Open Access Senior Thesis
Bachelor of Science
Image processing refers to the various operations performed on pictures that are digitally stored as an aggregate of pixels. One can enhance or degrade the quality of an image, artistically transform the image, or even find or recognize objects within the image. This paper is concerned with image processing, but in a very mathematical perspective, involving representation theory. The approach traces back to Cooley and Tukey’s seminal paper on the Fast Fourier Transform (FFT) algorithm (1965). Recently, there has been a resurgence in investigating algebraic generalizations of this original algorithm with respect to different symmetry groups. My approach in the following chapters is as follows. First, I will give necessary tools from representation theory to explain how to generalize the Discrete Fourier Transform (DFT). Second, I will introduce wreath products and their application to images. Third, I will show some results from applying some elementary filters and compression methods to spectrums of images. Fourth, I will attempt to generalize my method to noncyclic wreath product transforms and apply it to images and three-dimensional geometries.
Chang, William, "Representation Theoretical Methods in Image Processing" (2004). HMC Senior Theses. 160.