The shape of a three-dimensional (3D) object (respectively, one of its 2D cross-sections) can be represented by a 3D (respectively, 2D) binary matrix, in which 1 indicates the presence of the object at the corresponding point in space and 0 indicates its absence. If the object material is homogeneous (for example, it is a single piece of molded plastic), then physically obtained mesurements (for example, by X-rays) through the object indicate the number of 1s in the matrix that happen to fall on certain lines. Binary Tomography is concerned with the recovery of binary matrices from such line sums. It has applications, for example, in non-destructive testing, electron microscopy, and medical imaging.

A new direction of binary tomography is when the 1s in the matrix are interpreted as a set of points emitting rays (e.g., light or radioactivity) all of the same intensity and this emitted radiation is partially absorbed between its source and the detector by some material of a known uniform absorption coefficient. The problem in Binary Emission Tomography (BET) is to reconstruct the matrix from such absorbed line sums. BET is a young research field, it is full of mathematically fascinating questions and it has important potential applications (for example, the reconstruction of radioactive objects).

From the algorithmic point of view, BET is quite different from binary tomography. For example, the reconstruction of the so-called horizontally and vertically convex binary matrices from their row and column sums is an NP-hard problem in binary tomography, but the corresponding BET problem has polynomial time complexity (at least for certain absorption value s). On the other hand, the problem of uniqueness of the solution is more intractable in BET than in binary tomography.

In this presentation an overview of BET will be given, including its foundations (uniqueness, existence, complexity), reconstruction procedures, and applications. Since BET is a young area, many open problems can be posed that are interesting from a theoretical point of view and are also relevant to some applications.