WAVELET TRANSFORM TOOLS FOR
ENGINEERING AND MEDICINE
PAUL S ADDISON
SCHOOL OF THE BUILT ENVIRONMENT
NAPIER UNIVERSITY
EDINBURGH
SCOTLAND
This page summarises both current and previous research projects of mine which employ wavelet transform-based analysis methods. The wavelet transform has become a valuable analysis tool due to its ability to elucidate simultaneously both spectral and temporal information within the signal. This overcomes the basic shortcoming of Fourier analysis, which is that the Fourier spectrum contains only globally averaged information, so leading to location specific features in the signal being lost. Applications of wavelet analysis are now widespread and cover many fields of scientific research including medical science, geophysics, engineering, image analysis, fluid turbulence and financial analysis.
[More information on my book: THE ILLUSTRATED WAVELET TRANSFORM HANDBOOK]
LOW OSCILLATION COMPLEX WAVELETS
We are currently exploring the practical application of low-oscillation complex wavelets as powerful feature detection tools for data analysis. These wavelets, which have been largely ignored to date in the scientific literature, allow for a decomposition which is more ‘temporal than spectral’ in wavelet space. This is shown to be useful for the detection of small amplitude, short duration signal features which are masked by much larger fluctuations.
Preprint of the paper : ‘Low-Oscillation Complex Wavelets’,
P.S. Addison, J.N. Watson and T. Feng, Journal of Sound and Vibration, 2002,
Vol.254(4), 733-762. (in pdf format) 
WAVELET TRANSFORMS FOR LOW STRAIN INTEGRITY TESTING OF FOUNDATION PILES
P.S. Addison, Prof. A. Sibbald and Tong Feng
Funding: EPSRC (Grant GR/M21881)
Low strain integrity testing of foundation piles is a measurement of the pile head response to an instrumented hammer blow. The compressive stress wave produced by the hammer impact reflects from features within the pile giving a trace characteristic of its shape, construction and environment. Conventionally the resultant trace is interpreted using two complementary techniques: the Sonic Echo (time domain) and the Transient Dynamic Response (frequency domain). The wavelet transform allows for the decomposition of the response signal while retaining temporal information. Recent work indicates the 'unfolding' of the one dimensional signal into two dimensions can facilitate feature spotting while the use of a wavelet filter bank rather than conventional Fourier types can reduce its signal to noise ratio.
The work will potentially benefit all areas of construction and testing by providing a single improved test method for the interpretation of results so overcoming some of the basic shortfalls of Fourier analysis and sonic echo techniques. The images below show a sonic echo NDT trace taken from a foundation pile together with its associated wavelet transform plot. The scalogram highlights the location of the end of the pile as an obvious island in the top right hand quadrant of the plot. The investigators are currently using the wavelet transform as both a feature detection technique and a filtering device.
The images below show some recent pile integrity testing using our newly developed wavelet based system. The left image shows the testing of uninstalled piles with known defects and the right hand image shows the in-situ testing of piles on-site.
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WAVELET ANALYSIS OF MEDICAL SIGNALS
P.S. Addison, Jamie Watson and Nopadol Uchaipichat
Over recent years, wavelet transforms have played an increasingly important role in the medical signal analysis. Wavelet transform analysis has been applied to a wide variety of biomedical signals including: the ECG, EEG, EMG, Echocardiograms, MRI Images, clinical sounds-arterial bruits, heart sounds, breath sounds, respiratory patterns, blood pressure trends, and DNA sequences. Research by ourselves and a number of collaborators from the UK and Europe has led to new insights into the underlying structure of a number of cardiac arrhythmias including ventricular fibrillation (VF) and atrial fibrillation (AF).
The two plots below show a short segment of ECG containing normal sinus rhythm together with its associated wavelet transform scalogram. The QRS complex of the waveform is evident from the conical structures in the scalogram, converging to the high frequency components of the RS spike. The P and T waves are also labelled in the plot. This figure highlights the wavelet transform's ability to pick out short duration, high frequency components in the time-frequency plane. An equivalent short time Fourier transform (STFT) spectogram smears this short duration information due to its fixed width window.
Ventricular fibrillation (VF) is the primary cardiac arrhythmia associated with sudden cardiac death. In the literature, VF is often described as a signal which is 'uncoordinated', 'random', 'chaotic', 'noisy' etc. The two plots below show the ECG and corresponding wavelet transform for a segment of ventricular fibrillation. High frequency spiking is evident in the scalogram plot. This and other regular, coherent structure has been found by our group using wavelet-based interrogation tools.
The figure below shows a 7 minute sequence of ventricular fibrillation (VF) with CPR initiated after 5 minutes (evident by the red, high energy band appearing at 5 minutes in the lower right hand quadrant). Distinct banding can also be seen in the pre-CPR sequence. Closer inspection of the trace over smaller time windows reveals a rich structure within the VF signal. For more information see the recent references on the home page.
WAVELET ANALYSIS OF TURBULENT FLOW FIELDS
P.S. Addison and Kevin Murray
The turbulent flow fields downstream of a variety of flow obstacles placed in an open channel flow are measured using laser Doppler anemometry. The time series obtained are analysed using a variety of wavelets. The project seeks to develop an iterated transform which should adapt to the flow structures at various resolution levels in the flow. The advantage of such a transform would be that the analysis of time series would not be confined to a single transform used over all scales which must, at present, be selected using an a priori knowledge of the flow field.
The plots below show the velocity time series taken from the vortex shedding flow behind a cylinder in an open channel and its corresponding wavelet transform scalogram plot (Mexican Hat). The scalogram shows up both the vortex shedding (as regular peaks and troughs) in the lower part of the plot and larger coherent flow structures towards the top of the plot.
ANALYSIS OF CONCRETE CRACKING USING FRACTAL AND WAVELET TECHNIQUES
P.S. Addison and Lewis Dougan
This work uses both fractal geometric and wavelet analysis techniques to quantitatively analyse concrete cracking. It can be shown that through fractal analysis a complete geometric description of the spatial cracking phenomenon can be produced in the form of an effective Fokker-Planck diffusion equation. In addition, it has been shown how crack patterns may be synthesised using a family of random fractal functions known as fractional Brownian motions. Recently, the research work has turned its attention to using the wavelet transform to interpret concrete crack profiles and surfaces to infer matrix composition and subsurface defects.
The plot below on the left contains a SEM Image of a crack profiles at 4010´
magnification. The plot below on the right contains wavelet and Fourier power
spectra of a synthetic crack surface indicating fractal scaling. These methods
have been compared to the box counting method and the variable bandwidth method
for fractal dimension estimation.
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Good places to start a wavelet search:
http://www.amara.com/current/wavelet.html
http://www.mathsoft.com/wavelets.html
or subscribe to Wavelet Digest:
http://www.wavelet.org/wavelet/index.html
[More information on my book: THE ILLUSTRATED WAVELET TRANSFORM HANDBOOK]
[Go to my 'ENGINEERING WITH FRACTAL GEOMETRY' page.]
[Go to my book: FRACTALS AND CHAOS.]