Adaptive local boundary conditions to improve Deformable Image Registration

TL;DR

The paper tackles boundary-condition sensitivity in Deformable Image Registration (DIR) under partial field of view by introducing voxel-wise adaptive Robin boundary conditions that interpolate between Dirichlet and Neumann types based on boundary flow. It formalizes a generic DIR framework with $\mathcal{S}(\mathbf{T}) = \|\nabla \mathbf{T}\|^2$ and an energy $\mathcal{E}(\mathbf{T})$, augmented by a boundary term driven by two hyperparameters $a$ and $\gamma$ through $\mathbf{A}_s=\beta(\mathbf{g}_s)$, $\mathbf{B}_s=1-\mathbf{A}_s$, $\mathbf{C}_s=\gamma\mathbf{g}_s$, and a sigmoidal $\beta$. The boundary flows $\mathbf{g}$ are estimated from motion inverse consistency on the boundary, updated within a multi-resolution DIR pipeline, and the data fidelity uses a patch-based gradient similarity term $\mathcal{D}^{I,J}(\mathbf{T})=\exp(-\mathcal{C})$ with $\mathcal{C}$ computed over a $5\times5\times5$ neighborhood. Hyperparameters are chosen by grid search to minimize $\mathcal{E}(\mathbf{T})$, and the approach is validated on mono-modal CT thorax (TRE improvements up to 12% in some cases) and multi-modal CT-to-MRI abdomen (Dice scores approaching the best achievable), demonstrating that boundary-tailored conditions can significantly improve registration without prior image/motion assumptions.

Abstract

Objective: In medical imaging, it is often crucial to accurately assess and correct movement during image-guided therapy. Deformable image registration (DIR) consists in estimating the required spatial transformation to align a moving image with a fixed one. However, it is acknowledged that, boundary conditions applied to the solution are critical in preventing mis-registration. Despite the extensive research on registration techniques, relatively few have addressed the issue of boundary conditions in the context of medical DIR. Our aim is a step towards customizing boundary conditions to suit the diverse registration tasks at hand. Approach: We propose a generic, locally adaptive, Robin-type condition enabling to balance between Dirichlet and Neumann boundary conditions, depending on incoming/outgoing flow fields on the image boundaries. The proposed framework is entirely automatized through the determination of a reduced set of hyperparameters optimized via energy minimization. Main results: The proposed approach was tested on a mono-modal CT thorax registration task and an abdominal CT to MRI registration task. For the first task, we observed a relative improvement in terms of target registration error of up to 12% (mean 4%), compared to homogeneous Dirichlet and homogeneous Neumann. For the second task, the automatic framework provides results closed to the best achievable. Significance: This study underscores the importance of tailoring the registration problem at the image boundaries. In this research, we introduce a novel method to adapt the boundary conditions on a voxel-by-voxel basis, yielding optimized results in two distinct tasks: mono-modal CT thorax registration and abdominal CT to MRI registration. The proposed framework enables optimized boundary conditions in image registration without any a priori assumptions regarding the images or the motion.

Figures (5)

• Figure 1: Generic multi-resolution framework for DIR incorporating proposed voxel-wise adaptive boundary conditions. An illustrative input image dimension of $256 \times 256 \times 256$ is displayed as a typical example. The associated pseudo-code is detailed in Appendix \ref{['app:code']}. The minimization of the energy $\mathcal{E}(\mathbf{T})$ was conducted on down-sampled versions of the images, progressively refining until reaching the original image dimension (blue blocks). The computation of incoming/outgoing flow fields on the image boundaries (i.e.$\textbf{g}$) underwent updates throughout the registration process, at the beginning of each resolution step (yellow blocks). The system of equations (Eq. (\ref{['eq:PBEv']}a) and (\ref{['eq:PBEv']}b)) was solved in inner voxels (which corresponds to $\Omega \setminus \partial\Omega$), the one voxel wide perimeter (which corresponds to $\partial\Omega$) being computed separately according to the boundary conditions (Eq. (\ref{['eq:PBEv']}c), see Appendix \ref{['app:BC_AR']}).
• Figure 2: Example of mono-modal CT thorax registration results (case 8). Mean intensity projection of error maps are reported using HN (a), HD (b) and the proposed adaptive Robin conditions (c) (colorbar in millimeters).
• Figure 3: Typical example of hyperparameter optimization (i.e.$\gamma$ and $a$) for the proposed adaptive Robin boundary conditions (mono-modal CT thorax registration/case 7). (a): grid search optimizing the minimized DIR energy $\mathcal{E}(\mathbf{T})$ as expressed in Eq. \ref{['eq:VarForm']} (minimum energy yields optimal hyperparameters); (b): $\mathrm{TRE}$ metric calculated using Eq. (\ref{['eq:TRE']}) for image registration assessment (minimum $\mathrm{TRE}$ yields optimal registration outcomes).
• Figure 4: Example of multi-modal CT to MRI abdomen registration (case 4) showing the fix (a), moving (b) and registered images with HD (c), HN (d) and adaptive Robin (e). The segmentation contours are overlaid on all images to highlight the degree of alignment before and after registration (green: fix masks, red: registered masks). The labels include the liver, the spleen and both kidneys.
• Figure 5: Typical example of hyperparameter optimization (i.e.$\gamma$ and $a$) for the proposed adaptive Robin boundary conditions (multi-modal CT to MRI abdomen registration/case 4). (a): grid search optimizing the minimized energy $\mathcal{E}(\mathbf{T})$ as expressed in Eq. \ref{['eq:VarForm']} (minimum energy yields optimal hyperparameters); (b): $Dice$ metric calculated using Eq. (\ref{['eq:dice']}) for image registration assessment (maximum $Dice$ yields optimal registration outcomes).