**abstract:**
Tomographic techniques, in which objects are viewed from multiple directions, are used in electron microscopy and SPECT. Although the goals are different the mathematics is related. The author has developed local algorithms for these problems that recover singularities or boundaries of the objects as well as object shapes and contours.

In an important type of electron microscopy the molecules are all different (rather than copies of one single molecule in different orientations), and one wants to image the individual molecules. Therefore, any algorithm for this problem must reconstruct each molecule independently. The problem is local because the electron beam is only wide enough to penetrate a small part of the object. Furthermore, the data are limited angle since the object cannot be rotated through a full 180 deg. For these reasons, some data are missing, and inversion is unstable. The author will explain how singularities can be added (similar to the added singularities in cone-beam CT). The author will present a refined local algorithm and reconstructions (pictures) using his algorithm. Development and testing of this algorithm was done jointly with Ozan Oktem. The author plans to discuss a related algorithm of his for slant-hole SPECT. In this case, the emission detectors are all slanted in the same direction with respect to this axis, and they rotate about the center of the detector array. Thus, the data are given over lines parallel a cone with this same angle. The author's algorithm will be outlined. This work is joint with students Sohhyun Chung and Tania Bakhos.

The singularity detection properties of the algorithms will be described.

Thu 18 Oct, 11:30 - 12:30, Aula Dini

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