There are various situations where one has to deal with the problem of fitting a surface to a set of noisy measurement data, corresponding to points that are arbitrary distributed in a bounded area of the plane. For many applications it is important that the fitted surface preserves certain geometric properties of the surface undelying the data. In some cases, the shape preservation is essential for the fitted surface to be physically meaningful. In others, the use of a constrained surface fitting method can avoid improper bumps and holes that might be present in a surface fitted with the unconstrained method.
In this thesis algorithms are presented for smoothing scattered data with a convexity, nonnegativity or monotonicity preserving surface or with a surface subject to boundary constrains.
As for the approximating functions, Powell-Sabin (PS) splines on conforming triangles are used. These piecewise quadratics with C1-continuity are expressed as a linear combination of locally supported basis functions (B-splines) which are fairly easy to construct. Necessary and sufficient shape preserving conditions for the PS-splines are then formulated in terms of the B-spline coefficients. Due to the local support of the B-splines, these constraints are sparse, which can be exploited to efficiently solve the problem. In the presented algorithms a local refinement procedure for the trangulation problem is included that automatically takes account of the difficulties in the function underlying the data.
Numerical examples are supplied to illustrate the profit of surface fitting by means of constrained Powell-Sabin splines.