Deformable Registration Using Average Geometric Transformations for Brain MR Images

Guojun Liao; Kehong Yuan; Yongpei Zhu; Zicong Zhou

arxiv: 1907.09670 · v1 · pith:OWJHDRNBnew · submitted 2019-07-23 · 💻 cs.CV

Deformable Registration Using Average Geometric Transformations for Brain MR Images

Yongpei Zhu , Zicong Zhou , Guojun Liao , Kehong Yuan This is my paper

Pith reviewed 2026-05-24 18:05 UTC · model grok-4.3

classification 💻 cs.CV

keywords deformable registrationbrain MRIVoxelMorphJacobian determinantcurl vectorimage registrationdeep learningatlas construction

0 comments

The pith

Adding Jacobian determinant and curl vector as input channels improves VoxelMorph registration of brain MR images.

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

The paper introduces a two-stage training process for the VoxelMorph network that first produces an initial registration field, then extracts its Jacobian determinant and curl vector to serve as extra input channels for a second training round. It also constructs a fixed atlas image by averaging the transformations across a set of brain scans. The authors test the approach on the ADNI and MRBrainS18 datasets and report higher Dice overlap scores together with a larger fraction of non-negative Jacobian values than the original single-pass VoxelMorph. A reader would care because accurate, automatic alignment of brain scans supports diagnosis, longitudinal studies, and quantitative analysis without manual landmark placement.

Core claim

The central claim is that feeding the Jacobian determinant and curl vector of a diffeomorphic registration field as additional channels into a second training pass of VoxelMorph, while using an average transformation atlas as the fixed image, produces deformation fields that register brain MR images more accurately than the baseline method.

What carries the argument

A second training pass of VoxelMorph that receives Jacobian determinant and curl vector channels derived from an initial diffeomorphic field, together with an atlas formed by averaging transformations across training images.

If this is right

Deformation fields exhibit greater smoothness and fewer folding artifacts.
Registered images show higher overlap with the atlas as measured by Dice scores.
The averaged atlas provides a consistent reference for comparing multiple patient scans.
The geometric-channel approach can be inserted into existing VoxelMorph pipelines with only an extra training stage.

Where Pith is reading between the lines

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

The same geometric channels might be useful for registering images from other modalities such as CT.
Explicit geometric inputs could reduce the number of training epochs needed to reach a given accuracy level.
The method points toward hybrid pipelines that combine learned features with classical differential geometry constraints.
If the improvement holds, it would lower the barrier to building population-level brain atlases from routine clinical scans.

Load-bearing premise

That supplying Jacobian determinant and curl vector as extra channels will improve the learned deformation field without introducing artifacts or overfitting to the training distribution.

What would settle it

Running the two-stage model on a held-out test set and finding lower average Dice scores or a higher fraction of negative Jacobian locations than the single-pass VoxelMorph baseline.

Figures

Figures reproduced from arXiv: 1907.09670 by Guojun Liao, Kehong Yuan, Yongpei Zhu, Zicong Zhou.

**Figure 2.** Figure 2: The proposed framework of our method. 3.3 Construction of Atlas(Template) Here, we want to construct an unbiased template using average of transformations according to Fig.4. Step1: for our dataset with N subjects, we take one subject yi out as the initial template, then register yi to all N images yj for j = 1, 2, ..., i, ..., N to get all deformation filed φij with Voxelmorph CNN. Step2: find the average… view at source ↗

**Figure 3.** Figure 3: Generated images based on the deformation field: (a)RGB image of deformation field [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Average of transformation method. well. Let S k F ,S k M(φ) be the voxels for M(φ) and F respectively.DS is computed by: DS(Sk F , S k M(φ) ) = 2 ∗ S k F ∩ S k M(φ) |S k F | + |S k M(φ) | (3) Dice score is between 0 and 1, and Dice score is closer to 1 indicates that the structures are more identical. And vice, a score of 0 indicates that there is no overlap. We also evaluate the regularity of the registra… view at source ↗

**Figure 5.** Figure 5: Comparison of example MR slices (Two anatomical planes) for different experiments: [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison of registration filed φ for different experiments(RGB image and warped grid): (a)(b) φ from VoxelMorph-diff, (c)(d) φ from VoxelMorph-registration, (e)(f)φ from VoxelMorphdeformation. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: Results of anatomical segmentation for different experiments: (a)anatomical labels [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

**Figure 8.** Figure 8: Analysis of velocity sampling and uncertainty for different experiments(top is veloc [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 9.** Figure 9: Results of atlas construction: (a)registration field after first iteration, (b) registration field [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

read the original abstract

Accurate registration of medical images is vital for doctor's diagnosis and quantitative analysis. In this paper, we propose a new deformable medical image registration method based on average geometric transformations and VoxelMorph CNN architecture. We compute the differential geometric information including Jacobian determinant(JD) and the curl vector(CV) of diffeomorphic registration field and use them as multi-channel of VoxelMorph CNN for second train. In addition, we use the average transformation to construct a standard brain MRI atlas which can be used as fixed image. We verify our method on two datasets including ADNI dataset and MRBrainS18 Challenge dataset, and obtain excellent improvement on MR image registration with average Dice scores and non-negative Jacobian locations compared with MIT's original method. The experimental results show the method can achieve better performance in brain MRI diagnosis.

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.

Desk Editor's Note private letter to a colleague

Small VoxelMorph extension that adds JD/CV channels in a second pass plus an average atlas, but leaves test-time channel generation and all numbers unspecified.

read the letter

The main takeaway is that this paper runs VoxelMorph once, pulls Jacobian determinant and curl vector from its output field, then trains a second copy of the same network with those quantities as extra input channels while also building an average atlas to serve as the fixed image. They evaluate on ADNI and MRBrainS18 and state that the result improves Dice and non-negative Jacobian counts over the original MIT code. That specific combination of average geometric descriptors with a second training stage on VoxelMorph is not in the cited prior work, so the pipeline itself counts as new even if the individual pieces are known. The paper does a clean job of stating the geometric motivation and choosing two standard brain MRI datasets for testing. The citation pattern is appropriate and there is no circularity in the evaluation since it uses held-out sets. The soft spots are straightforward and fairly large. The abstract contains no numerical Dice values, no standard deviations, no p-values, and no ablation that separates the contribution of the geometric channels from the atlas step or the extra training. More critically, the stress-test point stands: because JD and CV are computed from the deformation field, the second network needs those channels at inference time for any new pair. The description gives no procedure for generating them on the fly, no mention of a cascade, and no confirmation that the reported gains hold when the channels are not supplied from the same data used in training. Without those details the method cannot be checked or reproduced. This is aimed at people already running VoxelMorph who might want to try adding geometric features. A reader looking for a ready-to-use improvement or a clear ablation study will not find enough here. I would not bring it to reading group and would not send it for peer review until the inference mechanism and the actual results tables are added.

Referee Report

3 major / 2 minor

Summary. The paper proposes a deformable registration method for brain MR images that first computes Jacobian determinant (JD) and curl vector (CV) from a VoxelMorph deformation field, feeds these as additional input channels for a second training pass of the same CNN architecture, and constructs a standard atlas via average transformations. It claims this yields improved average Dice scores and non-negative Jacobian determinants on the ADNI and MRBrainS18 datasets relative to the original MIT VoxelMorph method.

Significance. If the geometric-channel augmentation and inference procedure can be shown to work without circularity or data leakage, the approach would offer a lightweight way to enforce diffeomorphic properties in learning-based registration; however, the current manuscript supplies no quantitative results, ablations, or inference protocol, so the practical significance cannot yet be assessed.

major comments (3)

[Abstract, §3] Abstract and §3 (method): The inference procedure for the JD/CV channels is unspecified. Because these quantities are functions of the predicted deformation field, the second network cannot receive its required inputs for an unseen pair unless a cascaded first-then-second network procedure or a dataset-level average is used; neither mechanism is described, nor is any ablation isolating the contribution of the geometric channels provided.
[Abstract, results] Abstract and results section: The central claim of 'excellent improvement' on Dice scores and non-negative Jacobian locations is asserted without any numerical values, standard deviations, statistical tests, or comparison tables, rendering the empirical support unverifiable.
[§3] §3: No description is given of how the average transformation atlas is constructed or whether it is used only at training time or also at test time; this detail is load-bearing for reproducibility of the reported registration accuracy.

minor comments (2)

[Abstract] The abstract states 'average Dice scores' but does not specify which anatomical labels or overlap metric is used.
[§3] Notation for the curl vector (CV) and its multi-channel concatenation with JD is introduced without an equation or diagram.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight areas where the manuscript requires greater clarity and detail. We will revise the paper to address all three major comments by expanding the method description, adding quantitative results and tables, and clarifying the atlas construction and usage. Point-by-point responses follow.

read point-by-point responses

Referee: [Abstract, §3] Abstract and §3 (method): The inference procedure for the JD/CV channels is unspecified. Because these quantities are functions of the predicted deformation field, the second network cannot receive its required inputs for an unseen pair unless a cascaded first-then-second network procedure or a dataset-level average is used; neither mechanism is described, nor is any ablation isolating the contribution of the geometric channels provided.

Authors: We agree the inference procedure for the JD/CV channels was insufficiently specified. The method employs a cascaded procedure: an initial VoxelMorph network predicts the deformation field on an input pair, after which the Jacobian determinant and curl vector are computed from that field and supplied as additional input channels to a second network of identical architecture. This cascaded process is used at inference time for unseen pairs. We will add an explicit description of the cascaded inference protocol together with ablations that isolate the geometric-channel contribution in the revised manuscript. revision: yes
Referee: [Abstract, results] Abstract and results section: The central claim of 'excellent improvement' on Dice scores and non-negative Jacobian locations is asserted without any numerical values, standard deviations, statistical tests, or comparison tables, rendering the empirical support unverifiable.

Authors: We acknowledge that the abstract and results section do not present the specific numerical Dice scores, standard deviations, statistical tests, or comparison tables needed to substantiate the claims. We will insert detailed quantitative results, including mean Dice scores with standard deviations, p-values, and side-by-side tables versus the original VoxelMorph baseline, in the revised results section. revision: yes
Referee: [§3] §3: No description is given of how the average transformation atlas is constructed or whether it is used only at training time or also at test time; this detail is load-bearing for reproducibility of the reported registration accuracy.

Authors: We agree that the construction and usage of the average transformation atlas must be described. The atlas is formed by averaging the deformation fields produced by registering all training images to a chosen reference; the resulting average field is then applied to create a standard atlas image that serves as the fixed image during both training and test-time registration. We will supply the precise construction procedure, mathematical formulation, and usage statement in the revised §3. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical pipeline evaluated on held-out data

full rationale

The paper describes an empirical extension to VoxelMorph that adds Jacobian determinant and curl vector channels computed from an initial registration field, followed by atlas construction via averaging and evaluation on ADNI and MRBrainS18 test sets. No equation, claim, or result reduces by construction to a fitted parameter defined from the same quantity, no self-citation chain bears the central result, and no uniqueness theorem or ansatz is imported from prior author work. Performance metrics are reported as external verification rather than internal re-derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated.

axioms (1)

domain assumption Diffeomorphic registration fields are smooth and invertible so that Jacobian determinant and curl are well-defined and meaningful.
Implicit in the use of JD and CV as input channels for a diffeomorphic registration network.

pith-pipeline@v0.9.0 · 5668 in / 1316 out tokens · 37142 ms · 2026-05-24T18:05:39.599426+00:00 · methodology

discussion (0)

Reference graph

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