A powerful extension of neural networks, the so-called functional network, has been applied to the surface reconstruction problem. The introduced approach is very general; the data points may come from any kind of surface, and the approximating surface can be written in terms of any arbitrary family of basis functions. Because of this property, this approach opens a promising new line of research, as functional networks might be applied to many other challenging problems in surface modeling. This paper briefly describes the B-spline surface, and applies the functional network methodology to fit sets of given three-dimensional (3D) data points in the B-spline surface. The difference between neural and functional networks is also discussed.
The authors describe the problem and define the state equations in the following way: Look for the most general family of parametric surfaces P(s,t), such that their isoparametric curves s = and t = are linear combinations of the sets of functions: f(s) = {f0(s), f1(s),..., fm(s)} and f*(s) = {f*0(s), f*1(s),..., f*m(s)} respectively. This provides another way of looking for surfaces P(s,t), such that they satisfy the system of functional equations where the sets of coefficients {&agr;j(s); j=1,2,...,n} and {&bgr;i(t); i=1,2,...,m} can be assumed, without loss of generality, to be sets of linearly independent functions. This generalizes the neural networks paradigm, which includes all of the new features described by the functional network, simply with the expression where the Pij are elements of an arbitrary matrix P. Therefore, P(s,t) is a tensor-product surface. This functional network is applied to solve the surface reconstruction problem. To test the quality of the model, the authors present interesting examples, and compare them with a sufficient amount of data from four surfaces.
