Wavelets are mathematical functions that are used to focus data into different frequency components, based on space and scaling. Essentially, it is the study of analysis according to scale. They are used extensively in computer science, particularly in compressing image data. According to Amara, wavelets are preferred over traditional Fourier transform methods because of their ability to analyze situations with signal discontinuities.
The Facts
Wavelets are functions that are used to represent data or other functions based on particular mathematical requirements. A "mother wavelet" can produce a family of wavelets confined to a particular finite interval. As a result, "daughter wavelets" can then be formed through mathematical translation and contraction.
History
Wavelets and approximations of functions are not new concepts in mathematics. Functions have been approximated since the early 1800's, according to Amara. During that time, Joseph Fourier discovered a technique for superposing cosines and sines as a way to represent functions. Sines and cosines are not to scale because they stretch out to infinity and as a result are called non-local signals. As time went by, mathematicians and scientists sought out different ideas and functions (other than sines and cosines) to approximate more imperfect and real world signals such as sharp spikes that only exist in a localized region. As a result, wavelets were developed to study and approximate these type of finite domain signals with sharp discontinuities.
Features
In Fourier theory, a signal can be expressed as a sum of a series of cosines and sines. This sum is called a Fourier expansion. According to Polyvalens, a Fourier expansions have only one frequency resolution (and no time resolutions). As a result, Fourier expansions produce a signal in which all frequencies are present but there is no parameter of time of when they are present.
Function
Wavelets and wavelet theory are a modern solution to the disadvantage of Fourier expansions discussed above. Wavelet analysis, according to Amara, uses a window to shift along the signal and to calculate the spectrum of each position. This process is repeated many times to provide a collection of time-scale representations of the signal for analysis.
Applications
Wavelet theory has many applications. According to Amara, it is used to study signals in various fields including acoustics, astronomy, music, magnetic resonance imagine, nuclear engineering, optics, fractals, turbulence, earthquake prediction, and pure mathematics.
In particular, other applications include computer ad human vision algorithms. According to UDel, David Marr began work in the 1980's on artificial vision for robotics. Marr studied different intensity changes in image scales for optimal detection. This later became Marr's theory of image processing and today is often referred to as the Marr wavelet.
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