# Rough Set Neural Network Feature Extraction and Pattern Recognition of Shaft Orbits Based on the Zernike Moment

Authors: Xinfeng Ge; Jing Zhang; Ye Zhou; Jianguo Cai; Hui Zhang; Hongchang Hua; Dong Chen; Ming Zhao; Jinqi Du; Yuan Zheng
Journal: Shock and Vibration
 (2021)

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## Abstract

In the shaft axis monitoring of hydrogenerating unit condition monitoring and fault diagnosis, the shaft orbit is intuitive and comprehensively reflects the unit operation state, and different shaft orbits correspond to different fault types, which can accurately indicate a system vibration fault. Shaft orbit identification has important significance for vibration fault diagnosis. In getting the feature extraction and pattern recognition of a shaft orbit, the Zernike moment is better than the Hu moment; it has the advantages of a small calculation error and a high recognition rate. A rough set neural network (RS-BP hybrid model) of shaft orbit recognition is established, which uses just 13 moment eigenvalues reserved by the rough set feature selection algorithm as input variables; it has the same calculation error and recognition rate and reduces the calculation time step. The simulation of the recognition of shaft orbits shows that the hybrid model has achieved good results in the identification of shaft orbits.


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## Body

## 1. Introduction

Because of the complex operating conditions involved, there are many factors that cause the instability of hydropower units, including hydraulic, mechanical, and electromagnetic factors. Among these three important factors, hydraulic vibration is the excitation source that has a crucial influence on the vibration of the unit. If the low-frequency oscillation is induced by the vibration of the power system, it will affect the stable operation of the power grid [1, 2]. Through hydraulic excitation, each component of the unit exhibits different vibration characteristics. During the shafting monitoring, the shaft orbit directly and comprehensively reflects the operation state and vibration fault of the unit, and different shaft orbits usually reflect different faults. The hydroelectric unit condition monitoring system can obtain many operating characteristics of the unit from the axis orbit of the unit, so the recognition of the shaft orbit plays an important role in the fault diagnosis.In recent years, many experts have performed theoretical research and simulation analysis on hydrogenerator shaft orbits, using different methods to research the recognition of shaft orbits [3–7]. At present, the recognition of shaft orbits mainly includes precession direction recognition and shape recognition. Shape recognition of shaft orbits mainly includes feature extraction and pattern recognition. The feature extraction methods include Fourier descriptors [8–11], Hu moments [12–15], and Zernike moments [16], as well as classification of the data with an advanced pattern recognition method [17]. Kauppinen et al. [18] tried to use autoregressive modeling and Fourier descriptors of closed contours to recognize the typical shape of shaft orbits. This showed better performance for Fourier-based methods, especially for images containing noise. Fu et al. [8] combined a BP neural network and modified Fourier descriptors to classify the shaft orbits for fault diagnosis of the turbine rotor. The simulation results were for the automatic identification of three typical shaft orbits. In order to obtain the characteristics of the axis orbit, the edge acquisition algorithm is typically used, and, by contrasting each edge extraction algorithm, the applicability of the corresponding algorithm is obtained. Then the algorithm is improved according to the corresponding boundary features. An intelligent model, such as the neural network model, is established to achieve the intelligent classification of faults in the mechanical system.Fourier descriptors have the advantage of feature extraction for completely closed curves, but the shaft orbits’ coincidence performance for the hydropower unit is so poor that it is hardly closed, so this paper uses Hu moments and Zernike moments to extract features of the shaft orbit.For intelligent pattern recognition, this paper adopts the common BP neural network and applies the rough set theory to simplify the neural network with overly large input samples and a complex network structure.


## 2. Formation and Characteristics of the Shaft Orbit

In the hydrogenerating unit stability test and application of the monitoring system, two eddy current displacement sensors are orthogonally installed in the guide bearing (upper and lower guide) horizontal section. This ensures that the vibration signal of two mutually perpendicular directions can be measured. Then the high-frequency noise of the vibration signal is eliminated using a low pass filter, and finally a center orbital path is formed using the analog-to-digital converter. As shown in Figure1, two shaft swing sensors are installed in the unified plane of the rotating shaft. The position of the rotor center can be determined by the outputs xt and yt of the two sensors at the t moment, and then the position of the rotor center at different time can be transformed into a curve, which is the vibration track of the rotor in the bearing, called the shaft orbit.Figure 1
Measurement of the shaft orbit.The shaft orbit of different shapes reflects the operation of different units, and its shape is an important basis for judging the running state and failure of the shaft, which implies the fault information of the unit. Table1 shows the shaft orbit shapes and the potential failure types, and Figure 2 shows typical faults.Table 1
Shaft orbit shapes and potential failure types.
No.Shaft orbitsFailure type1EllipseUnbalanced or main axis is not straight2Banana and outer “8”Misalignment3Petal shapeVortex band of draft tube4Inner “8”Oil whirl5Two separate ellipsesBistability6Protruding burrStatic and dynamic component frictionFigure 2
Shaft orbit of typical faults. (a) Ellipse. (b) Banana. (c) Outer “8.” (d) Petal shape. (e) Inner “8.” (f) Two separate ellipses. (g) Protruding burr.
(a)(b)(c)(d)(e)(f)(g)


## 3. Moment Feature Extraction and Model Establishment

### 3.1. Hu Moments and Improvement

#### 3.1.1. Hu Moments

Hu moment theory has been widely used in the field of image recognition. It is a method of extracting image features based on a statistical feature. Two-dimensional moments are often used in digital image recognition. In fact, it extends the order moments of one-dimensional signals to two-dimensional space. By using normalization, we obtain the moment’s characteristic quantities that are sensitive to the change of shape, while the translation, rotation, and scaling of two-dimensional images remain unchanged. Hu [19] constructed 7 moments with rotation, translation, and scale invariance in 1962. In the field of image recognition, a statistical feature extraction method was established using moment theory, and it was widely used. In 1978, Wong and Hall [20] proved that the invariance of the Hu moment could be guaranteed when the scaling factor ρ≤2 and the rotation angle of the trajectory graph θ≤45°. For a two-dimensional function fx,y∈R2, which is defined on the plane o−xy, the p+q order mixed origin moment is defined according to the following formula:(1)p,q=0,1,2...Thep+q order mixed central moment is defined according to the following formula:(2)μpq=∫−∞+∞∫−∞+∞x−x¯py−y¯qfx,ydxdy,p,q=0,1,2...where x¯,y¯ is the gray center of gravity for the image fx,y.(3)x¯=m10m00,y¯=m01m00.The normalized central moment is defined as(4)ηpq=μpqμ00r.Under the constraints,r=p+q+2/2;p+q=0,1,2,3….Ming-kuei Hu proposed the construction method of moments in 1962. By using the above central moments, 7 moment functions are constructed. These seven expressions correspond to translation, scaling, and rotation.(5)ϕ1=η20+η02,ϕ2=η20−η022+4η112,ϕ3=η30−3η122+η03−3η212,ϕ4=η30+η122+η03+η212,ϕ5=η30−3η12η30+η12η30+η122−3η03+η212+3η21−η03η03+η213η30+η122−η03+η212,ϕ6=η20−η02η30+η122−η03+η212+4η11η30+η12η03+η21,ϕ7=3η21−η03η30+η12η30+η122−3η03+η212+3η12−η30η03+η213η30+η122−η03+η212.


#### 3.1.2. Moments in Discrete States

Thep+q order origin moment and central moment of a two-dimensional image function fx,y in a discrete state are defined according to the following formulas:(6)mp+q=∑i=1N∑j=1Mxipyjqfxi,yi,(7)μp+q=∑i=1N∑j=1Mxi−x¯pyj−y¯qfxi,yi.Under the constraints,(8)x¯=∑i=1N∑j=1Mxifxi,yi∑i=1N∑j=1Mfxi,yi,(9)y¯=∑i=1N∑i=1Myifxi,yi∑i=1N∑j=1Mfxi,yi.Normalized central moments in discrete states, which are defined by formulas (8) and (9), can clearly show that the moment functions in discrete states are also translation-independent. By observing the definition of the central moment, it is found that the central moment may be negative. Because the change range of moment s is large, data compression can be achieved by the logarithmic processing of moments. Since the moments may appear negative, we need to take the absolute value before doing logarithmic operations. The actual moments can be obtained by means of a logarithm:(10)Mi=log10φi,i=1,2,3,4,5,6,7.


#### 3.1.3. Improvement of Moments

If the scaling factor of the trajectory graph isρ, the relationship between the moment ϕi′ in discrete states after scaling and the moment ϕi of the original graph is shown with the following formula:(11)ϕ1′=ρ2ϕ1,ϕ2′=ρ4ϕ2,ϕ3′=ρ6ϕ3,ϕ4′=ρ6ϕ4,ϕ5′=ρ12ϕ5,ϕ6′=ρ8ϕ6,ϕ7′=ρ12ϕ7.In order to make these moment functions independent of scaling, we can consider the elimination of the scaling factor and useϕ2′ as the benchmark to find the final expression of the rest moment functions obtained by this moment function.(12)M1∗=ϕ1′ϕ2′=ϕ12ϕ2,M3∗=ϕ3′2ϕ2′3=ϕ32ϕ23,M4∗=ϕ4′2ϕ2′3=ϕ42ϕ23,M5∗=ϕ5′ϕ2′3=ϕ5ϕ23,M6∗=ϕ6′ϕ2′2=ϕ6ϕ22,M7∗=ϕ7′ϕ2′3=ϕ7ϕ23.


### 3.2. Zernike Moment

Zernike moment is one kind of special complex moments, which are based on orthogonal functions called Zernike polynomials. Compared with HU moment, the computation of Zernike moments is more complex, but they have greater advantages in terms of their expressive power and noise sensitivity. Zernike moments have the characteristics of rotation invariance and have been widely used in the field of target recognition.


#### 3.2.1. Definition of Zernike Moments

Zernike [21] presents a set of orthogonal polynomials Vnmρ,θ, which are orthogonal in the unit circle ρ≤1. The polynomial function in polar coordinates is obtained as follows:(13)Vnmρ,θ=Rnmρexpimθ.Under the constraints,n=0,1,2… and m=0,±1,±2,....., m≤n, where n−m is even number, ρ is vector from origin point to pixel point x,y, θ is the anticlockwise angle between the vector ρ and the x-axis, and Rnmρ is real-valued radial polynomial.(14)Rnmρ=∑s=0n−m/2−1s,n−s!ρn−2ss!n−2s+m/2!n−2s−m/2!.Radial polynomialsRnmρ satisfy the following relationship:(15)∫01RnlrRmlrρdρ=12n+1δnm,where δnm is the Kronecker delta.The orthogonality conditions of Zernike polynomials are as follows:(16)∫02π∫01Vnl∗rVmkρ,θρdρdθ=12n+1δnmδlk,where ∗ denotes the complex conjugate. The following formula can be obtained from formula (16): (17)Vnm∗ρ,θ=Vn,−mρ,θ.Some important properties ofRnmρ can be obtained from the following formula: (18)Rnm1=1,Rnnr=ρn,R00ρ=1.Ann order Zernike polynomial with a repetition rate of M is defined according to the following formula: (19)Anm=n+1Π∬fρ2≤1ρ,θVnm∗ρ,θρρdθ,ρ2≤1.For a digital image, the integral is replaced by a summation, namely,(20)Anm=n+1π∑x∑yfx,yVnm∗ρ,θdxdy,x2+y2≤1.To calculate the Zernike moments of an axis shaft orbit, it is necessary to move the center of the axis trajectory image to the coordinate circle and map the pixel points of the trajectory into the unit circle. Through the formula, it can be seen thatVnm∗ρ,θ can extract the features of images, and the high-frequency characteristics are obtained by Vnm∗ρ,θ for larger n values. The low-frequency characteristics are obtained by Vnm∗ρ,θ for smaller n values. Zernike moments can be used to construct high moments. In theory, higher moments can contain more image information, so pattern recognition will be better.


#### 3.2.2. Rotation of Zernike Moments

When the rotation angle of an image isa, the expression of its Zernike moments is as follows: (21)Anm′=n+1Π∬fρ2≤1ρ,θ1Rnmρexp−jmθ1−αρdρdθ=Anmexp−jmα.From the upper formula, it is shown that the Zernike moments’ phases change, but the modulus remains unchanged. So,Anm can be regarded as the target rotation invariance feature.


### 3.3. Rough Set Neural Network

Rough set theory (RS theory) [22] was proposed by Pawlak, a mathematician of Poland in 1982. It is an effective method for dealing with uncertain, inconsistent, and imprecise problems. It can discover hidden knowledge from uncertainty and reveal its potential law. It defines knowledge from a new point of view, regards knowledge as a division of the universe, and holds that knowledge is granular. In recent years, the research on rough set theory in the field of artificial intelligence has been a hot topic, and this theory has been widely used. Knowledge reduction [23–25] is one of the core concepts of rough set theory.


#### 3.3.1. Rough Set Principle

In rough set theory, knowledge is expressed in the form of an information tableS=U,R,V,f, where U is a collection of objects, also called a domain, R=C∪D, CnD = O, C is called the conditional attribute set, D is called the decision attribute set, V=r∈RUVr is a collection of attribute values, Vr represents the attribute value range of an attribute r∈R, and f:U×R⟶V is an information function that specifies the attribute value of each object x in U.The simplification of a decision table is meant to simplify the conditional attributes in the decision table. The simplified decision table also has the function of simplifying the decision table before, and the simplified decision table has less conditional attributes. The rough set is used to describe the knowledge, and the reduction of redundant attributes can be used to simplify the knowledge. Finally, a complex decision table is reduced to the minimum conditional attribute decision table with no redundant attributes and the correct classification is ensured.


#### 3.3.2. Rough Set Neural Network (RS-BP Hybrid Model)

The RS-BP hybrid model is a hybrid model combining artificial neural network and rough set theory. The training time is too long when the neural network and the sample are both large. This inherent disadvantage is one of the main factors which restricts the further practicability of the neural network. Although various algorithms for improving training speed have emerged, the problem is far from being completely solved. Simplifying the training sample set and eliminating redundant data are another way to improve training speed. The rough set is used to simplify the neural network in order to train the sample set, and the redundant data is eliminated under the premise of retaining important information. Simulation results show that the training speed is 4.77 times better, and good results are achieved [26].The characteristic parameters of the vibration are extracted through the condition monitoring system and a field test in the fault diagnosis process of the hydroelectric generating set. In order to describe the vibration mode of the hydrogenerator set as much as possible, more units are set up to extract the characteristic quantities of these units, some of which are related and some of which are independent. These feature parameters are often incomplete and redundant, which makes the modeling of subsequent fault diagnosis complex. The rough set theory can be used to reduce the knowledge of the training samples of neural network and get fewer attribute values while ensuring the goal of not missing important information.


## 3.1. Hu Moments and Improvement

### 3.1.1. Hu Moments

Hu moment theory has been widely used in the field of image recognition. It is a method of extracting image features based on a statistical feature. Two-dimensional moments are often used in digital image recognition. In fact, it extends the order moments of one-dimensional signals to two-dimensional space. By using normalization, we obtain the moment’s characteristic quantities that are sensitive to the change of shape, while the translation, rotation, and scaling of two-dimensional images remain unchanged. Hu [19] constructed 7 moments with rotation, translation, and scale invariance in 1962. In the field of image recognition, a statistical feature extraction method was established using moment theory, and it was widely used. In 1978, Wong and Hall [20] proved that the invariance of the Hu moment could be guaranteed when the scaling factor ρ≤2 and the rotation angle of the trajectory graph θ≤45°. For a two-dimensional function fx,y∈R2, which is defined on the plane o−xy, the p+q order mixed origin moment is defined according to the following formula:(1)p,q=0,1,2...Thep+q order mixed central moment is defined according to the following formula:(2)μpq=∫−∞+∞∫−∞+∞x−x¯py−y¯qfx,ydxdy,p,q=0,1,2...where x¯,y¯ is the gray center of gravity for the image fx,y.(3)x¯=m10m00,y¯=m01m00.The normalized central moment is defined as(4)ηpq=μpqμ00r.Under the constraints,r=p+q+2/2;p+q=0,1,2,3….Ming-kuei Hu proposed the construction method of moments in 1962. By using the above central moments, 7 moment functions are constructed. These seven expressions correspond to translation, scaling, and rotation.(5)ϕ1=η20+η02,ϕ2=η20−η022+4η112,ϕ3=η30−3η122+η03−3η212,ϕ4=η30+η122+η03+η212,ϕ5=η30−3η12η30+η12η30+η122−3η03+η212+3η21−η03η03+η213η30+η122−η03+η212,ϕ6=η20−η02η30+η122−η03+η212+4η11η30+η12η03+η21,ϕ7=3η21−η03η30+η12η30+η122−3η03+η212+3η12−η30η03+η213η30+η122−η03+η212.


### 3.1.2. Moments in Discrete States

Thep+q order origin moment and central moment of a two-dimensional image function fx,y in a discrete state are defined according to the following formulas:(6)mp+q=∑i=1N∑j=1Mxipyjqfxi,yi,(7)μp+q=∑i=1N∑j=1Mxi−x¯pyj−y¯qfxi,yi.Under the constraints,(8)x¯=∑i=1N∑j=1Mxifxi,yi∑i=1N∑j=1Mfxi,yi,(9)y¯=∑i=1N∑i=1Myifxi,yi∑i=1N∑j=1Mfxi,yi.Normalized central moments in discrete states, which are defined by formulas (8) and (9), can clearly show that the moment functions in discrete states are also translation-independent. By observing the definition of the central moment, it is found that the central moment may be negative. Because the change range of moment s is large, data compression can be achieved by the logarithmic processing of moments. Since the moments may appear negative, we need to take the absolute value before doing logarithmic operations. The actual moments can be obtained by means of a logarithm:(10)Mi=log10φi,i=1,2,3,4,5,6,7.


### 3.1.3. Improvement of Moments

If the scaling factor of the trajectory graph isρ, the relationship between the moment ϕi′ in discrete states after scaling and the moment ϕi of the original graph is shown with the following formula:(11)ϕ1′=ρ2ϕ1,ϕ2′=ρ4ϕ2,ϕ3′=ρ6ϕ3,ϕ4′=ρ6ϕ4,ϕ5′=ρ12ϕ5,ϕ6′=ρ8ϕ6,ϕ7′=ρ12ϕ7.In order to make these moment functions independent of scaling, we can consider the elimination of the scaling factor and useϕ2′ as the benchmark to find the final expression of the rest moment functions obtained by this moment function.(12)M1∗=ϕ1′ϕ2′=ϕ12ϕ2,M3∗=ϕ3′2ϕ2′3=ϕ32ϕ23,M4∗=ϕ4′2ϕ2′3=ϕ42ϕ23,M5∗=ϕ5′ϕ2′3=ϕ5ϕ23,M6∗=ϕ6′ϕ2′2=ϕ6ϕ22,M7∗=ϕ7′ϕ2′3=ϕ7ϕ23.


## 3.1.1. Hu Moments

Hu moment theory has been widely used in the field of image recognition. It is a method of extracting image features based on a statistical feature. Two-dimensional moments are often used in digital image recognition. In fact, it extends the order moments of one-dimensional signals to two-dimensional space. By using normalization, we obtain the moment’s characteristic quantities that are sensitive to the change of shape, while the translation, rotation, and scaling of two-dimensional images remain unchanged. Hu [19] constructed 7 moments with rotation, translation, and scale invariance in 1962. In the field of image recognition, a statistical feature extraction method was established using moment theory, and it was widely used. In 1978, Wong and Hall [20] proved that the invariance of the Hu moment could be guaranteed when the scaling factor ρ≤2 and the rotation angle of the trajectory graph θ≤45°. For a two-dimensional function fx,y∈R2, which is defined on the plane o−xy, the p+q order mixed origin moment is defined according to the following formula:(1)p,q=0,1,2...Thep+q order mixed central moment is defined according to the following formula:(2)μpq=∫−∞+∞∫−∞+∞x−x¯py−y¯qfx,ydxdy,p,q=0,1,2...where x¯,y¯ is the gray center of gravity for the image fx,y.(3)x¯=m10m00,y¯=m01m00.The normalized central moment is defined as(4)ηpq=μpqμ00r.Under the constraints,r=p+q+2/2;p+q=0,1,2,3….Ming-kuei Hu proposed the construction method of moments in 1962. By using the above central moments, 7 moment functions are constructed. These seven expressions correspond to translation, scaling, and rotation.(5)ϕ1=η20+η02,ϕ2=η20−η022+4η112,ϕ3=η30−3η122+η03−3η212,ϕ4=η30+η122+η03+η212,ϕ5=η30−3η12η30+η12η30+η122−3η03+η212+3η21−η03η03+η213η30+η122−η03+η212,ϕ6=η20−η02η30+η122−η03+η212+4η11η30+η12η03+η21,ϕ7=3η21−η03η30+η12η30+η122−3η03+η212+3η12−η30η03+η213η30+η122−η03+η212.


## 3.1.2. Moments in Discrete States

Thep+q order origin moment and central moment of a two-dimensional image function fx,y in a discrete state are defined according to the following formulas:(6)mp+q=∑i=1N∑j=1Mxipyjqfxi,yi,(7)μp+q=∑i=1N∑j=1Mxi−x¯pyj−y¯qfxi,yi.Under the constraints,(8)x¯=∑i=1N∑j=1Mxifxi,yi∑i=1N∑j=1Mfxi,yi,(9)y¯=∑i=1N∑i=1Myifxi,yi∑i=1N∑j=1Mfxi,yi.Normalized central moments in discrete states, which are defined by formulas (8) and (9), can clearly show that the moment functions in discrete states are also translation-independent. By observing the definition of the central moment, it is found that the central moment may be negative. Because the change range of moment s is large, data compression can be achieved by the logarithmic processing of moments. Since the moments may appear negative, we need to take the absolute value before doing logarithmic operations. The actual moments can be obtained by means of a logarithm:(10)Mi=log10φi,i=1,2,3,4,5,6,7.


## 3.1.3. Improvement of Moments

If the scaling factor of the trajectory graph isρ, the relationship between the moment ϕi′ in discrete states after scaling and the moment ϕi of the original graph is shown with the following formula:(11)ϕ1′=ρ2ϕ1,ϕ2′=ρ4ϕ2,ϕ3′=ρ6ϕ3,ϕ4′=ρ6ϕ4,ϕ5′=ρ12ϕ5,ϕ6′=ρ8ϕ6,ϕ7′=ρ12ϕ7.In order to make these moment functions independent of scaling, we can consider the elimination of the scaling factor and useϕ2′ as the benchmark to find the final expression of the rest moment functions obtained by this moment function.(12)M1∗=ϕ1′ϕ2′=ϕ12ϕ2,M3∗=ϕ3′2ϕ2′3=ϕ32ϕ23,M4∗=ϕ4′2ϕ2′3=ϕ42ϕ23,M5∗=ϕ5′ϕ2′3=ϕ5ϕ23,M6∗=ϕ6′ϕ2′2=ϕ6ϕ22,M7∗=ϕ7′ϕ2′3=ϕ7ϕ23.


## 3.2. Zernike Moment

Zernike moment is one kind of special complex moments, which are based on orthogonal functions called Zernike polynomials. Compared with HU moment, the computation of Zernike moments is more complex, but they have greater advantages in terms of their expressive power and noise sensitivity. Zernike moments have the characteristics of rotation invariance and have been widely used in the field of target recognition.


### 3.2.1. Definition of Zernike Moments

Zernike [21] presents a set of orthogonal polynomials Vnmρ,θ, which are orthogonal in the unit circle ρ≤1. The polynomial function in polar coordinates is obtained as follows:(13)Vnmρ,θ=Rnmρexpimθ.Under the constraints,n=0,1,2… and m=0,±1,±2,....., m≤n, where n−m is even number, ρ is vector from origin point to pixel point x,y, θ is the anticlockwise angle between the vector ρ and the x-axis, and Rnmρ is real-valued radial polynomial.(14)Rnmρ=∑s=0n−m/2−1s,n−s!ρn−2ss!n−2s+m/2!n−2s−m/2!.Radial polynomialsRnmρ satisfy the following relationship:(15)∫01RnlrRmlrρdρ=12n+1δnm,where δnm is the Kronecker delta.The orthogonality conditions of Zernike polynomials are as follows:(16)∫02π∫01Vnl∗rVmkρ,θρdρdθ=12n+1δnmδlk,where ∗ denotes the complex conjugate. The following formula can be obtained from formula (16): (17)Vnm∗ρ,θ=Vn,−mρ,θ.Some important properties ofRnmρ can be obtained from the following formula: (18)Rnm1=1,Rnnr=ρn,R00ρ=1.Ann order Zernike polynomial with a repetition rate of M is defined according to the following formula: (19)Anm=n+1Π∬fρ2≤1ρ,θVnm∗ρ,θρρdθ,ρ2≤1.For a digital image, the integral is replaced by a summation, namely,(20)Anm=n+1π∑x∑yfx,yVnm∗ρ,θdxdy,x2+y2≤1.To calculate the Zernike moments of an axis shaft orbit, it is necessary to move the center of the axis trajectory image to the coordinate circle and map the pixel points of the trajectory into the unit circle. Through the formula, it can be seen thatVnm∗ρ,θ can extract the features of images, and the high-frequency characteristics are obtained by Vnm∗ρ,θ for larger n values. The low-frequency characteristics are obtained by Vnm∗ρ,θ for smaller n values. Zernike moments can be used to construct high moments. In theory, higher moments can contain more image information, so pattern recognition will be better.


### 3.2.2. Rotation of Zernike Moments

When the rotation angle of an image isa, the expression of its Zernike moments is as follows: (21)Anm′=n+1Π∬fρ2≤1ρ,θ1Rnmρexp−jmθ1−αρdρdθ=Anmexp−jmα.From the upper formula, it is shown that the Zernike moments’ phases change, but the modulus remains unchanged. So,Anm can be regarded as the target rotation invariance feature.


## 3.2.1. Definition of Zernike Moments

Zernike [21] presents a set of orthogonal polynomials Vnmρ,θ, which are orthogonal in the unit circle ρ≤1. The polynomial function in polar coordinates is obtained as follows:(13)Vnmρ,θ=Rnmρexpimθ.Under the constraints,n=0,1,2… and m=0,±1,±2,....., m≤n, where n−m is even number, ρ is vector from origin point to pixel point x,y, θ is the anticlockwise angle between the vector ρ and the x-axis, and Rnmρ is real-valued radial polynomial.(14)Rnmρ=∑s=0n−m/2−1s,n−s!ρn−2ss!n−2s+m/2!n−2s−m/2!.Radial polynomialsRnmρ satisfy the following relationship:(15)∫01RnlrRmlrρdρ=12n+1δnm,where δnm is the Kronecker delta.The orthogonality conditions of Zernike polynomials are as follows:(16)∫02π∫01Vnl∗rVmkρ,θρdρdθ=12n+1δnmδlk,where ∗ denotes the complex conjugate. The following formula can be obtained from formula (16): (17)Vnm∗ρ,θ=Vn,−mρ,θ.Some important properties ofRnmρ can be obtained from the following formula: (18)Rnm1=1,Rnnr=ρn,R00ρ=1.Ann order Zernike polynomial with a repetition rate of M is defined according to the following formula: (19)Anm=n+1Π∬fρ2≤1ρ,θVnm∗ρ,θρρdθ,ρ2≤1.For a digital image, the integral is replaced by a summation, namely,(20)Anm=n+1π∑x∑yfx,yVnm∗ρ,θdxdy,x2+y2≤1.To calculate the Zernike moments of an axis shaft orbit, it is necessary to move the center of the axis trajectory image to the coordinate circle and map the pixel points of the trajectory into the unit circle. Through the formula, it can be seen thatVnm∗ρ,θ can extract the features of images, and the high-frequency characteristics are obtained by Vnm∗ρ,θ for larger n values. The low-frequency characteristics are obtained by Vnm∗ρ,θ for smaller n values. Zernike moments can be used to construct high moments. In theory, higher moments can contain more image information, so pattern recognition will be better.


## 3.2.2. Rotation of Zernike Moments

When the rotation angle of an image isa, the expression of its Zernike moments is as follows: (21)Anm′=n+1Π∬fρ2≤1ρ,θ1Rnmρexp−jmθ1−αρdρdθ=Anmexp−jmα.From the upper formula, it is shown that the Zernike moments’ phases change, but the modulus remains unchanged. So,Anm can be regarded as the target rotation invariance feature.


## 3.3. Rough Set Neural Network

Rough set theory (RS theory) [22] was proposed by Pawlak, a mathematician of Poland in 1982. It is an effective method for dealing with uncertain, inconsistent, and imprecise problems. It can discover hidden knowledge from uncertainty and reveal its potential law. It defines knowledge from a new point of view, regards knowledge as a division of the universe, and holds that knowledge is granular. In recent years, the research on rough set theory in the field of artificial intelligence has been a hot topic, and this theory has been widely used. Knowledge reduction [23–25] is one of the core concepts of rough set theory.


### 3.3.1. Rough Set Principle

In rough set theory, knowledge is expressed in the form of an information tableS=U,R,V,f, where U is a collection of objects, also called a domain, R=C∪D, CnD = O, C is called the conditional attribute set, D is called the decision attribute set, V=r∈RUVr is a collection of attribute values, Vr represents the attribute value range of an attribute r∈R, and f:U×R⟶V is an information function that specifies the attribute value of each object x in U.The simplification of a decision table is meant to simplify the conditional attributes in the decision table. The simplified decision table also has the function of simplifying the decision table before, and the simplified decision table has less conditional attributes. The rough set is used to describe the knowledge, and the reduction of redundant attributes can be used to simplify the knowledge. Finally, a complex decision table is reduced to the minimum conditional attribute decision table with no redundant attributes and the correct classification is ensured.


### 3.3.2. Rough Set Neural Network (RS-BP Hybrid Model)

The RS-BP hybrid model is a hybrid model combining artificial neural network and rough set theory. The training time is too long when the neural network and the sample are both large. This inherent disadvantage is one of the main factors which restricts the further practicability of the neural network. Although various algorithms for improving training speed have emerged, the problem is far from being completely solved. Simplifying the training sample set and eliminating redundant data are another way to improve training speed. The rough set is used to simplify the neural network in order to train the sample set, and the redundant data is eliminated under the premise of retaining important information. Simulation results show that the training speed is 4.77 times better, and good results are achieved [26].The characteristic parameters of the vibration are extracted through the condition monitoring system and a field test in the fault diagnosis process of the hydroelectric generating set. In order to describe the vibration mode of the hydrogenerator set as much as possible, more units are set up to extract the characteristic quantities of these units, some of which are related and some of which are independent. These feature parameters are often incomplete and redundant, which makes the modeling of subsequent fault diagnosis complex. The rough set theory can be used to reduce the knowledge of the training samples of neural network and get fewer attribute values while ensuring the goal of not missing important information.


## 3.3.1. Rough Set Principle

In rough set theory, knowledge is expressed in the form of an information tableS=U,R,V,f, where U is a collection of objects, also called a domain, R=C∪D, CnD = O, C is called the conditional attribute set, D is called the decision attribute set, V=r∈RUVr is a collection of attribute values, Vr represents the attribute value range of an attribute r∈R, and f:U×R⟶V is an information function that specifies the attribute value of each object x in U.The simplification of a decision table is meant to simplify the conditional attributes in the decision table. The simplified decision table also has the function of simplifying the decision table before, and the simplified decision table has less conditional attributes. The rough set is used to describe the knowledge, and the reduction of redundant attributes can be used to simplify the knowledge. Finally, a complex decision table is reduced to the minimum conditional attribute decision table with no redundant attributes and the correct classification is ensured.


## 3.3.2. Rough Set Neural Network (RS-BP Hybrid Model)

The RS-BP hybrid model is a hybrid model combining artificial neural network and rough set theory. The training time is too long when the neural network and the sample are both large. This inherent disadvantage is one of the main factors which restricts the further practicability of the neural network. Although various algorithms for improving training speed have emerged, the problem is far from being completely solved. Simplifying the training sample set and eliminating redundant data are another way to improve training speed. The rough set is used to simplify the neural network in order to train the sample set, and the redundant data is eliminated under the premise of retaining important information. Simulation results show that the training speed is 4.77 times better, and good results are achieved [26].The characteristic parameters of the vibration are extracted through the condition monitoring system and a field test in the fault diagnosis process of the hydroelectric generating set. In order to describe the vibration mode of the hydrogenerator set as much as possible, more units are set up to extract the characteristic quantities of these units, some of which are related and some of which are independent. These feature parameters are often incomplete and redundant, which makes the modeling of subsequent fault diagnosis complex. The rough set theory can be used to reduce the knowledge of the training samples of neural network and get fewer attribute values while ensuring the goal of not missing important information.


## 4. Simulation and Pattern Recognition of the Shaft Orbit

Using Matlab to simulate the shaft orbits of the generator set when the fault occurs, formula (22) can be obtained:(22)xt=A1sinωt+α1+A2sin2ωt+α2,yt=B1cosωt+β1+B2cos2ωt+β2,where ω is the angular frequency; A1,A2,α,1α2, A1,A2,α,1α2 are the amplitudes and initial phases of the first and second frequency components of xt and yt. By changing the 8 parameters A1,A2,α,1α2,B1,B2,β1,β2, the shaft orbits of the hydroelectric unit are obtained, as shown in Figure 3. The 30 simulated shaft orbits on the left are training samples, and the 10 orbits on the right are test samples.Figure 3
Simulation of axis orbits (ellipse, banana, inner 8, foreign 8, petal). (a) Training sample. (b) Testing sample.
(a)(b)In this paper, Hu moments and Zernike moments are used to extract the features of shaft orbits. 7 features of the Hu moments are extracted and there are 6 improved Hu moments. Zernike moments can be used to construct high moments, and higher moments contain more image information.In this paper, 49 features of Zernike moments can be obtained by choosing 0∼12 order Zernike moments. The Hu moments, improved Hu moments, and the data of the Zernike moments, after the rough set knowledge reduction is extracted from Figure3, are shown in Tables 2–4. Tables 3 and 4 show the average values of the Hu moments and the improved Hu moments of typical shaft orbits.Table 2
Hu moments of the simulated axis orbit.
No.Hu1Hu2Hu3Hu4Hu5Hu6Hu710.1870.3634.3094.6029.0674.5109.73520.2500.2252.9643.0866.1112.9898.30730.1160.1773.9315.0389.6105.1579.76440.1800.1042.8934.1017.8324.1047.68950.2701.4423.1622.9846.3933.9286.10860.2260.0742.1712.1954.4232.2364.74170.2310.2713.4613.2806.8643.1896.75180.2250.4302.7962.8565.6832.6477.08990.2790.1900.2682.9574.4123.1604.500100.0561.3910.3552.1133.4872.8363.509110.0780.5140.3861.5012.4511.8363.219120.1390.0700.2920.9511.5741.0232.577130.1560.1030.4301.1001.8941.2192.318140.2790.1900.2682.9574.4123.1604.500150.0561.3910.3552.1133.4872.8363.509160.0561.3910.3552.1133.4872.8363.509170.1621.6031.0990.8741.9452.7492.104180.1161.1921.2661.0853.5241.9362.262190.1500.8471.2341.0072.1571.5192.571200.2110.4620.9330.4401.1710.6731.488210.1660.2641.1320.5681.4250.7122.159220.1610.0640.7060.5681.2100.6002.048230.1660.2641.1320.5681.4250.7122.159240.1610.0640.7060.5681.2100.6002.048250.2510.3032.3003.8808.6913.7446.970260.2580.2590.8102.4954.6622.3664.169270.2670.2300.3842.0124.1851.9093.212280.2880.0510.1711.4032.1521.5912.188290.2460.0790.0232.9654.5073.0934.809300.2530.0290.0653.1394.7993.2344.856310.2620.0310.1542.9055.4003.1554.282320.2680.1130.2143.0154.4163.0875.631330.1310.0550.5101.1252.1381.4852.057340.1300.1120.1981.3122.2051.4012.232350.0950.4090.0331.3172.0431.9252.205360.0990.6240.1182.0173.1103.0743.124370.1310.0550.5101.1252.1381.4852.057380.1300.1120.1981.3122.2051.4012.232390.0950.4090.0331.3172.0431.9252.205400.1080.7030.1532.2973.3922.9683.863Table 3
M moments of the simulated axis orbit.
No.M1M3M4M5M6M710.019.7110.2910.165.2410.8220.276.606.856.793.448.9830.068.3910.6110.145.5110.3040.266.108.518.144.318.0051.982.001.642.071.041.7860.384.564.614.642.384.9670.197.747.377.683.737.5680.026.887.006.973.518.3890.751.115.343.842.783.93101.503.460.050.690.050.67110.670.771.460.910.811.68120.350.371.691.370.882.37130.211.172.512.201.422.63140.751.115.343.842.783.93151.503.460.050.690.050.67161.503.460.050.690.050.67171.932.613.062.860.462.70181.421.051.410.050.451.32191.150.070.530.380.170.03200.880.480.510.210.250.10210.601.470.340.630.181.37220.391.220.941.020.471.86230.601.470.340.630.181.37240.391.220.941.020.471.86250.205.518.679.604.357.88260.262.405.775.442.884.95270.301.464.714.882.373.90280.530.192.962.311.692.34290.410.286.174.743.255.04300.480.046.364.883.294.94310.560.405.725.313.094.19320.650.775.694.082.865.29330.320.852.091.971.371.89340.370.062.291.871.181.90350.601.291.410.821.110.98360.822.112.161.241.831.25370.320.852.091.971.371.89380.370.062.291.871.181.90390.601.291.410.821.110.98400.922.412.491.281.561.75Table 4
Zernike moments of the simulated axis orbit.
No.A5.3A6.6A6.4A6.2A8.8A8.4A8.2A9.7A10.8A12.10A12.8A12.6A12.4155.06.039.635.31.246.933.62.036.659.126.217.85.6224.649.322.414.29.25.05.65.612.016.223.03.815.2314.25.25.225.81.48.228.32.931.06.08.027.611.3412.611.83.225.60.810.423.51.37.71.314.322.02.1518.319.81.515.11.316.114.03.610.05.221.613.410.3617.528.916.25.30.67.89.10.73.917.714.83.314.7732.520.42.24.41.31.614.90.58.99.89.812.623.4830.14.95.021.81.03.728.90.511.02.98.721.85.5919.210.16.610.75.09.85.89.44.512.46.27.211.21027.19.73.829.513.91.328.04.811.714.05.117.515.11124.333.319.28.86.614.79.37.43.219.724.66.520.71214.425.326.17.76.613.15.75.412.129.36.010.015.61328.07.56.617.46.16.922.26.913.28.67.112.223.71432.07.413.517.740.241.84.07.64.17.420.311.612.41526.215.47.521.113.10.927.93.910.46.57.610.19.01627.19.73.829.513.91.328.04.811.714.05.117.515.1171.910.716.016.41.94.718.617.06.323.913.710.58.21815.26.825.818.613.820.326.426.32.620.619.725.86.2196.52.338.26.37.716.712.511.612.027.614.15.37.9207.814.714.15.39.110.32.20.90.68.61.47.412.0219.614.715.25.66.84.64.011.113.618.88.21.57.22217.118.016.63.97.19.64.38.91.916.64.62.30.72318.85.320.920.725.110.014.228.314.617.731.629.920.42416.47.225.05.82.420.214.516.15.94.54.727.726.3252.838.37.44.813.817.20.90.55.212.317.116.417.12612.035.45.713.84.54.318.42.411.122.322.21.013.22724.68.87.416.33.66.721.62.95.72.115.616.614.82818.026.27.47.613.618.38.10.68.37.69.03.91.72916.316.25.211.412.134.713.24.85.02.823.820.321.83026.114.74.54.23.45.713.41.76.85.414.214.517.83127.529.110.42.08.814.66.512.511.412.45.21.27.83219.413.99.010.811.86.68.011.112.923.86.95.213.43319.418.814.75.94.67.46.23.51.720.411.32.58.03411.023.820.81.64.83.110.19.412.316.15.210.71.9354.320.725.64.98.95.27.117.113.618.120.55.213.63610.69.827.33.91.48.211.620.111.917.78.24.50.93719.418.814.75.94.67.46.23.51.720.411.32.58.0385.911.519.24.213.412.71.25.28.27.426.012.91.6395.811.611.93.46.519.09.87.35.821.824.51.52.3406.713.225.110.90.57.09.917.07.323.68.04.44.5The Hu moments and the Zernike moments are chosen as input variables, and the type of shaft orbit is used as the output in order to establish a neural network. The shapes of the shaft orbits can be identified by training the samples. For 49 Zernike moments, the input vector node is 49, and the network structure is relatively complex, which will cause the training time to become longer. The rough set feature selection algorithm is used to reduce the input and remove redundant information. Finally, 13 Zernike moments’ features are selected: A5.3, A6.6, A6.4, A6.2, A8.8, A8.4, A8.2, A9.7, A10.8, A12.10, A12.8, A12.6, and A12.4.In order to fully reflect the influence of different feature quantities on pattern recognition, different feature input parameters are used for the pattern recognition of shaft orbits. The first method uses 7 Hu moments as input variables. The second method uses 6 improved Hu moments as input variables. The third method takes the 49 moments as input variables. The fourth method uses the 13 moments eigenvalues reserved by the rough set feature selection algorithm as input variables. The neural network models of the four methods all adopt a three-layer BP neural network structure. The number of neurons in each input layer is as follows: 6, 7, 49, and 13. The hidden layer uses 10 neurons, and the output layer is 5 neurons, namely, five shaft orbits. The training step = 20000, the descent gradient = 0.1, and the error target = 0.001. The flow chart of the fourth method is shown in Figure4. Input samples and test samples are shown in Figure 4. The data for the sample training are shown in Tables 1, 5, and 6 . The results of training and recognition for the four neural networks are shown in Table 7.Figure 4
Flowchart of the rough set neural network model.Table 5
Average values of Hu moments.
Axis orbitϕ¯1ϕ¯2ϕ¯3ϕ¯4ϕ¯5ϕ¯6ϕ¯7Ellipse0.2110.3863.2113.5186.9983.5957.523Banana0.1370.6550.3391.9763.1502.3633.455Inner 80.1620.5951.0260.7101.7581.1882.105Outer “8”0.2620.1370.5152.7274.8522.7724.514Petal0.1150.3100.2191.4782.4091.9582.497Table 6
Average values of improved Hu moments.
Axis orbitM¯1∗M¯3∗M¯4∗M¯5∗M¯6∗M¯7∗Ellipse0.3966.4977.1117.0733.6457.599Banana0.9041.8642.0631.7781.1052.066Inner 80.9191.1991.0090.8520.3301.325Outer “8”0.4221.3825.7565.1542.9744.817Petal0.5401.1172.0261.4801.3381.567Table 7
Statistics of training and recognition results of different recognition methods.
Serial numberInputTime stepErrorRecognition rate (%)Model 1Hu moments (7 variables)200000.0010480Model 2Improved Hu moments (6 variables)200000.0013180Model 3Zernike moments (49 variables)131530.00190Model 4Zernike moments (13 variables)93580.00190It can be seen from Table7 that Zernike moments are better than Hu moments for pattern recognition, and the former have the advantages of a small calculation error and a high recognition rate. Calculating the effects of Model 3 and Model 4, both have the same recognition accuracies and convergence rates, but the latter significantly reduces the calculation step. We can see that using the rough set method to advance the information reduction pretreatment is effective, and it can increase the convergence speed of the effect. The combination of Zernike moments, the rough set, and three neural networks can efficiently and accurately identify the shaft orbits, while the neural network with the Zernike moments and the rough set preprocessing can achieve good results in the identification of shaft orbits. Zernike moments can extract the features of shaft orbits, and the rough set method is used to solve the redundancy problem, while the neural network solves the problem of sensitivity to noise. Therefore, by integrating the advantages of Zernike moments, rough sets, and neural networks, the efficiency and accuracy of pattern recognition can be greatly improved.


## 5. Conclusions

In this paper, the feature extraction and pattern recognition of the shaft orbit of a hydroelectric generating unit are studied. The main content and conclusions are as follows:(1)
In terms of shaft orbits feature extraction and pattern recognition, the Zernike moment is better than the Hu moment and it has the advantages of a small calculation error and a high recognition rate.(2)
A rough set neural network (RS-BP hybrid model) of shaft orbit recognition is established, which uses just 13 moment eigenvalues reserved by the rough set feature selection algorithm as input variables, has the same calculation error and recognition rate, and reduces the calculation time step. The simulation of the recognition of shaft orbits shows that the hybrid model has achieved good results in the identification of shaft orbits.



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*Source: 6680640-2021-04-29.xml*