Wavelets are a set of mathematical functions used to decompose a signal continuously into its frequency components, the resolution of each component being equal to its scale.
A wavelet transform is the decomposition of a function based on wavelet functions. wavelets are transmitted and scaled samples of a function with finite length and highly damping oscillation. Compared to the Fourier transform, it can be said that the wavelet has a very good localization property.
For example, the Fourier transform of a sharp peak has a large number of coefficients, because the basic functions, the Fourier transform, are sine and cosine functions whose amplitude is constant over the whole interval.
Wavelet functions, on the other hand, are functions in which most of their energy is concentrated in a small interval and diminishes rapidly. Therefore, with proper selection of mother wavelets, better compression is performed compared to Fourier transform.
The mother wavelet is horizontally and vertically deformed through the effect of the signal.
The wavelet coefficients, which are equivalent to the Fourier decomposition amplitudes, are thus a measure of the correlation between the signal shape and the contour of the deformed mother wavelet that follows the signal.
As wavelets are constructed over a scale change operator, we can better understand their effectiveness using a fractal analysis, and more generally using any multi-scale approach.
They should become an incontrovertible tool across the whole of scientific geography, with its interest in multi-scalar and multi-scale phenomena.
More specifically, the signal − whether it is a chronicle series or an image − is decomposed into one general part and some details representing some irregularities.
The general part is then itself decomposed into a new general part and more details, and so on. This is an iteration model similar to that of the construction of a fractal.
This connection explains the effectiveness of wavelets in the study of fractals.
What other characteristics of wavelets can be used in fractal analysis?
According to the following discussion, discontinuity in wavelet signals can be considered as a kind of fractal.
In the figure below, how does this discontinuity in the signal contribute to fractal analysis?
- Badges
- Science topic
- Similar topics
- Correction