NationMaster - Encyclopedia: Continuous wavelet transform

In mathematics and signal processing, the continuous wavelet transform (CWT) of a function $f$ is a wavelet transform defined by Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... The wavelet transform is a transformation to basis functions that are localized in scale and in time as well (where the Fourier transform is only localized in frequency, never giving any information about where in space or time the frequency happens). ...

$gamma(tau, s) = int_{-infty}^{+infty} f(t) frac{1}{sqrt{|s|}} overline{psi left( frac{t - tau}{s} right)} dt$

where $τ$ represents translation, $s$ represents scale and $ψ$ is the "mother" wavelet. $overline{psi}$ is the complex conjugate of $ψ$ . In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

The original function $f$ can be reconstructed with the inverse transform

$f(t) = frac{1}{C_psi} int_{-infty}^{+infty} int_{-infty}^{+infty} gamma(tau, s) frac{1}{sqrt{|s|}} psileft( frac{t - tau}{s} right) dtau frac{ds}{s^2}$

where

$C_psi = int_{-infty}^{+infty} frac{left| hat psi(zeta) right|^2}{left| zeta right|} dzeta$

is called the admissibility constant and $hat{psi}$ is the Fourier transform of $ψ$ . For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition: In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

$0 < C_psi < +infty$ .

It is possible to show that the admissibility condition implies that $hatpsi(0) = 0$ , so that a wavelet must integrate to zero.

The function $ψ$ serves as the prototype for the "daughter" wavelets the signal is convolved with. For this reason, it is called the "mother" wavelet. The daughter wavelets are scaled and shifted copies of the mother wavelet:

$psi_{s,tau}(t) = frac{1}{sqrt{|s|}} psi left( frac{t-tau}{s} right)$ .

1 Computation
2 Applications
- 2.1 Determination of the fractal dimension
- 2.2 Time-frequency analysis
3 See also
4 References

Computation

The continuous wavelet transform of a discretised signal is typically computed over the temporal domain (translation) of the signal and a range of scales equivalent to the Nyquist range. Computation can either be performed using direct inner products (possibly taking advantage of the sparseness of the wavelet) or via the FFT. In the latter case, the continuous wavelet transform is noted to be a convolution at each scale, which can be performed efficiently via a discrete Fourier transform using the FFT.

Applications

Determination of the fractal dimension

Looks at extrema of the CWT with respect to translation in order to quantify the fractal dimension of a function.

Time-frequency analysis

Relates extrema of the CWT with respect to scale to conventional Fourier components in order to decompose a signal in terms of both time and frequency simultaneously. Continuous wavelets used for time-frequency analysis are designed to mimic the complex sinusoidal basis functions of the Fourier transform.

CWT-based time-frequency analysis has many benefits over other time-frequency methods (such as the short-time or windowed Fourier transform, Wigner-Ville and Choi-Williams distributions).^[1]

Time-frequency analysis has applications in many subjects including physics (quantum mechanics, seismic geophysics, turbulence), chemistry (diffraction), biology (EEG, ECG, protein- and DNA-sequence analysis), engineering (electrical transient response, impulse-shock response for non-destructive testing, fatigue analysis), finance, climatology and speech recognition.

References

Robi Polikar, The Engineer'S Ultimate Guide to Wavelet Analysis, The Wavelet Tutorial (1999)
Ingrid Daubechies, Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics), (1992) Soc for Industrial & Applied Math.

^ Paul S. Addison, The Illustrated Wavelet Transform Handbook, Taylor & Francis, 2002. ISBN 978-0750306928

Category: Wavelets Ingrid Daubechies (born August 17, 1954) is a Belgian physicist and mathematician. ...

Results from FactBites:

Continuous Wavelet Transform (3571 words)
	The CWT is a convolution of the data sequence with a scaled and translated version of the mother wavelet, the psi function.
	For the FFT fast convolution to be free of wraparound effects that arise as a consequence of non-periodicity in both the data and the response function (daughter wavelet), zero padding is needed equal to the half the length of the non-zero elements in the daughter wavelet's frequency response.
	The wavelet critical limit gradients are the following colors by default: 8-level grayscale from 10 to 50%, 8-level cyanscale from 50% to 90%, 8-level greenscale from 90% to 95%, 8-level yellowscale from 95% to 99%, and 8-level redscale from 99% to 99.9%.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m

Contents

Computation GA_googleFillSlot("encyclopedia_square");

Applications

Determination of the fractal dimension

Time-frequency analysis

See also

References

Computation