Some Basis Function Methods for Surface Approximation
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis considers issues in surface reconstruction such as identifying approximation methods that work well for certain applications and developing efficient methods to compute and manipulate these approximations.
The first part of the thesis illustrates a new fast evaluation scheme to efficiently calculate thin-plate splines in two dimensions. In the fast multipole method scheme, exponential expansions/approximations are used as an intermediate step in converting far field series to local polynomial approximations. The contributions here are extending the scheme to the thin-plate spline and a new error analysis. The error analysis covers the practically important case where truncated series are used throughout, and through off line computation of error constants gives sharp error bounds.
In the second part of this thesis, we investigates fitting a surface to an object using blobby models as a coarse level approximation. The aim is to achieve a given quality of approximation with relatively few parameters. This process involves an optimization procedure where a number of blobs (ellipses or ellipsoids) are separately fitted to a cloud of points. Then the optimized blobs are combined to yield an implicit surface approximating the cloud of points. The results for our test cases in 2 and 3 dimensions are very encouraging. For many applications, the coarse level blobby model itself will be sufficient. For example adding texture on top of the blobby surface can give a surprisingly realistic image.
The last part of the thesis describes a method to reconstruct surfaces with known discontinuities. We fit a surface to the data points by performing a scattered data interpolation using compactly supported RBFs with respect to a geodesic distance. Techniques from computational geometry such as the visibility graph are used to compute the shortest Euclidean distance between two points, avoiding any obstacles. Results have shown that discontinuities on the surface were clearly reconstructed, and the