Surface registration, the task of aligning several multidimensional point sets, is a necessary task in many scientific fields. In this work, a novel statistical approach is developed to solve the problem of nonrigid registration. While the application of an affine transformation results in rigid registration, using a general nonlinear function to achieve nonrigid registration is necessary when the point sets require deformations that change over space. The use of a local likelihood-based approach using windowed Gaussian processes provides a flexible way to accurately estimate the nonrigid deformation. This strategy also makes registration of massive data sets feasible by splitting the data into many subsets. The estimation results yield spatially-varying local rigid registration parameters. Gaussian process surface models are then fit to the parameter fields, allowing prediction of the transformation parameters at unestimated locations, specifically at observation locations in the unregistered data set. Applying these transformations results in a global, nonrigid registration. A penalty on the transformation parameters is included in the likelihood objective function. Combined with smoothing of the local estimates from the surface models, the nonrigid registration model can prevent the problem of overfitting. The efficacy of the nonrigid registration method is tested in two simulation studies, varying the number of windows and number of points, as well as the type of deformation. The nonrigid method is applied to a pair of massive remote sensing elevation data sets exhibiting complex geological terrain, with improved accuracy and uncertainty quantification in a cross validation study versus two rigid registration methods.

表面对准是对齐多个多维点集的任务，是许多科学领域中的必要任务。在这项工作中，开发了一种新颖的统计方法来解决非刚性注册的问题。虽然仿射变换的应用导致刚性配准，但是当点集需要随空间变化的变形时，必须使用常规的非线性函数来实现非刚性配准。使用加高斯加窗过程的基于局部似然性的方法的使用提供了一种灵活的方法来准确估计非刚性变形。通过将数据分成许多子集，此策略还使注册海量数据集变得可行。估计结果产生空间变化的局部刚性配准参数。然后将高斯过程表面模型拟合到参数字段，从而可以在未估计的位置（特别是在未注册的数据集中的观察位置）预测转换参数。应用这些转换会导致全局的，非刚性的注册。对变换参数的惩罚包括在似然目标函数中。结合表面模型的局部估计值的平滑处理，非刚性配准模型可以防止过拟合的问题。在两个模拟研究中测试了非刚性配准方法的有效性，其中改变了窗口的数量和点的数量以及变形的类型。非刚性方法应用于显示复杂地质地形的一对大型遥感海拔数据集，