Sidec Technologies AB, Kista, Sweden
This CIAM presentation will deal with the mathematical research problems related to optimisation theory that arise at Sidec Technologies in the work on electron tomography. Part of the seminar overlaps with the previous CIAM seminar held on Wednesday April 11 (see abstract below). From a mathematical point of view, the reconstruction problem in electron tomography amouts to the solution of an inverse scattering problem. Due to the severe ill-posedness of this reconstruction problem, a regularisation method must be used to obtain reconstructions of any practical value at all, and a good reconstruction is likely to require a carefully chosen regularisation.
We will start with a very brief introduction to electron tomography followed by a brief introduction to regularisation theory with an emphasis on variational regularisation methods. These can be formulated as solving an optimization problem which is defined by a regularisation functional and a data discrepancy functional. The former enforces uniqueness by selecting a unique element among the least squares solutions and it also acts as a stabiliser by enforcing a smoothing. It is e.g. wellknown that 2-norms yield smooth and continuous solutions and 1-norms allow for non-smooth solutions.
Further prior knowledge about the object to be reconstructed can be encoded as side conditions to the optimisation problem.
We will review the variational regularisation method used by Sidec Technologies and the difficulties encountered in solving the resulting optimisation problem. We then turn our attention to point enhancement of sparse representations, i.e. variational regularisation with the 1-norm applied on the basis coefficients of a sparse othonormal bases/frame of the object to be reconstructed. This technique is currently enjoying a great deal of interest since it has been shown to yield superresolution. Again, in applying this technique one is faced with difficult optimisation problems.