[Paper Review] On Reducing Negative Jacobian Determinant of the Deformation Predicted by Deep Registration Networks

Dongyang Kuang|arXiv (Cornell University)|Jun 28, 2019

Medical Image Segmentation Techniques17 references4 citations

TL;DR

This paper proposes two training mechanism improvements—cycle consistent design and a refinement module—to reduce negative Jacobian determinant in deep unsupervised image registration networks, significantly decreasing folding locations in deformations without sacrificing registration accuracy or altering network architecture.

ABSTRACT

Image registration is a fundamental step in medical image analysis. Ideally, the transformation that registers one image to another should be a diffeomorphism that is both invertible and smooth. Traditional methods like geodesic shooting approach the problem via differential geometry, with theoretical guarantees that the resulting transformation will be smooth and invertible. Most previous research using unsupervised deep neural networks for registration have used a local smoothness constraint (typically, a spatial variation loss) to address the smoothness issue. These networks usually produce non-invertible transformations with ``folding'' in multiple voxel locations, indicated by a negative determinant of the Jacobian matrix of the transformation. While using a loss function that specifically penalizes the folding is a straightforward solution, this usually requires carefully tuning the regularization strength, especially when there are also other losses. In this paper we address this problem from a different angle, by investigating possible training mechanisms that will help the network avoid negative Jacobians and produce smoother deformations. We contribute two independent ideas in this direction. Both ideas greatly reduce the number of folding locations in the predicted deformation, without making changes to the hyperparameters or the architecture used in the existing baseline registration network.

Motivation & Objective

• Address the issue of non-invertible deformations in unsupervised deep learning-based medical image registration, particularly those with negative Jacobian determinant indicating folding.
• Overcome the limitations of standard regularization, which often fails to eliminate folding even with high hyperparameter tuning.
• Improve geometric fidelity of predicted transformations without modifying network architecture or introducing new loss functions.
• Maintain high registration accuracy while enhancing smoothness and invertibility of deformation fields.

Proposed method

• Introduce a cycle consistent design that trains the network to perform forward and backward transformations, enforcing invertibility by minimizing the discrepancy between the original and reconstructed input after forward-backward pass.
• Implement a refinement module that iteratively corrects the predicted displacement field using a separate network, refining the deformation to reduce folding.
• Apply the cycle consistent loss during training without modifying the original loss function, using a shared encoder-decoder structure for both forward and backward mappings.
• Use a refinement network with residual blocks and Leaky ReLU activations to iteratively improve the displacement field, with four refinement steps per prediction.
• Train the entire system using Adam optimizer with a fixed learning rate, preserving the baseline U-Net architecture for deformation prediction.
• Evaluate performance using dice score for registration accuracy and the percentage of voxels with negative Jacobian determinant as a measure of folding.

Experimental results

Research questions

• RQ1 Can training mechanisms be designed to reduce negative Jacobian determinant in deep unsupervised registration networks without altering loss functions or architecture?
• RQ2 Does enforcing cycle consistency during training improve the invertibility and smoothness of predicted deformations?
• RQ3 Can iterative refinement of the displacement field reduce folding while maintaining high registration accuracy?
• RQ4 How do these methods compare to standard regularization in terms of geometric fidelity and generalization across datasets?
• RQ5 To what extent can these methods reduce folding without degrading registration performance?

Key findings

• The cycle consistent design reduced the percentage of voxels with negative Jacobian determinant from 1.97% (baseline) to 0.13% on average across the test set.
• The refinement module reduced the negative Jacobian percentage to 0.20%, with a lower standard deviation (0.02%) compared to the baseline (0.73%).
• Despite significant improvements in geometric properties, both methods maintained high registration accuracy, as shown by dice scores comparable to the baseline.
• Visualizations of warped grids confirmed that both methods effectively eliminated folding while preserving smoothness in non-folding regions.
• The cycle consistent approach outperformed the refinement module in reducing folding, eliminating over 90% of the problematic locations compared to the baseline.
• The proposed methods are compatible with existing registration networks and can be combined with other smoothness-enhancing techniques without architectural changes.