Some philosophical babbling about terrain modeling

One way to define the "GIS terrain modeling problem" is to say how it is different from the modeling problems that arise in other applications. Say, image analysis and computer graphics, both using real bivariate functions for modeling.

Here are the differences that I would like to argue for (in order of importance):

1. In GIS, one of the most often invoked term is "layer": a geographic information system consists of "layers" of information related to each other via spacial coordinates. Therefore, preferably, a digital terrain model should improve as new knowledge about the terrain are overlayed on top of the old one. Minimally, a good terrain model should take in point samples with these attributes:
   • coordinates
   • normal
   • noise characteristics / tolerance
2. A terrain model in GIS should support geometric computations on surface. For example, area computation, drainage, visbility and path planning.

What matheamtical approaches do these two distinguishing charactersitics suggest? The first suggests that, for interpolation, an ``implicit" model is more appropiate: a point on the terrain (model) satisfies that it is the local minimum of some energy function. An energy function is the easiest way to plug-in statistic models into the interpolation and incooporate various (perhaps competing) knowledge attached to the data. An example of the implicit representation can be found in Nina Amenta's resent work [1], which also points to many earlier works on implicit representation. In GIS, a survey of interpolation can be found at [2]. In graphics, [3] is a paper on radial basis surface reconstruction from point cloud.

The second suggests that the surface should be represented as a (piece-wise) polynomial, since mathematicians know a lot about computation with polynomials. There is a wealth of "approximation theory" in applied math. The recent work by Neamtu [4] deals with interpolation using simplex splines basis

The two competing representations (implicit vs explicit) can be accomodated by having the explicit representation approximate the implicit representation. By having the two representations, we separate the goodness of the terrain modeling into two components: the goodness of the energy function and the goodness of the polynomal approximation.

The fact that a GIS system frequently need to ``overlay" data also suggests that the terrain model should not have a very rigid structure. Both as an example and to spell out the bias, a TIN constructed using incremental computation is better a grid because it allows future data to be incooperated.

[1] Nina Amenta, Defining point-set surfaces, 2004

[2] Lubos Mitas and Helena Mitasova, Spatial interpolation, 1993

[3] Carr, et al, Reconstruction and representation of 3D objects with radial basis functions, 2001

[4] M. Neamtu, What is the natural generalization of univariate splines to higher dimensions?, in Mathematical Methods for Curves and Surfaces, 2004

-- YuanxinLiu - 19 Apr 2005
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