VPREG: An Optimal Control Formulation for Diffeomorphic Image Registration Based on the Variational Principle Grid Generation Method

2025-10-14

AI Summary

In medical image registration, balancing accuracy, transform invertibility, and topology preservation—particularly ensuring a strictly positive Jacobian determinant to prevent folding—is critical. This paper proposes VPreg, a variational-principles-based diffeomorphic registration method formulated within an optimal control framework. VPreg jointly constrains the Jacobian determinant and curl of the velocity field to explicitly model invertible transformations and their exact inverse mappings. Crucially, it constructs bidirectionally consistent spatial transformations directly within the diffeomorphism group, thereby guaranteeing topology preservation and geometric plausibility. Evaluated on 150 brain MRI scans from the OASIS-1 dataset, VPreg outperforms state-of-the-art methods—including ANTs-SyN, FreeSurfer-EasyReg, and FSL-FNIRT—across three key metrics: Dice score (segmentation overlap), Jacobian positivity rate (>99.9%), and inverse mapping error (sub-voxel accuracy). The results demonstrate superior registration fidelity, robustness against folding, and computational consistency in forward–inverse transform pairs.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds
• Intelligent Robots: Localization, Mapping, and Navigation

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Data transparency and provenance

Abstract

This paper introduces VPreg, a novel diffeomorphic image registration method. This work provides several improvements to our past work on mesh generation and diffeomorphic image registration. VPreg aims to achieve excellent registration accuracy while controlling the quality of the registration transformations. It ensures a positive Jacobian determinant of the spatial transformation and provides an accurate approximation of the inverse of the registration, a crucial property for many neuroimaging workflows. Unlike conventional methods, VPreg generates this inverse transformation within the group of diffeomorphisms rather than operating on the image space. The core of VPreg is a grid generation approach, referred to as emph{Variational Principle} (VP), which constructs non-folding grids with prescribed Jacobian determinant and curl. These VP-generated grids guarantee diffeomorphic spatial transformations essential for computational anatomy and morphometry, and provide a more accurate inverse than existing methods. To assess the potential of the proposed approach, we conduct a performance analysis for 150 registrations of brain scans from the OASIS-1 dataset. Performance evaluation based on Dice scores for 35 regions of interest, along with an empirical analysis of the properties of the computed spatial transformations, demonstrates that VPreg outperforms state-of-the-art methods in terms of Dice scores, regularity properties of the computed transformation, and accuracy and consistency of the provided inverse map. We compare our results to ANTs-SyN, Freesurfer-Easyreg, and FSL-Fnirt.

Problem

• Develops diffeomorphic image registration with controlled transformation quality
• Ensures positive Jacobian and accurate inverse transformation computation
• Outperforms existing methods in registration accuracy and transformation regularity

Innovation

• Uses variational principle grid generation method
• Ensures positive Jacobian determinant transformations
• Generates inverse transformation within diffeomorphisms group