Multiclass feature selection using computational homology for Lamb wave

Abstract

We investigate a new approach for characterizing class separability based on topological measures for use with identifying flaw severity in plates. Multi-mode Lamb waves are propagated across a flat-bottom hole of varying depth. Lamb wave tomography reconstructions are first generated to locate and size the flaw at each depth. As the flaw depth increases, scattering and mode conversion effects dominate the raw time-domain signals, obscuring information about flaw severity. Pattern classification provides an alternate means for processing the ultrasonic waveforms to identify flaw severity. High-dimensional feature spaces are generated from the Lamb wave signals using the dynamic wavelet fingerprinting technique. In order to achieve high classification accuracy, an optimal feature space is required. An intelligent feature selection routine is explored here that identifies favorable class distributions in multidimensional feature spaces using computational homology theory. Betti numbers and formal classification accuracies are calculated for each feature space subset to establish a correlation between the topology of the class distribution and the corresponding classification accuracy.

Keywords

Cubical homology feature selection structural health monitoring

Introduction

Lamb waves are a promising structural health monitoring (SHM) technique due to their through-thickness multi-mode propagation (Raghavan and Cesnik, 2007; Rose, 2002). Lamb waves are confined by a structure’s boundaries, giving sensitivity to material discontinuities at both surfaces and in the interior of thin materials including plates and pipes (Advani et al., 2010; Giurgiutiu and Cuc, 2005; Pruell et al., 2009; Yu and Giurgiutiu, 2008). Lamb waves also propagate long distances (Alleyne et al., 2009; Carandente and Cawley, 2012) and beneath material coatings (Hinders and Bingham, 2010) allowing for rapid inspection of large areas. Lamb waves provide a useful tool for interrogating structural defects in plates because the group velocity of individual Lamb wave modes is determined by the product of material thickness and transducer frequency used. Their multi-mode structure is often too complex for direct interpretation since high-frequency-thickness values give rise to multiple Lamb wave modes, often with overlapping group velocities (Leckey et al., 2012).

Many SHM applications of guided waves sidestep the complexities of multi-mode signals in some manner. Rather than attempting direct interpretation, one popular approach involves signal comparison with a damage-free baseline measurement to identify changes in a material over time (Wang et al., 2010). This approach is often impractical for use in real-world situations, where environmental and operational conditions result in deviations from the baseline data, which require corrections that may be larger than the waveform distortion caused by the flaw. Another common restriction is to use low-frequency-thickness values for inspection, reducing the number of Lamb wave modes that propagate (Raghavan and Cesnik, 2007). Low-frequency-thickness regimes promise easier analysis of mode arrival-time shifts, as lower order modes tend to be spread apart in group velocity (Bingham and Hinders, 2009). Since each mode propagates with a unique displacement/stress profile (Rose, 1999), it is still beneficial to use multi-mode Lamb wave signals where each mode is sensitive to different defects and has different dispersion characteristics (Leonard, 2005).

Using signals that contain higher order Lamb wave modes requires advanced analysis techniques in order to harness the complex multi-modal behavior. Lamb wave tomography (LWT) can produce quantitative maps of flaws using both single-mode and multi-mode Lamb wave signals (Hou et al., 2004; Leonard et al., 2002; Leonard and Hinders, 2005). By extracting Lamb wave mode arrival times, a slowness map is generated that highlights changes in mode speed that relate to material changes in the plate. These LWT reconstructions provide accurate location and sizing of defects (Hinders et al., 2002). Recent work has resulted in several novel reconstruction techniques that reduce the computational requirements and/or number of transducers required to accurately map an area (Morii et al., 2011; Zhao et al., 2011). Due to overlapping group/phase velocities, however, a direct correlation between mode arrival time and material thickness may not always exist for multi-mode Lamb wave signals. Additional complexities are introduced by more severe defects, where scattering and mode conversion effects dominate the received waveforms. Several techniques have been explored to analyze these strongly scattering defects (Cegla et al., 2008; Diligent et al., 2002; Kim et al., 2010; McKeon and Hinders, 1999; Malyarenko and Hinders, 2001; Moreau et al., 2011, 2012), including Mindlin plate theory approximations (Cegla et al., 2008; McKeon and Hinders, 1999), diffraction tomography (Malyarenko and Hinders, 2001), and using specific transducer polarizations to identify mode conversion effects (Kim et al., 2010).

While LWT reconstructions do not always provide information on flaw severity, they are able to locate and size even strongly scattering flaws. This provides a mechanism to automatically identify the subset of waveforms that have interacted with the material defect present. Pattern classification (Bishop, 2006; Duda et al., 2001) then provides an alternative method to associate features of these waveforms with flaw depth. Statistical pattern classification algorithms generate decision boundaries in a multidimensional feature space based on given training data. New data are then assigned class labels based on their position in the feature space relative to these boundaries. In most applications of pattern classification, each set of objects with a common shared quality is grouped into an individual class. High class separability in the feature space is required if unknown data are to be accurately assigned a class label, but the distribution of individual classes within a feature space is often irrelevant.

Corrosion is a continuous gradual destruction of material where the severity increases with time. Discretizing this continuous material loss into individual flaw depth bins allows each depth to be assigned a unique class label. It is not practical to imagine a training set that includes all possible flaw depths, let alone all possible flaw shapes, positions, and so on. It is therefore highly unlikely that any new data will belong to an existing class of a training set. Rather, it will almost always fall between existing classes and should be labeled as one of these existing class bounds. It follows that the class distribution within a feature space should be sequential with respect to flaw severity, where one class borders the next as flaw severity is increased. An interpolation of sequential classes could recreate the continuous class distribution associated with corrosion.

Reducing a large feature space to only those features that result in sequential class distribution is a feature selection problem. Appropriate features are usually unknown a priori, and as a result, many features are generated without any knowledge of their relevancy (Blum and Langley, 1997). There are three general techniques for feature selection: wrapper methods, embedded methods, and filter methods (Saeys et al., 2007). Wrapper methods use formal classification to rank individual feature space subsets, applying an iterative search procedure that trains and tests a classifier using different feature subsets for accuracy comparison. This continues until a given stopping criterion is met (Dash and Liu, 1997). This approach is computationally intensive, and there is often a trade-off among algorithms between computation speed and the quality of results that are produced (Fan and Lv, 2010; Guyon and Elisseeff, 2003; Raymer et al., 2000). Additionally, these methods have a tendency to overtrain themselves, where data in the training set are perfectly fitted and result in poor generalization performance (Romero et al., 2003). Similar to wrapper methods, embedded methods perform feature selection while constructing the classification algorithm itself. The difference is that the feature search is intelligently guided by the learning process itself. Filter methods perform their feature ranking by looking at intrinsic properties of the data, without the input of a formal classification algorithm. Traditionally, these methods are univariate and therefore do not account for multifeature dependencies.

Our research explores a new approach to the problem of intelligent feature space reduction. By constructing a representative data set of waveforms that have interacted with a flaw ranging in severity, we seek to identify a feature space where the class distribution is both well separated and sequential. Our feature selection approach can be most appropriately considered a multivariate filter method, as it considers features in combination with each other and does not directly depend on the classification model used. A main difference here is that we redefine what it means for a feature to be “relevant.” Typically, relevant features are identified by their contribution toward class separability. While still important, we add a requirement that the features in combination must result in sequential class connectedness. Here, optimal subsets of a high-dimensional feature space are selected based on topological measurements of the class distribution within the feature space itself. We explore the topological structure of the class distribution in the feature space using computational homology. Specifically, we are trying to find a correlation between topological connectedness of multiclass feature space distributions and their corresponding classification accuracy.

Computational homology has been previously implemented in a variety of analysis applications. Hyde et al. (1995) generated isosurface reconstructions of the atomic structure of thermally treated alloys, identifying a correlation between the number of cavities in each surface and the amount of thermal aging undergone by the alloy. More recent implementations span a range of applications including image processing and recognition (Allili et al., 2001; Niethammer et al., 2002; Zelawski, 2005; Ziou and Allili, 2002), sensor network analysis (De Silva and Ghrist, 2006), material science (Teramoto and Nishiura, 2010), biomedical analysis (Mrozek et al., 2012), and image-based pattern classification (Gameiro et al., 2005; Gameiro and Pilarczyk, 2007). The majority of current applications of computational homology in the sciences revolve around the analysis of 1-dimensional (1D), 2-dimensional (2D), or 3-dimensional (3D) spaces, most commonly in image analysis. We intend to break this mold by using the same algorithms to analyze high-dimensional data sets for favorable class separability.

Our work builds on that of Leonard et al. (2002), where the inverse problem of LWT is solved to provide a technique used to locate and size flaws in plates. Additionally, we build on the preliminary work of Miller and Hinders (2013), where tomography is used in combination with formal pattern classification techniques to attempt to characterize flaw severity. Our analysis explores the use of powerful computational homology algorithms toward a general solution to the inverse problem of damage characterization, where damage-scattered Lamb waves are used to quantify the flaw severity.

Cubical homology

Computational homology is a field of mathematics involving the application of homology theory to geometric data sets generated from physical measurements. We consider a specific implementation of this theory that requires the topological spaces under consideration to be built out of d-dimensional unit cubes with indices on the integer lattice (Kaczynski et al., 2004). This cubical structure extends neatly to experimental data sets, where d-dimensional data sets can be represented as d-dimensional rectangles whose width is determined by some measure of error bounds in each dimension. Recent algorithm development under this cubical set constraint has led to an increase in the application of computational homology for experimental data analysis (Mrozek and Batko, 2009; Mrozek et al., 2008; Peltier et al., 2007).

We seek to generate a quantitative measure of how much overlap exists for a multiclass distribution in a d-dimensional feature space. Homology theory provides us a definitive measure of n-dimensional connectedness for $n = 0, \dots, d$ in terms of the homology groups $H_{n} (X)$ . The elements of $H_{n}$ are equivalence classes of n-cycles that do not bound any $n + 1$ chains, which is another way of characterizing n-dimensional holes. The rank of $H_{n} (X)$ is defined as the nth Betti number $β_{n} (X)$ . Therefore, for $n = 0, \dots, d - 1$ , the nth Betti number of a d-dimensional space is a coarse measurement of the number of disconnected n-dimensional surfaces in that space. For example, $β_{0}$ measures the number of individual connected components, $β_{1}$ measures the number of circular “loops” (2D holes), $β_{2}$ measures the number of spherical voids (3D holes), and so on. For a more formal introduction into computational homology, see Hatcher (2001) and Kaczynski et al. (2004).

The mathematical tools used here are not novel. Rather, we present a unique application of them in the context of feature selection. Several publicly available software packages exist for the computation of homologies and their Betti numbers, including Computational Homology Project (CHomP, 2012), the GAP homology package (Dumas et al., 2008), JavaPlex (Tausz et al., 2011), Dionysus (Morozov, 2012), and the RedHom library (Institute of Computer Science at Jagiellonian University, 2012). In our analysis, we use the software CHomP. This software allows for the rapid computation of homology groups based on input of elementary cubical complexes. Specifically, the algorithm used here computes the homologies over the ring of integers.

A simple example of how Betti numbers correlate to the spatial distribution of classes within a feature space is presented in Figure 1. Here, three objects represented as squares are presented in a 2D space for three different levels of overlap: no overlap, slight overlap, and severe overlap. Each object can be thought of as a bounding box representation of an individual class. Because this space is 2D, $β_{n} = 0$ for all values of $n > 1$ . As the amount of object overlap increases, so does the number of 2D holes ( $β_{1}$ ), while the number of individual connected components ( $β_{0}$ ) decreases. Similarly, classification accuracy will decrease as class overlap increases within a feature space. This results in an inverse correlation between $β_{1}$ and classification accuracy. In a 3D feature space, each class would be bounded by a cube. The number of cavities (3D holes) ( $β_{2}$ ) would then provide the relevant measure of class overlap. This concept can be extended to d-dimensional feature spaces, where each class is represented by a d-dimensional hypercube. In d-dimensions, it is the value of $β_{d - 1}$ that we expect to provide a relevant measure of class overlap within the feature space.

Figure 1.

Three objects (classes) in a two-dimensional feature space ( $f_{1}$ vs $f_{2}$ ) for three different levels of class overlap: (a) no overlap ( $β_{0} = 3$ , $β_{1} = 3$ ), (b) slight overlap ( $β_{0} = 1$ , $β_{1} = 5$ ), and (c) severe overlap ( $β_{0} = 1$ , $β_{0} = 7$ ). Betti number $β_{0}$ corresponds to the number of connected objects in the space, while $β_{1}$ corresponds to the number of one-dimensional holes (enclosures) in the space. If the three objects have no overlap (a), then there exist three separate connected components ( $β_{0} = 3$ ), each of which is a one-dimensional hole ( $β_{1} = 3$ ). As the objects begin to overlap slightly (b), the number of connected objects decreases ( $β_{0} = 1$ ) and the number of one-dimensional holes increases ( $β_{1} = 5$ ). Further overlap (c) again increases the number of one-dimensional holes ( $β_{0} = 1$ , $β_{0} = 7$ ).

Experimental setup

The data used in this study are collected from a 305 mm × 305 mm × 3.15 mm aluminum plate. A shallow, rounded rectangle void 76 mm in length and 30 mm in width is introduced and then incrementally increased in depth by milling (Miller and Hinders, 2013). Waveforms are transmitted and received using two 2.15-MHz, 6.4-mm-diameter contact transducers in a pitch-catch configuration. The transmitter is excited by a Matec TB1000 tone-burst card, generating a five-cycle sine wave at 2.15 MHz. The received signal is amplified by the Matec card and then digitized using a GaGe CS8012A analog-to-digital (A/D) card. Each transducer is fit with an 11.5-mm-diameter acrylic delay line using a glycerin couplant at both the transducer and the plate surface.

The transducers are stepped individually about the perimeter of a region of interest in a double-crosshole geometry (Leonard et al., 2002), mimicking a four-legged array of transducers surrounding an area of interest. Each transducer is attached to a linear slide, and movement through 100 locations at 2 mm per step is controlled by a Velmex VP9000 motion controller. The transmitter is advanced one position along a particular edge of the region of interest, while the receiver is stepped through all positions on the opposing edge. The transmitter is then advanced to the next position, and the receiver again steps through all positions on the opposing edge. Each projection uses a set of 100² individual waveforms, with two projections per full scan. A full double-crosshole scan is collected at six flaw depths $D_{x}$ , where $x = 1, \dots, 6$ , where each flaw depth corresponds to the depth of the void (summarized in Table 1).

Table 1.

Thickness of the flaw region corresponding to each double-crosshole scan, $D_{1}, \dots, D_{6}$ .

Data set	$D_{1}$	$D_{2}$	$D_{3}$	$D_{4}$	$D_{5}$	$D_{6}$
Flaw depth (mm)	0.00	0.67	1.34	1.86	2.47	3.15
Percent thickness loss	0	21	43	59	78	100

The group velocity dispersion curve for aluminum is shown in Figure 2, with the frequency–thickness product of the full plate thickness identified by a dotted vertical line at 6.78 MHz mm. Most of the propagating modes at this frequency–thickness have group velocities that are very close together. In addition, the slower $A_{0}$ , $S_{0}$ , $A_{1}$ , and $S_{1}$ mode dispersion curves are relatively flat at 6.78 MHz mm and so are not very sensitive to thickness loss. This overlapping of mode velocities also introduces difficulties in signal analysis due to modal interference.

Figure 2.

Group velocity dispersion curve for aluminum, with symmetric (solid black line) and antisymmetric (dashed black line) modes shown. Using a 2.15-MHz inspection frequency and a 3.16-mm plate thickness results in a frequency–thickness product of 6.78 MHz mm, highlighted by the black vertical dotted line. This frequency–thickness product is above the cutoff value for multiple modes ( $S_{0}$ , $A_{0}$ , $S_{1}$ , $A_{1}$ , $S_{2}$ , $A_{2}$ , $S_{3}$ , $A_{3}$ ), indicating that they will propagate in the material, while below the cutoff value for infinite higher order modes ( $A_{4}$ , etc.).

A typical multi-mode waveform is shown in Figure 3, with predicted mode arrival times highlighted. For shallow flaws, changes in the fastest $S_{2}$ mode velocity result in mode arrival-time shifts. These simple changes in arrival time can be associated with a corresponding material thickness change using the dispersion curves. Deeper flaws introduce scattering and mode conversion effects, which dominate the received signals and prohibit direct correlation with material thickness (Shkerdin and Glorieux, 2013). A more sophisticated analysis technique is therefore required to analyze this complex deep flaw interaction with multi-mode waveforms.

Figure 3.

A raw waveform from the unflawed plate is shown here with predicted mode arrival times, indicated for both symmetric (solid vertical lines) and antisymmetric modes (dashed vertical lines). The portion of the signal that is of interest often lies buried in noise (from 25 to 45 µs here), while the higher amplitude portion of the signal (45–60 µs) contains an overlap of both slower, low-velocity modes and faster edge-reflected modes due to the finite size of the plate sample. For the deeper flaw, mode conversion and scattering effects prevent an accurate prediction of arrival times in addition to reducing the signal-to-noise ratio of the waveform.

LWT

Tomographic reconstructions are generated for each class $D_{x}$ using the raw double-crosshole scan data (Leonard et al., 2002) as seen in Figure 4. It should be noted how similar the reconstructions are for the three deepest flaws, an indication of scattering and mode conversion effects. This type of reconstruction is a gray-scale image of the scanned area, where each pixel corresponds to the average velocity of the first-arriving wave packets, which propagate through that point as determined by the reconstruction algorithm. The gray-scale color map of each image is therefore a normalized range of mode velocities for the entire scanned region. By normalizing the color map of each reconstruction this way, areas of the scanned region with differing material properties (in this case a change in thickness) are highlighted as bright regions in the reconstruction. For each reconstruction image, an over/under color intensity threshold is applied that automatically allows the flaw to be highlighted and sized. This is done for all the images, building a database of extracted flaw dimensions. The overlapping area for all extracted flaw locations is determined, that is, a common area where every scan indicates a flaw is present. Every raypath from each scan that crosses this common flaw region is then identified and segregated into a “flaw-only” subset of waveforms. This process serves as a convenient way to build a data set for use with the classification algorithms, discarding the approximately 18,000 waveforms per scan that do not interact with the flaw.

Figure 4.

Tomographic reconstructions for classes: (a) $D_{1}$ (0% material loss), (b) $D_{2}$ (21% material loss), (c) $D_{3}$ (43% material loss), (d) $D_{4}$ (59% material loss), (e) $D_{5}$ (78% material loss), and (f) $D_{6}$ (100% material loss). (g) Additionally, the layout of the scan area is shown with the flaw highlighted in gray. The raypaths used for classification are overlaid on the scan area for reference.

Due to the computation time required to perform feature extraction, the complete set of flaw-only raypaths is reduced further by selecting 50 of these raypaths at random. A visualization of the raypaths selected can be seen overlaid in the scan area seen in Figure 4(g). This random selection is used to minimize geometric bias that could be associated with the shape of the specific flaw used in the training set. Individual waveforms corresponding to these raypaths are identified from each full double-crosshole scan, compiling a data set to use for classification. A total of 300 waveforms are therefore included in the final data set. Individual waveforms are denoted $w_{i, j} (t)$ , where $i = 1, \dots, 50$ corresponds to the raypath number relative to the scan and $j = 1, \dots, 6$ corresponds to the class number.

Feature extraction techniques

Several different methods are used to extract feature values from each of the 300 waveforms: higher order statistics, Mellin transform statistics, wavelet packet decomposition, and dynamic wavelet fingerprinting (DWFP). All these methods provide a specific set of values that have direct physical relation back to the raw data, and each has shown promise in various applications for damage detection (Bertoncini et al., 2012; Bingham and Hinders, 2010; Harley et al., 2012; Mattson and Pandit, 2006; Sun and Chang, 2002). In total, 78 individual features are extracted from each waveform in our data set, resulting in a $d = 78$ dimensional feature space.

Several statistical features commonly used in ultrasonic signal analysis are generated from the raw waveforms (Cox and Hinkley, 1979). Standard time-domain features include the mean of the raw waveform, the variance, the Shannon entropy, the second central moment, the skewness, and the kurtosis of each waveform. Statistical moments provide insight by highlighting outliers due to any specific flaw-type signatures found in the data.

The Mellin transform is an integral transform closely related to the Fourier transform that can represent a signal in terms of a physical attribute similar to frequency known as scale. This transform has the key property of scale invariance. Small variations in mode velocity due to environmental changes, such as temperature fluctuations or applied strain, can be approximated as a uniform time-scaling effect on the transmitted signals. By considering the Mellin domain, features can be compared that are robust to environmental effects. While there are complexities associated with discretizing the fast Mellin transform algorithm, a MATLAB-based implementation is used (De Sena and Rocchesso, 2007). Once the Mellin transform is computed, features are extracted from the resulting Mellin domain for classification. These features include the mean of the Mellin transform, as well as the standard deviation (SD), variance, second central moment, Shannon entropy, kurtosis, and skewness of the mean-removed Mellin transform (Harley et al., 2012).

In the DWFP technique (Hou and Hinders, 2002), a continuous wavelet transform is applied to each of the original time-domain waveforms. The resulting coefficients are then used to generate binary time-scale images of fingerprint objects that are coincident in time with the time-domain signal. A variety of measures are then extracted from each object in the DWFP image. These include boundary measurements fitting second- and fourth-order polynomials, area of the on- and off-pixels, solidity, as well as several measures that are generated from an ellipse fitting the second moments of the fingerprint. As these values correspond to individual objects in the image, they are localized in time at each object’s centroid. The measured values are then interpolated in time to create a continuous set of feature values. An example of this process is shown in Figure 5. A distance metric is then used to identify times corresponding to high interclass separation between measurements with low intraclass SD. Measurements at time values for which the largest normalized distance metric values occur are extracted and used as features for classification. Wavelet-based measurements provide the ability to decompose noisy and complex information and patterns into elementary components, making them favorable for multi-mode ultrasonic signal analysis where subtle features can be highlighted with the optimal choice of a mother wavelet.

Figure 5.

A visual summary of the DWFP (Hou and Hinders, 2002) feature extraction technique. (a) A typical ultrasonic signal and (b) a close-up portion of the DWFP output. From each “fingerprint” object, a variety of measures are extracted and interpolated in time, including (c) the area of the on-pixels for each object and (d) the coefficients of a second-order polynomial ( $p_{1} \times x^{2} + p_{2} \times x + p_{3}$ ) fit to the boundary of each fingerprint object, with coefficients $p_{1}$ (solid line), $p_{2}$ (dashed line), and $p_{3}$ (dotted line).

Another wavelet-based feature used in classification is generated by wavelet packet decomposition (Learned and Willsky, 1995). A wavelet packet transform (WPT) is applied to each low-pass filtered waveform with a specified mother wavelet and number of decomposition levels. This generates a tree of coefficients similar in nature to the wavelet transform. From the WPT tree, a vector is computed containing the percentages of energy corresponding to the terminal nodes of the tree, known as the wavelet packet energy. Because the WPT is an orthonormal transform, the entire energy of the signal is preserved in these terminal nodes (Feng and Schlindwein, 2009). Singular value decomposition is then applied to the energy matrix, returning the eigenvector with the highest energy. The corresponding elements from the WPT are used as features. In the case of multiple nodes being selected, all corresponding WPT elements are included in the feature set. Wavelet packet decomposition uses redundant basis functions and can therefore provide arbitrary time–frequency resolution details, improving upon the wavelet transform when analyzing signals containing close, high-frequency components. Wavelet packet decomposition features are included to provide insight into the generation of high-frequency modes as a result of mode conversion at flaw boundaries.

Utilizing CHomP

Before we can use the CHomP algorithms, we first need to transform our data set from several clusters of points in the feature space into a form that can be read by the computational homology algorithms. The goal is to turn these data into a cubical set, where each class is represented by a hyper-rectangle embedded in the space $R^{d}$ . The first step in this transformation is to identify each class’s centroid and its SD in each dimension of a d-dimensional feature space. Each class can then be represented by a d-dimensional hyper-rectangle (referred to here as a d-rectangle), centered at the centroid and bound by the SD in each dimension. Each d-rectangle is composed of $2 \times d$ individual ( $d - 1$ )-dimensional faces. On this set of d-rectangles in a given feature space, we apply a homology-preserving map that transforms the bounds defined by the SDs onto an integer lattice. Next, each $2 \times d$ individual ( $d - 1$ )-face is divided into a set of d-dimensional elementary cubes, which themselves are the finite product of d-dimensional elementary intervals, one of which is degenerate and $d - 1$ of which are nondegenerate. An example of this process for a 2D feature space ( $d = 2$ ) can be seen in Figure 6. The number of elementary cubes required to define each ( $d - 1$ )-face depends on the size of that face within the integer lattice representation. In other words, if a transformed hyper-rectangle has a dimension larger than unit length, it is defined in terms of several smaller elementary components.

Figure 6.

An example of converting a six-class distribution in a two-dimensional feature space into a cubical set. (a) The first step in computing the homology is to establish the class bounds in the feature space, defined here as the standard deviation in each dimension centered at the class centroid. (b) A mapping is then applied that maintains the homology of the space and sets these bounds to the integer lattice $Z$ . (c) Finally, faces of each class are divided into elementary cubes of dimension $d - 1$ .

To summarize, each d-rectangle is represented by a set of $2 \times d$ individual ( $d - 1$ )-faces, each of which is composed of elementary cubes, where each elementary cube is defined as a Cartesian product of one degenerate and $d - 1$ nondegenerate elementary intervals.

For each feature space, the set of elementary intervals defining the hyper-rectangles in $R^{d}$ is written into a temporary file, which is used as input for CHomP. The resulting output of Betti numbers is extracted from the CHomP program and saved in an array, along with the corresponding features that make up the given feature space. The class distribution in this feature space is then used to train and test the formal classification functions described previously, with the classifier accuracy saved in the same array. This process is repeated for every feature space subset of dimension d, building an array of feature space subsets and their corresponding classification accuracies and Betti numbers. A correlation can be made from these values between the selected feature space’s classification accuracy and any trends present in the Betti numbers.

There are computational restrictions that need to be taken into account before the CHomP algorithms are used. For the most part, the restrictions are based on the fact that each class has to be composed of elementary cubes, defined on the set of integers. It follows that a large number of classes results in a large number of elementary components required to build the resulting hyper-rectangles in the feature space. This exponentially increasing number of elementary cubes required to define a higher dimensional feature space translates to an exponential increase in the memory required to define the space. Additionally, the number of elementary cubes required in each dimension depends on number of classes present, which defines the range in each dimension for the feature space. For example, a six-class feature space will have 12 boundaries in each dimensions (an upper and lower bound per class). This further increases the memory required to define a class distribution as a cubical set. It is easy to see how the size of this problem increases drastically as the number of classes and dimensions increase.

Classifier design

We implement a supervised statistical pattern classification routine here, where new patterns are assigned class labels using the natural structure of the training data. By using a statistical approach where each pattern is represented as a point in a multidimensional feature space, we retain the physical interpretation of each feature. That is, while abstract, each feature can be connected directly back to the waveforms themselves. For example, the number of ridges in each fingerprint corresponds to waveform peak height. Since the goal of this analysis is to determine flaw severity, we assign class labels to each of the six flaw depths $D_{x}$ for $x = 1, \dots, 6$ , where waveforms corresponding to each flaw depth are grouped in a unique class labeled $ω_{x}$ .

There is a generally accepted phenomenon in high-dimensional spaces known as the curse of dimensionality, which identifies the concept that a finite-sized data set becomes sparse when represented in too high of a dimensional space. A rule of thumb commonly used in pattern classification is that the number of training samples per class (n) should be at least 10 times as many dimensions (d) as there are in the feature space ( $n / 10 > d$ ) (Jain et al., 2000). This suggests that we can find a feature subset of dimension $d = 5$ or less for the case discussed here with acceptable performance. Therefore, our analysis explores all possible feature spaces of dimensions $d = 2, \dots, 5$ . Comparing all possible five-dimensional feature space combinations ( $(_{5}^{78}) > 21 \times 10^{6}$ ) would take over a century to compute on a single machine. This could be reduced through a combination of serial and parallel processing techniques, but even with our available computing cluster, it would still require 6 months of continuous dedicated runtime.¹ This is not a realistic request, nor is it a worthwhile exercise when we can use other well-established feature selection techniques as an intermediate step. To make the computation time reasonable, we must first reduce the original 78-dimensional feature set. We therefore reduce the size of our original feature set down from $d = 78$ to $d = 10$ features using a sequential floating backward selection (SFBS) feature selection routine, reducing the computation time to approximately 3 h on our computing cluster. The SFBS routine is a wrapper-type feature selection technique first introduced by Pudil et al. (1994) that has been shown to provide close to optimal feature space reduction based on class separability with reasonable computation cost. The SFBS routine does not take into account the sequential nature of the classes, however, which is the underlying issue at hand in this analysis. We therefore only use the SFBS routine to reduce the initial dimensionality of the feature space and rely on the proposed techniques to finalize the feature selection process. With the resulting 10-dimensional SFBS feature space as a base, we calculate both the classification accuracy and the Betti numbers of the class distribution for all possible feature subsets of dimension $d = 2, \dots, 5$ . Dimension $d = 1$ is not considered here as $β_{0} = 1$ and $β_{n} = 0$ when $n > 1$ for all feature spaces of dimension $d = 1$ due to class overlap in our data set.

We have chosen to use a quadratic discriminant classification (QDC) algorithm here because it was shown to perform best in the work performed in preliminary results (Miller and Hinders, 2013), where a more complete exploration of multiple classifier performance was considered for a similar incremental damage analysis. It was shown that the QDC results were among the best in both classifier development and validation testing.

The full six-class data set is split into training (80%) and testing (20%) subsets using random sampling (preserving class distributions) for classification. For each feature subspace under consideration, the waveforms selected as the training set are submitted to the classifier using only the features identified by the feature space. The testing subset is then submitted to the trained classifier, again using only those features defined by the feature space, and the resulting predicted labels are stored in an array. Classification accuracy is then determined to be the number of labels predicted to be within ±1 flaw depth of their actual label, presented here as a percentage of the total waveforms tested.

Results

We first present results from the computational homology analysis to identify trends present between class overlap and corresponding classification accuracy in a given feature space. The Betti numbers for all feature space subsets of dimension $d = 2, \dots, 5$ sorted by QDC classifier accuracy can be seen in Figure 7. A series of subplots is given where each subplot contains the Betti number $β_{n}$ (y-axis) as a function of classifier accuracy (x-axis) for all possible feature space combination of dimension d. Each row of subplots corresponds to Betti numbers of dimension $β_{n}$ , where $n = 1, \dots, d - 1$ from bottom to top, while each column corresponds to feature space dimension $d = 2, \dots, 5$ from left to right. Results for $β_{0}$ are left out as $β_{0} = 1$ for all feature space subsets. As stated in the “Cubical Homology” section, we expect the values of $β_{d - 1}$ to provide a measure of class separability by being inversely proportional to the classification accuracy of a given feature space. Generally, the higher the number of d-holes, the more class overlap present in the d-dimensional feature space and the lower the expected classification accuracy.

Figure 7.

Betti number results sorted using QDC classification accuracy. Each row corresponds to Betti number $β_{n}$ , where $n = 1, \dots, d - 1$ , from bottom to top, while the columns represent feature space dimension d, where $d = 2, \dots, 5$ , from left to right. Each subplot contains the Betti number $β_{n}$ (y-axis) as a function of classifier accuracy (%) (x-axis) for all possible feature space subsets of dimension d.

Since $β_{0} = 1$ for all feature space subsets, it follows that there is always overlap between classes in the feature space. This verifies that a sophisticated analysis technique is required for analysis of these data. If the values of $β_{0}$ were consistently larger than 1, there would be disjoint classes in the feature space, and a simple clustering analysis would be more appropriate.

For plots of $β_{d - 1}$ over all dimensions $d = 2, \dots, 5$ , it appears that the classification accuracies are separated into two distinct ranges: a group of feature spaces that correlate to higher accuracies and a group of feature spaces that correlate to lower accuracies, separated by a gap in accuracy. This effectively splits the Betti numbers into two disjoint sections. These sections, when isolated, display the general higher Betti number–lower accuracy relationship we were expecting to find. An example of this is shown in Figure 8, where the subplot of $β_{d - 1}$ for $d = 5$ from Figure 7 is enlarged. The two disjoint sections of data are highlighted, with bounds of each individual cluster of feature spaces indicated by vertical lines. The gap in classification accuracy between these two clusters can be easily seen here. Individual linear best-fit lines are included to highlight the general trend of the data. The linear fit for the feature spaces in the lower range of accuracies, bounded by the dotted lines, has a best-fit slope of −0.84. Similarly, the linear fit for the feature spaces in the higher range of accuracies, bounded by the dashed lines, has a best-fit slope of −1.61. These both indicate the negative correlation between Betti number $β_{d - 1}$ and classification accuracy we expected to find.

Figure 8.

A plot of $β_{d - 1}$ for $d = 5$ as a function of classification accuracy for all feature space subsets of dimension $d = 5$ . There is a gap in the classification accuracies dividing the feature space subsets into a group of lower accuracy feature spaces and a second distinct group of higher accuracy feature spaces. These two sections are bounded here by dotted and dash-dot vertical lines, receptively. Individual linear best-fit lines are applied to each group, shown here as dashed lines. The linear fit for the lower range of accuracies has a best-fit slope of −0.84. Similarly, the linear fit for the higher range of accuracies has a best-fit slope of −1.61. While a correlation across the entire accuracy range is lacking, both these accuracy groups indicate the negative correlation between Betti number and classification accuracy we expected to find.

The cause of this gap in classification accuracies is unknown, but it may be related to the concept of nested classes in the feature space being topologically identical to disjoint classes. In other words, only classes that overlap each other contribute to the increase in n-holes and therefore Betti number $β_{n - 1}$ ; disjoint or nested classes do not. If a class is nested within the bounds of a separate class, it follows that it would be difficult to distinguish between the two classes using formal classification techniques. Nested classes therefore result in a lower classification accuracy while maintaining an identical topological measure when compared to similar disjoint classes.

We have shown that there is no reason to believe that lower Betti numbers will directly correspond to higher classification accuracy. This lack of correlation is possibly due to nested classes within the feature spaces. We therefore require an intermediate step to isolate only the higher accuracy cluster of feature spaces. Isolating this group of feature space subsets would allow us to then utilize the negative correlation between Betti number $β_{d - 1}$ and classification accuracy to rank a feature space’s potential accuracy.

We thus explore lower dimensional topological connectedness as well as several geometric measurements of the class distribution from within each feature space. That is, we combine a threshold on $β_{n}$ , where $n < d - 1$ , with three other geometric thresholds that are calculated directly from the class centroid locations within the feature space. The geometric measures used are the average tortuosity of, average angle between, and average distance between the centroid distributions. Any feature spaces that do not meet all these thresholds are removed from consideration.

We include the maximum average distance between centroids under the assumption that classes that are spaced far apart from each other in a feature space will be easier to distinguish between, increasing classifier accuracy. We first calculate the corresponding vectors between successive n-dimensional centroids, $C_{j} = [c_{j, 1}, c_{j, 2}, \dots, c_{j, n}]$ , where $j = 1, \dots, 6$ classes, using

v_{i} = C_{i + 1} - C_{i}

(1)

= (c_{i + 1, 1} - c_{i, 1}, c_{i + 1, 2} - c_{i, 2}, \dots, c_{i + 1, n} - c_{i, n})

(2)

for $i = 1, \dots, | j | - 1$ , where $| j |$ indicates the magnitude of $j$ . We then take the Euclidean norm of each vector, $∥ v_{i} ∥$ , and average them to get a single average centroid distance value for each feature space

\begin{matrix} \bar{∥ v ∥} = \frac{1}{| j | - 1} \sum_{i} ∥ v_{i} ∥ \end{matrix}

(3)

These values are sorted by highest average distance. This technique breaks down when the classes under consideration overlap significantly within the feature space due to larger intraclass spread.

If we construct vectors between successive class centroids, we can calculate the minimum average angle between successive vectors. This measure assumes that a linear spread of classes within a feature space would result in new data of intermediate flaw severity to lie correctly between its bounding classes. For example, if data corresponding to a flaw of 35% thickness loss are introduced to a classifier, it would ideally fall somewhere in the feature space between classes $D_{2}$ (21% thickness loss) and $D_{3}$ (43% thickness loss). If there is not a linear distribution of classes in the feature space, there is no guarantee that an alternate class would occupy this area of the feature space, resulting in misclassification of the new data. From equations (2) and (3), we calculate the angle between successive vectors using the dot product

\begin{matrix} θ_{m} = \overset{- 1}{\cos} (\frac{v_{m} \cdot v_{m + 1}}{∥ v_{m} ∥ ∥ v_{m + 1} ∥}) for m = 1, \dots, | i | - 1 \end{matrix}

(4)

From these, we calculate the average angle between successive vectors

\begin{matrix} \bar{θ} = \frac{1}{| i | - 1} \sum_{m} θ_{m} \end{matrix}

(5)

We sort these values by lowest average angle between class centroids in order to identify a feature space where the classes are most linearly distributed. We also calculate the tortuosity of each class centroid curve, made up of individual vectors between each successive pair of classes. Tortuosity ( $τ$ ) is defined here as the ratio of the length of the curve connecting the successive class centroids to the distance between the ends of it

\begin{matrix} τ = \frac{\sum_{i} ‖ v_{i} ‖}{‖ C_{| j |} - C_{1} ‖} where 1 \leq τ < inf \end{matrix}

(6)

where again we have that $i = 1, \dots, | j | - 1$ . We sort tortuosity in ascending order, where a value of $τ = 1$ corresponds to a linear sequential centroid curve. An example of all three of these measures can be seen in Figure 9.

Figure 9.

An example of a four-class data distribution in a two-dimensional feature space, with classes $i (Δ)$ , $i + 1 (\nabla)$ , $i + 2 (◇)$ , and $i + 3 (□)$ . Class centroids are $C_{i}$ , $C_{i + 1}$ , $C_{i + 2}$ , and $C_{i + 3}$ , respectively. Vectors $v_{i} = C_{i + 1} - C_{i}$ , $v_{i + 1} = C_{i + 2} - C_{i + 1}$ , and $v_{i + 2} = C_{i + 3} - C_{i + 2}$ are shown here. The angle between vectors $v_{i}$ and $v_{i + 1}$ is $θ_{i}$ , while that between vectors $v_{i + 1}$ and $v_{i + 2}$ is $θ_{i + 1}$ . The tortuosity ( $τ$ ) is defined here as the sum of the length of vectors that make up the curve, $∥ v_{i} ∥ + ∥ v_{i + 1} ∥ + ∥ v_{i + 2} ∥$ , divided by the distance between end points of the curve, $∥ C_{i + 3} - C_{i} ∥$ .

For each feature space of dimension $d = 2, \dots, 5$ , we compare the three sets of geometric measurement values against their QDC classification accuracy. A threshold is applied for each metric that includes the highest feature space classification accuracies but removes as many of the lower classification accuracy feature spaces as possible. Each feature space’s Betti numbers $β_{n}$ for $n = 0, \dots, d - 1$ are then compared against their corresponding classification accuracies, and a similar threshold is applied for $β_{n}$ with $n < d - 1$ . Finally, each feature space’s $β_{d - 1}$ values are plotted against their corresponding classification accuracies and highlight any feature spaces that meet all these threshold limits. Ideally, only those feature spaces corresponding to the higher accuracy cluster of classification accuracies described above remain, which are then sorted by $β_{d - 1}$ values to identify the feature spaces with the highest classification accuracy.

For brevity, only results for $d = 4$ are presented in Figures 10 to 12; however, the trends seen here are found to generalize across all dimensions ( $d = 2, \dots, 5$ ). After a preliminary analysis of the results, specific threshold values are selected for the three geometric measures that could be applied across all dimensions $d = 2, \dots, 5$ ; feature space subsets are retained when the tortuosity metric follows $τ < 4.7$ , the average angle between centroid values follows $[77 < \bar{θ} < 107]$ , and the average distance between class centroid values follows $\bar{∥ v ∥} > 0.12$ . Due to the limited size of the data sets, a formal threshold analysis to determine ideal threshold levels is not performed in this study. Instead, thresholds are selected “by eye” based only on the data at hand in an attempt to provide a proof of concept for this technique. For the Betti number results, since the Betti numbers are dimensionally dependent (e.g. values of $β_{2}$ cannot fundamentally be compared between $d = 3$ - and $d = 4$ -dimensional spaces), individual thresholds are selected for each dimension d.

Figure 10.

Values of the three geometric feature space versus QDC classification accuracy for all feature space subsets of dimensions $d = 4$ . Average distance between class centroids ( $\bar{∥ v ∥}$ , top), average angle between class centroids ( $\bar{θ}$ , upper middle), and tortuosity relative to class centroids ( $τ$ , lower middle) are shown. Thresholds represented by the black dotted lines have been applied at $\bar{∥ v ∥} > 0.12$ , $[77 < \bar{θ} < 107]$ , and $τ < 4.7$ , respectively, to isolate data corresponding to the highest accuracies. Feature spaces that meet each of thresholds are highlighted by black stars ( $*$ ), while those which meet all three are highlighted in the plot of $β_{d - 1}$ (bottom) as black circles ( $○$ ). It can be seen that the majority of feature spaces identified by this method correspond to higher classification accuracies; however, several of the feature spaces corresponding to lower classification accuracies are also included.

Figure 11.

Betti numbers $β_{n}$ for $n = 0, \dots, d - 1$ plotted against QDC classifier accuracy (%) for all possible feature space subsets of dimensions $d = 4$ . A threshold has been applied to $β_{n}$ for $n < d - 1$ , shown by black dotted lines in each subplot. Feature space subsets that meet each individual threshold are indicated by black stars ( $*$ ), while those that meet all three are highlighted in the plot of $β_{d - 1}$ (bottom) as black circles ( $○$ ). It can be seen that thresholding the lower dimensional Betti numbers $β_{n}$ for $n < d - 1$ only removes a portion of the feature spaces from consideration; however, those removed correspond to lower accuracies.

Figure 12.

Plots of $β_{d - 1}$ for $d = 2, \dots, 5$ , where feature space subsets that meet all three of the geometric thresholds ( $\bar{∥ v ∥}$ , $\bar{θ}$ , and $τ$ ) in addition to all of the Betti number thresholds on $β_{n}$ for $n = 0, \dots, d - 2$ are highlighted by black circles ( $○$ ). It can be seen that almost all the feature spaces that correspond to the highest accuracy values are included by the applied thresholding, while the majority of the lower accuracy feature spaces are removed from consideration.

Figure 10 shows the three geometric measures plotted against their QDC accuracy, with applied thresholds represented by black dotted lines. Any data points that meet each threshold are highlighted by a black star. The bottom plot is that of $β_{d - 1}$ , where feature space subsets that meet all three of the geometric thresholds are indicated by black circles. In Figure 11, a similar threshold analysis is applied for Betti numbers $β_{n}$ for $n < d - 1$ . A threshold is applied to each $β_{n}$ , and all feature spaces that meet said threshold are highlighted by black stars. For the $β_{d - 1}$ plot, any feature space that meets all of the $β_{n}$ for $n < d - 1$ thresholds is represented by a black circle.

Several trends are observed in these results that are consistent across all dimensions. For the tortuosity metric, the group of feature spaces that have the highest accuracy are all clustered together below the $τ = 4.7$ threshold. Those corresponding to lower classification accuracies tend to produce significantly higher values of $τ$ , which are mostly filtered out with the applied threshold. For each dimension, there still exist several lower accuracy feature space points that also meet the threshold limit. These results indicate that class centroid tortuosity is a promising technique for identifying feature spaces that result in high classification accuracy; however, alone it is not perfect.

The average angle between centroid values also has a trend that is consistent across all dimensions. Most of the feature spaces corresponding to high classification accuracy are clustered within the $[77 \circ < \bar{θ} < 107 \circ]$ range we have identified. Higher dimensional feature spaces show a second cluster of average angle values up around 130°; however, we have chosen not to include this range of angles in our threshold because it significantly increases the number of lower accuracy feature spaces as well. Because we can still identify the highest accuracy values in all dimensions, we believe that this metric also shows potential for use in parallel with other metrics for removing the lower accuracy feature spaces from consideration.

The average distance between class centroid values ( $\bar{∥ v ∥}$ ) has less of an ability to separate feature spaces that correlate to higher accuracies from those that correlate to lower accuracies. The expectation was that the higher values of $\bar{∥ v ∥}$ correlate to better class separation; however, the range of $\bar{∥ v ∥}$ values for high-accuracy feature spaces is similar to that of the low-accuracy values. Feature space subsets corresponding to the lower extreme of classification accuracies do exhibit a lower $\bar{∥ v ∥}$ accuracy relationship that we can exploit with a threshold, so we include this metric in our analysis. As is apparent in the $d = 4$ case seen in Figure 10, only a handful of feature spaces are removed.

Betti number values are unique to each dimensional space as the number of nonzero $β_{n}$ is related to dimension d by $β_{n} \neq 0$ for $n \leq d - 1$ . We therefore determine the threshold values for each dimension on an individual basis. The same principle applies, however, in that we are looking for a straightforward amplitude threshold to apply, which separates many of the higher accuracy feature spaces from the lower accuracy ones. We apply these thresholds only to $β_{n}$ for $n < d - 1$ since we have already seen that $β_{d - 1}$ displays behavior that prevents us from using a simple selection criterion such as a threshold. For $d = 4$ , no threshold can be applied to the constant $β_{0}$ values, while $β_{1}$ includes variations that allow thresholding to be applied, identifying many of the lower accuracy feature spaces.

There can also exist a trade-off between how many lower accuracy feature spaces are removed and how many higher accuracy spaces are kept. It can be seen that if the threshold limit in $β_{2}$ is raised to include all the higher accuracy spaces, a much larger number of lower accuracy spaces would be included as well. We have chosen a threshold value that only removes a small number of higher accuracy spaces here, as seen in Figure 11.

Feature selection results are summarized in Figure 12 for all dimensions $d = 2, \dots, 5$ . A plot of $β_{d - 1}$ is shown for each dimension, where feature spaces that meet all applied thresholds are highlighted by black stars. The $d = 4$ case can again be seen here, combining the thresholding results from Figures 10 and 11. It can be seen that the majority of feature spaces corresponding to the highest classification accuracies are identified in each feature space dimension using this combination of geometric and Betti number thresholding. Similarly, the majority of feature spaces corresponding to the lower classification accuracies have been removed from consideration. By identifying the high-accuracy feature spaces using these geometric measures, a negative correlation between $β_{d - 1}$ and classification accuracy can be applied to identify feature space subsets that return high classification accuracy.

Conclusion

We have presented a feature selection technique using computational homology for multi-mode Lamb wave damage characterization. We first generated reconstructions of double-crosshole plate scans using LWT to identify and size the flaw at each depth. Waveforms that pass through the flaw region were automatically identified, and a subset was extracted for use with the pattern classification analysis. We generated a large number of features using a variety of signal analysis techniques, including DWFP, wavelet packet decomposition, statistical features, and Mellin domain features. We then introduced a feature selection technique that uses topological measures of the class distribution within the feature space to identify those most appropriate for Lamb wave flaw severity analysis.

We have created a filter-type feature selection routine that is able to identify subsets of a larger feature space that correlate with the highest possible classification accuracies of each subset dimension. We have approached the problem of the sequential ordering associated with a physical process that is a function of time, such as progressive damage in materials. To do this, we have measured the class overlap using a topological measure known as Betti numbers in addition to several geometric measures of the class distribution within each potential feature subspace. We have shown that applying thresholds to these measures allows for the identification of high-accuracy feature subspaces. From these remaining feature spaces, we have shown that a class distribution’s Betti numbers in feature spaces of dimension $d - 1$ have a negative correlation to classification accuracy.

Our work provides insight into the potential use of computational homology algorithms for studying the topological structure of class distributions within various feature spaces. For applications where the sequential ordering of classes is important, as it is in Lamb wave SHM, comparing the Betti number $β_{d - 1}$ of each d-dimensional feature space under consideration shows potential for identifying feature subspaces that return the highest classification accuracies.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was performed, in part, using computational facilities at the College of William and Mary, which were provided with the assistance of the National Science Foundation, the Virginia Port Authority, Sun Microsystems, and Virginia’s Commonwealth Technology Research Fund.

Notes

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Multiclass feature selection using computational homology for Lamb wave–based damage characterization

Abstract

Keywords

Introduction

Cubical homology

Experimental setup

LWT

Feature extraction techniques

Utilizing CHomP

Classifier design

Results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

Notes

References