International Symposium (NDT-CE 2003)Non-Destructive Testing in Civil Engineering 2003
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Wavelet algorithms for non destructive testingJochen H. Kurz, Hans-Jürgen Ruck, Florian Finck,
Christian U. Grosse, Hans-Wolf Reinhardt
University Stuttgart, Institute of Construction Materials, Pfaffenwaldring 4, 70550
Stuttgart, Germany, Phone: +49 - 711 - 6856792, Fax: +49 - 711 - 6856796
Email address: email@example.com (Jochen H. Kurz)
AbstractThe application of wavelets on time series in a general sense is more than 15 years old. Applications that are making use of wavelets include signal and image processing, optics, fractals, seismology and acoustics. Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques miss. This makes wavelets interesting and useful for acoustic emission analysis. The acoustic emission signal caused by microscopic movements contains a lot of information about the fracture process but also usually a significant amount of noise. The wavelet shrinkage and thresholding methods allow recovering the signal from noisy data. De-noising is possibly the most popular practice of wavelets on data because after decomposing the signal the further process is rather intuitive. The basic methods used for signal decomposition are the discrete and the continuous analysis. The use of discrete or continuous decomposition techniques depends on the complexity of the signal. In fact, de-noising is a powerful but not the only application of wavelets on time series. They can be used for statistical methods like density estimation and detection of self-similar behavior in time series. Wavelets show an alternative and fast way of getting information from acoustic emission signals where classical interpretation is often difficult and time consuming.
Key words: wavelets, acoustic emissions, de-noising, filtering, scalegram
1 IntroductionThe wavelet transform is a mathematical tool which is able to examine a signal simultaneously in time and frequency. The method itself came up in the mid - 1980s where it was originated to investigate seismic signals (Goupillaud et al., 1984). But in many aspects wavelets are a synthesis of older ideas with new elegant mathematical results and efficient computational algorithms (Percival andWalden, 2002). Fields of application are e. g. astronomy, acoustics, signal and imaging processing, fractals and seismology.
Transforms often result in simplified problem solving analysis (Brigham, 1974). One such transform analysis technique and furthermore really widespread is the Fourier transform. Another one is the wavelet transform described in the following in detail. The Fourier transform is able to reveal the frequencies present in a signal. But it is not possible to say when they are present. An effort to correct this deficiency is to analyze only a small section of the signal at a time - a technique called windowing the signal, also called the Short-Time Fourier Transform (STFT) which maps a signal into a two-dimensional function of time and frequency. The STFT represents a sort of compromise between the time- and frequency-based views of a signal. It provides some information about both when and at what frequencies a signal event occurs. However information is optained with limited precision, and that precision is determined by the size of the window (Misiti et al., 2000). This is the starting point for the Wavelet transform. Using the Wavelet transform a timescale joint representation is possible where scale is proportional to frequency. By decomposing a time series into time-scale (equal to frequency) space, one is able to determine both, the dominant modes of variability and how those modes vary in time (Torrence and Compo, 1998). In addition, this scale based analysis is often easier to interpret because the knowledge of the variance in time makes the understanding of process development simpler.
As mentioned above, the wavelet transform offers a variety of features which are interesting for time series analysis. Especially the application to acoustic emission data is very successful (Motz et al., 2002). One severe problem in acoustic emission analysis is that huge data sets (often more than 1000 acoustic emission events) with a low signal to noise ratio are detected (Figure 1).
Acoustic emission data, e. g. from concrete, normally contain a lot of high frequency noise, mainly caused by the measurement equipment (preamplifier etc.) and the surrounding. Due to the testing process itself a low frequent signal, caused by the testing device (loading machine), often superimposes the acoustic emission signal additionally. This makes the onset detection difficult. The wavelet transform is a fast and efficient way of filtering and especially de-noising data (Motz et al., 2002; Ruck and Reinhardt, 2002; Misiti et al., 2000). Due to the large number of events the use of an automatic onset detection algorithm is preferable. The use of such an algorithm requires high quality data. That means filtering and de-noising is a pre-condition for automatic data processing. Furthermore, the wavelet transform offers a variety of other application forms on time series. One example is the use of so called scalegrams (Scargle et al., 1993) which measure the variance of the wavelet coefficients as a function of the time scale. Another field of application is the detection of self similarities and long term evolution processes (Misiti et al., 2000). This listing about using wavelets leads to a closer look at the mathematical concept of the wavelet transform.
2 Mathematical ConceptsAs mentioned above, wavelets are a synthesis of older ideas with new elegant mathematical results and efficient computational algorithms (Percival and Walden, 2002). Wavelets are mathematical functions that have to be well localized (other requirements are of technical matter). The mathematical steps used for the wavelet transform can be summarized as follows:
These four points represent the principle of the wavelet transform. General concepts known from Fourier analysis form the application base. Therefore, the basic concepts of convolution and filtering of finite sequences and some basics about orthonormal functions are presented first before the discrete (DWT) and the continuous wavelet transform (CWT) will be discussed. Due to the fact that our focus of application here is set on the discrete wavelet transform the mathematical concept presented in the following is adopted from Percival and Walden (2002).
2.1 Convolution/FilteringThe first three steps of the basic concept (above) of the wavelet transform show that a periodized filter is needed. Concerning acoustic emission data this filter is applied to a finite sequence. The process is a convolution of two sequences. Suppose two infinite sequences ut and xt where the first one is the filter and the second one is the signal. The convolution is defined as:
or in flow diagram:
Regarding a finite sequence xt and a periodized filter ut which is a circular filter of length N the convolution is defined as:
Here a sequence of N variables is transformed into a new sequence of length N. In essence the filter is cut into blocks of coefficients and these are overlain and then added to form the periodized filter. Using a flow diagram this can be expressed as:
2.2 Orthonormal FunctionsUsing orthonormal transforms, reconstructing the time series from its transform can easily be managed. The series and its transform can be considered to be two representations of the same mathematical entity. Let be a N x N real valued matrix satisfying the orthonormality property vv* vv = where vv* is the transposed matrix and ., is the identity matrix. The orthonormality property can be restated in terms of the inner product as:
That means the columns and the rows of are an orthonormal set of vectors. Such a matrix can be used for an orthonormal transformation in both directions, as it is applied in the DWT.
2.3 Discrete and Continuous Wavelet TransformThe DWT is an orthonormal transform of the form:
The first two points of the above enumeration show that a periodized filter is needed which is indeed the wavelet. The N X N matrix consists of these periodized filters. The third point indicates that the filter is rescaled for every new cycle. That means the matrix consists of the wavelet and the scaling filter. The wavelet filter is a high pass filter with a nominal pass-band while the scaling filter is a low pass filter with a different nominal pass-band. Applying them on the time series X, the Wavelet coefficients W are gained which are a collection of time-scale representations of the signal, all with a different solution. In other words: the wavelet is scaled and shifted and then moved along the time series.
The main difference between the discrete and the continuous wavelet transform is that the DWT can be thought of as a subsampling of the wavelet coefficients with dyadic scales. That means each vector of W contains 2j elements, j = 1......J. Each element is one wavelet coefficient. The rows of that produce the wavelet coefficients (the wavelet filter) for a particular scale are circularly shifted versions of each other. The amount of the shift between adjacent rows is 2 elements, j = 1......J. Furthermore, changes concerning the wavelet scale (the scaling filter) are also of dyadic order in the same range as for the wavelet filter.
In contrast to the DWT the CWT can be formally written as:
Here the function X is decomposed into a set of basis functions Y w,n (t) called the wavelets. The variables w,n are the new dimension, scale and translation, after the wavelet transform. The CWT is calculated by continuously shifting a continuously scalable function over a signal. It is clear that the principal ideas behind the DWT and the CWT are identical.
3 Application to Acoustic Emission DataWavelets offer a variety of possible applications on time series. De-noising and compression are possibly the most popular practices of wavelets on data. Due to the time-scale joint representation of the wavelet transform a lot of information from the time series can be found separated in the coefficients. This information can be analyzed using different statistical approaches (e. g. detecting self similarities and long term evolution processes (Percival and Walden, 2002)). Some of them with a relevance for the application on acoustic emission data are discussed in the following.
3.1 Filtering and De-NoisingThe wavelet filter technique is easy to handle because it is obvious from a glance at the coefficients which one is containing the signal's low frequency part. For the wavelet decomposition to give an example a biorthogonal wavelet of the order 4.4 was used (Fig.2). The comparison of the original and the filtered signal on the right side of Fig. 2 shows that the low frequency part resulting from the testing procedure is completely extracted. The remaining signal contains now still some noise which is generally not of any periodic structure.
De-noising is quite similar to the wavelet filtering. After revealing the coefficients which contain the noise it is possible to extract the noise by thresholds. Again a biorthogonal wavelet of the order 4.4 and individual thresholds were used (Fig.3). The onset detection at the de-noised signal is easier than at the original signal. That makes the use of an automatic picking algorithm possible.
3.2 ScalegramsBeside the features presented above for which the success is directly visible there are several statistical methods based on the wavelet transform applicable. One of these features is making a so called scalegram (Scargle et al., 1993). A scalegram is produced by taking the mean square wavelet coefficient over a scale. That means it is a measure for the energy over a scale of a signal.
4 Discussion and ConclusionAcoustic emissions are defined as the spontaneous release of localized strain energy in stressed material (Grosse, 2002). As mentioned before, often huge data sets are gained and the data e. g. from concrete normally contain a lot of noise, mainly caused by the measurement equipment and the surrounding. Due to the testing process itself a low frequent signal also caused by the testing apparatus often superimposes the acoustic emission signal additionally. This makes the onset detection really difficult. Using the wavelet transform an additional tool for filtering and denoising time series exists. Due to the time-scale joint representation of the wavelet transform the procedure is relatively easy to handle. The results show immediately (Fig. 2 and 3) the filtering and de-noising success. Furthermore, the DWTalgorithm is fast and efficient. The fast fourier transform (FFT) needs Nlog2N multiplications whereas the DWT needs only N multiplications for the same number of equations. The non-phase shifted results and the fast algorithm are the base for using this procedure in an automatic data processing routine.
The wavelet decomposition of a time series in a time-scale joint representation offers a variety of applications on statistical signal analysis e.g. detection of self similarities, analysis of long memory processes, signal estimation or the analysis of the variance of the wavelet coefficients as a function of time scale as it is described by a scalegram (Fig. 4). The signal based acoustic emission analysis is generally pertained to the influence of noise. But all the information about the fracture process and its behavior over the testing time span is contained in the signal. Therefore, as mentioned above the elimination of noise is a central part for the application of wavelet algorithms on time series especially for acoustic emission analysis. But also a separated analysis of the different decomposition levels using a scalegram reveals a lot about the different behavior of the signal part and the noise part of a signal over the time scale. The scalegram of the detail coefficients D1 (Fig. 4) which have got a low signal to noise ratio shows a straight lower bound which results from the fact that no lower coefficient exists and shows a diffuse upper bound. The detail coefficients D4 which have got a high signal to noise ratio show a diffuse lower bound but show a straight upper bound although higher detail coefficients exist. This upper bound follows a slightly increasing curve (note the logarithmic scale of the variance in Fig. 4). The increase seems to represents the strain hardening effect which occurs during the pressure test on steel fiber reinforced concrete. The material behavior also confirms the theory on granular systems where the same effect is theoretically predicted Hidalgo et al. (2002). Extracting information from the wavelet coefficients using e.g. scalegrams is an approach which always needs to be synchronized with theoretical models or different experimental results. It is possible to reveal changes on different timescales but without certainty on the correctness of the visualized effect wavelet decomposition can also lead into the wrong direction.
Finally it can be stated that the wavelet transform is a relatively easy to handle tool for time series analysis which provides fast and efficient algorithms. Only a few powerful applications especially on acoustic emission analysis were shown here. But the time-scale joint representation of the transformed signal gives an idea about what else is possible using the wavelet transform.
5 AcknowledgementsThe funding of this work by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.