Undergraduate Projects

This project investigated how changing the group ring can introduce alternative forms of convolution. The dihedral group ring has a block matrix representation composed of two unique matrices where one matrix is a circulant matrix and the other a reverse circulant matrix. This matrix derived from the dihedral group ring was implemented in python to be applied and analysed. It was observed that there are differences in output between the circulant convolution and the dihedral convolution. This is largely dependent on the design of the filter as certain filters will produce a dihedral operation that will contain the circular convolution. It was found that the convolution theorem used in the fast fourier transform can be applied to the dihedral convolution with the use of permutations. The goal is to further examine the applications of the resulting operation in signal processing and within convolution neural networks.