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In this paper we study three-dimensional cone beam local tomography. We analyze the local tomography function , which was proposed earlier in [A.K. Louis and P. Maass, IEEE Trans. Medical Imaging, 12 (1993), pp. 764--769]. Let f be an unknown density distribution inside an object being scanned. We find a relationship between the wave fronts of and f and compute the principal symbol of the operator which maps f into . Our results prove the fact, which was first noted in Louis and Maass, that one can recover most of the singularities of f knowing . It is shown that these are precisely the singularities of f that are visible from the data. A simple and efficient algorithm for finding values of jumps of f knowing local cone beam data is proposed. The nature of artifacts inherent in cone beam local tomography is studied.