Computational and Applied Math Proseminar

Friday, September 15, 4:00 p.m. GWC 604

Dennis Cates

Dept. Math. & Stats

Improved Results in Edge Detection Using Fourier Data with Applications

Abstract This work is an investigation of detecting jump discontinuities in piecewise smooth functions in one dimension, and edges (or feature boundaries) in two dimensions, which are realized by their Fourier spectral data. In two dimensions, 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. We employ an algorithm, known as the concentration method, that directly uses Fourier information to extract the jump locations of a function. We introduce improvements to reduce the inherent oscillations and which also enables the method to work in noisy environments.

The first technique modifies the concentration method to include the benefits of filtering. A second uses convolution to enhance the jump location by determining the strongest correlation between the concentration method waveform of the piecewise smooth function and the corresponding waveform experienced for a simple unit step function. We also introduce a zero crossing based concentration factor that can be used to create a more compactly supported formulation for localizing the edges.

Additionally, this work describes the development of an algorithm that segments a two dimensional image from its Fourier spectral data. An edge map is generated directly from the Fourier coefficients without first reconstructing the image in pixelated form. The edge map is processed with a segmentation algorithm that was 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. One and two dimensional numerical examples in noise and noise-free environments will be presented, as well as tests on a simulated MRI brain image.

For further information please contact: mittelmann@asu.edu