We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data.

Volume intersection algorithms are used to reconstruct incomplete objects from their silhouettes. An imagined light source is moved about the data and the cumulative amount of "light" seen at each point an space is interpreted as indicating the likelihood that the point is inside the object. The object data need not be uniformly distributed nor exclusively surface data. Explicit distinction between noise, surface and interior data is avoided. The novel concept of a localised viewing region is introduced to overcome the inherent inability of volume intersection algorithms to reconstruct concave surfaces. Algorithms for 2D pixel and 3D voxel data are described and applied to 3D ultrasound data.

Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from three-dimensional (3-D) medical graphics. The specific application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial basis functions impose few restrictions on the geometry of the interpolation centers and are suited to problems where the interpolation centers do not form a regular grid. However, their high computational requirements have previously limited their use to problems where the number of interpolation centers is small (< 300). Recently developed fast evaluation techniques have overcome these limitations and made radial basis interpolation a practical approach for larger data sets. In this paper radial basis functions are fitted to depth-maps of the skull's surface, obtained from X-ray computed tomography (CT) data using ray-tracing techniques. They are used to smoothly interpolate the surface of the skull across defect regions. The resulting mathematical description of the skull's surface can be evaluated at any desired resolution to be rendered on a graphics workstation or to generate instructions for operating a computer numerically controlled (CNC) mill.

A mathematical procedure based on Newton's method is described that enables surface measurements to be registered, or normalized, with respect to spatial position, orientation, and, optionally, scale in three dimensions. An operator is required to identify homologous landmarks on the computer graphics images of surfaces to be registered. In this application, where the method is used to measure changes in facial shape, these landmarks are restricted to parts of the surface that have remained unchanged between the surfaces to be registered. Error in the registration of landmarks is minimized in a least-squares sense; hence multiple landmarks are favored to minimize the effect of individual errors produced by the measuring system and the operator. Examples are presented using measurements of the head taken with an optical surface scanner and a conventional X-ray computed tomography scanner.