Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 759035, 12 pages

http://dx.doi.org/10.1155/2015/759035

## Defect Detection and Localization of Nonlinear System Based on Particle Filter with an Adaptive Parametric Model

^{1}School of Mechanical Engineering, Jiangnan University, Wuxi 214122, China^{2}Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Wuxi 214122, China

Received 31 July 2015; Accepted 17 November 2015

Academic Editor: Xinggang Yan

Copyright © 2015 Jingjing Wu 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

A robust particle filter (PF) and its application to fault/defect detection of nonlinear system are investigated in this paper. First, an adaptive parametric model is exploited as the observation model for a nonlinear system. Second, by incorporating the parametric model, particle filter is employed to estimate more accurate hidden states for the nonlinear stochastic system. Third, by formulating the problem of defect detection within the hypothesis testing framework, the statistical properties of the proposed testing are established. Finally, experimental results demonstrate the effectiveness and robustness of the proposed detector on real defect detection and localization in images.

#### 1. Introduction

Fault detection, as a subfield of control engineering, is to monitor a stochastic system and identify when a fault occurred and the information of fault such as the type and its location. Fault detection approaches play a fundamental role to improve manufacturing process and application process in a variety of industry contexts [1–7], for example, quality control in process monitoring [1–3], product manufacturing [4], medium restoration [5], and facilities maintenance [8]. Advantages of low costs, high automation, and high quality of defect detection techniques resulted in growing interests in recent years. Fault/defect detection for quality inspection includes the use of cameras, eddy current, ultrasonic, X-ray sensors and other sensors, which offer the measurements to be analyzed for extraction of the information about the fault. With the high demands for quality control using cameras in industry, visual inspection systems attract more attention in recent years. In a similar vein to fault detection of linear or nonlinear systems, defect detection in visual inspection systems can be taken as monitoring the variation of the measurements from cameras. In this paper, we devote ourselves to solving the visual defect detection problems from an unusual perspective of fault detection.

Currently, two rough categories of approaches are available for defect detection, that is, signal processing-based and model-based methods. The signal processing algorithms perform mathematical and statistical analysis tools on the measurements to extract faults [8–13]. The developed methods like Gabor filter and wavelet transform have been proven to be effective solutions to locating the defects with less prior information in nonlinear systems. In recent years, defect detectors on the basis of Gabor filter [10] and wavelet transform [11] are efficient to find defects for web fabrics with stable repetition of textures, since it is easier to find defects in frequency domain. Techniques of the signal processing also cover a number of computational intelligence approaches like fuzzy logic and neural network [13], which provide some effective solutions for fault detection in various industrial problems. However, the data-driven methods suffer from large computational load and storage space.

The second category is an evolving collection of methodologies aiming to exploit the models of the system in temporal and spatial space to decide the occurrence of fault/detection [3, 5–7, 14]. Taking advantages of temporal or spatial models with the prior information, the model-based method can achieve robustness even in a scene with heavy amount of noises. This paper focuses on techniques of the defect detection of this category.

Various model-based approaches for defect/fault detection have been proposed in the past decades [3, 5, 6, 15]. Since the system state completely represents the system’s internal (hidden) status and condition containing the fault information, the intrinsic problem in the model-based methodologies is the estimation of the system states from measurements of sensors. One of the most popular components of model-based algorithms is the Bayesian method [16], which yields the posterior distribution of the system state containing the hidden state such as the occurrence and the fault type. Bayesian filter and its variations have been effective solutions to visual inspection of surface defects of materials and fault detection in process control [3], for example, chemical process. For the linear Gaussian system, Kalman filter [15–17] has been exploited to detect surface defects in visual inspection systems. Particle filter, as an implementation algorithm of Bayesian filter for nonlinear and non-Gaussian systems, has been used for fault detection of chemical process [3, 18]. However, state estimation and fault detection still remain a challenge due to the absence of suitable models for the actual application context. The linear Gaussian models in [15] cannot meet the requirements of the defect detection and localization in the actual visual inspection systems, due to the complicated intensity distributions of background pixels.

In this paper, a new intelligent defect detection algorithm without prior knowledge based on PF is presented. We formulate the visual defect detection problem to estimate the hidden states using PF and decide the occurrence or locations of defects in 2D images by chi-square test. In our method, intensity of the 2D image (the same vein as the control system) along each row or column line is assumed as a time series or a random process . To detect faults/defects in the image (the system), reasonable state and measurement models are proposed firstly for particle filter. Then, with the proposed models, particle filters along rows and columns are implemented to estimate states and the measurement innovations (residuals). Finally, abrupt changes in the measurement innovations are used to locate defects in the inspection images by chi-square test. Tests on the real database demonstrate the effectiveness of the proposed algorithm.

#### 2. Problem Formulation of Defect Detection

The defect detection in a visual inspection system is to identify the occurrence of the defect and its location from the measurements of camera sensors, that is, the digital image of the inspected object. A measurement in a digital image is the gray level or intensity of the pixel. Then the pixel intensity along each row or column line in an image of pixels is assumed as a time series or a random process or , where is one-dimensional coordinate of the relevant pixel on the scan line (i.e., the time step of the random process).

When the inspected target is free from defects, the intensity of pixels along each scan line takes on small variations. Therefore, the gray level along a scan line in a defect-free image can be defined as a white Gaussian state series or . Due to subtle intensity changes for defect-free areas, the state sequence can be modeled by the following discrete time linear Gaussian dynamical model; that is, where is the state in the time series, is the state transition model, and is the zero-mean white Gaussian noise sequences with variance . For defect inspection problem in this paper, the dynamical mode uses the random walk model with . Each state in a Gaussian series can also be written aswhere denotes a Gaussian distribution with expectation and variance .

However, due to the influence of defects, uneven illumination, and the geometrical structure of the inspected object, the actual measurements usually follow nonlinear model, which can be formulated by a transition function approximating the Markov transition relation between the state estimate of the defect-free and the actual measurement of a pixel (defect-free or defective). Therefore, the measurement model for an arbitrary pixel can be defined as where denotes the transition function from the state estimate to the measurement and is zero-mean white Gaussian noise sequences with variance .

In terms of the established models, Bayesian filter can be exploited to yield the state estimate of for each pixel along a scan line of an image by Bayes recursion below [17]:where is the transition density defined by (1) or (2) and is the likelihood function defined by (3). The posterior density includes all information of the state at time and the state estimate of the th pixel in a scan line can be obtained by maximum a posteriori (MAP) criterion.

Once the state estimate at time step (i.e., intensity of the th pixel along the scan line) is obtained, the difference between the predicted measurement and the actual measurement is defined as residual , which can be calculated byHere, can be derived by (1) and (3) with the formwhere is the predicted state using (1). It can be seen from (6) and (7) that a residual reveals the difference between the actual measurement of a pixel and the defect-free pixel, which can be used to decide the occurrence of a fault/defect. When the measurement is a Gaussian variable, the statistic is chi-square distributed. Then a chi-square test [17] will declare detection of a defect ifwhere is the level of confidence, indicates the degrees of freedom, and is the detection threshold.

#### 3. Particle Filter Based on Adaptive Parametric Model

##### 3.1. Adaptive Parametric Measurement Model

As explained in Section 2, it is important to warrant that the defect information is revealed by residual of the state and the measurement. To attain this purpose, the state model about the defect-free object based on random walk model is presented in (1) and (2). Furthermore, it is also necessary to design an accurate measurement model which can closely approximate the real measurement contaminated by noise, defects, and other disturbances. As shown in Figure 1, the measurements along a scan line in a complex visual inspection system follow multimodel distribution.