Thesis (DSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2009. / The replacement of the standard discrete cosine transform (DCT) of JPEG with the windowed modifed DCT (MDCT) is investigated to determine whether improvements in numerical quality can be achieved. To this end, we employ an existing algorithm for optimal quantisation, for which we also propose improvements. This involves the modelling and prediction of quantisation tables to initialise the algorithm, a strategy that is also thoroughly tested. Furthermore, the effects of various window functions on the coding results are investigated, and we find that improved quality can indeed be achieved by modifying JPEG in this fashion.