Development of a Symbolic Computer Algebra Toolbox for 2D Fourier Transforms in Polar Coordinates

Abstract

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.