Topological 3D Spatial Interpolation Based on the Interval-Valued Homotopy Continuation

Abstract

Estimating unknown values using its surrounding measured values is called spatial interpolation, a vital tool for estimating continuous spatial data such as the earth’s surface. Construction of the Digital Elevation Model is one of the most common applications of spatial interpolation methods. There are various global and local interpolation techniques, including Kriging, Inverse Distance Weighted (IDW), Thiessen polygons (TIN), Natural Neighbor (NN), and Spline interpolation. This paper introduces the interval-valued homotopy continuation for 3D spatial data interpolation. Straight lines or algebraic curves can be reconstructed using homotopy continuation between any pairs of 3D data. The novel method of the interval-valued homotopy to restore the topographic surface between spatial data is developed in MATLAB programming language. For a dataset of ASTER GDEM, the presented mathematical algorithm shows better results compared to TIN and IDW methods in terms of Mean Squared Error, Mean Absolute Error, and Root Mean Squared Error with values of 5.2897, 1.53, and 2.299 m, respectively.