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A high-resolution positron tomograph consisting of two high-density avalanche chambers mounted on a rotating gantry has been installed in the Nuclear Medicine Department of the Cantonal Hospital in Geneva. Positron annihilation data are collected from six detector positions and back-projected in real time to form a single three-dimensional image. Up to 16 transverse or longitudinal sections may be imaged simultaneously. The derivation of the appropriate deconvolution filter is described in this paper, and the first images taken in the transverse mode are presented. It is shown that quantitative information can be obtained from such images.