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Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks

Tony C. W. Mok, Albert C. S. Chung

arXiv:2003.09514v3cs.CV

TL;DR

CNN-based deformable registration is fast but often does not guarantee topology preservation, local smoothness, or inverse transformations. The paper presents an unsupervised symmetric CNN method operating in diffeomorphic-map space and evaluates it on 3D brain MR scans. It reports state-of-the-art accuracy and running time while maintaining desirable diffeomorphic properties.

  • Problem

    Existing CNN-based registration methods often overlook topology preservation and invertibility, while global smoothness regularization may not ensure local orientation consistency.

  • Method

    The method learns symmetric deformation fields within diffeomorphic-map space and estimates paired transformations that can produce forward and inverse mappings.

  • Results

    On a large-scale dataset of over 400 brain scans, the method achieves state-of-the-art registration accuracy while producing transformations more consistent with diffeomorphic properties.

  • Takeaways & Limitations

    The approach maintains topology-preserving and invertible transformations while providing fast registration for 3D brain MR images.

  • Takeaways & Limitations

    The experiments assume affine and scaling transformations have already been factored out through preprocessing into MNI152 space.

Abstract

from arXiv · show

Diffeomorphic deformable image registration is crucial in many medical image studies, as it offers unique, special properties including topology preservation and invertibility of the transformation. Recent deep learning-based deformable image registration methods achieve fast image registration by leveraging a convolutional neural network (CNN) to learn the spatial transformation from the synthetic ground truth or the similarity metric. However, these approaches often ignore the topology preservation of the transformation and the smoothness of the transformation which is enforced by a global smoothing energy function alone. Moreover, deep learning-based approaches often estimate the displacement field directly, which cannot guarantee the existence of the inverse transformation. In this paper, we present a novel, efficient unsupervised symmetric image registration method which maximizes the similarity between images within the space of diffeomorphic maps and estimates both forward and inverse transformations simultaneously. We evaluate our method on 3D image registration with a large scale brain image dataset. Our method achieves state-of-the-art registration accuracy and running time while maintaining desirable diffeomorphic properties.

1. Introduction

The paper introduces a fast, unsupervised symmetric registration method that uses CNNs within diffeomorphic maps to preserve topology and invertibility. It demonstrates strong accuracy and diffeomorphic behavior on a large 3D brain MR dataset.

  • Motivation: Deformable registration establishes nonlinear correspondence between images, supporting anatomical comparison across subjects, sensors, and acquisition times.It can also enable automatic brain-structure delineation by registering scans to a well-delineated atlas.
  • Motivation: Traditional registration methods iteratively optimize energy functions, making them computationally intensive and time-consuming in practice.
  • Method: The proposed CNN method learns symmetric deformations in diffeomorphic-map space and outputs paired maps toward the images’ geodesic middle ground.Forward mappings are obtained by composing the estimated maps with the inverse of the other map.
  • Contributions: The method guarantees topology preservation and transformation invertibility, while orientation-consistent regularization penalizes regions with negative Jacobian determinants.
  • Evaluation: On over 400 T1-weighted brain scans, the method achieves state-of-the-art registration accuracy while producing transformations more consistent with diffeomorphic properties.

2. Background

Deformable registration minimizes image dissimilarity while regularizing deformation smoothness, but displacement-field parameterizations do not guarantee invertibility. The paper instead uses stationary velocity fields and diffeomorphic flows to ensure smooth, one-to-one, topology-preserving mappings.

  • Deformable registration: Deformable registration seeks an optimal displacement field that minimizes dissimilarity between a fixed image and warped moving image while maintaining smoothness.
  • Deformable registration: The experiments assume affine and scaling transformations have already been factored out, with brain scans affinely registered to MNI152 space during preprocessing.
  • Diffeomorphic registration: Direct displacement-field parameterizations do not guarantee a true inverse or one-to-one mapping, especially for large deformations.
  • Diffeomorphic registration: The method implements diffeomorphic deformation with a stationary velocity field, whose differentiable and invertible maps preserve topology.
  • Diffeomorphic registration: Exponentiating the stationary velocity field produces a time-1 deformation that is diffeomorphic and invertible using the same flow field.

3. Related Work

Earlier registration methods relied on iterative optimization or displacement fields, while CNN-based methods enabled fast unsupervised learning. The paper identifies persistent gaps in topology preservation, local orientation consistency, and inverse consistency.

  • Classical methods: Classical methods iteratively optimize constrained deformation models to minimize an energy function.
  • Classical methods: Traditional approaches regularize displacement-field smoothness with energy functions or Gaussian filtering, including Demons, B-splines, DRAMMS, MRF, and SPM methods.
  • Learning-based methods: Supervised CNN registration methods commonly use ground-truth deformation fields or anatomical segmentation maps as learning guidance.
  • Learning-based methods: Unsupervised CNN methods combine spatial transformers with differentiable similarity functions to learn dense mappings between image pairs.
  • Remaining gaps: Most CNN-based methods parameterize registration with displacement vectors and overlook topology preservation and invertibility.
  • Remaining gaps: Global smoothness regularization does not ensure local orientation consistency, and existing methods generally neither consider nor guarantee inverse transformations.

4. Method

The method uses a fully convolutional network to estimate symmetric diffeomorphic deformations in both directions, warping X and Y toward a mean shape before composing transformations. Similarity losses enforce bidirectional alignment, while selective Jacobian regularization encourages local orientation consistency and topology preservation.

  • Symmetric Diffeomorphic Neural Network: The network jointly estimates deformation fields from X and Y without assigning fixed or moving identities, supporting symmetric registration.It predicts two dense, nonlinear velocity fields from the paired inputs.
  • Symmetric Diffeomorphic Neural Network: Time 0.5 diffeomorphic fields warp both images toward their mean shape, and composing them yields transformations between X and Y.The fields are integrated with scaling and squaring, while inverse transformations follow from diffeomorphic mappings.
  • Network Architecture: The FCN uses a five-level encoder-decoder with skip connections to estimate the two velocity fields from concatenated 3D image inputs.Encoder features are downsampled, decoder features are upsampled, and feature maps are linked through skip connections.
  • Symmetric Similarity: The symmetric similarity loss combines mean-shape and pairwise dissimilarity terms, encouraging alignment in both transformation directions.The mean-shape term compares both images warped toward M, while the pairwise term compares mappings between X and Y.
  • Local Orientation Consistency: Selective Jacobian determinant regularization penalizes locally negative determinants rather than applying only a global smoothness penalty.Negative determinants indicate local orientation reversal and loss of one-to-one mapping; the regularizer uses a ReLU-equivalent activation.

5.1. Data and Pre-processing

The evaluation uses 425 preprocessed T1-weighted brain MRI scans and an atlas-based task registering 725 test-set volume pairs per method.

  • 5.1. Data and Pre-processing: 425 T1-weighted brain MRI scans from OASIS were resampled to 256 × 256 × 256 at 1mm×1mm×1mm and preprocessed with FreeSurfer.Preprocessing included motion correction, skull stripping, affine spatial normalization, and subcortical structure segmentation; the scans were then center cropped.
  • 5.1. Data and Pre-processing: Five randomly selected test-set MR volumes served as atlases, and each method registered the remaining test volumes to them, totaling 725 pairs per method.For SYMNet, X was set to the atlas and Y to the moving subject.
  • 5.1. Data and Pre-processing: Figure 3 compares atlas, moving, warped, and deformation-field slices for DIF-VM, VM, and the proposed method, marking non-positive Jacobian regions in red.Red circles identify artifacts on the left and right putamen in the DIF-VM result.

5.2. Measurement

Registration quality is assessed through anatomical overlap and local deformation behavior, using DSC and Jacobian determinants to evaluate correspondence and diffeomorphic properties.

  • 5.2. Measurement: The evaluation warps each subject’s anatomical segmentation to the atlas and then computes overlap using the resulting deformation field.This procedure is used because an ideal ground-truth nonlinear deformation field is not well-defined.
  • 5.2. Measurement: DSC measures spatial overlap between atlas and warped-volume anatomical segmentations, with higher scores indicating higher anatomical correspondence.The analysis includes 26 anatomical structures, and DSC ranges from 0 to 1.
  • 5.2. Measurement: Figure 4 presents anatomical-structure Dice-score boxplots for SyN, DIF-VM, VM(λ = 10), and the proposed method, combining hemispheres for visualization.The figure includes structures such as brain stem, thalamus, putamen, hippocampus, ventricles, CSF, and cerebral cortex.
  • 5.2. Measurement: Positive Jacobian determinants indicate locally diffeomorphic regions, whereas negative determinants indicate violations of topology preservation and invertibility.The Jacobian captures local deformation behavior including shearing, stretching, and rotation.

5.3. Baseline Methods

The study compares the proposed method with SyN and two recent unsupervised CNN-based registration methods, using pairwise training data and varied VM regularization weights.

  • 5.3. Baseline Methods: SyN is compared with VM and DIF-VM, two recent unsupervised deformable registration methods, as well as the proposed approach.SyN is described as a top-performing algorithm among 14 typical nonlinear deformation algorithms.
  • 5.3. Baseline Methods: VM and DIF-VM were trained from scratch using pairwise image-volume pairs from the training set, excluding atlases from training.This differs from the experiment settings reported in the referenced prior methods.
  • 5.3. Baseline Methods: VM was trained with different regularizer weights to study the effect of regularization on registration performance.The comparison includes VM variants alongside the other baseline methods.

5.4. Implementation

SYMNet is implemented in PyTorch and trained with stochastic gradient descent, with parameters selected through grid search and model selection based on validation Dice score.

  • 5.4. Implementation: SYMNet uses PyTorch with stochastic gradient descent, a learning rate of 1e−4, and momentum of 0.9.The network was trained on a GTX 1080Ti GPU.
  • 5.4. Implementation: The best reported parameter setting is λ1 = 1000, λ2 = 3, and λ3 = 0.1, with all parameters tuned by grid search.The selected model achieved the highest Dice score on the validation set.

5.5. Results

Experiments show that SYMNet combines strong registration accuracy with diffeomorphic deformation fields and substantially faster runtime than traditional SyN.

  • Registration performance: Learning-based methods outperform SyN in average DSC, but VM produces many voxels with non-positive Jacobian determinant.Thus, registration accuracy alone does not ensure diffeomorphic results.
  • Registration performance: SYMNet achieves the overall best average DSC while keeping voxels with non-positive Jacobian determinant close to zero.This combines high overlap accuracy with the desirable diffeomorphic property.
  • Registration performance: SYMNet achieves the best DSC across all anatomical structures among methods with diffeomorphic properties.The comparison is shown through the DSC distributions for individual anatomical structures.
  • Orientation-consistent loss: The local orientation-consistency loss guides correction of violations caused by discrete deformation representations and velocity-field interpolation.Compared with VM’s global regularization, it maintains diffeomorphic solutions without excessively sacrificing registration accuracy.
  • Runtime analysis: Learning-based methods register a subject MR volume to an atlas in less than a second, whereas SyN is substantially slower.The runtime comparison uses an Intel i7-7700 CPU and NVIDIA GTX1080Ti GPU, excluding affine normalization.

6. Conclusion

The paper presents a fast CNN-based symmetric diffeomorphic registration method that aligns image pairs to their mean shape and uses a local orientation-consistency loss. On a large-scale brain MR dataset, it outperforms traditional and unsupervised learning-based methods in registration accuracy and deformation-field quality.

  • The method learns symmetric deformation fields that align image pairs to their mean shape within diffeomorphic maps.
  • A local orientation-consistency loss uses the Jacobian determinant to further guarantee desirable diffeomorphic properties.
  • On a large-scale brain MR dataset, the method outperforms traditional and learning-based methods in registration accuracy and deformation-field quality.
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