Physics Maths Engineering
Arbitrary Order Total Variation for Deformable Image Registration
Jinming Duan,
Xi Jia,
Joseph Bartlett,
Wenqi Lu
Peer Reviewed
Abstract
In this work, we investigate image registration in a variational framework and focus on regularization generality and solver efficiency. We first propose a variational model combining the state-of-the-art sum of absolute differences (SAD) and a new arbitrary order total variation regularization term. The main advantage is that this variational model preserves discontinuities in the resultant deformation while being robust to outlier noise. It is however non-trivial to optimize the model due to its non-convexity, non-differentiabilities, and generality in the derivative order. To tackle these, we propose to first apply linearization to the model to formulate a convex objective function and then break down the resultant convex optimization into several point-wise, closed-form subproblems using a fast, over-relaxed alternating direction method of multipliers (ADMM). With this proposed algorithm, we show that solving higher-order variational formulations is similar to solving their lower-order counterparts. Extensive experiments show that our ADMM is significantly more efficient than both the subgradient and primal-dual algorithms particularly when higher-order derivatives are used, and that our new models outperform state-of-the-art methods based on deep learning and free-form deformation. Our code implemented in both Matlab and Pytorch is publicly available at https://github.com/j-duan/AOTV.
Key Questions
What is image registration, and why is it important?
Image registration is the process of aligning two or more images into a common coordinate system. It is crucial in medical imaging, satellite imagery, and computer vision for tasks like comparing scans, tracking changes over time, or combining data from different sources.
What is the variational framework used in this study?
The study uses a variational framework to model image registration. This framework combines the Sum of Absolute Differences (SAD) for measuring image similarity and a novel arbitrary-order Total Variation (TV) regularization term to preserve discontinuities in the deformation field.
What makes this variational model unique?
The model is unique because it introduces arbitrary-order TV regularization, which allows for higher-order derivatives. This makes it robust to outlier noise while preserving sharp discontinuities in the deformation field, improving registration accuracy.
Why is optimizing this model challenging?
The model is non-convex, non-differentiable, and involves arbitrary-order derivatives, making optimization difficult. To address this, the study proposes a linearization step and a fast, over-relaxed Alternating Direction Method of Multipliers (ADMM) algorithm.
What is ADMM, and how does it help?
ADMM is an optimization algorithm that breaks down complex problems into simpler, point-wise subproblems. In this study, ADMM makes solving higher-order variational models as efficient as solving lower-order ones, significantly speeding up computation.
How does this method compare to deep learning approaches?
The proposed method outperforms state-of-the-art deep learning and free-form deformation techniques in terms of accuracy and efficiency, especially when higher-order derivatives are involved. It also avoids the need for large training datasets.
What are the practical applications of this method?
This method is useful in medical imaging (e.g., aligning MRI or CT scans), satellite imagery (e.g., tracking environmental changes), and computer vision (e.g., object tracking or 3D reconstruction).
How efficient is the proposed ADMM algorithm?
The ADMM algorithm is significantly faster than traditional subgradient and primal-dual methods, especially for higher-order models. This makes it practical for real-world applications where speed and accuracy are critical.
Is the code for this method available?
Yes, the code is publicly available on GitHub in both Matlab and Pytorch. You can access it Arbitrary Order Total Variation (AOTV) for Deformable Image Registration.
What are the key advantages of this approach?
The key advantages are:
- Robustness to outlier noise.
- Preservation of discontinuities in the deformation field.
- Efficient optimization using ADMM, even for higher-order models.
- Superior performance compared to deep learning and free-form deformation methods.
What are the future directions for this research?
Future research could explore extending this method to 3D image registration, integrating it with deep learning for hybrid models, or applying it to real-time applications like video processing.
