On the Degrees of Freedom of Gridded Control Points in Learning-Based Medical Image Registration

Wen Yan; Qianye Yang; Yipei Wang; Shonit Punwani; Mark Emberton; Vasilis Stavrinides; Yipeng Hu; Dean Barratt

eess.IVMar 15, 2026

On the Degrees of Freedom of Gridded Control Points in Learning-Based Medical Image Registration

Authors: Wen Yan, Qianye Yang, Yipei Wang, Shonit Punwani, Mark Emberton, Vasilis Stavrinides, Yipeng Hu, Dean Barratt

Organizations: UCL Hawkes Institute; Department of Medical Physics and Biomedical Engineering, University College London, Gower St, London WC1E 6BT, London, U.K. · Centre for Medical Imaging, Division of Medicine, University College London, Gower St, London WC1E 6BT, London, U.K. · Division of Surgery and Interventional Science, University College London, 10 Pond St, London NW3 2PS, U.K. · Cancer Institute, Urology Department, UCL Hospital, University College London, 16-18 Westmoreland St, London W1G 8PH, U.K ·

Abstract

Many registration problems are ill-posed in homogeneous or noisy regions, and dense voxel-wise decoders can be unnecessarily high-dimensional. A sparse control-point parameterisation provides a compact, smooth deformation representation while reducing memory and improving stability. This work investigates the required control points for learning-based registration network development. We present GridReg, a learning-based registration framework that replaces dense voxel-wise decoding with displacement predictions at a sparse grid of control points. This design substantially cuts the parameter count and memory while retaining registration accuracy. Multiscale 3D encoder feature maps are flattened into a 1D token sequence with positional encoding to retain spatial context. The model then predicts a sparse gridded deformation field using a cross-attention module. We further introduce grid-adaptive training, enabling an adaptive model to operate at multiple grid sizes at inference without retraining. This work quantitatively demonstrates the benefits of using sparse grids. Using three data sets for registering prostate gland, pelvic organs and neurological structures, the results suggested a significant improvement with the usage of grid-controled displacement field. Alternatively, the superior registration performance was obtained using the proposed approach, with a similar or less computational cost, compared with existing algorithms that predict DDFs or displacements sampled on scattered key points.

Figures & tables

Figure 1: Demonstration of an illustrative example of false-positive correspondence (orange stars), possibly caused by similar intensity patterns (e.g., unsupervised loss similarity) or label errors (e.g., segmentation noise). A sparse low-resolution grid helps filter out such a noise, as stronger “true” correspondences (four corners of the shaded ROIs) dominate. In contrast, a finer high-resolution grid may amplify the noise, distorting additional control points within the ROIs and leading to inaccurate registration.

Figure 2: Illustration of GridReg. The encoder extracts multi-scale 3D feature maps; at each scale, a skip-projection flattens features into 1D tokens that are fused with grid-cell queries via (local) attention. The resulting features are mapped to a sparse control-point displacement field via Bayesian integration, and a dense displacement field (DDF) is obtained by interpolation (e.g., trilinear, B-spline, or transposed convolution).

Figure 3: Examples of anatomical landmarks annotated for the same patient at two different time points. The first and second rows show 5 landmarks at the first and second time points, respectively.

Figure 4: Examples of 7 landmarks labeled by experts in brain dataset.

Dataset	Metric	Original	ANTs SyN	VoxelMorph	TransMorph	KeyMorph	GridReg(Ours)
Prostate	Dice $\uparrow$	0.70 $\pm$ 0.10	$0.81\pm 0.05$	0.86 $\pm$ 0.02	0.84 $\pm$ 0.03	0.86 $\pm$ 0.05	0.88 $\pm$ 0.02 ‡
	$D^{ldmk}_{CD}\downarrow$ (mm)	9.60 $\pm$ 4.09	9.09 $\pm$ 8.09	8.76 $\pm$ 8.59	7.73 $\pm$ 6.98	3.98 $\pm$ 2.14	4.54 $\pm$ 2.77 †
	$D^{msk}_{CD}\downarrow$ (mm)	8.76 $\pm$ 4.04	3.38 $\pm$ 0.78	2.70 $\pm$ 0.54	3.05 $\pm$ 0.84	2.56 $\pm$ 0.66	1.87 $\pm$ 1.16 ‡
	GPU(Mb)	/	/	208	377	274	146
Pelvis	Dice $\uparrow$	$0.53\pm 0.17$	0.69 $\pm 0.09$	0.73 $\pm$ 0.16	0.73 $\pm$ 0.09	0.79 $\pm$ 0.07	0.78 $\pm$ 0.09 †
	$D^{msk}_{CD}\downarrow$ (mm)	10.65 $\pm 5.85$	6.03 $\pm 2.78$	5.99 $\pm 5.68$	4.47 $\pm 2.95$	4.14 $\pm$ 2.47	4.52 $\pm$ 2.88 †

Table 1: Performance and memory across methods. Values are mean

\pm

SD over identical holdout test cases. Proposed methods appear in the last columns (GridReg). For each dataset, we compared GridReg to each baseline using paired, one-sided t-tests and controlled multiplicity within the family using BH-FDR (

5\%

); we report q-values. Bold indicates the best mean.

{\ddagger}

denotes GridReg is significantly better than all baselines (

q<0.05

);

{\dagger}

denotes it is significantly better than three of baselines (

q<0.05

). Slash

/

indicates that a result is not available.

Figure 5: Visualisation of prostate registration. The first column and the second column show four examples of the fixed and moving images, respectively. The next four columns show the registration results from (A)GridReg, (B) VoxelMorph, (C) KeyMorph, (D)TransMorph and (E) ANTs SyN, respectively.

Figure 6: Visualisation of brain registration. The first column and the second column show four examples of the fixed and moving images, respectively. The next four columns show the registration results from (A)GridReg, (B) VoxelMorph, (C) KeyMorph, (D)TransMorph and (E) ANTs SyN, respectively.

Setting	Grid	Base-NC	Upsample	Projector	Bayesian	Dice	$D_{CD}^{lmdk}$ (mm)	Cost(Mb)
Config 1	5	32	trilinear	$\checkmark$	$\checkmark$	0.86 $\pm$ 0.05 ∗	5.29 $\pm$ 2.85	146
Config 2	10	32	trilinear	$\checkmark$	$\checkmark$	0.89 $\pm$ 0.03	4.89 $\pm$ 3.19	148
Config 3	10	8	trilinear	$\checkmark$	$\checkmark$	0.87 $\pm$ 0.04	5.88 $\pm$ 3.27	144
Config 4	10	16	trilinear	$\checkmark$	$\checkmark$	0.88 $\pm$ 0.02	4.54 $\pm$ 2.77	146
Config 5	10	16	trilinear	$\times$	$\checkmark$	0.86 $\pm$ 0.03 ∗	6.06 $\pm$ 3.23 ∗	145
Config 6	10	16	deconv	$\checkmark$	$\checkmark$	0.89 $\pm$ 0.04	4.18 $\pm$ 2.88	171

Table 3: Ablation study of proposed method with varying configuration of grid number, base channel of encoder, upsampling method and usage of projection module. The ablation studies are performed on holdout test set in Dataset 1. The symbol * represents that the best results have a statistical difference between Config 4 and other methods (all

p-values<

0.050).

Figure 7: This figure shows four examples of uncertainty maps generated by the Bayesian Grid transformer. A lighter area in the Uncertainty map represents lower uncertainty, while a darker area represents higher uncertainty.

Dataset	Metric	GridReg Bayesian	GridReg	KeyMorph	VoxelMorph	TransMorph
Prostate	$\log\det(J)$	$0.0352\pm 0.0193^{\dagger}$	$0.0446\pm 0.0228^{*}$	$13.3064\pm 32.1452$	$1.9258\pm 1.0610$	$0.0975\pm 0.0532$
Prostate	$\det(J)<0$ (%)	$0$	$0$	$3.8$	$0.4$	$0.03$
Brain	$\log\det(J)$	$0.0022\pm 0.0015^{\dagger}$	$0.0038\pm 0.0014^{*}$	$55.2012\pm 133.4529$	$1.0793\pm 0.3254$	$15.9102\pm 6.5335$
Brain	$\det(J)<0$ (%)	$0$	$0$	$11.0$	$0.2$	$4.5$

Table 4: Jacobian statistics across methods.

\log\det(J)

is reported as mean

\pm

standard deviation; folding rate is the percentage of voxels with

\det(J)<0

. For each dataset, we report paired, one-sided t-tests with BH-FDR(5%) with confidence

\alpha=0.05

\dagger

and ∗ denote GridReg Bayesian and GridReg is significantly better than other methods, respectively.

Figure 8: (1)Effect of Bending Energy Weight on Segmentation Accuracy (DSC) and Contour Distance (CD); (2)-(3)Comparisons on Dataset1 showing Dice and centroid distance between separately trained and auto-adapted models with the same grid size range from 5 to 15.

Medical Image Registration

Medical image registration aligns scans across time, patients, or imaging modalities to track anatomy and support diagnosis, treatment, and image-guided procedures.

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We present MMD-Reg, a novel correspondence-free approach to point-cloud registration that is differentiable and has linear computational complexity in the number of points. We model registration as a nonlinear least-squares problem based on the Maximum Mean Discrepancy, approximated using random Fourier features. The resulting objective can be solved efficiently with standard methods such as Levenberg-Marquardt, and the solution is differentiable via the implicit function theorem. This allows MMD-Reg to be used as a differentiable optimization layer within end-to-end trainable models, supporting registration under challenging conditions such as poor initial alignment and partial overlap. We demonstrate this Neural MMD-Reg formulation by integrating the layer with a set transformer, training the resulting model in supervised and unsupervised settings, and comparing its performance against recent learning-based methods. We also evaluate standalone MMD-Reg, comparing its accuracy and scalability against widely used non-learning-based registration methods.

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