Mathematical Morphology‐Based Fault Diagnosis to Rotating Machinery: A Review

1. Introduction

Rotating machinery is a significant component of modern industrial system, and plays important role in industrial applications. As a complex system, it is composed of various rotating parts to different aims, for example bearings are to support axis and/or radius loads, while gears are generally to change input speed or motion direction, etc. In most cases, their working conditions are quite harsh, such as high temperature and pressure, and time-varying in load and/or speed. Hence, the failure of rotating components is unavoidable, impairing the reliability and applicability of the mechanical equipment. Condition monitoring to the mechanical systems, especially the key rotating parts, has become an efficient and available solution scheme in theoretical studies and engineering applications [1,2].

The vibration measurement is most sensitive to the mechanical component faults, directly showing the information associated with abnormal conditions. However, the collected vibration signals are quite often nonstationary and nonlinear due to signal amplitude or/and frequency demodulation induced by time-varying effect. Meanwhile, additive-noise disturbance from background environment is embedded into such signals, increasing the difficult to detect these faults. Targeted this problem, different signal processing methods are considered, such as short time Fourier transform (STFT) [3,4,5,6,7] wavelet analysis [8,9,10,11,12], empirical mode decomposition (EMD)-based method [13,14,15,16,17] and variational mode decomposition (VMD) [18,19,20,21,22,23], etc. As we know, these methods have individual merits and demerits, for instance STFT can locate the change of interested signal components simultaneously in time and frequency but cannot obtain the optimal resolution because of Heisenberg uncertainty principle; the EMD-based method is a special model without basis function definition and suitable to process non-stationary signal but has mode mixing, end effect and the sensitivity to noise; VMD amends EMD but depends on the mode number and penalty coefficient highly.

It should be said that mathematical morphology is an interesting and robust theory. Firstly, it completes the successful crossing from image area to signal one. If saying that the crossing benefit only nonlinear filtering, the self-extension and the horizontal-extension (i.e., combining morphology with other theories) show the robustness, because the former unify the qualitative analysis with quantitative analysis for a specified target like fault diagnosis, and the latter retain the individual merit between two theories, such as morphological wavelet that has not only the nonlinearity of morphology, but also the multiresolution of wavelet. The theory framework is showed in Figure 1. The merit melting maybe gives better explaining to a specified problem. As a result, morphology model provide a multi-view to rotating machine fault diagnosis.

Like most signal processing methods, mathematical morphology has its own working mechanisms. Firstly, it just involves simple computation rules, such as addition, subtraction and extrema solving, which are totally different to inner-product operation in Fourier transform (FT) and wavelet transform, allowing the change of the local details of signal to be observed visually. This brings more controllability in actual applications and is easily performed on hardware platform with lower computation cost. Besides, as a referred object or an information carrier, structuring element (SE) in morphology is more or less identical to the basic functions in Fourier and wavelet transform or window functions in STFT, and is designed by sampling point stacking as a specified shape. Obviously, the stacking definition to SE is completely different to the mathematical formula definition in conventional signal processing area. To a signal shape that is explicit to us, the stacking pattern, relatively speaking, can well describe both the integrity and the local variation of a signal, while the formulation one emphasizes the commonality for the shape. Therefore, the processing mechanisms maybe offers a more flexible solving scheme to improve fault diagnosis effect.

For these reasons, enormous publications on mathematical morphology applied in rotating machine fault detection area occur in academic journals, conference proceedings and technical reports, etc. This paper makes an attempt to summarize and retrospect the current studies, allowing interested researchers to acquire relatively apparent level on mathematical morphology. The remaining part of the paper is structured as follows. Section 2 presents morphological foundation about morphological operators, structuring element and four morphology-based methods. Section 3 introduces the reference review on morphological applications in rotating machinery fault diagnosis, followed by a brief discussion and potential prospects in Section 4. Section 5 describes some concluded remarks.

2. Mathematical Morphology

Morphological theory is developed based on set theory, topology, stochastic geometry, lattice theory, and nonlinear partial differential equations, etc. [24,25,26]. In image processing field it can complete image segmentation [27,28,29,30,31], image feature extraction [32,33,34,35,36,37], edge detection [38,39,40,41] and image enhancement [42,43,44,45]. Due to the nonlinear filtering and strong signal demodulation behavior, it is gradually extended into 1D signal processing [46,47,48].

Mathematical morphology is an interactive behavior between structuring element and a signal as certain computation rules, with a goal that is to remove irrelevant contents and retain pivotal detail. It involves two important factors: morphology operator and structure element. The former defines the mathematical computation rules, while the latter is a reference object or an information carrier that is expected to be extracted from a processed signal or image.

2.1. Morphological Operator

Morphology theory includes four basic operators, i.e., dilation, erosion, opening and closing. Let f(n) be one-dimensional signal with a domain at F = (0, 1, 2, …, N-1), and s(n) be a structuring element with at a domain at S= (0, 1, 2, …, M-1) (M << N). Then dilation and erosion are formulated as:

$$\left(\mathrm{f}\oplus \mathrm{s}\right)\left(\mathrm{n}\right)=\max \left\{\mathrm{f}\left(\mathrm{n}-\mathrm{m}\right)+\mathrm{s}\left(\mathrm{n}\right)\right\} \mathrm{m}\in \mathrm{0,1},2\dots \mathrm{M}-1$$

(1)

$$\left(f\ominus s\right)\left(n\right)=\min \left\{f\left(n+m\right)-s\left(n\right)\right\} m\in \mathrm{0,1},2\dots M-1$$

(2)

where $\oplus$ and $\ominus$ represent dilating and eroding, respectively. Because dilation and erosion are similar to addition and subtraction in arithmetic, they can extent or shrink the morphological shapes of image. It is noteworthy that the two operators in practical applications cannot restore original image, thus closing and opening are defined as:

$$\left(f·s\right)\left(n\right)=(f\oplus \mathrm{s}⊝\mathrm{s})\left(\mathrm{n}\right)$$

(3)

$$\left(f\circ \mathrm{s}\right)\left(n\right)=(f⊝s\oplus s)\left(n\right)$$

(4)

where $\circ$ and • stand for opening operation and closing operation, respectively. Toward a one-dimensional signal, morphological operations have specified computation rules, for instance, dilation and erosion separately solve local max and minimum values. Based on Eqs.(3) and (4), closing detects local max then minimum and opening is inverse performance.

On the other hand, the dynamic feature of signal is complex during sampling time, but basic operators have single performance to identify such features (the reason will be explained subsequently). In order to improve the situation, except for the four basic operators, morphological gradient operators are constructed as a combination between them. For example, the difference between closing and opening (DCO) and the difference between dilation and erosion (DDE) are given as follows:

$$DCO\left(f\right)=\left(f\oplus s⊝s\right)\left(n\right)-(f⊝s\oplus s)\left(n\right)$$

(5)

$$DDE\left(f\right)=\left(f\oplus s\right)\left(n\right)-(f⊝s)\left(n\right)$$

(6)

In order to distinguish the difference in extracting signal feature for these operators, a simulation analysis is considered. Here, a linear flat SE with the length of 5-sampling-point is utilized to simplify the discussion.

Figure 2(a) is the result of dilation, showing that the signal peaks are smoothed and the valleys are extended upward, i.e., the signal waveform is almost shifted upward. In fact, the performance depends on three factors, the computation rule of dilation operator, the SE length and the location of the dilated point. For example, when a data point near a peak is dilated, because the peak is biggest in the local area that is controlled by the SE length (the relation between the length and the locality will be explained in Section 2.2), this point is thus replaced by the peak. Following the SE movement, others near the peak will be also replaced and the peak extension happens, i.e., the peak area is flatted. Conversely, when one point near a valley is dilated, it possibly is replaced by a local max. Furthermore, when the valley bottom is dilated, there must be a bigger point in this local area, thus it will be replaced with the bigger point, that is, the whole valley area is shifted upward. If there are not local maxima around the currently dilated point, it keeps original status, for example the previous sampling points in Figure 2(a). According to the analysis, the dilation behavior to a signal is summarized as smoothing the signal peaks, retaining signal valleys, and making the signal waveform moved upwards. Differently, Erosion is to detect local minimum, resulting in a reverse performance as illustrated in Figure 2(b).

Next, closing is analyzed. According to Eq.(3), it is a cascading model between dilation and erosion, thus the closing output in Figure 2(c) corresponds to eroding the waveform in Figure 2(a). And the date points at the special flatted and volley areas need be concerned, because normal points can be derived based on the performance of dilation and erosion to signal. For instance, when one point at the flatted peak is eroded, because there must be a minimum, it is replaced by the minimum. Following multi-erosion, the flatted peak area is gradually restored to original status. On the contrary, when a point near a valley is eroded, because the valley bottom is smallest in the local area, this point will be replaced by the valley bottom, that is, the valley region is smoothed during the eroding procedure. Consequently, closing operation to a signal can extract peaks and smooth valleys, and opening is the inverse performance displayed in Figure 2(d).

Among gradient operators, the operator DCO, relatively speaking, is analyzed easily. According to Eq.(5), the signal waveform in Figure 2(e) can be regarded as the difference computation between Figure 2(c) and Figure 2(d). Because the two figures are almost similar each other, only smaller peaks are obtained in Figure 2(e), where the flatted regions in Figure 2(c) are preserved, and the negative flatted regions Figure 2(d) are converted into the positive ones due to the difference computation. It indicates that the DCO operator can keep oscillation tendency in the signal. In comparison with DCO, DDE has more complex behavior presented in Figure 2(f), where the waveform oscillations quite often happen. This is mainly because the extrema in the signal are strongly random, meanwhile solving extrema through dilation and erosion is controlled by the SE length, thereby leading to the strong random oscillation.

According to the analysis above, it should be said that the morphological operation provide a microcosmic and intuitive view to observe or process a signal. As a result, it brings more controllability in theoretical study and actual applications.

2.2. Structuring Element

Structuring element acts as a moving window with specified shape to probe interested component in signal or image. It is designed as a matrix containing 1’ component and 0’ component. Thereinto, the former called neighborhood is located at some positions in the matrix to shape a specified figure with the assistance of the latter. In fact, structuring element itself is a given information carrier, then these morphological features in an object that are identical to the SE will be identified by the morphology processing.

Theoretically, SE is simpler in shape and smaller in size than those of the analyzed object, and comprises three physical properties, shape, length and height. The shape represents what geometric outline is concerned to us, and can be triangle, circle, disk, and rectangle, etc. If required, irregular shapes also can be defined. Moreover, the shape should contain detailed quantification in time scale and space scale, i.e., length and height. The former shows the shape in the time duration, and the latter indicates space extension. In general, the length is an integer number, and height is a real one. To practical applications, structuring element will be determined by the three parameters, but it does not mean that they all must be defined. In most cases, the first two properties need be considered, but the height can be neglected (i.e., setting as 0) because it can simplify morphological calculation. For this reason, it is grouped into two types, flat structuring element and non-flat structuring element. When flat SE is used, morphological operation just involves local extrema searching, while non-flat one is employed, the height values is introduced into morphological calculation by subtraction or addition, and then the obtained result performs extrema searching.

In order to show the difference when SE of different type is utilized, two examples are presented here. Assume that operator is dilation, and a flat SE is set as {1 1 1}, where the underlined element is an original point, which will be matched with data points from the first point to last one in a signal, i.e., the whole morphological operation is conducted when it is matched with the last. For example, when the origin is matching with the point x_i of the signal X in Figure 3, it means that x_i is dilating, and the current dilation is the max among three points (i.e., x_i-1, x_i, x_i+1) because the SE length is three-sampling-point. Until such matching happens at the last point, the eventual dilation result is obtained. On the other hand, if SE is {1 1 0}, the output of dilating x_i is the max between the two points x_i-1 and x_i, rather than the three points. Because the point x_i+1 corresponds to the 0’ component in the SE, it cannot be involved into the current dilation. As a result, 1’s component in structure element depends on how many data point will be introduced in a morphological calculation, while 0’s component assists 1’s component to construct a specified shape.

When a non-flat SE is defined as [{1, 1, 1}, {h₀, h₁, h₂}], where h_i is the corresponding height. In the same way, when the point x_i is dilated, firstly x_i adding h_i is performed, i.e., the new three points x_i-1+h₀, x_i+h₁, x_i+1+h₂ are obtained, and the current dilation output is the max among them. It indicates that when a non-flat SE is used, the morphology operation is firstly addition/subtraction (when using erosion) between data points and the SE heights and then solves extrema.

It is clear that SE determines the computation range in a morphological processing, and highly dominates the result when compared with morphological operators. As stated in Introduction, a merit of morphology is nonlinear filtering performance. Based on the viewpoint, it should be known that a small SE is beneficial to preserve morphological features but unbeneficial to remove noise, and vice versa.

2.3. Introduction to Morphology-Based Methods

Mathematical morphology is a broad systemic framework, following the development, four methods based on the morphology are introduced, i.e., single-scale morphology (SSM), multi-scale morphology (MSM), morphological pattern spectrum (MPS) and morphology wavelet (MW).

2.3.1. Single-Scale Morphology

SSM means that just one structuring element is utilized. As we know, the SE definition is very flexible. To obtain better processing, sometimes different SEs will be used, but the optimal one is eventually considered by certain selection criterion. In this case, it still belongs to the SSM analysis.

Relatively speaking, the single-scale analysis is simpler in applications, for instance when flat SE is considered, the length is only defined and its reasonability is certainly quite crucial. Otherwise, the analysis easily suffers from under- or over-processing. In general, SSM is suitable to process the signal that does not contain multiple signal components, because it is hard to well match the morphology features between single SE and a complex signal. As a result, how to construct proper SE in SSM is most important to the expected processing result.

2.3.2. Multi-Scale Morphology

In a real environment, the measured signal is complicated, possibly containing different signal components. Based on the morphological features matching, it is expected that using different scales allows each signal component to be isolated into individual corresponding scale. Accordingly, the multi-scale technique has comprehensive analysis ability and is suited to deal with such complex signal.

In the Multi-scale model, varying scales are achieved by self-dilation of unit scale. In this situation, it produces two advantages, enhancing computation efficiency and simplifying morphology calculation.

Let γ (γ=1,2,3…γ_max) be the scale parameter, and s be unit SE. The SE at scale γ can be given as

$$\gamma s=\underset{\gamma -1 times}{\underbrace{s\oplus s\oplus \dots s\oplus s}}$$

(7)

According to the definition above, the dilation and erosion operators of signal $f(x)$ at scale γ are written as

$$f\oplus \gamma s=\underset{\gamma -1 times}{\underbrace{f\oplus s\oplus \dots s\oplus s\left(n\right)}}$$

(8)

$$f\ominus \gamma s=\underset{\gamma -1 times}{\underbrace{f⊝s⊝\dots s⊝s\left(n\right)}}$$

(9)

And the closing and opening operators of signal $f(x)$ at scale γ can be separately formulated as:

$$f\cdot \mathsf{\gamma }\mathrm{s}=\mathrm{f}\oplus \mathsf{\gamma }\mathrm{s}⊝\mathsf{\gamma }\mathrm{s}\left(\mathrm{n}\right)$$

(10)

$$f\circ \gamma s=f⊝\gamma s\oplus \gamma s\left(n\right)$$

(11)

Similarly, morphological gradient operators DCO and DDE at scale γ are represented as

$${DCO\left(f\right)}_{\mathsf{\gamma }}=\left(f·\mathsf{\gamma }s\right)\left(n\right)-(f\circ \mathsf{\gamma }s)\left(n\right)$$

(12)

$${DDE\left(f\right)}_{\mathsf{\gamma }}=\left(f\oplus \mathsf{\gamma }s\right)\left(n\right)-(f⊝\mathsf{\gamma }s)\left(n\right)$$

(13)

Toward the multi-scale analysis, two problems need to be noticed. Firstly, the used scales are generally continuous, and according to the effect of the SE scale to morphology result (i.e., small scale is beneficial to retain the morphological features of signal but unbeneficial to clear noise), the scales also require proper definition, otherwise an over- or under- analysis will happen. On the other aspect, the core to MSM is the diversity among all scales used, each signal component in a signal has stronger correlation with corresponding scale, and the final signal is reconstructed by the results of the whole or part of the used scales.

2.3.3. Morphological Pattern Spectrum

In image processing field, the shape representation and the shape-size description are key tasks, because the former shows qualitatively that an image contains what important information, and the latter replies quantitatively that how many such one is covered in the image. However, SSM and MSM work through structuring element moving inside an object, just answering that the SE is contained or not in the object, thus essentially they are qualitative analysis. As mentioned above, the quantitation of specific information facilitates more accuracy estimation and understanding to image, thereby Maragos presented morphological pattern spectrum [49,50], which is also called granulometric analysis in image processing area by a sieving procedure to extract size distributions [51,52,53,54].

Morphology pattern spectrum is composed of two terms, morphology pattern and spectrum. Morphology pattern represents a particular prototype shape that is maybe contained in one object, and spectrum is a quantized evaluation to the shape pattern. Obviously, when the “spectrum” is discussed, it allows us to associate with the idea spectrum in Fourier transform. In fact, there exists certain similarity between them. In order to understand MPS well, the Fourier transform (FT) is described here.

As we know, the FT of a signal f(t) is defined as $F\left(\mathsf{\omega }\right)={\int }_{-\infty }^{\infty }f\left(t\right)e^{-i\omega t}dt$, noticing that the signal spectral component is calculated through a modulation f(t)e^-iωt between f(t) and complex sinusoid e^-iωt. For instance, when a frequency ω is defined, the spectral component is obtained by measuring the area of the modulation. Following ω varying over a certain interval, the whole spectral will be produced. It is clear that the complex signal e^-iωt works as a probing pattern with a series of spectrum component ω, and they can be extracted by the interactions between the complex signal and a processed signal. According to the transform, it can tell us that ω exists or not in the signal, but the details like shape and shape-size of e^-iωt is missed.

In order to construct MPS by FT, corresponding concepts should be modified. for example, one-dimension signal s(t) is transformed into two-dimension image X, the complex signal e^-iωt corresponds to structuring element S_n, where subscript n defines S as a specified shape with certain size, and converting the signal modulation as the shape-size transformation of X. Then the area of a transformed image is calculated to obtain MPS of X by measuring the size distribution for X relative to S_n. For example, the area measuring to X by opening is defined as a normalized version:

$$F\left(r\right)=A(X\circ rS)/A\left(X\right)$$

(14)

where notation A(•) means area solving, and numerator and denominator stand for the areas of opening and the original image X, respectively. As we know, the area of the image A(X) is fixed, but the area A(X∘rS) varies following the change of r. Considering from probability theory, the function F(r) represents the probabilistic measures of the size distribution relate to S of size r in image X. Besides, the probing pattern rS will translate in the image, thus such area solving shows cumulative distribution property.

Based on the area measurement above, MPS can be defined as differential size distribution:

$$MP{S}_X\left(r,S\right)=-dA\left(X\circ \mathrm{r}\mathrm{S}\right)/dr r\ge 0$$

(15)

It represents the union that contains all patterns of X that are relative to size r. Furthermore, varying S or r, it can obtain different information, for example, if S is shaped as various figures with a fixed r, MPS indicates that the image most resembles which one. If S is fixed but r changes, it means all size components in the image. Additionally, Convexity of S implies that MPS is nonnegative for all r ∈ R due to $X ° rS\supseteq X\circ lS$ if r ≤ l, thus MPS is the non-negativity when increasing r.

On the other hand, probing pattern e^-iωt in FT has a positive frequency component ω. In fact, the probing pattern also can be e^iωt, corresponding to negative frequency component -ω. It is the same as MPS, that is, when negative scale –r is used in closing, MPS is written as:

$$MP{S}_X\left(-\gamma ,s\right)=dA(X·rs)/dr r>0$$

(16)

Viewing from image processing and set theory, the positive scale is to process the umbra of an image, whereas the negative scale is to process the complementary of the umbra of the image.

Due to the consecutiveness of r according to Eq.(16), it is hard to use an analytical way to compute MPS, thus the discrete-form MPS is given as:

$$MP{S}_f\left(+n,s\right)=A[f\circ ns - f\circ (n+1)s ] 0\le n\le N$$

(17)

$$MP{S}_f\left(-n,s\right)=A[f·ns-f·(n-1)s ] 1\le n\le K$$

(18)

where N is the max size n, and K is the minimum size n. In two equations, $A\left(f\right)={\sum }_{(x,y)}f(x,y)$, and notation (a - b)(x) = a(x) – b(x) refers to an algebraic difference between a(x) and b(x). A(f) is calculated by the difference between closing or opening using two approaching scales.

As we know, MPS is a further development in mathematical morphology, whose basis is set theory. Based on this point, it can be considered as a mapping relationship between two compact sets, i.e., set $A=\{H_1,K_2,H_3,\dots H_n\}$ and structuring elements set $\mathsf{\Theta }={\{S}_1,S_2,S_3\dots S_m\}$. Then, the area measuring by SE in set $\mathsf{\Theta }$ constructs a mapping on set A that is defined as $P_{\mathsf{\Theta }}\left(H_i\right)=\left\{{\mathsf{{\rm X}}}_{i1},{\mathsf{{\rm X}}}_{i2},{\mathsf{{\rm X}}}_{i3},\dots {,\mathsf{{\rm X}}}_{im}\right\},$ where morphological pattern spectrum ${\mathsf{{\rm X}}}_{ij}$ for Hi in set A is produced by set Sj. It means that if and only if $P_{\mathsf{\Theta }}\left(H_i\right)\ne P_{\mathsf{\Theta }}\left(H_m\right)$ when i ≠m, $P_{\mathsf{\Theta }}$ is an invertible mapping on set A, that is, the shapes in A is completely distinguished by the measurement.

When a shape pattern is utilized, the probabilistic measures reflect its size distribution by area solving. It means that the defined pattern is qualitative information, and the measure represents quantitative descriptor. On the other hand, the pattern information of rotating parts with different faults is otherness, thus the pattern spectrum provides a new view as classification tool to rotating part faults.

2.3.4. Morphological Wavelet

In a real world, the features of signal are in general both scale- and resolution-varying and highly nonstationary in space. Researchers recognized that multiresolution signal decomposition is fairly significant to theoretical developments and practical applications, because it makes these features represented at different scales, with efficient and clear understanding to such signal and with high computational advantages.

As a classical multiresolution technique, traditional wavelet transforms (TWT), termed first-generation wavelet, is based on Fourier transform and z transform to design a filter bank with special properties that are homologous with the multiresolution, for example such applications in the signal and image processing applications [8,9,10,11,12]. It is known that TWT is linear processing method like signal scaling, but the linear model is may not be compatible with attributes of interest, for example blurring the edge information of signal or image. Hence, scholars made contribution to attempt a theoretical extension from linear to nonlinear for the multiresolution analysis, expecting that the results obtained by this analysis varied in non-intuitive manner.

The extension is nonlinearity and wavelet-type multiresolution. The former means using nonlinear operators, such as dilation and erosion in mathematical morphology. At the same time, the multiresolution analysis is multi-level decomposition, i.e. a signal is analyzed and understood within different signal spaces, and should be an inverse process. Viewing from these points, the extension concerns two aspects, non-redundant to signal decomposition and perfect reconstruction to signal reconstruction. To satisfy the requirements, it actually experiences many trials.

In 1991, Pei and Chen [55,56] firstly put forward non-redundant nonlinear sub-band decomposition by using morphological theory, but it cannot perform perfect signal reconstruction. For this reason, Egger and Li [57] developed an improved method by means of median-type operation. And a similar study is found in literature [58]. Subsequently, Queiroz et.al. [59] proposed a type of nonlinear wavelet decomposition for lowing image code complexity. In 1997, Cha and Luis [60] used morphological opening operator to construct a nonlinear wavelet transform having perfect signal reconstruction. During the exploration process, these studies provide significant reference for future development, but do not present a general framework in morphology area to such nonlinear development. Differently, the traditional wavelet is done well at the same period, i.e., lifting scheme termed second-generation wavelet [61,62,63]. In order to understand the morphological wavelet well, lifting scheme is simply discussed here.

The so-called lifting is further development to these existing wavelets, i.e., using a new tool to construct them for analyzing signals that are defined in arbitrary domains, without the assistance of Fourier and z transform. It is divided into three steps, split, predicting and update. Firstly, a signal A₀ is split into two subsets, A₁ and B₁, i.e., the former is even samples with high approximation to the original signal, and the latter is odd samples with less information. Theoretically, a signal has some correlation structures in its sampling domain, thus it is instinctive to utilize a specified way to replace B₁ with A₁, i.e., prediction B₁= P(A₁), where P is predicting operator. In practice, it is hard to exactly predict B₁ by means of A₁, then the difference between B₁ and its prediction P(A₁) is considered as.

$$B_1^{\prime }=B_1-P\left(A_1\right)$$

(19)

After n steps are conducted repeatedly, the original signal A₀ can be replaced with a subset {A_n, B_n, B_n-1,…,B₁}. Noting that the analysis is down-sampling, the last step will obtain a random sample, possibly appearing aliasing. It means that the globe property of the signal can’t be preserved, e.g., mean. In order to deal with the problem, an update operator is introduced:

$$A_1^{\prime }=A_1-U\left(B_1\right)$$

(20)

where U is updating operator. The procedure is the lifting scheme, and interested readers can refer [64,65,66,67,68,69,70,71]. Lifting scheme provides a general systemic framework that constructs those exiting wavelets. What’s more, it also provides alternative idea for the nonlinear extension.

Inspired by the lifting scheme, Goustias and Heijmans [72,73] proposed an axiomatic framework morphological wavelet, in which morphological pyramid based on complete lattice is constructed as a type of nonlinear model to satisfy the multiresolution decomposition. The model presents two wavelet-type decompositions, nonlinear scheme called coupled wavelet decomposition and linear one called uncoupled wavelet decomposition. In practice, the morphological wavelet can be regarded as a special case of lifting scheme, namely prediction and update utilize morphological operators.

1. Coupled wavelet decomposition

To nonlinear signal decomposition cases, coupled wavelet decomposition is conducted by three operators, covering two analytical operators (one for the approximation signal and one for the detail signal) to perform multiresolution analysis and one synthesis operator to reconstruct signal.

Assume that V_j and W_j are signal space and detail space at level j, respectively. Then the signal decomposition is performed by analytical operators as increased direction through mapping level j into j+1, i.e., the analysis operator ${\psi }_j^{\uparrow }:V_j⟶V_{j+1}$, and the detail operator ${\omega }_j^{\uparrow }:V_j⟶W_{j+1}.$ The analysis operator can obtain approximation or simplification of signal with many signal properties, while the detailed operator obtain a refinement. Reversely, the signal synthesis can be conducted by synthesized operator as decreased direction ${\mathsf{\Psi }}_j^{\downarrow }:V_{j+1}\times W_{j+1}⟶V_{j.}$ The one-stage decomposition is depicted in Figure 4 (a).

On the other hand, a robust signal decomposition method requires certain constraint conditions to guarantee the non-redundancy in decomposition and the perfection in reconstruction. Therefore, when Eq.(21) and Eq.(22) are hold, the two requirements can be guaranteed. At the same time, the two conditions implicitly show that the mapping of operators is injective upward and surjective downward.

$${\mathsf{\Psi }}_j^{\downarrow }\left({\psi }_j^{\uparrow }\left(x\right), {\omega }_j^{\uparrow }\left(x\right)\right)=x, if \mathrm{x}\in V_j$$

(21)

$$\left\{\begin{array}{c}{\psi }_j^{\uparrow }\left({\mathsf{\Psi }}_j^{\downarrow }\left(x,y\right)\right)=x, if x\in V_{j+1},y\in W_{j+1}\\ {\omega }_j^{\uparrow }\left({\mathsf{\Psi }}_j^{\downarrow }\left(x,y\right)\right)=y, if x\in V_{j+1},y\in W_{j+1}\end{array}\right.$$

(22)

According to the decomposition scheme, a given signal $x_0\in V_0$ can be decomposed in the recursive way as:

$$x_0\to \{x_0,y_0\}\to \{x_1,y_1\}\to \{x_2,{y_2,y}_1\}\to \dots \{x_k,{y_k,y_{k-1},\dots y}_1\}\to \dots$$

(23)

On the other hand, the original signal can be completely rebuilt from these sub-signals x_k and y₁, y₂, $\dots$ y_k by the synthesized scheme:

$$x_j={\mathsf{\Psi }}_j^{\downarrow }\left(x_{j+1},y_{j+1}\right), j=k-1,k-2,\dots 0$$

(24)

Hence, the signal decomposition scheme as Eqs. (20)-(23) is called the coupled wavelet decomposition scheme.

2. Uncoupled wavelet decomposition

In morphology wavelet, uncoupled wavelet scheme is linear decomposition as a special case of the nonlinear decomposition, and the decomposition procedure is the similar as that of nonlinear one, only the signal synthetization will use a binary addition operation $\dotplus$ to make Eq. (25) hold when ${\psi }_j^{\downarrow }:V_{j+1}⟶V_j$ and ${\omega }_j^{\downarrow }:W_{j+1}⟶V_j$. The two operators ${\psi }_j^{\downarrow }$ and ${\omega }_j^{\downarrow }$ are taken to the signal synthesizing and the detail synthesizing, respectively.

$${\mathsf{\Psi }}_j^{\downarrow }\left(x,y\right)={\psi }_j^{\downarrow }\left(x\right)\dotplus {\omega }_j^{\downarrow }\left(x\right) x\in V_{j+1},y\in W_{j+1}$$

(25)

Meanwhile, in the linear case Eq. (26) is perfect reconstruction condition, and Eq. (27) and Eq. (28) represent non-redundant decomposition conditions.

$${\psi }_j^{\downarrow }{\psi }_j^{\uparrow }\left(x\right)\dotplus {\omega }_j^{\downarrow }{\omega }_j^{\uparrow }\left(x\right)=x, if x\in V_j$$

(26)

$${\psi }_j^{\uparrow }({\psi }_j^{\downarrow }\left(x\right)\dotplus {\omega }_j^{\downarrow }\left(x\right))=x, if x\in V_{j+1},y\in W_{j+1}$$

(27)

$${\omega }_j^{\uparrow }({\psi }_j^{\downarrow }\left(x\right)\dotplus {\omega }_j^{\downarrow }\left(x\right))=y, if x\in V_{j+1},y\in W_{j+1}$$

(28)

For a signal x₀ $\in$V₀, the reconstruction in recursive synthesis scheme is:

$$x_j={\psi }_j^{\downarrow }\left(x_{j+1}\right)\dotplus {\omega }_j^{\downarrow }\left(y_{j+1}\right) j=k-1, k-2, \dots , \mathrm{1,0}$$

(29)

It means that the signal x_j at level j is reconstructed as mapping-down by the standard addition of an approximation signal and a detail signal at level j+1. As a result, the signal representation process represented as Eqs. (22) - (27) is the uncoupled wavelet decomposition scheme illustrated in Figure 4 (b).

Morphological wavelet completes the nonlinear extension of TWT by using mathematical morphology, keeping the multiresolution analysis and emphasizing shape and edge preservation. It affords a flexible and versatile analysis to signal.

3. Application of Mathematical Morphology to Rotating Machine

As stated in Introduction, rotary mechanical parts, such as bearing and gear, are crucial main components of rotating machinery, and their working statuses quite often work as key referenced indicators that can directly evaluate the mechanical equipment with or without faults. Hence, diagnosis and classification to part faults are always academic study topic.

Once a defect happens on these rotating components, the vibration response embodies impulsive feature, whose strength depends on the defect size, load, input speed and other objective factors. Moreover, it probably triggers the systemic resonance of mechanical equipment or rotating components, resulting in the energy increase around these system inherent frequencies. On the other hand, the vibrational signal shows two characteristics. One is the modulation in amplitude, frequency and phase, or the combination between them under complicated running conditions. The modulation carriers can be the shaft or cage rotational frequency in bearing, and the meshing frequency or inherent frequencies of rotating parts in gear. Another characteristic is noisy contamination to the vibration signal from signal collecting environment, especially when the occurred failure is incipient, and it affects the diagnosis performance of the used methods significantly.

Aiming at bearing and gear fault diagnosis, the morphology-based approaches have conducted many theory explorations and achieve successful investigations. This section will present a synthetical summary on rotating machinery fault diagnosis, with a classification as single-scale morphology, multi-scale morphology, morphological pattern spectrum and morphological wavelet. Besides, a sub-classification in each method is given according to fault objects like bearing, gear and other mechanical fault detection.

3.1. Single-Scale Morphology-Based Rotating Parts Fault Diagnosis

SSM just uses an operator and a fixed SE to complete a morphological operation and is relatively simple in practical applications. Under the condition, the morphology filtering behavior highly relies on the physical properties of structuring element.

3.1.1. SSM to Detect Bearing Defect

Nikolaou and Antoniadis [74] firstly employed SSM to extract the impulsive signals from defective bearings. They analyzed the different properties of morphology operators to impulsive signals and experimentally defined a flat structure element length. Dong et al. [75] defined four fixed SE length and just used one of them for different bearing defects. The similar method that selects the length of flat SE is founded in Ref [76,77,78,79] when diagnosing bearing failure. In these studies, structuring elements are all defined by prior knowledge. In this case, the sampling frequency of raw signals and the noisy interference intensity all affected on morphological results, easily resulting in over- or under- filtering phenomena.

In order to deal with the problem, many scholars attempt to utilize some adaptive methods to construct SE. For instance, Yu et al. [80] defined the optimal length, when the amplitude of the first five feature frequencies of bearings is largest. Hu et al. [81] presented SE construction method according to the relationship between the bearing fault impulse and a harmonic function with the resonant frequency, harmonic waveform in a period. Differently, Hu et al [82] defined the SE length by the relationship between the vibration sampling rate and the frequency response. Wang et al. [83] believed that the impulses of defective bearing have attenuated attribute, and SE is thus modelled as an impulse attenuation function formulated by the max amplitude, natural frequency and decay rate of the structure element. Yan and Jia [84] presented morphological slice bispectrum fused with cuckoo searching for diagnosing bearing failure. Van et al. [85] used particle swarm optimization algorithm to obtain the SE length.

On the other hand, when the different lengths of SE are used in real applications, the filtered signals have different statistical properties in time- or frequency- or time-frequency domain, thus some evaluated indexes are introduced to determinate the optimal one of the SE lengths. For example, Dong et al. [86] and Raj et al. [87] presented the optimized algorithms separately based on signal to noise and kurtosis to obtain the optimal length of flat SE. Zhang et al. [88] calculated alpha stable distribution to optimize the flat SE length and demonstrated the optimization superiority in capturing bearing outer and inner fault characteristic frequencies when compared with the optimization based on kurtosis. Noting that the aforementioned studies only employed one indictor to define the structuring element length, sometimes considering the higher reliability, some researchers tend to use a combination way between multiple indicators to achieve the task. For instance, Osman et al. [89] and Lv et al. [90] designed a fusion indicator separately based on kurtosis and Renyi entropy, and kurtosis and Teager energy operator to select the optimal SE length, and the results demonstrated that it is effective and robust to incipient bearing fault detection. Li et al. [91,92] considered characteristic frequency intensity coefficient working with third-order cumulant slice spectrum and diagonal slice spectrum for defining the SE length in bearing fault feature extraction.

3.1.2. SSM to Detect Gear Defect

Feng et al. [93] applied four sampling points as the SE length in SSM to filter background noise from raw signals and calculated multi-fractal entropy to classify gear working statuses. Chen et al. [94] empirically defined the linear SE length to demodulate the gear fault features. Gryllias et al. [95] proposed SSM to perform gearing fault detection, in which SE is constructed a linear shape with the length of 0.6 times impulse cycle period.

Lin et al. [96] developed a hybrid approach based on morphological filtering, wavelet theory and empirical mode decomposition to detect the incipient gear fault. Guo et al. [97] presented a combined scheme based on single scale morphology, where the optimal SE length is adaptively determined according to kurtosis and modulation signal bispectrum.

3.2. Multi-Scale Morphology-Based Rotating Parts Fault Diagnosis

MSM acts as an extension to SSM, thus it is more suitable to handle complex real signal. Essentially speaking, the multiscale analysis still considers establishing the corresponding relationship between the SE scales and the morphological feature of signal components. Based on the morphology filtering peculiarity (small scales are conducive to retaining useful information but non-conducive to suppressing noise, and vice versa), the effectiveness of MSM also depends on the reasonability to the used scales.

3.2.1. MSM to Detect Bearing Defect

Li et al. [98] employed multiscale morphology with gradient operator and experimentally determined the length scales of flat SEs. Feng et al. [99] considered a hybrid scheme based on MSM and manifold learning to classify bearing flaws. Gong et al. [100] proposed a repeated multiscale morphology to deal with surplus noise at some small scales after conventional multiscale analysis. Refs [101,102] defined the SE scales from unit scale to 0.6 times impulse period when extracting bearing fault feature frequencies. Gong et al. [103] introduced an iterative asymmetric multiscale morphology targeted at the amplitude demodulation of vibration signatures of defective bearings.

Based on the MSM theory in Section 2.3.2, it can be noticed that this method uses a series of scales to perform morphological processing, thus the last scale is defined properly. To the Refs [98,99,100,101,102,103], the problem is solved by the experimental way. In fact, according to the morphological filtering characteristic, the scale or length of SE is not bigger than the repetitive cycle of rotating parts fault, thus Li et al. [104] proposed the scale as f_s/f -2, where f_s is the sampling frequency of bearing vibration signals, and f is bearing fault feature frequencies. The same method is found in Refs [105,106,107,108].

Although the last scale of structuring element is obtained by math computation based on sampling rate and rotating feature frequency, the MSM methods still lack certain adaptation. In order to handle the problem, Zhang et al. [109] employed an adaptive local-peak algorithm to define the length and height scales. Li et al. [110] combined Zhang’s study with bandwidth empirical mode decomposition to detect bearing characteristic frequency and its harmonic, and Patel et al. [111] melted Zhang’s work with adaptive noise cancellation to improve bearing defect detection. Shuai et al. [112] designed based on local-peak-interval scale definition with support vector regression to distinguish roller bearing fault types. Cui et al. [113] presented information-entropy threshold to obtain optimal scales. Shen et al. [114] and Li et al. [115] developed a bearing fault detection method termed time-varying-scale, in which all flat-SE length scales are given by the time distance between two adjacent impulse peaks in local range. Deng et al. [116] studied particle swarm method to optimize the weight of each scale in the reconstructed signals, and finally roller bearing fault diagnosis effect is improved. Yan et al. [117] presented a scale definition method based on bearing feature frequencies.

Note that the MSM studies above all use a SE with one defined scale to perform time-shifting morphological operation into an original signal, until all SEs are used. Differently, Wang et al. [118,119] proposed a novel structuring element scale definition in more microcosmic way, in which the shape and length of SE is specified based on local minima in the two adjacent waveforms of the raw signals. What’s more, such SE will conduct morphological operation just in local area, without the time-shifting operation in the whole signal and without signal reconstruction by the results of all SEs. In other words, one SE is defined and operated in local area. When it is completed in this area, another new one is used in the same way. Although it is completely different from the conventional multiscale analysis, structural elements are varied and the model is grouped into the multiscale morphology.

3.2.2. MSM to Detect Gear Defect

Li et al. [120] explored multiscale morphology analysis combined with empirical mode decomposition to identify the seeded wear faults on sun, planet and ring gears. Li et al. [121] investigated the multiscale filtering capacity of eight morphological operators, and found that it surpasses continuous wavelet transform in extracting gear feature characteristics. Guo et al. [122] defined the flat SE scales into a certain range decided by the signal sampling frequency and gear fault feature frequency, then used the multiscale morphological method to diagnose sun gear chipping in a planet gearbox. Wang et al. [123] utilized three fixed scales to diagnose the gearbox faults, Facing gearbox defect recognizing, Cai et al. [124] similarly defined the SE scales the is the same as Ref [104].

To the flexible definition of the SE scales, Refs [125,126,127] integrated the adaptive scale definition method in Ref [109] with other approaches to discriminate gearbox faults. Zhang et al. [128] and Yu et al. [129] developed a multiscale morphology to diagnose gear fault, where the SE scales are chose adaptively based on kurtosis. Liu et al. [130] developed a scale optimization method based on the parameter changing of impact signals to locate gear faults. Yan et al. [131] constructed a special indictor melting signal-to-noise ratio with fault feature ration to choose the SE scales when identifying wind turbine gearbox faults. Liu et al. [132] considered Chebyshev window as structuring elements, and defined the length and height scales based on the signal features of planetary gearbox. Zhuang et al. [133] proposed the back-propagation learning algorithm to update the SE scale for detecting gearbox failure. In order to remove impulse interference of the vibration signals of gearbox, Cao et al. [134] designed the SE length selection based on max maximizing L-kurtosis.

3.3. Morphological Pattern Spectrum-Based Rotating Parts Fault Diagnosis

Different to SSM and MSM, MPS construct a one-to-one mapping relationship between the quantifiable spectrum index and the defined pattern, it is able to distinguish rotating parts working statue under different working conditions.

3.3.1. MPS to Detect Bearing Defect

Chen et al. [135] calculated morphological pattern spectrum with double-dot SEs, and used an improved support vector machine (SVM) algorithm to classify the outer fault, the inner race fault, and the ball fault on rolling ball bearing. In order to improve bearing fault classification accuracy, Gao et al. [136] proposed an optimized method based on MSP and S transform to bearing faults, and found that it can successfully distinguish normal state and abnormal state with three fault levels. Sun et al. [137] calculated morphological pattern spectrum obtained by dilation, with support vector machine to classify bearing faults. Zhu et al. [138] noticed that in a signal from bearing fault the detailed information is similar but the position is different, thus introduced the feature vector of hit and miss at local impact feature position to enhance classifier recognition performance. Li et al. [139] found that MPS is obtained by using the difference between open minus close, and least square support vector machine to recognize different wheelset bearing fault types. Gao et al. [140] developed the MPS of the time-frequency representations of signal as the input feature vectors of the classifier to identify bearing states.

Those investigations mentioned above are applications on basic MPS. In fact, pattern spectrum can be further developed to special target. Hao et al. [141,142] believed that the combination between morphological pattern spectrum entropy and barycenter scale location of spectrum is more effective than the combination between kurtosis and enveloping demodulating spectrum when differentiating six bearing defects. Wang et al. [143] noticed that original MPS cannot recognize damaged and normal bearings, and then developed the sample entropy and Lempel-Ziv complexity of MPS curve as bearing fault classification index. Yan et al. [144] exploited multiscale pattern gradient spectrum entropy based on MSP as the input feature sets of extreme machine learning, and the bearing defect classification demonstrated the effectiveness of this combined approach. Yu et al. [145] developed MPS entropy and pattern spectrum values as the bearing feature parameters of proximal SVM.

MPS is utilized as not only bearing fault classification tool, but also bearing performance estimation indicator. For instance, Zhao et al. [146,147] presented a new idea entitled high-order differential morphological spectrum entropy based on MPS and performed it on bearing performance trend prediction. Li et al. [148] extended the conventional definition of MSP to the general space, and simulation and practical applications demonstrated that the improvement is capable of evaluating bearing performance. Gao et al. [149] studied the relationship between generalized pattern spectrum entropy and performance degradation course, showing that they have a good correlation. Similarly, Wang et al. [150] found that it is accordant and preferable for the relevance between improved pattern spectrum entropy and bearing degradation degree.

3.3.2. MPS to Detect Gear Defect

Li et al. [151] constructed morphological pattern spectra obtained by separately using four basic operators as the input of K nearest neighbor classifier, Bayes classifier and least-square support vector machine, and finally observed that the MPS obtained by erosion are best for gear fault classification. Li et al. [152] characterized the time-frequency representation based on S transform by morphological pattern spectrum to recognize five gear states in a gearbox. Barbieri [153] considered a correlation analysis according to two pattern spectra produced by signals with and without gear fault, and integrated it with wavelet transform, single value decomposition, and stabilization diagram to discriminate the defects of gear and bearing installed in gearbox.

3.4. Morphological Wavelet -Based Rotating Parts Fault Diagnosis

Morphological wavelet is a nonlinear multiresolution signal decomposition scheme, with high computation efficiency and retaining local features of signal. It is capable of providing a different view to rotating part detect diagnosis.

3.4.1. MW to Detect Bearing Defect

Hao and Chu [154] discussed a morphological undecimated wavelet decomposition, keeping the length of original signals and showing the less distortion in time-domain signal, and the actual bearing defects applications indicated that this method well extract impulses. Wang et al. [155] designed a mixed scheme based on lifting morphological wavelet and ensemble empirical mode composition for extracting fault features of defective rolling element bearings. Li et al. [156] proposed an adaptive morphological gradient lifting wavelet, in which the average filter and morphological gradient filter are used to update the approximating signal, and the results showed that the proposed algorithm outperforms linear wavelet transform in bearing fault diagnosis. Li et al. [157] analyzed the geometrical structures of the bearing vibrations dispersed various morphological scales, and developed morphological stationary wavelet to extract bearing fault features. Chen et al. [158] performed a MW decomposition method based on open and closing operators for impulse feature extraction of damaged bearings.

Han et al. [159] carried out morphological wavelet and least squares support vector machine to classify the various type of bearing failure. Meng et al. [160] used morphological wavelet and morphological filter to diagnose the outer, inner, rolling ball and mixed bearing faults. Khakipour et al. [161] performed morphological wavelet based on morphology gradient operator to detect bearing flaws and proved its superiority by comparing with morphological harr wavelet and the morphological undecimated wavelet decomposition. Li et al. [162] focused their study in using undecimated MW lifting scheme based on morphological convolution operator to identify bearing defects. Guo et al. [163] presented a combined scheme based on morphological undecimated wavelet and morphological filtering to extract the weak fault feature of rolling bearing. Li et al. [164] used the different morphology operators in morphology undecimated wavelet to well suppress noisy signals, and the results indicated that it diagnoses the wheelset bearing compound faults accurately.

Li et al. [165] took full advantage of the nonlinear characteristics of morphology Harr wavelet, and combined it with Perona-Malik filtering for detect the bearing inner race failure. Duan et al. [166] exploited an empirical morphology undecimated wavelet to process the faulty bearing vibration signals, and found that it effectively detect the fault with high computation efficiency. In addition, Wang et al. [167] presented a morphology undecimated wavelet scheme, in which an improved gradient based on closing and opening is considered as analysis operator, and showed that the improvement can avoid statistic bias in bearing fault vibrations.

Zhang et al. [168] proposed the combination between dilation and erosion as and one between opening and closing superlatively as synthesis operator and analysis operator, and the results demonstrated that the undecimated decomposition is suited for bearing faults diagnosis. Lin et al. [169] constructed morphological wavelet package decomposition method, and at the same time set soft threshold to remove the noisy signals for bearing fault feature detection. Yang et al. [170] presented an integrated method based on morphology wavelet and S transform, demonstrating that MW can inhibit noise and the harmonic components of axis rotating frequency to bearing fault classification.

3.4.2. MW to Detect Gear Defect

Li et al. [171] utilized morphological gradient wavelet to detect gear fault feature frequencies. Zhang et al. [172] presented multi-scale morphological undecimated wavelet decomposition and grey incidence to identify the gear fault patterns. Hong et al. [173] employed morphological mean wavelet transform to gear fault diagnosis and found that it has the sensitivity to local extrema of signal and the high effectiveness to reduce noise. Zhang et al. [174] developed morphological Harr wavelet to deal with original signals and used permutation entropy as the fault eigenvalue for detecting various gear fault types. Cai et.al [175] proposed a morphological wavelet method to cancel the strong noisy signals when diagnosing a gear fault. Ding et al [176] introduced morphological Harr wavelet to purify the vibration signals of faulty gearbox and used Permutation entropy to classify fault type. Zhang et al. [177] developed max-lifting morphological wavelet for extracting gear features, in which dilation is used as predicted and updated operator.

Tong et al. [178] employed weighted morphological un-decimated wavelet decomposition and correlated kurtosis to gear defects, using difference morphological gradient operator based on opening and closing. Shen et al. [179] found that morphological wavelet de-noising provide good behavior in gear fault features. Li et al. [180] believed that the traditional MW is disadvantage in dealing with sudden impulses of damage gear because of using a fixed filter, and proposed adaptive morphological undated lifting wavelet.

3.5. Morphology-Based Methods to Other Mechanical Defective Objects

Li et al. [181] investigated adaptive multiscale morphology to recognize railway wheel flat, where the optimal scales are defined by spectrum kurtosis crooner criterion. Jiang et al. [182] studied local mean decomposition and adaptive multiscale morphology to demodulate the defective hydraulic pump signals and believed that the combined method is anti-noise and has stronger demodulation ability. Li et al. [183] introduced an improved morphology pattern spectrum into time-frequency analysis and the results illustrated that this approach is effective to characterize the vibration signals of engine with five working states.

3.6. Comparison and Analysis of Operators and SE in Applications

Through these representative applications, which are listed in Table 1 to present a more direct view for interested readers, it shows that mathematical morphology is applicable and robust in the rotating machine fault diagnosis area. Meanwhile, it is also noticed that these studies concentrate two aspects: morphological operator and structuring element, which benefit enhancing the morphology performance in practical applications. As a result, morphological operator selection and structuring element definition are further analyzed here.

3.6.1. Morphology Operator Analysis

Morphological operator selection, relatively speaking, is handled easily. Apparently, basic operators have individual behavior to signal morphological feature recognition, for example, closing operator can capture positive impulses and restrain negative ones. Meanwhile, basic operators can constitute various combined morphological gradients, providing more choices for actual applications.

In general, if it focuses on the extracting morphological feature in time-domain, closing and opening, and the combination between them are a better candidate, because they possess the recovery capacity to an original signal. When an application pays attention to isolating frequency-domain feature components, gradient operator should be considered because they are able to retain positive and negative feature components in a signal, with lesser loss to them. In some special cases, more complex combination patterns between basic operators, such as product, convolution and cross-correlation, are utilized for attain superior result [184]. Hence, the selectable candidates are relatively limited in actual application, i.e., basic and gradients operators.

3.6.2. Structuring Element Analysis

To a structuring element, the first should be considered is shape. In general, the shape almost has no effect to process a 1D signal, e.g., a vibration signal sampled from rotating machine. The reason is space dimension mismatching between structuring element and signal. In order to explain the problem, a diamond-shaped SE presented in Figure 4 is considered, in which black dot and gray dot represent 1 and 0 respectively, and the black one circled by a red rectangle is the origin. Let a 1D signal be {x₀, x₁, …, x_i,…, x_n-1, x_n}. When the point x_i is processed and firstly overlapped with the origin, other data points covered by the neighborhoods are detected. Obviously, these points only at the left and right sides of x_i can be introduced into the morphological operation, because there are no others at the top and bottom area of x_i. It means that when the diamond SE is used, just the black dots linked by a blue line in Figure 5 work as effective neighborhood to the 1D signal and will control the morphological computation range. Accordingly, the complex shape has on influence to morphological processing of one-dimensional signal. Here, it must be emphasized that the shape has no effect to the morphological result, just when structural element is flat. On the contrary, if the SE is non-flat, its length and height together construct a specified figure, such as triangle SE and Gaussian SE, now the height parameter is introduced in morphological, thus such shape will affect the morphological result.

Furthermore, as mentioned above, once the SE height is considered, their specified values will take part in a morphology computation, i.e., the length and height parameters work together to the morphology result. However, it can be noticed that in the aforementioned morphology applications the overwhelming cases use only the length parameter, and a few cases [109,110,111,125,126,127] use the two parameters together. This is mainly because most cases just concern the nonlinear filtering of mathematical morphology, that is, using the length controls the filtering range, thereby obtaining a good filtering performance. In addition, the SE length describes the time-scale representation for feature information like their cycle of rotating part faults, only if the inputting speeds and the mechanical structure configuration are determined, it can be exactly and easily achieved by theoretical calculation. However, the SE height corresponds to a spatial representation to a signal with apparent randomness. Therefore, it is more difficult to define the height than to define the length, and non-flat SE is less utilized in the mechanical fault diagnosis field.

On the other hand, the essence of morphology theory is the geometrical structure interacting and matching between SE and a processed object. In this situation, if the length is only considered, the flat SE and the object lack higher spatial matching. Obviously, a better way should use non-flat SE, in the sense that such morphology application can take full advantage of two merits: non-linear filtering and closer shape mapping between the two objects. If so, much more characteristic information could be extracted. Certainly, the key is that how to determine the length and height of structuring element. Although a few tries are conducted, it still needs further study.

4. Discussion and Development Orientation

Under the framework of mathematical morphology, four methods are introduced and their relevant applications on detecting rotating machine flaws are presented in this study. As an open study topic, there still are several unsolved problems and potential developments in mathematical morphology:

(1) when single- and multi-scale analysis use SE, especially large-scale ones, the morphological processing easily suffer from edge distortion in time domain following the shift of SE. Essentially speaking, it is determined by the nonlinear filtering mechanism, i.e., local extrema search. So, considering from signal restoring in time domain with retaining this nonlinear filtering character, how to alleviate the time-domain distortion as much as possible should be further studied.

(2) In morphology theory, the core is how to construct the structure element adaptively. It can be known that once the scales are defined, they will not change in one morphological operation. However, dynamic characteristics of a real signal always change following sampling time. Obviously, the localized variability of SE is a good ability that relieves the dependence to the scale constraint, highlight the dynamic characteristics of signal, and enhance the feature extracting capacity.

(3) One of the advantages of morphology is low computation cost, and it is very essential to real-time monitoring of mechanical equipment. Notice that Morphology processing is conducted by using the point-wise shift of SE inside original signal, sometimes the shifting is maybe unnecessary to such processing, at least in specific range, for example the interval between two impulses produced by rotating part faults. In a permissible condition without the loss in time- or frequency domain, the non-pointwise shift could be considered, and it will further improve the morphology computation efficiency.

(4) Morphology pattern spectrum is a fault classification tool. In actual applications, noise disturbance will affect this processing accuracy, thus in some cases MPS frequently works with other pattern classification algorithms to improve the classification performance. On the other aspect, MPS can be regarded as the moment of first-order of a distribution function. It is evident that there exist the different high order moments between feature shapes and noise in a signal. Based on such consideration, it can inhibit the noise interference to strengthen feature shape classification performance.

(5) Nonlinear multiresolution morphological wavelet is sensitive to certain types of noise, developing more robust methods that are specifically designed to mitigate the impact of noise could enhance the reliability of morphological wavelet-based fault detection. This could involve new mathematical formulations or filtering techniques that preserve important nonlinear features while suppressing noise.

(6) Morphology processing is conducted by one computation in SMS and MPS and multi-computations in MSM and MW, but they are unrelated each other, thus the current morphology-based approaches rely on the defined SE highly. If a novel morphological computation model can be figured out, the dependence to SE can be reduced in a certain degree. For instance, the gradient descent algorithm in optimization theory is to gradually obtain the optimal solution. Maybe, it is a good reference to mathematical morphology.

5. Concluding Remarks

A literature surveys and summarization on mathematical morphology and actual applications for rotating machine fault diagnosis is presented in this paper. Section 2 describes two basic elements, structuring element and morphology operators, and four morphology-based methods, containing single-scale morphology, multi-scale morphology, morphological pattern spectrum and morphological wavelet. Their typical applications on rotating machine fault diagnosis and classification are given with a clear classification as fault object and diagnostic method, and the effect of operator and SE to morphology result is analyzed in Section 3. Finally, several potential research directions are proposed. The review paper attempts to introduce diagnosing methods based on mathematical morphology as much as possible and summarize the related applications in the rotatory machine fault detection field. It is believed that it can offer a comprehensive reference to researchers who are interested to rotating machine fault diagnosis based on mathematical morphology.