Volume-preserving interpolation of a smooth surface from polygon-related data

Abstract

The interpolation of continuous surfaces from discrete points is supported by most GIS software packages. Some packages provide additional options for the interpolation from 3D line objects, for example surface-speci®c lines, or contour lines digitized from topographic maps. Demographic, social and economic data can also be used to construct and display smooth surfaces. The variables are usually published as sums for polygonal units, such as the number of inhabitants in communities or counties. In the case of point and line objects the geometric properties have to be maintained in the interpolated surface. For polygon-based data the geometric properties of the polygon boundary and the volume should be preserved, avoiding redistribution of parts of the volume to neighboring units during interpolation. The pycnophylactic interpolation method computes a continuous surface from polygonbased data and simultaneously enforces volume preservation in the polygons. The original procedure using a regular grid is extended to surface representations based on an irregular triangular network (TIN).Surfaces with only one possible height value at a location in the plane (2D surfaces) are usually generated from discrete geometric elements, such as points or lines with a height information. The height values between the data points or lines are interpolated following rules based on global or local properties of the data, mathematical models, experience, or expert knowledge about the formation of the real surface. The most common example for the model of a surface and its visualization is the surface of the earth as displayed on topographic maps. I thank Waldo Tobler for his contributions and the constructive discussion during the implementation and test of the TIN-based version of the pycnophylactic interpolation.

A Geostatistical Framework for Area‐to‐Point Spatial Interpolation

“…Similarly, the recently developed variant of pycnophylactic interpolation using triangular irregular (TIN) surface representation (Rase 2001) alleviates the regular grid restriction of Tobler's method.…”

A Quantile Regression Approach to Areal Interpolation

“…Ensuring the volume-preserving property requires solving the problem with quadratic programming methods, however (Yoo, Kyriakidis, and Tobler 2010). Tobler's pycnophylactic procedure using regular grids also has been extended to a surface representation based on a TIN model by Rase (2001). Other raster-based approaches use the centroid of a source region as the location of average density within the region and estimate the population density of other cells based on an inverse distance weighting (see Bracken and Martin 1989;Martin 1989;Bracken 1991;Martin and Bracken 1991).…”

A Template Based Method for Normalizing Thermal Images of the Human Body

“…Triangular based interpolation has the advantage of preserving the geometry of the lines that are combined [11], having a similar effect to the Nearest Neighbour method. The developed geometrical models based in triangulation and are presented in the Fig.…”