Scattered data modeling and deformation with transfinite constraints

Scattered data interpolation methods are used to construct surfaces through inter-polating ordinate values given at unstructured data points. Although effective in handling discrete data points, the methods are unable to interpolate transfinite constraints, such as curves. Local shape (such as convexity) modeling is difficult with scattered data interpolation methods. In this thesis, an interpolant is developed which takes advantage of both the flexibility offered by scattered data interpolation methods to model unstructured data, and the capability of piecewise polynomial curves to incorporate shape information into the interpolation scheme. The new interpolant is able to interpolate both scattered data and transfinite curves. A significant portion of this thesis is devoted to scattered data deformation. Scattered data deformation methods which enforce transfinite constraints are developed. For curve constraints, the deformation method is an extension of the scattered data interpolation with curve constraints. For surfaces and 3D volumes, the deformation is formulated as a general feature constrained optimization problem. Deformation Jacobians and curl are included in the optimization objective function to control the deformation behaviors. The feature constrained scattered data deformation method is applied to 3D brain image deformations. Issues of interests to brain image deformation are addressed.