A Bi-variant Variational Model for Diffeomorphic Image Registration with Relaxed Jacobian Determinant Constraints

TL;DR

The paper tackles large-deformation image registration by relaxing the pointwise Jacobian constraint into a positive relaxation function $f(\bm{x})$ with $\det(\nabla(\bm{\varphi}+\bm{u}))=f(\bm{x})$, promoting diffeomorphisms while avoiding folding. It introduces a bi-variant variational model that couples a similarity term with a regularizer on $\bm{u}$ and a penalty on $f$ plus a smoothness term for $f$, and enforces the constraint softly via a quadratic penalty, backed by a rigorous existence analysis. A penalty-splitting, multilevel algorithm is developed, including a deformation-correction mechanism to ensure grid integrity and diffeomorphism, with detailed discretization and Euler–Lagrange derivations. Numerical experiments in 2D and 3D demonstrate convergence, effective control of volume changes on average, and superior performance against diffusion-, curvature-, and several diffeomorphic models, especially for large local deformations. The method also shows strong generalization with average-parameter settings across diverse data, making it a robust tool for practical large-deformation image registration.

Abstract

Diffeomorphic registration is a widely used technique for finding a smooth and invertible transformation between two coordinate systems, which are measured using template and reference images. The point-wise volume-preserving constraint $\det(\nabla\bm{\varphi}(\bm{x})) =1$ is effective in some cases, but may be too restrictive in others, especially when local deformations are relatively large. This can result in poor matching when enforcing large local deformations. In this paper, we propose a new bi-variant diffeomorphic image registration model that introduces a soft constraint on the Jacobian equation $\det(\nabla\bm{\varphi}(\bm{x})) = f(\bm{x}) > 0$. This allows local deformations to shrink and grow within a flexible range $0<κ_{m}<\det(\nabla\bm{\varphi}(\bm{x}))<κ_{M}$. The Jacobian determinant of transformation is explicitly controlled by optimizing the relaxation function $f(\bm{x})$. To prevent deformation folding and improve the smoothness of the transformation, a positive constraint is imposed on the optimization of the relaxation function $f(\bm{x})$, and a regularizer is used to ensure the smoothness of $f(\bm{x})$. Furthermore, the positivity constraint ensures that $f(\bm{x})$ is as close to one as possible, which helps to achieve a volume-preserving transformation on average. We also analyze the existence of the minimizer for the variational model and propose a penalty-splitting algorithm with a multilevel strategy to solve this model. Numerical experiments demonstrate the convergence of the proposed algorithm and show that the positivity constraint can effectively control the range of relative volume without compromising the accuracy of the registration. Moreover, the proposed model generates diffeomorphic maps for large local deformations and outperforms several existing registration models in terms of performance.

Figures (15)

• Figure 1: Finite difference computation involved the Jacobian determinant $\det(\nabla\bm{\varphi})|_{o}$ of the deformation $o:=\varphi_{i,j}$ at the cell center $(i, j)$ for $d=2$.
• Figure 2: Grid folding correction.
• Figure 3: Registration of 2D images with penalty function $\phi_1$: (a) template images; (b) reference images; (c) registered results; (d) the hotmap of $f(\bm{x})$; (e) the hotmap of the Jacobian determinant; (f) the deformation grids. Parameters of the two sets of images. UT ($\phi_1$): $\tau_1=1.20$, $\tau_2= 1e-3$, $\tau_3=5e-2$, $\lambda=1.2$, $\rho=1.2$, $\gamma=120$; UT ($\phi_2$): $\tau_1=1.40$, $\tau_2= 1e-3$, $\tau_3=5e-2$, $\lambda=1.2$, $\rho=1.2$, $\gamma=100$; UT ($\phi_3$): $\tau_1=1.40$, $\tau_2=1e-3$, $\tau_3=5e-2$, $\lambda=1.2$, $\rho=1.2$, $\gamma=100$; Brain ($\phi_1$): $\tau_1=0.32$, $\tau_2= 1e-3$, $\tau_3=1e-2$, $\lambda=0.6$, $\rho=1.1$, $\gamma=100$; Brain ($\phi_2$): $\tau_1=0.32$, $\tau_2=1e-3$, $\tau_3=1e-2$, $\lambda=0.6$, $\rho=1.1$, $\gamma=100$; Brain ($\phi_3$): $\tau_1=0.32$, $\tau_2=1e-3$, $\tau_3=1e-2$, $\lambda=0.6$, $\rho=1.1$, $\gamma=100$.
• Figure 4: Registration results with and without penalty term using the deformaton correction: (a) template images; (b) reference images; (c) registered images with penalty term; (d) registered images without penalty term; (e)-(f) a visualization of the deformed grids with and without penalty term. Parameters of the three sets of images. Circle-Square: $\tau_1=0.3$, $\tau_2= 1e-2$, $\tau_3=1e-3$, $\lambda=0.8$, $\rho=1.08$, $\gamma=18$; Watermelon: $\tau_1=0.2$, $\tau_2= 1e-3$, $\tau_3=1e-3$, $\lambda=1.06$, $\rho=1.06$, $\gamma=16$; BrainMR: $\tau_1=0.4$, $\tau_2= 5e-3$, $\tau_3=1e-3$, $\lambda=0.4$, $\rho=1.16$, $\gamma=20$.
• Figure 5: Comparisons of the proposed, diffusion, and curvature models. (a) the reference and template images; (b) the deformed template images of the three models with the optimal parameters (IC: $\tau_1=3$, $\tau_2=1e-2$, $\tau_3 =1e-3$, $\lambda=1$, $\gamma=100$, $\rho=1.06$ for the proposed model; $\alpha=8800$ for diffusion model; $\alpha=200$ for curvature model); (c) the registration errors of $T(\bar{\bm{\varphi}})-R(\bm{x})$; (d) the Jacobian determinant hotmaps of the deformation fields; (e) the deformation grids.

Theorems & Definitions (11)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.4: Lower semi-continuity of $\mathcal{S}(\bm{\omega})$ and $\mathcal{C}(\bm{\omega})$
• Lemma 3.5: Lower semi-continuity of $\mathcal{L}_\lambda$
• Theorem 3.6: Existence
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Remark 4.4
• Remark 4.5