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Deformable Groupwise Image Registration using Low-Rank and Sparse Decomposition

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Abstract

Groupwise image registration describes the problem of simultaneously aligning a series of more than two images through individual spatial deformations and it is a common task in the processing of medical image sequences. Variational methods with data fidelity terms based on robust PCA (RPCA) have proven successful in accounting for structural changes in image intensity stemming, e.g., from the uptake of a contrast agent in functional imaging. In this article, we investigate the drawbacks of the most commonly used RPCA data term and derive an improved replacement that employs explicit constraints instead of penalties. We further present a multilevel scheme with theoretically justified scaling to solve the underlying fully deformable registration model. Our numerical experiments on synthetic and real-life medical data confirm the advanced adaptability of RPCA-based data terms and showcase an improved registration accuracy of our algorithm when compared to related groupwise approaches.

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Notes

  1. For ease of presentation, the remainder of this article will only be concerned with discrete images.

  2. From here on, we omit the explicit notation of the dependence of M on \(T_1, \ldots , T_N\) whenever it is clear from the context.

  3. Note that this somewhat abusive notation refers to an interpolation of \(T_i\) over a regular cell-centered grid that is offset by \(u^j\)—in practice, we use linear interpolations as described in [26, Chapter 3.3].

  4. A more detailed discussion of how to select the parameter \(\nu \) in practical applications will be given in Sect. 4.1.

  5. We consider the case \(d = 2\).

  6. To be more precise, that means \(\hat{u}_{i,j,c}^k := u_{i,j,c}^k + t_c\) for all \(i = 1, \ldots , m\), \(j = 1, \ldots , n\), \(c = 1, 2\) and \(k = 1, \ldots , N\).

  7. For simplicity, we assume \(2^{(n_{lev} - 1)} \mid m\) and \(2^{(n_{lev} - 1)} \mid n\).

  8. To this end, it might be worthwhile to investigate, to which degree the accuracy of the \(D_{\text {PCA2}}\) experiments in elastix and the pairwise \(D_{\text {NGF}}\) experiments in FAIR in Sect. 5 benefitted from their advanced interpolation models.

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Acknowledgements

The authors thank Allen D. Elster (MRIQuestions.com) for kindly providing the cardiac cine study used in this article as well as Jarle Rørvik and the Bergen Abdominal Imaging Research Group, Haukeland University Hospital Bergen, Norway, for the renal DCE-MRI cine. The authors further acknowledge support through DFG grant LE 4064/1-1 “Functional Lifting 2.0: Efficient Convexifications for Imaging and Vision” and NVIDIA Corporation.

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Haase, R., Heldmann, S. & Lellmann, J. Deformable Groupwise Image Registration using Low-Rank and Sparse Decomposition. J Math Imaging Vis 64, 194–211 (2022). https://doi.org/10.1007/s10851-021-01059-7

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