Voronoi tessellation-based lifting scheme in bounded regions

We study the Voronoi tessellation-based lifting scheme in two-dimensional regions where the spatial data is available in a finite and bounded two-dimensional region. The lifting scheme is a second-generation wavelet method that is used for the analysis of spatial data which we model as being an underlying `true' surface corrupted by noise. On the other hand, Voronoi tessellation is a standard technique to partition the space into smaller sub-regions called Voronoi cells that are used as an ingredient in the lifting scheme.

We investigate the statistical properties of Voronoi cells for homogeneous Poisson points in the infinite plane and bounded regions. The properties are the cell area, perimeter, and the number of cell edges. Our findings show that the distributions of cell properties differ substantially when boundaries are imposed. These differences are affected by proximity.

We emphasize the consequences of the boundaries on the Voronoi cells, and we devise a method that treats the spatial data in the finite region as if it is a subset of a larger region or an infinite plane. This approach predicts the true cell area that is actually clipped by a boundary line using regression-based models. The models are updated for general data cases, and have an overall promising performance.

Lifting scheme uses the features of Voronoi tessellation and the information obtained from the Voronoi cells. The ultimate goal of this thesis is to implement the devised method, which adjusts the cell area near boundaries, into the lifting scheme framework and compare its performance to the standard approaches. Various configurations are considered; standard and proposed weight methods, noisy test functions with different spatial characteristics, and randomly distributed, regular, and clustered point patterns. The proposed approach over-perform the existing options and even gives better performance over the standard spatial prediction techniques such as kriging in certain cases.