In the local basis-function approach, a reconstruction is represented as a linear expansion of basis functions, which are arranged on a rectangular grid and possess a local region of support. The basis functions considered here are positive and may overlap. It is found that basis functions based on cubic B-splines offer significant improvements in the calculational accuracy that can be achieved with iterative tomographic reconstruction algorithms. By employing repetitive basis functions, the computitisnal effort involved in these algorithms can be minimized through the use of tabulated values for the line or strip integralsover a single-basis function. The local nature of the basis functions reduces the difficulties associated with applying iocal wnstraints on reconstruction values, such as upper and lower limits. Sincea r e~l l s t~d i~n is specified everywhere by a set of coefficients, display of a coarsely represented image does not require an arbitrary choice of an interpolation function

An arbitrary source function cannot be determined fully from projection data that are limited in number and range of viewing angle. There exists a null subspace in the Hilbert space of possible source functions about which the available projection measurements provide no information. The null-space components of deterministic solutions are usually zero, giving rise to unavoidable artifacts. It is demonstrated that these artifacts may be reduced by a Bayesian maximum a posteriori (MAP) reconstruction method that permits the use of significant a priori information. Since normal distributions are assumed for the a priori and measurement-error probability densities, the MAP reconstruction method presented here is equivalent to the minimum-variance linear estimator with nonstationary mean and covariance ensemble characterizations. A more comprehensive Bayesian approach is suggested in which the ensemble mean and covariance specifications are adjusted on the basis of the measurements.

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One ob/eetive of the aerial radiometric surveys flown as part of the U.S. Department of Energy's National Uranium Resource Evaluation (NURE) program was to ascertain the spatial distribution of near-surface radioelement abundances on a regional scale. Some method for identifying groups of observations with similar ~-ray spectral signatures and radioelement concentration values was therefore required. It is shown in this paper that cluster analysis can identify such groups with or without a priori knowledge of the geology of an area. An approach that combines principal components analysis with convergent k-means cluster analysis is used to classify 6991 observations (eaeh observation comprising three radiometric variables) ]'rom the Precambrian rocks of the Copper Mountain, Wyoming area, This method is compared with a convergent k-means analysis that utitizes available geologic knowledge. Both methods identify four clusters. Three of the clusters represent background values for the Precambrian rocks of the area, and the fourth represents outliers {anoma-lously high 214Bi). A segmentation of the data corresponding to "geologic reality" as inter. preted by other methods has been achieved by perceptive quantitative analysis of aeriaI radiometric data. The techniques employed are composites of classical elustering methods designed to handle the special problems presented by large data sets.