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arxiv: 1907.09732 · v1 · pith:CEVL3ZJZnew · submitted 2019-07-23 · 📡 eess.IV · cs.CV· cs.NA· math.NA

Variational Registration of Multiple Images with the SVD based SqN Distance Measure

Kai Brehmer , Hari Om Aggrawal , Stefan Heldmann , Jan Modersitzki This is my paper

Pith reviewed 2026-05-24 17:13 UTC · model grok-4.3

classification 📡 eess.IV cs.CVcs.NAmath.NA
keywords image registrationmultiple imagesSVDSchatten q-normSqN distance measurevariational registrationhistological sections
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The pith

The SVD-based SqN distance measure aligns multiple images effectively and outperforms rank and feature-volume methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines distance measures for registering more than two images at once within a variational setting. It rests on the idea that singular values from a feature matrix assembled from the images can drive alignment. The authors compare the Schatten q-norm based SqN measure against a rank-based approach and a feature-volume approach. Tests on dynamic image sequences and stacks of histological sections indicate that SqN produces valid registrations and better outcomes than the other two methods. A reader would care because many practical imaging tasks require simultaneous alignment of image series rather than isolated pairs.

Core claim

The paper claims that information about the singular values of a feature matrix of images can be used for alignment, that the Schatten q-norm based SqN distance measure is a suitable similarity measure for multiple-image registration, and that it yields superior results to its rank-based and feature-volume competitors in the examined applications.

What carries the argument

The SqN distance measure, which applies the Schatten q-norm to the singular values of the feature matrix formed from the set of images.

If this is right

  • SqN supports registration of dynamic image sequences.
  • SqN delivers better alignment quality than rank-based or feature-volume measures in the tested cases.
  • SqN works for stacks of histological sections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the feature matrix construction carries over, SqN could be tried on other multi-frame tasks such as video frame alignment.
  • Performance might change when the number of images grows much larger than the examples shown.

Load-bearing premise

The singular values of the feature matrix encode enough information to determine correct alignment among the images.

What would settle it

A set of multiple images on which SqN registration produces visibly poorer alignment accuracy than the rank-based or feature-volume method.

Figures

Figures reproduced from arXiv: 1907.09732 by Hari Om Aggrawal, Jan Modersitzki, Kai Brehmer, Stefan Heldmann.

Figure 1
Figure 1. Figure 1: Three representative axial slices of a marmoset monkey brain dataset; [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Registration results for 3D reconstruction of the monkey brain datasets. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Registration results after random permutation of the axial slices. As ex [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Three representative 2D coronal slices of the 4D DCE-MRI dataset of a [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustrated are sagittal cuts through the stack of 2D slices from a 4D DCE [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

Image registration, especially the quantification of image similarity, is an important task in image processing. Various approaches for the comparison of two images are discussed in the literature. However, although most of these approaches perform very well in a two image scenario, an extension to a multiple images scenario deserves attention. In this article, we discuss and compare registration methods for multiple images. Our key assumption is, that information about the singular values of a feature matrix of images can be used for alignment. We introduce, discuss and relate three recent approaches from the literature: the Schatten q-norm based SqN distance measure, a rank based approach, and a feature volume based approach. We also present results for typical applications such as dynamic image sequences or stacks of histological sections. Our results indicate that the SqN approach is in fact a suitable distance measure for image registration. Moreover, our examples also indicate that the results obtained by SqN are superior to those obtained by its competitors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the SqN distance measure, defined via the Schatten q-norm on the singular values of a feature matrix assembled from multiple images, as a similarity criterion within a variational framework for multi-image registration. It relates SqN to two competitors (rank-based and feature-volume approaches), states the key assumption that singular-value information can drive alignment, and reports results on dynamic image sequences and histological section stacks, concluding that SqN is suitable and empirically superior.

Significance. If the empirical superiority is substantiated, SqN would supply a new, SVD-based similarity measure for simultaneous registration of image collections, with direct relevance to dynamic MRI, CT perfusion, and histology alignment tasks. The explicit formulation of the central assumption and the side-by-side comparison with existing methods are constructive features; the absence of free parameters in the core distance is also a strength.

major comments (2)
  1. [Results] Results section (and associated figures/tables): the superiority claim rests on visual examples without reported quantitative metrics (e.g., target-registration error, Dice scores, or landmark distances with standard deviations), statistical tests, or cross-validation details. This leaves the central empirical assertion unsupported by verifiable numbers and undermines the assertion that SqN outperforms its competitors.
  2. [§3] §3 (method comparison): the three distance measures are introduced and related, yet no explicit complexity analysis, parameter count, or convergence-rate comparison is supplied to justify why SqN should be preferred on theoretical grounds before the empirical section.
minor comments (2)
  1. [§2] Notation for the feature matrix and the precise definition of the Schatten q-norm (including the range of q) should be stated once in a dedicated subsection or table to avoid repeated inline definitions.
  2. [Figures] Figure captions for the registration examples should include the exact values of q used and the number of images in each stack.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below.

read point-by-point responses

  1. Referee: [Results] Results section (and associated figures/tables): the superiority claim rests on visual examples without reported quantitative metrics (e.g., target-registration error, Dice scores, or landmark distances with standard deviations), statistical tests, or cross-validation details. This leaves the central empirical assertion unsupported by verifiable numbers and undermines the assertion that SqN outperforms its competitors.

    Authors: We agree that the current presentation relies on visual assessment. In revision we will add quantitative metrics: Dice scores on the histology data (using available segmentations) and target-registration error on the dynamic sequences (using available landmarks), each with standard deviations across cases. Basic paired statistical comparisons between methods will also be included. revision: yes

  2. Referee: [§3] §3 (method comparison): the three distance measures are introduced and related, yet no explicit complexity analysis, parameter count, or convergence-rate comparison is supplied to justify why SqN should be preferred on theoretical grounds before the empirical section.

    Authors: Section 3 centers on the shared singular-value assumption and the explicit algebraic relations among the three measures. We will add a concise paragraph noting that all three require an SVD of an n-by-d feature matrix (n = number of images) and that the core SqN formulation contains no tunable parameters, while the rank and volume approaches introduce at least one. A full complexity or convergence-rate analysis is outside the paper’s scope; preference remains empirical. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central contribution is an empirical comparison of the SqN (Schatten q-norm) distance on singular values of a feature matrix against rank-based and feature-volume competitors for multi-image registration. It states its key assumption explicitly, introduces the variational formulation, and reports results on dynamic sequences and histological stacks. No equation reduces a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported via self-citation, and no ansatz is smuggled in; the superiority claim is presented as an observed outcome rather than a definitional necessity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that singular values of a feature matrix carry usable alignment information; no free parameters or invented entities are identifiable from the abstract alone.

axioms (1)
  • domain assumption Information about the singular values of a feature matrix of images can be used for alignment.
    Explicitly stated as the key assumption in the abstract.

pith-pipeline@v0.9.0 · 5711 in / 1148 out tokens · 24706 ms · 2026-05-24T17:13:02.035774+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    In: 2004 2nd IEEE In- ternational Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No

    Bhatia, K.K., Hajnal, J.V., Puri, B.K., Edwards, A.D., Rueckert, D.: Consistent groupwise non-rigid registration for atlas construction. In: 2004 2nd IEEE In- ternational Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821). pp. 908–911 Vol. 1 (2004) Variational registration with S qN 11 Time point 5 Time point 11 Time point 21 Fig. 4: Th...

    work page 2004

  2. [2]

    In: International Workshop on Biomedical Image Registration

    Brehmer, K., Wacker, B., Modersitzki, J.: A novel similarity measure for image sequences. In: International Workshop on Biomedical Image Registration. pp. 47–

    work page

  3. [3]

    PAMM 18(1), e201800370 (2018)

    Brehmer, K., Wacker, B., Modersitzki, J.: Simultaneous registration of image se- quences - a novel singular value based images similarity measure. PAMM 18(1), e201800370 (2018)

    work page 2018

  4. [4]

    In: Pajdla, T., Matas, J

    Cootes, T.F., Marsland, S., Twining, C.J., Smith, K., Taylor, C.J.: Groupwise diffeomorphic non-rigid registration for automatic model building. In: Pajdla, T., Matas, J. (eds.) Computer Vision - ECCV 2004. pp. 316–327. Springer Berlin Heidelberg, Berlin, Heidelberg (2004)

    work page 2004

  5. [5]

    IEEE Transactions on Computers 22(1), 67–92 (1973)

    Fischler, M.A., Elschlager, R.A.: The representation and matching of pictorial structures. IEEE Transactions on Computers 22(1), 67–92 (1973)

    work page 1973

  6. [6]

    Educational and Psychological Measurement 41(1), 11–21 (1981)

    Friedman, S., Weisberg, H.F.: Interpreting the first eigenvalue of a correlation matrix. Educational and Psychological Measurement 41(1), 11–21 (1981)

    work page 1981

  7. [7]

    NeuroImage 47(4), 1341–1351 (2009)

    Geng, X., Christensen, G.E., Gu, H., Ross, T.J., Yang, Y.: Implicit reference-based group-wise image registration and its application to structural and functional MRI. NeuroImage 47(4), 1341–1351 (2009)

    work page 2009

  8. [8]

    Scientific Reports 8(1) (2018)

    Guyader, J.M., Huizinga, W., Poot, D.H.J., van Kranenburg, M., Uitterdijk, A., Niessen, W.J., Klein, S.: Groupwise image registration based on a total correlation dissimilarity measure for quantitative MRI and dynamic imaging data. Scientific Reports 8(1) (2018)

    work page 2018

  9. [9]

    Haber, E., Modersitzki, J.: A multilevel method for image registration. SIAM J. Sci. Comput. 27(5), 1594–1607 (2006)

    work page 2006

  10. [10]

    In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006, pp

    Haber, E., Modersitzki, J.: Intensity gradient based registration and fusion of multi- modal images. In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006, pp. 726–733. Springer Berlin Heidelberg (2006)

    work page 2006

  11. [11]

    In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006

    Haber, E., Modersitzki, J.: Intensity gradient based registration and fusion of multi- modal images. In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006. vol. 3216, pp. 591–598 (2006)

    work page 2006

  12. [12]

    Medical Image Analysis 29, 65 – 78 (2016) 12 K

    Huizinga, W., Poot, D., Guyader, J.M., Klaassen, R., Coolen, B., van Kranenburg, M., van Geuns, R., Uitterdijk, A., Polfliet, M., Vandemeulebroucke, J., Leemans, A., Niessen, W., Klein, S.: Pca-based groupwise image registration for quantitative mri. Medical Image Analysis 29, 65 – 78 (2016) 12 K. Brehmer et al. Unregistered S qN4 SqN∞ Total Correlation NG...

    work page 2016

  13. [13]

    NeuroImage 23, S151–S160 (2004)

    Joshi, S., Davis, B., Jomier, M., Gerig, G.: Unbiased diffeomorphic atlas construc- tion for computational anatomy. NeuroImage 23, S151–S160 (2004)

    work page 2004

  14. [14]

    In: 22nd Annual Meeting of the International Society for Magnetic Resonance in Medicine (2014)

    Marschner, H., Pampel, A., M¨ uller, R., Bock, N.A., Weiss, M., Geyer, S., M¨ oller, H.E.: High-resolution quantitative magnetization transfer imaging of post-mortem marmoset brain. In: 22nd Annual Meeting of the International Society for Magnetic Resonance in Medicine (2014)

    work page 2014

  15. [15]

    SIAM (2009)

    Modersitzki, J.: FAIR: Flexible Algorithms for Image Registration. SIAM (2009)

    work page 2009

  16. [16]

    Springer Series in Operations Research, Springer, 2nd edn

    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research, Springer, 2nd edn. (2006)

    work page 2006

  17. [17]

    In: European Conference on Computer Vision

    Papadopoulo, T., Lourakis, M.I.: Estimating the Jacobian of the singular value decomposition: Theory and applications. In: European Conference on Computer Vision. pp. 554–570. Springer (2000)

    work page 2000

  18. [18]

    Medical Image Analysis 46, 15 – 25 (2018)

    Polfliet, M., Klein, S., Huizinga, W., Paulides, M.M., Niessen, W.J., Vandemeule- broucke, J.: Intrasubject multimodal groupwise registration with the conditional template entropy. Medical Image Analysis 46, 15 – 25 (2018)

    work page 2018

  19. [19]

    International Journal of Computer Vision 73(1), 5–39 (2006)

    Schmitt, O., Modersitzki, J., Heldmann, S., Wirtz, S., Fischer, B.: Image regis- tration of sectioned brains. International Journal of Computer Vision 73(1), 5–39 (2006)

    work page 2006

  20. [20]

    Medical Imaging, IEEE Transactions on 32(7), 1153–1190 (2013)

    Sotiras, A., Davatzikos, C., Paragios, N.: Deformable medical image registration: A survey. Medical Imaging, IEEE Transactions on 32(7), 1153–1190 (2013)

    work page 2013

  21. [21]

    Inf Process Med Imaging 22, 648–59 (2011)

    Yigitsoy, M., Wachinger, C., Navab, N.: Temporal groupwise registration for motion modeling. Inf Process Med Imaging 22, 648–59 (2011)

    work page 2011

  22. [22]

    Image and Vision Computing 21, 977–1000 (2003)

    Zitov´ a, B., Flusser, J.: Image registartion methods: a survey. Image and Vision Computing 21, 977–1000 (2003)

    work page 2003