a) (b) Figure 1: (a) Fitting a Radial Basis Function (RBF) to a 438,000 point-cloud. (b) Automatic mesh repair using the biharmonic RBF.
AbstractWe 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 nonuniformly 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.
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 PC-based system has been developed to automatically detect epileptiform activity in sixteen-channel bipolar EEG's. The system consists of three stages: data collection, feature extraction, and event detection. The feature extractor employs a mimetic approach to detect candidate epileptiform transients on individual channels, while an expert system is used to detect focal and nonfocal multichannel epileptiform events. Considerable use of spatial and temporal contextual information present in the EEG aids both in the detection of epileptiform events and in the rejection of artifacts and background activity as events. Classification of events as definite or probable overcomes, to some extent, the problem of maintaining high detection rates while eliminating false detections. So far, the system has only been evaluated on development data but, although this does not provide a true measure of performance, the results are nevertheless impressive. Data from 11 patients, totaling 180 minutes of sixteen-channel bipolar EEG's, have been analyzed. A total of 45-71% (average 58%) of epileptiform events reported by the human expert in any EEG were detected as definite with no false detections (i.e., 100% selectivity) and 60-100% (average 80%) as either definite or probable but at the expense of up to nine false detections per hour. Importantly, the highest detection rates were achieved on EEG's containing little epileptiform activity and no false detections were made on normal EEG's.
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