Wavelet Transformation

Abstract

Wavelet transformation is one of the most practical mathematical transformations in the field of image processing, especially image and signal processing. Depending on the nature of the multiresolution analysis, Wavelet transformation become more accessible and powerful tools. In this paper, we refer to the mathematical foundations of this transformation.

<strong>Introduction</strong>: The wavelet transform has been proven effective for image analysis, data compression and feature extraction. This linear transform is a powerful tool for time (space)-frequency analysis of signals, especially images. The expansion of a signal into several frequency channels generates a joint representation in time and frequency domains. The wavelet transform does provide a multiresolution decomposition and representation of an image at given resolutions. It is computed by expanding a signal into a family of functions which are the dilations and translations of a unique function called the basic wavelet. It is also interpreted as a decomposition into a set of frequency channels having the same bandwidth on a logarithmic scale.

<strong>Materials and Methods:</strong> Using a one-dimensional wavelet transform and its mathematical foundations, the effect on the image resolution was compared to the short time Fourier transform.

<strong>Results:</strong> In the transformation of the wavelet, the finite signal does not convert the Fourier transform, and thus the individual peaks or, in other words, the negative frequencies are not calculated.

<strong>Conclusion:</strong>

The continuous wavelet transform can be presented as an alternative to the short time Fourier transform, and aims to resolve the resolution problems.