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PULPo: Probabilistic Unsupervised Laplacian Pyramid Registration

Leonard Siegert, Paul Fischer, Mattias P. Heinrich, Christian F. Baumgartner

TL;DR

Deformable registration aims to align images while accounting for multiple viable deformations. PULPo formulates registration as a probabilistic generative model with hierarchical latent variables $\mathbf{z}$ across multiple resolutions, where stationary velocity fields are integrated to produce diffeomorphic deformations $\phi(\mathbf{z})$ and combined via a Laplacian pyramid; the likelihood is $p(\mathbf{f}|\mathbf{z},\mathbf{m})=\mathcal{N}(\mathbf{f}; \mathbf{m}\circ \phi(\mathbf{z}), \sigma_I^2 \mathbf{I})$, and an amortized variational posterior $q(\mathbf{z}|\mathbf{m},\mathbf{f})$ is optimized. Empirically, PULPo achieves competitive registration performance while delivering substantially better calibrated uncertainty than the probabilistic VoxelMorph baseline on OASIS-1 and BraTS-Reg, with pronounced uncertainty localized around challenging regions like tumors. The approach enables clinically meaningful interpretation of registration results, supporting tasks such as disease monitoring and neurosurgical planning by providing voxelwise uncertainty maps $\mathrm{var}(\mathbf{m}\circ \phi(\mathbf{z}))$. Overall, PULPo advances uncertainty-aware registration by integrating hierarchical probabilistic modeling with Laplacian-pyramid fusion for fast, diffeomorphic, multi-scale alignment.

Abstract

Deformable image registration is fundamental to many medical imaging applications. Registration is an inherently ambiguous task often admitting many viable solutions. While neural network-based registration techniques enable fast and accurate registration, the majority of existing approaches are not able to estimate uncertainty. Here, we present PULPo, a method for probabilistic deformable registration capable of uncertainty quantification. PULPo probabilistically models the distribution of deformation fields on different hierarchical levels combining them using Laplacian pyramids. This allows our method to model global as well as local aspects of the deformation field. We evaluate our method on two widely used neuroimaging datasets and find that it achieves high registration performance as well as substantially better calibrated uncertainty quantification compared to the current state-of-the-art.

PULPo: Probabilistic Unsupervised Laplacian Pyramid Registration

TL;DR

Deformable registration aims to align images while accounting for multiple viable deformations. PULPo formulates registration as a probabilistic generative model with hierarchical latent variables across multiple resolutions, where stationary velocity fields are integrated to produce diffeomorphic deformations and combined via a Laplacian pyramid; the likelihood is , and an amortized variational posterior is optimized. Empirically, PULPo achieves competitive registration performance while delivering substantially better calibrated uncertainty than the probabilistic VoxelMorph baseline on OASIS-1 and BraTS-Reg, with pronounced uncertainty localized around challenging regions like tumors. The approach enables clinically meaningful interpretation of registration results, supporting tasks such as disease monitoring and neurosurgical planning by providing voxelwise uncertainty maps . Overall, PULPo advances uncertainty-aware registration by integrating hierarchical probabilistic modeling with Laplacian-pyramid fusion for fast, diffeomorphic, multi-scale alignment.

Abstract

Deformable image registration is fundamental to many medical imaging applications. Registration is an inherently ambiguous task often admitting many viable solutions. While neural network-based registration techniques enable fast and accurate registration, the majority of existing approaches are not able to estimate uncertainty. Here, we present PULPo, a method for probabilistic deformable registration capable of uncertainty quantification. PULPo probabilistically models the distribution of deformation fields on different hierarchical levels combining them using Laplacian pyramids. This allows our method to model global as well as local aspects of the deformation field. We evaluate our method on two widely used neuroimaging datasets and find that it achieves high registration performance as well as substantially better calibrated uncertainty quantification compared to the current state-of-the-art.
Paper Structure (18 sections, 6 equations, 5 figures, 1 table)

This paper contains 18 sections, 6 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Neural Network Implementation
  4. Training
  5. Inference
  6. Experiments & Results
  7. Data
  8. OASIS-1
  9. BraTS-Reg
  10. Metrics
  11. Experiments
  12. Evaluation of Registration Performance
  13. Evaluation of Uncertainty Quantification
  14. Discussion and Conclusion
  15. Acknowledgments.
  16. ...and 3 more sections

Figures (5)

  • Figure 1: Overview: (Left) Our probabilistic unsupervised Laplacian pyramid registration approach (PULPo) hierarchically models the deformation field at different resolutions allowing for accurate registration and well-calibrated uncertainty quantification particularly in the presence of lesions. (Right) Graphical model for our method.
  • Figure 2: Architecture: Schematic network architecture with 3 latent levels. In practice, we used 4 latent levels.
  • Figure 3: Registration Results: The figure shows the moving image $\mathbf{m}$, the fixed image $\mathbf{f}$, the prediction $\hat{\mathbf{f}}$, the predicted DF $\phi$, the voxel variance $\text{var}(\mathbf{f})$ and the DF variance $\text{var}(\phi)$ of the registration of T1CE-scan 5 from the BraTS-Reg dataset. The intensities of $\text{var}(\mathbf{f})$ and $\text{var}(\phi)$ needed to be visually amplified for DIF-VM due to their very small values. In this figure, the value ranges $\text{var}(\mathbf{f})$ for DIF-VM/PULPo are [0,3.98e-6]/[0,0.042], and the value ranges $\text{var}(\phi)$ for DIF-VM/PULPo are [0,3.37e-5]/[0,1.407].
  • Figure 4: Additional registration results: The figure shows the moving image $\mathbf{m}$, fixed image $\mathbf{f}$, prediction $\hat{\mathbf{f}}$, predicted DF $\phi$, voxel variance $var(\mathbf{f})$ and DF variance $var(\phi)$ for two example registrations. (a) A coronal slice of the registration between subject 10 ($\mathbf{m}$) and subject 86 ($\mathbf{f}$) from OASIS. (b) A coronal slice of the registration for T1CE-scan 7.
  • Figure 5: Registration samples: The figure shows four samples, the image variance, and the DF variance for an example registration. (a) A coronal slice of subject 10 ($\mathbf{m}$) and subject 86 ($\mathbf{f}$) on OASIS. (b) A coronal slice of T1CE-scan 7 on BraTS-Reg.