Merging Point Data for InSAR Deformation Processing

Given a collection of points $S\subset {\mathbb{R}}^N$, which is partitioned into $M$ overlapping subsets $\{S_i{\}}_{i=1}^M$, and approximate data $\{D_i{\}}_{i=1}^M$ associated with the subsets, one may seek a consistent merged dataset $D$ that is derived from $\{S_i{\}}_{i=1}^M$ and $\{D_i{\}}_{i=1}^M$. This note presents a method for constructing $D$ under the assumption that $D$ represents discrete samples of a suitably smooth function $f:{\mathbb{R}}^N\to \mathbb{R}$ evaluated at the points in $S$. The method has two steps. The first step uses a least-squares solve to approximate the constant offsets for each $D_i$. The second step uses a sequence of discrete Dirichlet problems to resolve any remaining differences. We include a two dimensional example of this method applied to deformation measurements derived from Interferometric Synthetic Aperture Radar (InSAR).