Introduction to Wavelet Transform

Introduction to Wavelet Transform

Wavelet analysis is a kind of signal time-frequency analysis, before the advent of wavelet analysis, Fourier transform is the most widely used signal processing, the most effective analysis means. Fourier transform is a tool to transform the time domain into the frequency domain, in the physical sense, the essence of Fourier transform is to decompose the waveform into the superposition of different frequency sine wave. It is this important physical meaning of Fourier transform that determines the unique position of Fourier transform in signal analysis and signal processing. Fourier transform uses sinusoidal wave, which is infinitely stretched in two directions as orthogonal basis function, to show the periodic function into Fourier series, to show the non-periodic function as Fourier integral, to use Fourier transform to analyze the function spectrum, to reflect the time spectrum characteristic of the whole signal, and to reveal the characteristic of the stationary signal better.

Fourier transformation, as a global change, has certain limitations, such as not having the ability to localize and analyzing non-stationary signals. In practice, people began to make various improvements to the Fourier transformation to improve this limitation, such as STFT (short-time Fourier transform, also called Window Fourier transform, windowing Fourier transform, Gabor transform). Because of the sliding window function adopted by STFT, the fixed time-frequency resolution is fixed, and the adaptive capability is not available, and the wavelet analysis solves the problem very well. Wavelet analysis is a new branch of mathematics, it is the most perfect crystallization of pan function, Fourier analysis, harmonic analysis and numerical analysis, especially in the fields of signal processing, image processing, speech processing and many nonlinear sciences. It is considered to be another effective method of frequency analysis after Fourier analysis. Compared with Fourier transform, wavelet transform is a local region transformation in time and frequency domain, so it can extract information from signal effectively, and perform multi-scale refinement analysis of function or signal by operation function of scaling and moving, etc. (Multiscale analyze). It inherits and develops the idea of the localization of the short-time Fourier transform, and overcomes the disadvantage of the window size not changing with the frequency, and can provide a "time-frequency" window with the frequency change, which is the ideal tool for the signal time-frequency analysis and processing.

The term wavelet (Wavelet), as the name implies, "wavelet" is small area, the length is limited, the average value is 0 waveform. The so-called "small" means that it has attenuation, and called "wave" refers to its volatility, its amplitude and positive and negative phase of the oscillation form. Compared with the Fourier transform, the wavelet transform is the time (space) frequency localization analysis, it through the telescopic translation operation to the signal (function) gradually multi-scale refinement, finally reach the high-frequency time subdivision, low frequency division, can automatically adapt to the requirements of time-frequency signal analysis, so as to focus on the signal arbitrary details, It solves the difficult problem of Fourier transformation and becomes a major breakthrough in scientific method since the Fourier transformation. Some people refer to the wavelet transform as a "mathematical microscope".

From the perspective of image processing, there are several advantages of the wavelet transform:

⑴ wavelet decomposition can cover the entire frequency domain (provides a mathematically complete description)

⑵ wavelet transform can greatly reduce or remove the correlation between different characteristics by selecting the appropriate filter.

⑶ wavelet transforms with "Zoom" feature, high frequency resolution and low time resolution (wide Analysis window) available at low frequencies, low frequency resolution and high time resolution (narrow analysis window) at high frequency segments

A fast algorithm for ⑷ wavelet transform (mallat wavelet decomposition algorithm)

Introduction to Wavelet Transform