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History of Wavelet

 
 

1805

Joseph Fourier (French mathematician and physicist) asserted Fourier analysis in 1805. It’s one of the most important tool in mathematical analysis.

Fourier(1768-1830).jpg

Fig.1 Joseph Fourier (1768-1830)

For any 2π periodical function f(x) can be expressed as the following equations.

 

 

1909

    Alfréd Haar(a Hungarian mathematician) found another basis function to analysis a signal as follows.

  

The function basis h(x) is called Haar function, and it’s the first mention of “wavelets” at that time.          

 Fig.2 Haar wavelet

 

1930

    Paul Levy found Haar basis function is superior to the Fourier basis functions for studying small complicated details in the Brownian motion.

1946

    Dennis Gabor was Hungarian-born electrical engineer who won the Nobel Prize for Physics in 1971 for his invention of holography. He is considerd as the father of holography. His other work included research on high-speed oscilloscopes, communication theory, physical optics, and television.

    In order to study the spectral behavior of an analog signal from its Fourier

transform, full knowledge of the signal in the time-domain must be acquired. The deficiency of Fourier transform in time-frequency analysis was observed by Dennis Gabor, who, in1946, introduced a time-localization “window function” to extract local information of the Fourier transform of the signal. He suggested representing a signal with time and frequency as coordinates.The method is short time Fourier transform,and the discussion of STFT is Gabor transform.

Fig.3  Dennis Gabor(1900-1979)

 Fig.4  Alfréd Haar(1885-1933) 

 

1975

Ronald Coifman and Guido Weiss interpreted Lusin's theory in terms of atoms and molecules which were to form the basic building blocks of a function space. The first synthesis of these theories leading up to wavelet analysis.

 

1984

    Morlet tried Windowed Fourier Analysis while working for an oil company. Oil companies searched for underground oil by sending impulses into the ground and analyzing their echoes. These echoes could be analyzed to tell how thick a layer of oil underground would be. Fourier Analysis and Windowed Fourier Analysis were used to analyze these echoes. However, Fourier Analysis was a time-consuming process so Morlet began to look elsewhere for a solution.

    When he worked with Windowed Fourier Analysis he discovered that keeping the window fixed was the wrong approach.  He did exactly the opposite. He kept the frequency of the function (number of oscillations) constant and changed the window.  He discovered that stretching the window stretched the function and scrunching or squeezing the window compressed the function. This concept of Morlet’s was touched with the Haar wavelets from earlier.

    Morlet had made an impact on the history of wavelets; however, he wasn’t satisfied with his efforts. In 1981, Morlet teamed up with a man named Alex Grossman. They developed continuous wavelet transform in1984.

Fig.5  Jean Morlet

Fig.6  Morlet wavelet

 

1985

Yves Meyer discovered the first smooth orthogonal wavelet basis functions with better time and frequency localization.

 

1986

     Stephane Mallat( a former student of Yves Meyer) collaborated with Yves Meyer to develop multiresolution analysis theory (MRA), DWT, wavelet construction techniques.

 

Meyer.jpg

Fig.7  Yves Meyer

Fig.8  Stephane Mallat

 

1987

The final stage of developing wavelet, Ingrid Daubechies found

compact and orthogonal wavelets. Daubechies wavelets are a family of orthogonal wavelets defining a discrete wavelet transform and characterized by a maximal number of vanishing moments for some given support.

Daubechies wavelets were not only orthogonal (like Meyer’s) but which could be implemented using simple digital filtering ideas, in fact, using short digital filters. The new wavelets were almost simple to program.

Fig.9  Ingrid Daubechies

Fig.10  Daubechies wavelet

With the ideal of Mallat and Daubechies, wavelet transform could be computed on personal computers quickly and easily.

1993

David Edward Newland in 1993 introduced the harmonic wavelet transform, It combines advantages of the short-time Fourier transform and the continuous wavelet transform. It can be expressed in terms of repeated Fourier transforms.

 Fig.11  David Edward Newland

 

Post 1993

Wavelet transform is a very popular tool in various field and keeps developing.