Research On Surface Interpolation And Approximation From Scattered Data

The problem of constructing approximations based upon scattered data are encountered in many areas of scientific applications, like meteorological information, such as the amount of rainfall, or geological information, such as depth of underground formmations. This is done using interpolation techniques that estimate value on unexplored points of a region considering the values sampled on it. The main work of this thesis is to improve some approximation methods that are often used in practice. With the numerical examples, we prove the improved methods .are more convenient to compute or have more approximate quality.In chapter 1, with regard to the Shepard method, we use the truncated polynomials, the B-spline basis functions and exponential functions to construct the weight functions. They are of better properties of smoothness and decay, which can be adjusted freely. And so the surface can be fitted better by the improved method. It also can be applied in modified Shepard Mehthod[5].In practice, interpolation methods of radial basis functions are often required for approximation with very large number of data, in which the interpolation matrix is usually ill-conditioned. In order to solve this problem, we construct a localized method for interpolation with radial basis functions (global supported) based on the idea of multivariate spline. Moreover, we verify that this method is feasible through theoretic analysis and numerical experiments. These methods perform almost as good as the global supported radial basis functions interpolation methods do.In chapter 3, we use NURBS to approximate the scattered data. Paper[20]gave out the iterative algorithm of B-spline interpolation and approximation. In this chapter,we generalize this result and present an iterative algorithm of NURBS interpolation and approximation. Using this algorithm, we can get the approximat NURBS curve or surface directly without solving a linear system to compute the weights and control points. This algorithm is consistent with the algorithm in paper [20] and the latter is just the degenerate form of the former in essence.Furthermore, using the new methods introduced in former chapters, we improve the NURBS approximation method given out in paper[21], and then the computing is simpler. We also present an approach for scattered sampling and uniform mesh generation. Since the uniform mesh is easy to be parameterized.this approach is more convenient to be used. .The numerical examples in this thesis show us these methods are feasible, and the results are satisfying.