A method for local approximation of a planar deformation field

We present a method of approximation of a deformation field based on the local affine transformations constructed based on n nearest neighbors with respect to points of adopted grid. The local affine transformations are weighted by means of inverse distance squared between each grid point and observed points (nearest neighbors). This work uses a deformation gradient, although it is possible to use a displacement gradient instead – the two approaches are equivalent. To decompose the deformation gradient into components related to rigid motions (rotations, translations are excluded from the deformation gradient through differentiation process) and deformations, we used a polar decomposition and decomposition into a sum of symmetric and an anti-symmetric matrices (tensors). We discuss the results from both decompositions. Calibration of a local affine transformations model (i.e., number of nearest neighbors) is performed on observed points and is carried out in a cross-validation procedure. Verification of the method was conducted on simulated data-grids subjected to known (functionally generated) deformations, hence, known in every point of a study area.