Fall 2007

Wavelet and Fourier Analysis

Listed in: Mathematics and Statistics, as MATH-19


Tanya L. Leise (Section 01)


The first half of the course covers continuous and discrete Fourier transforms (including convolution and Plancherel's formula), Fourier series (including convergence and the fast Fourier transform algorithm), and applications like heat conduction along a rod and signal processing. The second half of the course is devoted to wavelets: Haar bases, the discrete Haar transform in 1 and 2 dimensions with application to image analysis, multiresolution analysis, filters, and wavelet-based image compression like JPEG2000. Three class hours per week plus a weekly one-hour computer laboratory. Requisite: Mathematics 13 and one of 21, 22, or 25. First semester. Professor Leise.

Submitted by Tanya L. Leise
"Madonna with Child" by Perugino (1446–1523). Wavelet decomposition can reveal what areas Perugino painted and what parts were probably painted by his apprentices (see me for the article if you are interested). Wavelet analysis has also been used to detect forgeries and to determine whether images are true photographs or computer generated. Wavelets have also been used in Pixar movies to add fine detail to computer animation.