This paper proposes a numerical algorithm for image registration using energy minimization and nonlinear elasticity regularization. Application to the registration of gene expression data to a neuroanatomical mouse atlas in two dimensions is shown. We apply a nonlinear elasticity regularization to allow larger and smoother deformations, and further enforce optimality constraints on the landmark points distance for better feature matching. To overcome the difficulty of minimizing the nonlinear elasticity functional due to the nonlinearity in the derivatives of the displacement vector field, we introduce a matrix variable to approximate the Jacobian matrix and solve for the simplified Euler-Lagrange equations. By comparison with image registration using linear regularization, experimental results show that the proposed nonlinear elasticity model also needs fewer numerical corrections such as regridding steps for binary image registration, it renders better ground truth, and produces larger mutual information; most importantly, the landmark points distance and L2 dissimilarity measure between the gene expression data and corresponding mouse atlas are smaller compared with the registration model with biharmonic regularization.

中文翻译：

---

使用非线性弹性平滑器和地标点约束将基因表达数据用于小鼠图谱注册。

本文提出了一种使用能量最小化和非线性弹性正则化的图像配准数值算法。显示了将基因表达数据注册到二维神经解剖小鼠图谱的应用。我们应用非线性弹性正则化以允许更大和更平滑的变形，并进一步对地标点距离强制执行优化约束以实现更好的特征匹配。为了克服由于位移矢量场导数的非线性而使非线性弹性函数最小化的困难，我们引入了一个矩阵变量来逼近雅可比矩阵并求解简化的欧拉-拉格朗日方程。通过与使用线性正则化的图像配准相比，实验结果表明，所提出的非线性弹性模型还需要较少的数值校正，例如二值图像配准的重新网格步骤，它呈现更好的地面真实性，并产生更大的互信息；最重要的是，与具有双调和正则化的配准模型相比，基因表达数据与相应的小鼠图谱之间的标志点距离和 L2 差异度量更小。