Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 702760, 11 pages

http://dx.doi.org/10.1155/2015/702760

## Mechanical Fault Diagnosis Using Color Image Recognition of Vibration Spectrogram Based on Quaternion Invariable Moment

College of Electrical Engineering, Nantong University, Nantong 226019, China

Received 4 May 2015; Revised 27 July 2015; Accepted 30 July 2015

Academic Editor: Yang Tang

Copyright © 2015 Liang Hua et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Automatic extraction of time-frequency spectral image of mechanical faults can be achieved and faults can be identified consequently when rotating machinery spectral image processing technology is applied to fault diagnosis, which is an advantage. Acquired mechanical vibration signals can be converted into color time-frequency spectrum images by the processing of pseudo Wigner-Ville distribution. Then a feature extraction method based on quaternion invariant moment was proposed, combining image processing technology and multiweight neural network technology. The paper adopted quaternion invariant moment feature extraction method and gray level-gradient cooccurrence matrix feature extraction method and combined them with geometric learning algorithm and probabilistic neural network algorithm, respectively, and compared the recognition rates of rolling bearing faults. The experimental results show that the recognition rates of quaternion invariant moment are higher than gray level-gradient cooccurrence matrix in the same recognition method. The recognition rates of geometric learning algorithm are higher than probabilistic neural network algorithm in the same feature extraction method. So the method based on quaternion invariant moment geometric learning and multiweight neural network is superior. What is more, this algorithm has preferable generalization performance under the condition of fewer samples, and it has practical value and acceptation on the field of fault diagnosis for rotating machinery as well.

#### 1. Introduction

Quality of products and safety of entire production system are affected by working condition of rotary machineries directly. The fault diagnosis is performed and the working condition is monitored so that faults can be detected as early as possible, which is a key issue in the field of engineering. Moreover, signal processing, extractions of the fault features, and pattern recognition play a key role in the process of realization. At present, working state analysis based on vibration signal of machine parts is a common research method of the rotary machinery diagnosis and monitoring. There are many deficiencies of traditional methods of fault diagnosis. For example, time-domain analysis supplies limited information, so it can be only applied to simple machinery. Frequency-domain analysis is based on an assumption of stationary signals, and it analyzes the frequency-domain information of signals but it neglects time-domain characteristics of signals. However, actual signals of mechanical faults are usually nonstationary. In other words, the frequency of signals always changes with time, so time-domain and frequency-domain analysis are no longer suitable to handle complex fault diagnosis. Time-frequency analysis method is an effective method to cope with nonstationary and nonlinear signals. It includes short time Fourier transform [1], Hilbert time-frequency spectrum analysis [2], Wigner-Ville distribution [3], Wavelet analysis [4], HHT time-frequency analysis, and high order spectral analysis [5, 6]. Time-frequency methods described above can solve problems of fault signal processing and features extraction successfully, but it is hard to analyze potential weak fault feature as vibration spectrum images are too subjective and the observation ability of operators varies. Only to use the combination of image features extraction and model recognition can avoid the subjective mistakes. Furthermore, the intelligent mechanical fault diagnosis is demanded urgently by the market, and researches to new feature extractions and recognition methods become a common trend.

With the development of information theories and machine-vision, fault diagnosis technologies based on image processing has improved a lot. The focus on intelligent image processing is how to exact information to characterize the image features from images. The methods of features extraction which are related to the vibration spectral images usually can be divided into three categories: spectral feature, texture feature, and shape feature. The spectral feature recognition is on the basis of grey level histogram with single spectral band or color histogram with multispectral bands. It is simple, intuitive, and robustness. Zhou et al. [7] and others identified the voice signals successfully according to combinations of spectral features and prosodic features. Combined with spectroscopy, structural, and semantic features, the classification for high-resolution remote sensing images is performed in the literature [8], and it resulted significantly with the support vector machine (SVM). However, different objects may have the same spectral feature, so these algorithms cannot describe specific object accurately. The texture features describe the spatial changes situation of the color, the gradation or the fine structure, and shape of the images [9]. Texture feature extraction proposed in literature [10] considered global nature of images, so it can describe regional features steadily. In addition, it has favorable rotation invariance and anti-interference capability. Haralick et al. [11] firstly proposed fourteen feature parameters to show the gray information of the images based on the texture features. The literature [12] effectively realized the image classification by utilizing the wavelet transform to extract the texture feature of the SAR images, selecting wavelet energy feature kurtosis which constitutes the feature vector for the SVM classifier. In literature [13], three-dimensional discrete wavelet texture was utilized to realize the image classification and achieved better result than the spectral-spatial classification method widely used. The disadvantages are that the more the changes of image pixel resolution, the more the deviation of texture feature. Shape features are not only an inherent feature of the rigid body but also an important feature of human object recognition. Shape features can be expressed by form factors and profile factors. Literature [14] applied a feature extraction based on anomaly detection and the local shape features to the high resolution video images of the vessel detection. In addition, some researchers have proposed some image shape feature detection operators. Gool et al. [15] constructed a line moment feature, which is not effected by translation and rotation; Belongie et al. [16] proposed SC (Shape Context) features for detecting edge points; Fergus et al. [17] used the Canny algorithm to select the image edge; Kadir et al. [18] used to calculate the color histogram of the local maximum entropy, and circular the feature extraction area. Shape feature extraction is researched on the necessary part of the image and the integrality of the target is grasped well. The disadvantage is that when the image’s target is deformed, the stability of the characterization would decline [19]. However, there is no target deformation on the mechanical vibration’s time-frequency images that we researched on; consequently, it is appropriate for the field of mechanical fault diagnosis. The “,” “,” and “” channels of color image can be processed independently as three gray images. However, the relationships between trichannels are ignored, and the processed image will get distortion. Quaternion is a branch of non-Euclidean geometry and is firstly put forward by mathematician Hamilton. In literature [20], some researchers raised up a color images processing method based on the quaternion. Compared with traditional methods, it can sufficiently describe the color relations of the images. Literature [21] rose up a quaternion geometric moment calculation method of the medical images to realize the registration of the skull images.

Images recognition technology is one of the hot spots in the area of artificial intelligence, which has the intuitive and extensive statistical significances. When this method is applied to the faults diagnosis, not only human experience evaluation can be liberated, but also accurate and timely fault warning can be achieved to prevent faults. Time-frequency analysis of the vibration signals can get 2-dimensional or 3-dimensional time-frequency distribution images which reflect the features, and image recognition technology can be applied to the color time-frequency images [22]. Image recognition methods are usually divided into statistical method [23], syntactic recognition method [24], neural networks method [25], template matching method [26], and geometrical transformation method [27]. Furthermore, multiagent system is used in image recognition widely [28–30]. Multiweight neural network is utilized to identify the faults in the paper. This method especially has unique advantages on solving the pattern recognition problems for the few samples, and it successfully solves a lot of few samples, nonlinearity, and high dimensional and other problems. It has the practical significance on the solutions of faults diagnosis.

This paper utilized image recognition technology to process the mechanical vibration spectral images. The main idea is to analyze mechanical fault signals by using pseudo Wigner-Ville distribution method and explore the feature information of different mechanical faults in time-frequency domain. In order to prevent the color image information losing and ensure the integrality and relevance of the image’s triphosphor, the images’ invariant moment in quaternion domain has been analyzed; at last quaternion invariable moment features were sent to multiweights neural network for training and testing, and recognition results were got by using unknown samples’ test. The experimental result proved that modeling for color images by using quaternion invariable moment created a new method for color images processing, and this method has some practical values as well.

#### 2. Generation of Mechanical Vibration Time-Frequency Spectral Image

Methodology of fault diagnosis based on mechanical vibration time-frequency color image recognition mainly consists of three steps: (1) generating of vibration spectral color image reflecting mechanical vibration fault’s feature; (2) extracting features of vibration spectral image accurately; (3) recognizing the extracted vibration spectral feature effectively. Wigner-Ville distribution can accurately define time-frequency structure of signals [31]. For a certain signal denoted here as , the Wigner-Ville distribution can be defined asThere is interference of cross terms in the transformation expressed as (1), and the cross terms will not disappear with the changes of time-frequency distance. In order to eliminate the interference of these cross terms, windowing processing is applied to (1), and a novel transformation, denoted here as pseudo Wigner-Ville distribution, can consequently be obtained and expressed as follows:Wherein, of (2) should meet some requirements listed as follows [5]:(I)Symmetry is .(II)Normalization is .(III)The length of window is limited, which can be set as , and then for any , is right.(IV)Fourier Transform for is a low-pass function.

Here, is selected as rectangular window. Because actual collected signal is discrete, should be discredited. Considering the length of denoted as is limited, that is, for any , it has , and then (2) can be adapted asEquation (3) is discredited in time domain; that is, , , where represents sampling frequency.

Then (3) can be adapted asThe length of discrete window is .

Equation (4) is discretized in frequency domain; that is, , , and ; then (4) can be adapted asEquation (5) is a formulation of discrete pseudo Wigner-Ville distribution. Pseudo Wigner-Ville distribution inherits good performance in time-frequency concentration of Wigner-Ville distribution. In addition, it can suppress cross-terms interference and has higher time-frequency resolution. So it can reflect time-frequency feature information of vibration signals of rotation machinery.

#### 3. Vibration Spectrum Feature Extraction

From the perspective of image pattern recognition analysis, it is a new way to extract effective vibration spectrum characteristics and combine with intelligent recognition method. To solve the key technology of image features extract and reflect accurately in the fault diagnosis of vibration spectra, this paper used quaternion invariant moment and gray level-gradient cooccurrence matrix for image feature extraction.

##### 3.1. Quaternion Geometric Moment of Image

When different faults happen, their vibration signals have different vibration information. In terms of the same fault, time-frequency distributions of the vibration signal in different cycle period are similar. However, under different fault conditions, time-frequency distribution of vibration varies significantly, especially in the features extraction method of quaternion invariable moment. Taking advantage of this nature, effective features can be extracted from the time-frequency spectral images of rotation machinery.

Quaternion theory [32] is an extension of plurality basic theory, which consists of one-dimensional real part together with three-dimensional imaginary part, and can be expressed aswhere , and are real numbers and , and are quaternion’s imaginary part element. If real part , is a pure imaginary quaternion and meets Hamilton rules; that is,Conjugate of quaternion is named as and can be expressed asThe mold of quaternion is denoted as and can be expressed asColor images can commonly be described as combinations of three primary colors, that is, Red, Green, and Blue (*RGB*) [33]. In the present work, three primary values of color image pixel values are represented by pure imaginary quaternion; therefore, any pixel denoted here as can be expressed aswhere , , and represent the channels’ values of , , and of image coordinates ; therefore, a color image can be expressed as a purely imaginary quaternion matrix. Compared with traditional methods, quaternion model can better reflect the integrity and relevance of the color image.

Regarding a time-frequency distribution of certain vibration for rotation machinery, its essence is a two-dimensional color image, expressed here as . From the uniqueness theorem proposed by Hu in literature [34], it can be deduced that if a two-dimensional image is piecewise continuous and has nonzero value in limited area, then any order moment will exist. Set of quaternion moment can be uniquely identified by ; otherwise, two-dimensional image can also be determined by set of quaternion moment ; that is, the set of quaternion moment that we obtained can describe information of this time-frequency image partly. Moreover, based on Hu, and combining quaternion as well, taking full advantage of color information of images, quaternion invariant moment reflects the integrity and relevance of the image.

Given a two-dimensional color image , then the corresponding two-dimensional geometric quaternion moment denoted here as can be indicated aswhere indicates the order of , and it exists by , , , and .

The image quaternion geometric moment is an expansion of grayscale images geometric moment concept [35]. is a quaternion vector and contains images’ color information. It not only describes global features of the image shape but also provides plenty of geometric feature information of images. Each order moment has different physical meanings.

###### 3.1.1. Quaternion Zeroth-Order Moment

In (11), if , is quaternion zeroth-order moment and represents color sum of entire image. We can consider it as the “mass” of the image; moreover, this “mass” takes information of both size and direction.

###### 3.1.2. Quaternion First-Order Moment

, are first-order moment of the color image regarding -axis and -axis, and color moment center denoted here as can be expressed as where indicates module of the quaternion moment, which expresses color intensity, and coordinate is a physical centroid of the image.

###### 3.1.3. Quaternion Central Moment

The geometric moment calculated with respect to centroid of the image is called central moment and expressed aswhere is the centroid of the image, and , . The second-order moments are a measure of variance of the image intensity distribution about the origin. The central moments , represent the variance about the mean (centroid ). is the covariance measure of them.

In order to make with scale and rotation invariance [36], (13) should be transferred to In 1962, Hu firstly proposed invariant moment in literature. Seven invariant moments are also a nonlinear combination of second-order and third-order central moment given in (14). They have performance of invariance in transformation of translation and rotation and can be expressed as follows:where matrix is a matrix extracted from a characteristic matrix of matrix with respect to each time-frequency color image.

##### 3.2. Gray Level-Gradient Cooccurrence Matrix Feature Extraction

The element of gray level-gradient cooccurrence matrix is defined as gray value and grade value pixels between normalization gray image and normalization grade image. In the process of normalization gray level-gradient cooccurrence matrix, the sum of each element is expressed as follows:And ; the above formula can be expressed as follows: where is the element of gray level-gradient cooccurrence matrix, is gray value, and is grade value.

Exacted 15 feature statistics, small gradient advantage, large gradient advantage, gray distribution heterogeneous, gradient distribution heterogeneous, energy, the average of gray, the average of gradient, gray mean square error, gradient mean square error, correlation, gray entropy, gradient entropy, mixed entropy, inertia, and inverse difference moment [37], are recorded as , and , respectively, and expressed in Table 1.