Mother wavelet selection

There are numerous mother wavelet functions available with Discrete Wavelet Transform decomposition, and each of these will provide varying results. Selecting an appropriate mother wavelet that matches the signal or galling wear feature is crucial for effectively characterising galling wear. Qualitative comparison between the signal and mother wavelet is a common method for selection in other applications such as power quality assessment and medical Electromyograms (Ngui et al., 2013). In these cases, the shape of the mother wavelet is compared visually to the feature of interest in the respective signals. Example mother wavelet functions of the Daubechies wavelet family and the Haar wavelet can be seen in Figure 5.2. Visual comparison is simple in cases where the morphology of target features are known, as is the case here with the galling wear features highlighted in Figure 5.1. For assessing galling, the shape of the selected wavelet function must resemble the increase and sudden drop in the 2D profile that occurs due to material pile-up on the edges of the gouge, as shown in Figure 5.6a. The Daubechies 2 mother wavelet represents a close approximation of these features. A large positive pulse immediately followed by a negative pulse corresponds well with the material build- up next to the wear gouge, as shown in Figure 5.6b. Because of these attributes the Daubechies 2 wavelet has been selected for the analysis of galling wear and preceding damage in this study. However, other galling wear features such as adhered material or adhesive pull-out scars may require selection of different mother wavelet functions for accurate measurement.

Figure 5.6: a) 2D profile galling wear cross section features. b) Selecting an appropriate mother wavelet function for galling wear feature of interest.

68

In cases where the feature morphology of the target wear damage is not well known, a quantitative technique for the selection of mother wavelet function may be required. For example, quantitative selection methods may be appropriate when attempting to quantify other forms of wear damage or wear in other applications. Minimum Description Length (MDL) criterion (Saito, 1994) is a quantitative technique that is commonly used in noise suppression and signal compression applications (Ngui et al., 2013), and will be highlighted as a quantitative alternative to visual comparison. The MDL criterion assumes that the ‘best’ wavelet function for describing a signal is given by the model that provides the most compact description of the data of interest. In situations where the morphology of the wear is not known, it is possible that application of MDL criterion with a range of mother wavelet function to a sample 2D profile of the wear feature of interest could assist with selection.

MDL criteria of a wavelet decomposition of the 2D surface profiles of galling wear regions has been calculated using equation (10) (Hamid and Kawasaki, 2002).

𝑀𝐷𝐿(𝑘, 𝑛) = 𝑚𝑖𝑛 {3 2 𝑘 log 𝑁 + 𝑁 2log‖𝛼̃𝑛− 𝛼𝒏 (𝑘) ‖2} , 0 ≤ 𝑘 ≤ 𝑁 (10)

Where N represents the number of wavelet coefficients or length of the decomposed signal, 𝑛 is the wavelet filter, 𝛼̃𝑛 represents the wavelet decomposition coefficients

of the 2D surface profile determined using wavelet filter n, and 𝛼_𝒏(𝑘)= Θ(𝑘)𝛼̃𝑛 where

Θ(𝑘) represents a hard-thresholding operation such that the k largest values of the

vector 𝛼̃_𝑛 are maintained as absolute values and the remaining values are set to zero.

The number of retained coefficients k where MDL reaches its minimum is deemed the optimal value. This process is repeated for different wavelet filters, n, utilising different mother wavelet functions in order to determine which mother wavelet provides the minimum MDL value for the tested 2D surface profile.

In order to compare the selection using MDL and visual comparison MDL values were determined for a series of test 2D surface profile of galling track regions. A series of wavelet functions available from the MATLAB wavelet toolbox were assessed with the series of test 2D surface profiles and the MDL criteria values were

69

determined, Table 5.1 shows the 5 ‘best’ mother wavelets for each of the test

profiles. These test profiles (label using the notation: die name – part number) exhibited a range of quantities and severities of galling wear damage on different materials in order to provide a representative sample for demonstrative purposes.

Table 5.1: Minimum Description Length values for test surface profiles of galling wear damage regions. See Appendix A for full wavelet names and functions.

Rank Galling feature profile regions MDL results for various mother wavelets #31 - 4200 #31 - 4600 #30 - 400 #30 – 1600 7 - 14

1 Haar 489.98 Haar 374.59 Haar 573.95 Haar 418.22 Haar 517.54

2 bior3.1 551.23 bior3.1 426.83 bior3.1 584.63 bior3.1 436.22 rbio3.1 523.14 3 db2 587.19 db2 450.49 db2 631.12 fk4 495.83 db2 528.96 4 fk4 613.52 fk4 453.18 fk4 651.47 db2 496.85 rbio2.2 535.17 5 rbio3.1 618.14 rbio3.1 472.17 bior2.2 664.94 rbio3.1 524.40 rbio1.3 537.70

The Daubechies 2 wavelet, which was selected based on visual comparison, ranks highly based on the MDL criteria for all the test 2D surface profiles of galling wear regions. However, based on the criteria, the Haar and Biorthogonal 3.1 wavelets both rank higher for the majority of the test cases. As shown in figure 4.2a, the Haar wavelet is a square shaped wavelet, and the Biorthogonal 3.1 wavelet is a complex function with numerous pulses. MDL criterion is based on efficiency of encoding the data, so even if a model or wavelet may match the data more accurately, it may not be as efficient in encoding as other functions and so will be ranked lower by MDL criterion. The demonstrated application of MDL criterion shows that it is a viable alternative for mother wavelet function selection in cases where morphology of the targeted wear damage is unknown. However, given the knowledge of the targeted galling wear features in 2D surface profiles, visual assessment will be used for mother wavelet function selection in this work.