In image-guided liver surgery, 3D-3D non-rigid registration methods play a crucial role in estimating the mapping between the preoperative model and the intraoperative surface represented as point clouds, addressing the challenge of tissue deformation. Typically, these methods incorporate a biomechanical model, represented as a finite element model (FEM), used to regularize a surface matching term. This paper introduces a novel 3D-3D non-rigid registration method. In contrast to the preceding techniques, our method uniquely incorporates the FEM within the surface matching term itself, ensuring that the estimated deformation maintains geometric consistency throughout the registration process. Additionally, we eliminate the need to determine zero-boundary conditions and applied force locations in the FEM. We achieve this by integrating soft springs into the stiffness matrix and allowing forces to be distributed across the entire liver surface. To further improve robustness, we introduce a regularization technique focused on the gradient of the force magnitudes. This regularization imposes spatial smoothness and helps prevent the overfitting of irregular noise in intraoperative data. Optimization is achieved through an accelerated proximal gradient algorithm, further enhanced by our proposed method for determining the optimal step size. Our method is evaluated and compared to both a learning-based method and a traditional method that features FEM regularization using data collected on our custom-developed phantom, as well as two publicly available datasets. Our method consistently outperforms or is comparable to the baseline techniques. Our code and datasets will be available at https://github.com/zixinyang9109/BCF-FEM.

翻译：在图像引导肝脏手术中，三维-三维非刚性配准方法对于估计术前模型与以点云形式表示的术中表面之间的映射关系至关重要，旨在解决组织形变带来的挑战。传统方法通常将生物力学模型（以有限元模型（FEM）表示）作为正则化项引入表面匹配项中。本文提出了一种新颖的三维-三维非刚性配准方法。与现有技术不同，本方法创新性地将FEM直接嵌入表面匹配项内部，确保估计的形变在整个配准过程中保持几何一致性。此外，我们无需在FEM中确定零边界条件及外力施加位置，这是通过在刚度矩阵中引入软弹簧单元并允许外力分布于整个肝脏表面实现的。为进一步提升鲁棒性，我们提出了一种针对力幅值梯度的正则化技术，该技术施加空间平滑性约束，有助于防止对术中数据中不规则噪声的过拟合。优化过程通过加速近端梯度算法实现，并辅以我们提出的最优步长确定方法以提升性能。我们在自主研发的体模采集数据及两个公开数据集上，将本方法与基于学习的方法以及采用FEM正则化的传统方法进行了对比评估。实验结果表明，本方法在各项指标上均优于或与基线方法相当。我们的代码与数据集将在https://github.com/zixinyang9109/BCF-FEM 公开。