  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.
