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Three-dimensional (3D) tomographic analysis of extreme ultraviolet (EUV) images is used to place empirical constraints on the corona’s temperature and density structure. The input data are images taken by the EUVI instrument on STEREO A and B spacecraft for Carrington Rotation 2069 (16 April to 13 May 2008). While the reconstructions are global, we demonstrate the capabilities of this method by examining specific structures in detail. Of particular importance are the results for coronal cavities and the surrounding helmet streamers, which our method allows to be analyzed without projection effects for the first time. During this rotation, both the northern and southern hemispheres exhibited stable polar crown filaments with overlying EUV cavities. These filaments and cavities were too low-lying to be well observed in white-light coronagraphs. Furthermore, due to projection effects, these cavities were not clearly discernible above the limb in EUV images, thus tomography offers the only option to study their plasma properties quantitatively. It is shown that, when compared to the surrounding helmet material, these cavities have lower densities (about 30%, on average) and broader local differential emission measures that are shifted to higher temperatures than the surrounding streamer plasma. [ABSTRACT FROM AUTHOR]