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A theory of imaging for detector systems with a very limited number of projections has been developed. The relationships between a matrix which determines the system, its eigenvectors and eigenvalues, and the physical characteristics of the detector system are analyzed in order to assist in the most effective design of an instrument. It is shown that reconstruction methods for complete data sets are essentially an extension of the methods developed for incomplete sets. The concept of mathematical sweeping to replace mechanical detector motion in incomplete detector systems is demonstrated.