Surface approximation of curved data using separable radial basis functions

The need to treat collectively data obtained from a number of different sources arises in a varied number of disciplines. Within the field of metrology, the science of measurement, it is often necessary to fuse together data which are known to contain differing levels of error. Often the intention is to collect the data in some structured way, however external influences can force the abscissae away from the intended path and produce data that lie along a number of curved paths in the xy - plane. In this paper we present an algorithm which overcomes the need to treat curved paths (or lines) of data as being generally scattered and allows the semi-structured form of the abscissae to be exploited. The method presented here has two key features. Firstly we relate the data collected along each curved path to a local approximation, enabling any source errors to be considered separately. We show how a set of secant lines can be defined from the locations of the centres chosen for each local approximation to provide a representation of the global spread of the abscissae. Secondly, we present a method for fast evaluation of the global approximation by creating a new set of 'approximated' data values lying at scattered locations on parallel lines, where the separable properties of the Gaussian function can be utilised to produce an efficient tensor product approximation. We further suggest how greater efficiency can be achieved by identifying opportunities for parallel processing.