Computational and Applied Math Proseminar

Monday, April 30, 1:30 p.m. GWC 604

Dennis Cates

Dept. Math. & Stats

Edge Detection Using Fourier Data with Applications (dissertation defense)

Abstract This work investigates jump discontinuity detection in piecewise smooth functions in one dimension, and edges (or feature boundaries) in two dimensions, which are realized by their Fourier spectral data. Fourier spectral data is often the source of image information. For example, magnetic resonance imaging (MRI) and synthetic aperture radar (SAR) sensors measure the values of the Fourier coefficients of an image. An algorithm known as the concentration method is employed to extract the jump locations of a function directly from Fourier information. This thesis introduces improvements to the concentration method that aid in reducing the inherent oscillations of this technique. They also enable the concentration method to work in noisy environments. The effect of random Gaussian noise on the concentration method is also analyzed. Quantitative measures that test the effectiveness of the edge detection method in the presence of noise are developed.

This work further describes a new algorithm that segments a two dimensional image from its Fourier spectral data. An edge map is generated directly from the Fourier coefficients by the concentration method. The edge map is processed with a segmentation algorithm that is designed to follow the Gestalt principles of feature visualization to generate closed contours around individual features within the image. This allows for extraction of any particular feature of interest for further analysis, which has the added advantage that costly reconstruction of the entire image is not needed. Examples in noise and noise-free environments are presented, as well as tests on a simulated MRI brain image.

This work also extends the concentration method to estimate the locations of jump discontinuities in the first derivative of a function in one and two dimensions. This extension is ideal for post-processing numerical solutions of partial differential equations, since shock locations and contact discontinuities must be identified for high order reconstruction.

For further information please contact: mittelmann@asu.edu