[Paper Review] Region-specific Diffeomorphic Metric Mapping

Zhengyang Shen, François‐Xavier Vialard+1 more|PubMed|Jun 1, 2019

Medical Image Segmentation Techniques44 references4 citations

TL;DR

This paper introduces Region-specific Diffeomorphic Metric Mapping (RDMM), a non-parametric image registration framework that uses a spatio-temporal regularizer advected by the estimated velocity field to enable region-specific regularization. Unlike fixed-regularization methods like LDDMM, RDMM adapts regularization to local deformation patterns, achieving state-of-the-art performance with improved modeling of large, localized deformations in medical imaging applications.

ABSTRACT

We introduce a region-specific diffeomorphic metric mapping (RDMM) registration approach. RDMM is non-parametric, estimating spatio-temporal velocity fields which parameterize the sought-for spatial transformation. Regularization of these velocity fields is necessary. In contrast to existing non-parametric registration approaches using a fixed spatially-invariant regularization, for example, the large displacement diffeomorphic metric mapping (LDDMM) model, our approach allows for spatially-varying regularization which is advected via the estimated spatio-temporal velocity field. Hence, not only can our model capture large displacements, it does so with a spatio-temporal regularizer that keeps track of how regions deform, which is a more natural mathematical formulation. We explore a family of RDMM registration approaches: 1) a registration model where regions with separate regularizations are pre-defined (e.g., in an atlas space or for distinct foreground and background regions), 2) a registration model where a general spatially-varying regularizer is estimated, and 3) a registration model where the spatially-varying regularizer is obtained via an end-to-end trained deep learning (DL) model. We provide a variational derivation of RDMM, showing that the model can assure diffeomorphic transformations in the continuum, and that LDDMM is a particular instance of RDMM. To evaluate RDMM performance we experiment 1) on synthetic 2D data and 2) on two 3D datasets: knee magnetic resonance images (MRIs) of the Osteoarthritis Initiative (OAI) and computed tomography images (CT) of the lung. Results show that our framework achieves comparable performance to state-of-the-art image registration approaches, while providing additional information via a learned spatio-temporal regularizer. Further, our deep learning approach allows for very fast RDMM and LDDMM estimations. Code is available at https://github.com/uncbiag/registration.

Motivation & Objective

• Address the limitation of spatially-invariant regularization in existing non-parametric registration methods like LDDMM.
• Develop a framework that allows spatially-varying regularization which evolves with the deformation via advection of the regularizer by the velocity field.
• Enable more realistic modeling of large deformations by adapting regularization strength to anatomical regions (e.g., lung vs. surrounding tissue).
• Provide a variational formulation that ensures diffeomorphic transformations and generalizes LDDMM as a special case.
• Integrate deep learning to predict the initial regularizer and velocity field for fast, end-to-end inference.

Proposed method

• Propose a variational formulation where the spatio-temporal regularizer is advected by the time-dependent velocity field, ensuring it tracks deforming regions.
• Define a multi-kernel Gaussian regularizer with spatially-varying weights that evolve over time via the advection equation.
• Formulate the registration energy as a functional depending on initial velocity and initial regularizer, minimizing geodesic distance in a Riemannian metric.
• Introduce three RDMM variants: (1) pre-defined region-specific regularizers, (2) jointly optimized regularizers, and (3) deep learning-based end-to-end prediction of regularizer and velocity.
• Use L-BFGS for optimization in non-parametric settings and train a multi-step affine network followed by a non-parametric network in deep learning settings.
• Apply intensity normalization and coordinate scaling to [0,1]^d for consistent interpretation of regularizer standard deviations.

Experimental results

Research questions

• RQ1 Can a spatio-temporal regularizer that evolves with deformation improve registration accuracy in regions with large, localized deformations?
• RQ2 How does region-specific regularization compare to fixed regularization in capturing anatomical changes in medical images?
• RQ3 Can a learned, spatially-varying regularizer from deep learning achieve fast and accurate registration while preserving diffeomorphic constraints?
• RQ4 Is the proposed RDMM framework mathematically consistent with diffeomorphic transformations and a generalization of LDDMM?
• RQ5 Does the inclusion of a range loss and OMT penalty ensure existence and uniqueness of solutions in the optimization process?

Key findings

• RDMM achieves registration performance comparable to state-of-the-art methods like SyN, VoxelMorph, and AVSM on synthetic 2D data and real 3D datasets (OAI knee MRI and lung CT).
• The deep learning-based RDMM variant enables very fast estimation of both RDMM and LDDMM transformations, significantly outperforming iterative optimization in inference speed.
• In the lung CT registration task, the region-specific regularizer effectively preserved small deformations in non-lung structures while allowing large deformations in the lung, as intended.
• The optimized regularizer variant demonstrated improved alignment in complex anatomical regions by learning appropriate regularization strength across space.
• Theoretical analysis confirms that the model ensures existence of solutions under strictly convex penalty functions, and that LDDMM is a special case of RDMM when regularization is spatially invariant.
• The inclusion of inverse consistency loss in deep learning models improved deformation symmetry and reduced registration errors.