Fitting Free Form Surfaces

The problem of reconstructing smooth surfaces from discrete scattered data arises in many fields of science and engineering and has now been studied thoroughly for nearly 40 years. The data sources include measured values (meteorology, oceanography, optics, geodetics, geology, laser range scanning, etc.) as well as experimental results (from physical, chemical or engineering experiments) and computational values (evaluation of mathematical functions, finite element solutions of partial differential equations or results of other numerical simulations). Due to the vast variety of data sources many different methods have been developed, each of them more or less suited to a specific problem. In the field of geology, meteorology, cartography, a. o., the problem can typically be stated as follows: given data points (xi, yi, zi) ∈ IR, find a scalar function F : IR → IR that approximates or interpolates the value zi at (xi, yi), i.e. F (xi, yi) ≈ zi. This problem is generally known as Scattered Data Interpolation (cf. Figure 1) and there exist many solutions to that problem which include Shepard’s methods [42], radial basis functions [26] and finite element methods. Good surveys of these methods and further references can be found in [1, 17, 35, 40].