Evaluation of New Gaussian Wavelet Functions in Signal Edge Detection

Edge detection is a fundamental tool in image processing, machine vision and computer vision, particularly in the areas of feature detection and feature extraction. The same problem of finding discontinuities in 1D signals is known as step detection and the problem of finding signal discontinuities over time is known as change detection. In this paper, a new set of wavelet basis functions for the edge detection issue is introduced in 1D space. First, we develop the Gaussian wavelet and present new bases by the derivation of Gaussian smoothing filter. It is proven that these filters have the necessities of the wavelet basis. After that, for proposed wavelet functions, three Canny criteria (signal-to-noise ratio, localization and low spurious response) and spatial and frequency width, which are surveys for edge detectors are discussed and formulated. For the better understanding the behavior of bases, the formulas are presented in the parametric form and compared with each other in relevant tables. The unit step and line edge are modeled as two particular types of edges and detected in the wavelet domain via introduced wavelet functions. Moreover, the effect of smooth filtering as a denoising preprocessing stage in the edge detection is discussed, and relevant formulas are derived