Diffeomorphic registration has become a powerful approach for seeking a smooth and invertible spatial transformation between two coordinate systems which have been measured via the template and reference images. While the pointwise volume-preserving constraint is effective for some problems, it is too stringent for many other problems especially when the local deformations are relatively large, because it may lead to a poor large-deformation for enforcing local matching.In this paper, we propose a novel bi-variant diffeomorphic image registration model with the soft constraint of Jacobian equation, which allows local deformations to shrink and grow in a flexible range.The Jacobian determinant of the transformation is explicitly controlled by optimizing the relaxation function. To prevent deformation folding and enhance the smoothness of deformation, we not only impose a positivity constraint in optimizing the relaxation function, but also employ a regularizer to ensure the smoothness of the relaxation function.Furthermore, the positivity constraint ensures that is as close to one as possible, which helps to obtain a volume-preserving transformation on average.We further analyze the existence of the minimizer for the variational model and propose a penalty splitting method with a multilevel strategy to solve this model. Numerical experiments show that the proposed algorithm is convergent, and the positivity constraint can control the range of relative volume and not compromise registration accuracy. Moreover, the proposed model produces diffeomorphic maps for large deformation, and achieves better performance compared to the several existing registration models.

翻译：微分同胚配准已成为在模板图像与参考图像所测量的两个坐标系之间寻求光滑可逆空间变换的有力方法。尽管逐点体积守恒约束对某些问题有效，但在局部形变较大时该约束过于严格，可能导致大形变下局部匹配效果较差。本文提出了一种带雅可比方程软约束的新型双变量微分同胚图像配准模型，允许局部形变在灵活范围内收缩与扩张。通过优化松驰函数显式控制变换的雅可比行列式。为防止变形折叠并增强形变光滑性，我们不仅对松驰函数优化施加正性约束，还采用正则化项确保松驰函数的光滑性。此外，正性约束使函数尽可能接近1，有助于获得平均意义上的体积守恒变换。进一步分析了变分模型极小值点的存在性，提出了一种结合多层策略的罚分裂方法求解该模型。数值实验表明，所提算法具有收敛性，且正性约束能控制相对体积范围而不降低配准精度。同时，该模型在大形变下可生成微分同胚映射，与现有多种配准模型相比实现了更优性能。