Single Photon Emission Computed Tomography  (SPECT) enjoys an importance burst due to the advance of theranosctics, a combination of cancer isotope therapy and diagnostics. Nuclear isotopes bound to tumour specific compounds find their ways to the proper place of irradiation by metabolism. Emitted short range radiation of charged particles (beta's and alpha's mostly) ensure a precise dose delivery to the cancer tissue while minimizing impact on organs at risk. Monitoring the delivery of radioisotopes is possible by the by-product gamma radiation. As the  radiation mixture is optimized for dose delivery rather than for imaging, the SPECT measured data is suboptimal in many respect: either a low gamma yield or unfavourable gamma energy yield necessitate a leap forward in the quality of image reconstruction. Low intensity imaging and precise scatter signal removal requires the tailoring of reconstruction techniques towards ultimate precision. 

Iterative statistical reconstruction schemes (Maximum Likelihood Expectation Maximization -MLEM, Primal-Dual Hybrid Gradient, etc) employs regularization to stabilise convergence and to direct the iteration towards favourable fixed points. The regularization is usually formulated as a constraint on e.g. the noise term: we prefer lower noise content of a reconstructed volume, or we would like to hinder the forming of image artefacts. A popular technique is the Total Variation (TV) norm (local sum of the absolute value of pixel intensity differences), that is often used in practice while its mathematical properties necessitate cumbersome approximations resulting in One Step Late (OSL) algorithms, or artefacts. TV norm with OSL-MLEM is notorious for creating chequerboard artefacts. Another example is the formation of the Gibbs-like artefact, where a certain spatial frequency above the resolution cut-off, become over-pronounced and adds a ringing to abrupt changes. 

The task of the student is to create from existing building blocks a 2D ML-EM reconstruction sandbox in python, where the above mentioned TV regularization and the forming of the Gibbs artefact can be studied with different noise terms present. The properties of the regularization and Gibbs artefacts should be characterized. Some theoretically founded and some experimentally conjured algorithms are present in the scientific literature, such as forward projection biasing, sampling volume sharpening and inter-iteration filtering.  The student shall experiment with these techniques, extend to  inhomogeneous resolution problems and formulate solutions with sound theoretical bases and palpable image quality enhancements.