

• The FTVNNR algorithm is used as-is, with the exception of minor change on line 38 of TVLR_opt.m which needs to be changed to
  [m, n, T, C] = size(F_gt);
  or similar, since in CT the data (sinogram) is rarely the same size as the desired reconstruction.

Toolboxes and references

LLR and MRLR algorithms use the Primal-Dual Fixed Point (PDFP) algorithm [6] for minimization.

The Discrete Wavelet Transform requires the Wavelet Toolbox.

The tomography forward operator is computed using the ASTRA Toolbox

W. Van Aarle, W. J. Palenstijn, J. Cant, E. Janssens, F. Bleichrodt, A. Dabravolski, J. De Beenhouwer, K. J. Batenburg and J. Sijbers, "Fast and flexible x-ray tomography using the ASTRA toolbox", Opt. Express 24 25129–47 (2016).

W. Van Aarle, W. J. Palenstijn, J. De Beenhouwer, T. Altantzis, S. Bals, K. J. Batenburg and J. Sijbers, "The ASTRA toolbox: a platform for advanced algorithm development in electron tomography", Ultramicroscopy 157 35–47, (2015).

The algorithm also uses Spot and HelTomo

E. Van den Berg and M. P. Friedlander, "Spot - a linear-operator toolbox", v1.2, (2013) http://cs.ubc.ca/labs/scl/spot

A. Meaney, "HelTomo - Helsinki Tomography Toolbox", v2.0.0, (2022) https://github.com/Diagonalizable/HelTomo

The Fast Total Variation and Nuclear Norm Regularization codes are related to [4,5] and are available on github.com/uta-smile/FTVNNR_Dynamic_MRI_MEDIA/

License

My codes are licensed under GNU General Public 3.0 license.

The FTVNNR codes have no formal license.

References

[2] Ricardo Otazo, Emmanuel Candes, and Daniel K. Sodickson. "Low‐rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components". Magnetic resonance in medicine 73.3 (2015): 1125-1136.

[3] Hao Gao, Jian-Feng Cai, Zuowei Shen, and Hongkai Zhao. "Robust principal component analysis-based four-dimensional computed tomography". Physics in Medicine & Biology, 56(11), 3181, (2011).

[4] Jiawen Yao, Zheng Xu, Xiaolei Huang, and Junzhou Huang. "An efficient algorithm for dynamic MRI using low-rank and total variation regularizations." Medical image analysis 44 (2018): 14-27.

[5] Jiawen Yao, Zheng Xu, Xiaolei Huang, and Junzhou Huang. "Accelerated dynamic MRI reconstruction with total variation and nuclear norm regularization." International Conference on Medical Image Computing and Computer-Assisted Intervention. Cham: Springer International Publishing, (2015).

[6] Peijun Chen, Jianguo Huang, and Xiaoqun Zhang. "A primal–dual fixed point algorithm for convex separable minimization with applications to image restoration." Inverse Problems 29.2 (2013): 025011.