Wavelet analysis

A wavelet is, roughly speaking, a (wave-like) function that is well localized in both time and frequency. A well-known example is the Mexican hat wavelet

$$ \tag{a1 } g( x) = ( 1- x ^ {2} ) e ^ {- x ^ {2} /2 } . $$

Another one is the Morlet wavelet

$$ \tag{a2 } g( x) = \pi ^ {- 1/4 } ( e ^ {- i \xi _ {0} x } - e ^ {- \xi _ {0} ^ {2} /2 } ) e ^ {- x ^ {2} /2 } . $$

In wavelet analysis scaled and displaced copies of the basic wavelet $ g $ are used to analyze signals and images. The continuous wavelet transform of $ s( t) $ is the function of two real variables $ a > 0 $, $ b $,

$$ \tag{a3 } S( a, b) = \frac{1}{\sqrt a } \int\limits \overline{g}\; {} _ {a,b } s( t) dt , $$

where

$$ g _ {a,b } ( t) = g \left ( t- \frac{b}{a} \right ) $$

and $ \overline{g}\; $ is the complex conjugate of $ g $. In terms of the Fourier transform $ \widehat{g} $ of $ g $ one has

$$ \tag{a4 } S( a, b) = \sqrt a \int\limits \overline{ {\widehat{g} }}\; ( a \omega ) e ^ {ib \omega } \widehat{s} ( \omega ) d \omega . $$

On the basic wavelet $ g $ one imposes the admissibility condition

$$ \tag{a5 } c _ {g} = 2 \pi \int\limits | \widehat{g} ( \omega ) | \frac{d \omega }{| \omega | } < \infty $$

(which implies $ \widehat{g} ( 0) = 0 $, i.e. $ \int g( t) dt = 0 $, if $ \widehat{g} ( \omega ) $ is differentiable). Assuming (a5), there is the inversion formula

$$ \tag{a6 } s( t) = c _ {g} ^ {-} 1 \int\limits \left [ \int\limits S( a, b) g _ {a,b } ( t) db \right ] \frac{da}{a ^ {2} } . $$

The wavelet transform is associated to the wavelet group $ \{ {T _ {ab} } : {a > 0, b \in \mathbf R } \} $, $ T _ {ab} ( x) = ax + b $, and certain subgroups $ \{ {T _ {ab} } : {a = 2 ^ {k} , k,b \in \mathbf Z } \} $ in much the same way that the Fourier transform is associated with the groups $ \mathbf R $ and $ \mathbf Z $. The early vigorous development of wavelet theory is mainly associated with the names of J. Morlet, A. Grosmann, Y. Meyer, and I. Daubechies, and their students and associates. One source of inspiration was the windowed Fourier analysis of D. Gabor, [a1].

An orthonormal wavelet basis is a basis of $ L _ {2} ( \mathbf R ) $ of the form

$$ \psi _ {j,k } ( x) = 2 ^ {j/2} \psi ( 2 ^ {j} x - k) ,\ \ j , k \in \mathbf Z . $$

A non-differentiable example of such a basis is the Haar system. Orthonormal bases with $ \psi $ of compact support and $ r $- times differentiable were constructed by Daubechies. These are called Daubechies bases. Higher differentiability, i.e. larger $ r $, for these bases requires larger support.

Wavelets seem particularly suitable to analyze and detect various properties of signals, functions and images, such as discontinuities and fractal structures. They have been termed a mathematical microscope. In addition, wavelets serve as a unifying concept linking many techniques and concepts that have arisen across a wide variety of fields; e.g. subband coding, coherent states and renormalization, Calderon–Zygmund operators, panel clustering in numerical analysis, multi-resolution analysis and pyramidal coding in image processing.

References

[a1]	D. Gabor, "Theory of communication" J. Inst. Electr. Eng. , 93 (1946) pp. 429–457
[a2]	Y. Meyer, "Les ondelettes" , A. Colin (1992)
[a3]	Y. Meyer, "Ondelettes et opérateurs" , I. Ondelettes , Hermann (1990)
[a4]	I. Daubechies, "Ten lectures on wavelets" , SIAM (1992)
[a5]	C.K. Chui, "An introduction to wavelets" , Acad. Press (1992)
[a6]	C.K. Chui, "Wavelets: a tutorial in theory and applications" , Acad. Press (1992)
[a7]	M.B. Ruskai (ed.) et al. (ed.) , Wavelets and their applications , Jones & Bartlett (1992)
[a8]	J.M. Combes (ed.) A. Grosmann (ed.) Ph. Tchamitchian (ed.) , Wavelets. Time-frequency methods and phase space , Springer (1989)
[a9]	P.G. Lemarié (ed.) , Les ondelettes en 1989 , Springer (1990)
[a10]	F. Argorel, A. Arnéodo, J. Elezgaray, G. Grasseau, "Wavelet transform of fractal aggregates" Physics Letters A , 135 (1989) pp. 327–336
[a11]	M. Holschneider, "On the wavelet transformation of fractal objects" J. Stat. Phys. , 50 (1988) pp. 963–993
[a12]	M. Holschneider, Ph. Tchamitchian, "Pointwise analysis of Riemann's nondifferentiable function" Invent. Math. , 105 (1991) pp. 157–176
[a13]	M. Antonini, M. Barlaud, I. Daubechies, P. Mathieu, "Image coding using vector quantization in the wavelet transform domain" , IEEE Int. Conf. on Acoustics, Speech, and Signal Processing , IEEE (1991) pp. 2273–2276
[a14]	G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms" Comm. Pure Appl. Math. , 44 (1991) pp. 141–183
[a15]	S.G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation" IEEE Trans. Pattern Analysis and Machine Intelligence , 11 (1989) pp. 674–693