Morlet wavelet

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Real-valued Morlet wavelet MorletWaveletMathematica.svg
Real-valued Morlet wavelet
Complex-valued Morlet wavelet Wavelet Cmor.svg
Complex-valued Morlet wavelet

In mathematics, the Morlet wavelet (or Gabor wavelet) [1] is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing [2] and vision. [3]

Contents

History

In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms , and which provide the best trade-off between spatial and frequency resolution. [1] These are used in the Gabor transform, a type of short-time Fourier transform. [2] In 1984, Jean Morlet introduced Gabor's work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the continuous wavelet transform. [4]

Definition

The wavelet is defined as a constant subtracted from a plane wave and then localised by a Gaussian window: [5]

where is defined by the admissibility criterion, and the normalisation constant is:

The Fourier transform of the Morlet wavelet is:

The "central frequency" is the position of the global maximum of which, in this case, is given by the positive solution to:

[6]

which can be solved by a fixed-point iteration starting at (the fixed-point iterations converge to the unique positive solution for any initial ).[ citation needed ]

The parameter in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction is used to avoid problems with the Morlet wavelet at low (high temporal resolution).[ citation needed ]

For signals containing only slowly varying frequency and amplitude modulations (audio, for example) it is not necessary to use small values of . In this case, becomes very small (e.g. ) and is, therefore, often neglected. Under the restriction , the frequency of the Morlet wavelet is conventionally taken to be .[ citation needed ]

The wavelet exists as a complex version or a purely real-valued version. Some distinguish between the "real Morlet" vs the "complex Morlet". [7] Others consider the complex version to be the "Gabor wavelet", while the real-valued version is the "Morlet wavelet". [8] [9]

Uses

Use in medicine

In magnetic resonance spectroscopy imaging, the Morlet wavelet transform method offers an intuitive bridge between frequency and time information which can clarify the interpretation of complex head trauma spectra obtained with Fourier transform. The Morlet wavelet transform, however, is not intended as a replacement for the Fourier transform, but rather a supplement that allows qualitative access to time related changes and takes advantage of the multiple dimensions available in a free induction decay analysis. [10]

The application of the Morlet wavelet analysis is also used to discriminate abnormal heartbeat behavior in the electrocardiogram (ECG). Since the variation of the abnormal heartbeat is a non-stationary signal, this signal is suitable for wavelet-based analysis.

Use in music

The Morlet wavelet transform is used in pitch estimation and can produce more accurate results than Fourier transform techniques. [11] The Morlet wavelet transform is capable of capturing short bursts of repeating and alternating music notes with a clear start and end time for each note.[ citation needed ]

A modified morlet wavelet was proposed to extract melody from polyphonic music. [12] This methodology is designed for the detection of closed frequency. The Morlet wavelet transform is able to capture music notes and the relationship of scale and frequency is represented as the follow:

where is the pseudo frequency to scale , is the center frequency and is the sampling time.

Morlet wavelet is modified as described as:

and its Fourier transformation:

Application

See also

References

  1. 1 2 A Real-Time Gabor Primal Sketch for Visual Attention "The Gabor kernel satisfies the admissibility condition for wavelets, thus being suited for multi-resolution analysis. Apart from a scale factor, it is also known as the Morlet Wavelet."
  2. 1 2 Mallat, Stephane (September 18, 2009). "Time-Frequency Dictionaries". A Wavelet Tour of Signal Processing, The Sparse Way.
  3. J. G. Daugman. Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. Journal of the Optical Society of America A, 2(7):1160–1169, July 1985.
  4. "Joint Time/Frequency Analysis, Q Quality factor and Dispersion computation using Gabor-Morlet wavelets or Gabor-Morlet transform" (PDF). Archived from the original (PDF) on 2013-06-09. Retrieved 2012-05-12.
  5. John Ashmead (2012). "Morlet Wavelets in Quantum Mechanics". Quanta. 1 (1): 58–70. arXiv: 1001.0250 . doi:10.12743/quanta.v1i1.5. S2CID   73526961.
  6. Lilly, J.M.; Olhede, S.C. (January 2009). "Higher-Order Properties of Analytic Wavelets". IEEE Transactions on Signal Processing. 57 (1): 158. arXiv: 0802.2377 . Bibcode:2009ITSP...57..146L. doi:10.1109/TSP.2008.2007607.
  7. "Matlab Wavelet Families". Archived from the original on 2019-08-10.
  8. Mathematica documentation: GaborWavelet
  9. Mathematica documentation: MorletWavelet
  10. "Definition of the Neurochemical Patterns of Human Head Injury in 1H MRS Using Wavelet Analysis" (PDF). Archived from the original (PDF) on 2014-07-28.
  11. Kumar, Neeraj; Kumar, Raubin (2020-01-29). "Wavelet transform-based multipitch estimation in polyphonic music". Heliyon. 6 (1): e03243. Bibcode:2020Heliy...603243K. doi: 10.1016/j.heliyon.2020.e03243 . ISSN   2405-8440. PMC   7000807 . PMID   32042974.
  12. Kumar, Neeraj; Kumar, Raubin; Murmu, Govind; Sethy, Prabira Kumar (2021-02-01). "Extraction of melody from polyphonic music using modified morlet wavelet" . Microprocessors and Microsystems. 80 103612. doi:10.1016/j.micpro.2020.103612. ISSN   0141-9331.
  13. Shao, Haidong; Xia, Min; Wan, Jiafu; Clarence, W. de Silva (February 2022). "Modified Stacked Autoencoder Using Adaptive Morlet Wavelet for Intelligent Fault Diagnosis of Rotating Machinery". IEEE/ASME Transactions on Mechatronics. 27 (1): 24–33. Bibcode:2022IATM...27...24S. doi:10.1109/TMECH.2021.3058061.
  14. Zulqurnain, Sabir; Muhammad, Umar; Muhammad, Asif Zahoor Raja; Haci, Mehmet Baskonus; Gao, Wei (2022). "Designing of Morlet wavelet as a neural network for a novel prevention category in the HIV system" . International Journal of Biomathematics. 15 (4) 2250012. doi:10.1142/S1793524522500127.
  15. Wang, B. O; J. F. Gomez-Aguilar; Zulqurnain Sabir; Muhammad Asif Zahoor Raja; Wei-Feng Xia; H. A. D. I. Jahanshahi; Madini O. Alassafi; Fawaz E. Alsaadi (2022). "Numerical computing to solve the nonlinear corneal system of eye surgery using the capability of Morlet wavelet artificial neural networks". Fractals. 30 (5): 2240147–2240353. Bibcode:2022Fract..3040147W. doi: 10.1142/S0218348X22401478 .
  16. Z. Cui; Y. Gao; J. Hu; S. Tian; J. Cheng (March 2021). "LOS/NLOS Identification for Indoor UWB Positioning Based on Morlet Wavelet Transform and Convolutional Neural Networks". IEEE Communications Letters. 25 (3): 879–882. Bibcode:2021IComL..25..879C. doi:10.1109/LCOMM.2020.3039251.
  17. Simon J. Ward; Rabeb Layouni; Sofia Arshavsky-Graham; Ester Segal; Sharon M. Weiss (2021). "Morlet Wavelet Filtering and Phase Analysis to Reduce the Limit of Detection for Thin Film Optical Biosensors". ACS Sensors. 6 (8): 2967–2978. arXiv: 2103.07524 . Bibcode:2021ACSSe...6.2967W. doi:10.1021/acssensors.1c00787. PMC   8403169 . PMID   34387077.