Chirplet transform

In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.[2][3]

Comparison of wave, wavelet, chirp, and chirplet[1]
Chirplet in a computer-mediated reality environment.

Similar to the wavelet transform, chirplets are usually generated from (or can be expressed as being from) a single mother chirplet (analogous to the so-called mother wavelet of wavelet theory).

Definitions

The term chirplet transform was coined by Steve Mann, as the title of the first published paper on chirplets. The term chirplet itself (apart from chirplet transform) was also used by Steve Mann, Domingo Mihovilovic, and Ronald Bracewell to describe a windowed portion of a chirp function. In Mann's words:

A wavelet is a piece of a wave, and a chirplet, similarly, is a piece of a chirp. More precisely, a chirplet is a windowed portion of a chirp function, where the window provides some time localization property. In terms of time–frequency space, chirplets exist as rotated, sheared, or other structures that move from the traditional parallelism with the time and frequency axes that are typical for waves (Fourier and short-time Fourier transforms) or wavelets.

The chirplet transform thus represents a rotated, sheared, or otherwise transformed tiling of the time–frequency plane. Although chirp signals have been known for many years in radar, pulse compression, and the like, the first published reference to the chirplet transform described specific signal representations based on families of functions related to one another by time–varying frequency modulation or frequency varying time modulation, in addition to time and frequency shifting, and scale changes.[2] In that paper,[2] the Gaussian chirplet transform was presented as one such example, together with a successful application to ice fragment detection in radar (improving target detection results over previous approaches). The term chirplet (but not the term chirplet transform) was also proposed for a similar transform, apparently independently, by Mihovilovic and Bracewell later that same year.[3]

Applications
(a) In image processing, periodicity is often subject to projective geometry (i.e. chirping that arises from projection). (b) In this image, repeating structures like the alternating dark space inside the windows, and light space of the white concrete, chirp (increase in frequency) towards the right. (c) The chirplet transform is able to represent this modulated variation compactly.

The first practical application of the chirplet transform was in water-human-computer interaction (WaterHCI) for marine safety, to assist vessels in navigating through ice-infested waters, using marine radar to detect growlers (small iceberg fragments too small to be visible on conventional radar, yet large enough to damage a vessel).[4][5]

Other applications of the chirplet transform in WaterHCI include the SWIM (Sequential Wave Imprinting Machine).[6][7]

More recently other practical applications have been developed, including image processing (e.g. where there is periodic structure imaged through projective geometry),[8][9] as well as to excise chirp-like interference in spread spectrum communications,[10] in EEG processing,[11] and Chirplet Time Domain Reflectometry.[12]

Extensions

The warblet transform[13][14][15][16][17][18] is a particular example of the chirplet transform introduced by Mann and Haykin in 1992 and now widely used. It provides a signal representation based on cyclically varying frequency modulated signals (warbling signals).

See also
Other time–frequency transforms

References
  1. From page 2749 of "The Chirplet Transform: Physical Considerations", S. Mann and S. Haykin, IEEE Transactions on Signal Processing, Volume 43, Number 11, November 1995, pp. 2745–2761.
  2. S. Mann and S. Haykin, "The Chirplet transform: A generalization of Gabor's logon transform", Proc. Vision Interface 1991, 205–212 (3–7 June 1991).
  3. D. Mihovilovic and R. N. Bracewell, "Adaptive chirplet representation of signals in the time–frequency plane," Electronics Letters 27 (13), 1159–1161 (20 June 1991).
  4. Mann, Steve, and Simon Haykin. "The chirplet transform: A generalization of Gabor’s logon transform." Vision interface. Vol. 91. 1991.
  5. WaterHCI Part 1: Open Water Monitoring with Realtime Augmented Reality, IEEE SPICES, INTERNATIONAL CONFERENCE ON SIGNAL PROCESSING, INFORMATICS, COMMUNICATION AND ENERGY SYSTEMS 2022 (IEEE SPICES 2022), 10 - 12 MARCH, 2022, Nalanchira, Trivandrum, Kerala, India, 6 pages
  6. Mann, Steve. "Time-Frequency" Perspectives”." Advances in Machine Vision: Strategies and Applications 32 (1992): 99.
  7. Mann, Steve, et al. "Water-Human-Computer-Interface (WaterHCI): Crossing the Borders of Computation, Clothes, Skin, and Surface."
  8. Mann, Steve. "Time-Frequency" Perspectives”." Advances in Machine Vision: Strategies and Applications 32 (1992): 99.
  9. Mann, Steve, and Simon Haykin. "Adaptive." Optical Engineering 31.6 (1992): 1243-1256.
  10. Bultan, Akansu; Akansu, A.N. (May 1998), "A novel time-frequency exciser in spread spectrum communications for chirp-like interference", Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 6, pp. 3265–3268, doi:10.1109/ICASSP.1998.679561, ISBN 0-7803-4428-6
  11. Cui, J.; Wong, W.; Mann, S. (17 February 2005), "Time–frequency analysis of visual evoked potentials using chirplet transform" (PDF), Electronics Letters, vol. 41, no. 4, pp. 217–218, Bibcode:2005ElL....41..217C, doi:10.1049/el:20056712, retrieved 2010-07-29
  12. "Example Programs - National Instruments". Archived from the original on 2012-02-14. Retrieved 2007-12-31.
  13. Mann, Steve, and Simon Haykin. "'Chirplets' and'warblets': novel time-frequency methods." Electronics letters 28, no. 2 (1992): 114-116.
  14. Mann, S., & Haykin, S. (1992, March). Time-frequency perspectives: the chirplet transform. In Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on (Vol. 3, pp. 417-420). IEEE.
  15. Angrisani, L., D'Arco, M., Moriello, R. S. L., & Vadursi, M. (2005). On the use of the warblet transform for instantaneous frequency estimation. Instrumentation and Measurement, IEEE Transactions on, 54(4), 1374-1380.
  16. Angrisani, L., Arco, M. D., Moriello, R. S. L., & Vadursi, M. (2004, August). Warblet transform based method for instantaneous frequency measurement on multicomponent signals. In Frequency Control Symposium and Exposition, 2004. Proceedings of the 2004 IEEE International (pp. 500-508). IEEE.
  17. Kazemi, S., Ghorbani, A., Amindavar, H., & Morgan, D. R. (2016). Vital-Sign Extraction Using Bootstrap-Based Generalized Warblet Transform in Heart and Respiration Monitoring Radar System.
  18. Zelinsky, N. R., & Kleimenova, N. G. Chirplet transform as the useful tool for study the time-frequency structure of geomagnetic pulsations.

Florian Bossmann, Jianwei Ma, Asymmetric chirplet transform--Part 2: phase, frequency, and chirp rate, Geophysics, 2016, 81 (6), V425-V439.

Florian Bossmann, Jianwei Ma, Asymmetric chirplet transform for sparse representation of seismic data, Geophysics, 2015, 80 (6), WD89-WD100.

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