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Péter Balázs: Continuous and discrete image reconstruction |
Computerized Tomography (CT) was originally a method of diagnostic radiology to obtain the density distribution within the human body based on X-ray projection samples. From a mathematical point of view it seeks to determine an unknown function defined over the 3D Euclidean space from weighted integrals over subspaces, called projections. Since the values of the function can vary over a wide range, a huge number of projections are needed to ensure an accurate reconstruction. In the first part of the talk we introduce two widely used methods to solve the abovementioned image reconstruction problem.
There are applications where the aim is to reconstruct objects from just a small number of their projections. In industrial non-destructive quality testing or in Electron Tomography it often occurs that the sample does not fit into the scanner in certain directions. In these cases x-ray projections of the object can be measured just from a limited angle, therefore CT reconstruction methods are no longer successfully applicable. However, there is still a chance to get an accurate reconstruction from just a small number of projections. By exploiting prior knowledge that the range of the image function is discrete and consists of only a small number of known values the reconstruction quality can be enhanced. This leads us to the field of Discrete Tomography that will be discussed in the second part of the talk.
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