UVM Theses and Dissertations
Format:
Print
Author:
Elfataoui, Mohamed
Dept./Program:
Electrical and Computer Engineering
Year:
2007
Degree:
PhD
Abstract:
Using complex representation of real signals has long been recognized as having many advantages. The simplest example is a polar representation, that is, the complex exponential. The extension of the complex exponential to the general case of a complex representation of an arbitrary real signal, is referred to as the analytic signal or more specifically, as the one-dimensional analytic signal. It has many fundamental and important applications in one-dimensional signal processing. The direct accessibility of the so-called local structure or local properties, of a one-dimensional real signal, that is, its instantaneous amplitude and instantaneous phase, from the modulus and argument of the analytic signal is perhaps one of its more important properties.
Given the significance and applications of the analytic signal corresponding to onedimensional real signals, there has been many attempts to extend this notion to twodimensional and higher-dimensional real signals. A direct extension of the one-dimensional case is not possible. Different approaches have been used. We describe them, noting their merits and drawbacks. We note that unlike their one-dimensional counterpart, the different approaches do not generate the two or higher-dimensional analytic signals that satisfy the Cauchy-Riemann equations.
In our work presented here, we use the theory of analytic functions to generate a two-dimensional analytic signal, which ideally, satisfies the Cauchy-Riemann equations. We introduce a practical implementaGon of the new method using Cauchy-Riemann equations and an approximation by smoothing using B-splines. For fast implementation, we propose a filter implementation using steerable filters and Gabor wavelets. A new definition of an edge is proposed. As an application, we apply our new two-dimensional analytic signal to feature detection. We compare our method with other classical methods.
A new edge detector using Cauchy's theorem is under investigation. Multiscale edge detection using the analytic function method and Gabor filters for the quadrature filters is also being investigated.
Given the significance and applications of the analytic signal corresponding to onedimensional real signals, there has been many attempts to extend this notion to twodimensional and higher-dimensional real signals. A direct extension of the one-dimensional case is not possible. Different approaches have been used. We describe them, noting their merits and drawbacks. We note that unlike their one-dimensional counterpart, the different approaches do not generate the two or higher-dimensional analytic signals that satisfy the Cauchy-Riemann equations.
In our work presented here, we use the theory of analytic functions to generate a two-dimensional analytic signal, which ideally, satisfies the Cauchy-Riemann equations. We introduce a practical implementaGon of the new method using Cauchy-Riemann equations and an approximation by smoothing using B-splines. For fast implementation, we propose a filter implementation using steerable filters and Gabor wavelets. A new definition of an edge is proposed. As an application, we apply our new two-dimensional analytic signal to feature detection. We compare our method with other classical methods.
A new edge detector using Cauchy's theorem is under investigation. Multiscale edge detection using the analytic function method and Gabor filters for the quadrature filters is also being investigated.