# On the degrees of freedom of gridded control points in learning‐based medical image registration

## TL;DR

This paper introduces GridReg, a new method for medical image registration that uses sparse control points to reduce computational cost while maintaining accuracy.

## Contribution

The novel contribution is a registration framework that uses sparse gridded control points with cross-attention and grid-adaptive training for efficient and accurate registration.

## Key findings

- GridReg achieves improved computational efficiency by predicting sparse-grid displacements.
- The method outperforms existing algorithms in registration accuracy with similar or lower compute cost.
- GridReg supports adaptive inference at multiple grid sizes without retraining.

## Abstract

Many registration problems are ill‐posed in homogeneous/noisy regions, and dense voxel‐wise decoders can be unnecessarily high‐dimensional. A sparse control‐point parameterization 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. In particular, as sparse as 5×5×5 control points are configured and compared with alternative approaches, including those using scattered control points and displacements sampled at every voxel, that is, dense displacement fields.

We present GridReg, a learning‐based registration frametwork 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: Each control point attends to encoder tokens within its local grid neighborhood to estimate its displacement, which is subsequently interpolated to a dense field. 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 experimental results suggest a much improved computational efficiency, due to the prediction of sparse‐grid‐sampled displacements. Alternatively, the superior registration performance was obtained using the proposed approach, with the similiar or less compute cost, compared with existing algorithms that predict DDFs (e.g., VoxelMorph/TransMorph) or displacements sampled on scattered key points (KeyMorph).

We conclude that predicting sparsely gridded displacements provides reduced computational cost and/or improved performance, independent of the encoder architecture, and can be readily implemented. Therefore, GridReg should potentially be considered for many registration tasks with adaptive grid sizes. The code is available via git@github.com:yanwenCi/GridReg.git.