Textural Fabric Defect Detection using Adaptive Quantized Gray-level Co-occurrence Matrix and Support Vector Description Data

INTRODUCTION

Fabric defect detection is one of the most important procedures effecting the manufacturing efficiency and quality in fabric industry. So far, most factories accomplish this study by human vision which makes the quality of fabric devoid of consistency and reliability. It is found that even a highly trained inspector can only detect about 70% of the defects at a speed of 15-20 m min^-1 (Sari-Sarraf and Goddard, 1999). As the development of image processing technology, many methods base on texture analysis were proposed for defect detection. Several statistical methods using texture features, such as fractal dimension (Conci and Proenca, 1998; Bu et al., 2009), morphological feature (Mallick-Goswami and Datta, 2000; Chandra et al., 2010) and co-occurrence matrix (Latif-Amet et al., 2000; Lin, 2010) were proposed to discriminate defects from normal texture. Cohen et al. (1991) used the Gauss Markov Random Field (GMRF) to model the texture pattern of non-defective fabric image and the defect is detected by the rejection of the model using hypothesis testing theory. Because of the high degree of periodicity of the fabric texture, the spectral approaches are also used for fabric defect detection. Chan and Pang (2000) used Fourier transform for this task which turned out to be only suitable for global defect detection because of its poor local resolution in the frequency domain. Defect detection methods with multiresolution decomposition using a bank of Gabor filters were proposed by Bodnarova et al. (2002) and Kumar and Pang (2002). As Gabor filter banks will lead to redundant features at different scales, the orthogonal wavelet transform (Nadhim, 2006) are also used for texture characterization (Loum et al., 2007), classification (Raju et al., 2008) and fabric defect detection (Yang et al., 2002; Mingde and Zhigang, 2011).

The defect detection can be considered as a one-class classification problem. Ohanian and Dubes (1992) and Randen and Husoy (1999) provide some occasions that, in the viewpoint of texture classification, GLCM outperforms other features such as Gabor filters feature, fractal feature, MRF feature. However GLCM is still degraded for its high computational complexity. In this study, an adaptive quantization method is proposed base on the texture model of mixture of two Gauss distribution. The quantization method reduces the gray-level from 256 levels to several levels which greatly reduces the computation complexity of GLCM features. The SVDD classifier is used as a detector for defect detection. Several machine learning based texture classification methods have been proposed using neutral network (Kumar, 2003; Jianli and Baoqi, 2007; Chandra et al., 2010; Nagarajan and Balasubramanie, 2008; Mahi and Izabatene, 2011), Support Vector Machine (SVM) (Kumar and Shen, 2002; Chu et al., 2011) and SVDD (Bu et al., 2009, 2010). The neutral network and support vector based methods considered the defect detection as a two-class classification problem which required both defective and non-defective samples for training. However, the requirement of large quantities of defective samples is usually not desirable for online inspection in industrial occasions. Similar to SVM, the SVDD classifier is also a kernel function based classifier which does not suffer the problem of local minima. However, it is a one-class classifier which requires only the non-defective sample for training, thus it is adopted in our algorithm.

FABRIC TEXTURE MODEL

Figure 1a and b show two samples of non-defective texture of plain and twill fabrics, respectively. It can be see that the plain and twill texture are, respectively, made up of one and two gray tones which is also indicated in their histograms in Fig. 1c and d by solid lines. The fabric texture model is built on the histogram of gray-level image of fabric texture. The histogram of the plain fabric texture tends to obey the Gauss distribution and the histogram of the twill fabric texture tends to obey a mixture of two Gauss distributions. So, the Probability Density Function (PDF) of gray-level in the plain fabric image can be modeled as:

(1)

where, μ and σ are the mean and standard deviation of the Gauss distribution, respectively. And the PDF of gray-level in the twill fabric image can be modeled as:

(2)

where, P₁+P₂ = 1 and μ₁≤μ₂. P₁, μ₁, σ₁ and P₂, μ₂, σ₂ are the proportions, means and standard deviations of two Gaussians in the mixture, respectively.

Fig. 1(a-d):	Non-defective samples of (a) plain, (b) twill fabrics and (c, d) corresponding histogram and fitting results

In order to build a consistent model for both plain and twill fabrics, only Eq. 2 is used and Eq. 1 is considered as a particular case of Eq. 2 with P₁ = 1, P₂ = 0, μ₁ = μ₂ and σ₁ = σ₂. And parameters of mixed Gauss distribution in Eq. 2 can be estimated by curve fitting. The fitting results of both plain and twill fabric image histograms are illustrated in Fig. 1c and d with dash lines, respectively.

GRAY-LEVEL CO-OCCURRENCE MATRIX AND ADAPTIVE QUANTIZATION

Spatial gray-level co-occurrence estimates image properties related to second-order statistics. It turns out that GLCM has become one of the most well-known texture features and is widely used for texture characterization (Shang et al., 2011; Sheng et al., 2010). A co-occurrence matrix is a square matrix whose elements correspond to the relative frequency of occurrence of pairs of gray level values of pixels separated by a certain distance in a given direction. The GxG gray-level co-occurrence matrix P_d for a displacement vector d = (dx, dy) is defined as follows. The entry (I, j) of P_d is the number of occurrences of the pair of gray levels I and j which are a distance d apart. Formally, it is given as:

(3)

where, I denotes an image of size UxV with G gray values (x₁, y₁) (x₂, y₂), UxV (x₂, y₂) = (x₁+dx, y₁+dy) and |.| is the cardinality of a set. The co-occurrence matrix reveals certain properties about the spatial distribution of the gray levels in the texture image.

Generally, for a gray-level image whose pixel is represented by an 8-bit integer (i.e., G = 256), its GLCM is a matrix of size 256x256. Extracting features from such a large matrix is quite computational expensive. An intuitive way to reduce the size of GLCM is to reduce the gray level G of original image by equal quantization (Latif-Amet et al., 2000). However, during the equal quantization if the gray level of the defect is close to the normal texture, it is probably quantized to the same level as the normal texture which makes it hard to distinguish. Figure 2a and b illustrate a defective fabric image and its equal quantization result with 16 levels, respectively. The quantized image is histogram equalized for visual convenience. It can be seen that the defect in Fig. 2b becomes ambiguous and hard to detect.

In this study, an adaptive quantization method is proposed based on the fabric texture model elaborated in earlier. As shown in Fig. 1, the histogram of gray-level of normal texture obeys a mixture Gauss distribution which contains two Gaussians. Due to the randomization of texture intensity, to which Gaussian a single pixel belongs and the distance between its gray-level and the center of its belonging Gaussian which is considered as Gaussian Central Distance (GCD), is of more importance than in which exact gray-level the pixel is. The pixel in the image is quantized by examining its GCD. Prior to the calculation of GCD, we should decide which of the two Gaussians the pixel belongs to. We consider that if gray-level g<μ₁, then it belongs to the first Gaussian and if g>μ₂ then it belongs to the second Gaussian. For the gray-level μ₁<g<μ₂, the following probability criterion is used to determine its belonging Gaussian:

(4)

where, ω_i, I = 1and 2, denote first and second Gaussian, respectively, P (ω_i) is the prior probability of each Gaussian, i.e., P (ω₁) = P₁ and P (ω₂) = P₂ in Eq. 2.

Fig. 2(a-c):	Comparison of quantization methods

If P (ω₁|g)>P (ω₂|g), then g is considered to belong to first Gaussian, otherwise it belongs to the second Gaussian. This classification method is known as Bayes classification which minimizes the misclassification rate. Formally the adaptive quantization method consists of following steps:

Figure 3 gives an example of adaptive quantization procedure. Figure 3a and b illustrate the division results of two Gaussians of a mixture distribution with λ = 1, N = 2. Each Gaussian is divided into 7 intervals. The final quantization result is illustrated in Fig. 3c. The intervals [0, 54), [54, 64), [64, 74) and [74, 84) of the first Gaussian and [99, 125), [126, 151), [151, 176) and [176, 255] of the second Gaussian are completely voted intervals. Thus each of them directly forms a quantized level in Fig. 3c. The intervals [84, 94) of the first Gaussian and [74, 99) of the second Gaussian are non-completely voted intervals and they overlap with each other. So they are merged into [74, 99]. As [74, 84) is a completely voted interval, the merged interval is truncated into [84, 94) which forms a new a quantized level in Fig. 3c. Finally, 256 gray-levels are quantized to 9 levels.

Fig. 3(a-c):	Example of adaptive quantization procedure

The pixel is quantized according to its GCD which is measured by the standard deviation of the belonging Gaussian rather than its gray-level. Pixels belonging to the same Gaussian with similar GCD rather than similar gray-levels tend to be quantized to the same level. Particularly, quantized level 0 and L-1 are two special levels, where L denotes the total number of quantized levels. It can be seen that quantized level 0 is out of the confidence interval of the first Gaussian which means, in the sense of hypothesis testing, the pixels within this quantized level do not belong to the normal texture. As the pixels in this level are much darker than the normal texture, they are called darkness exceptional pixels. Similarly the pixels in quantized level L-1 are called lightness exceptional pixels. The advantage of adaptive quantization method is that it can preserve most useful information of the original image while reduce the 256 gray-levels to several quantized levels. Besides, as the quantization is based on GCD of the texture model, the defective region does not obey the texture model and thus contains more lightness or darkness exceptional pixels, so the quantization method can also emphasize the presence of the defects. Figure 2c shows the result of adaptive quantization of Fig. 2a with only 9 quantized levels which is even better than Fig. 2b with 16 levels.

SUPPORT VECTOR DESCRIPTION DATA (SVDD)

Support vector data description is a powerful kernel method that has been commonly used for novelty detection (Tax and Duin, 2004). It provides a solution to the one-class classification problem without any negative sample in training. By mapping the data into a higher dimensional space, the objective of SVDD is to find, in this space, a spherically shaped boundary around the training dataset such that the sphere can enclose as many samples as possible while having minimum volume. The sphere is characterized by its center c and radius R>0. Let {x_i}, I = 1, 2,.., M, be a set of training examples, x_iεR^d, d being the dimension of the input space and M is the number of training samples. The minimization of the sphere volume is achieved by minimizing its square radius R². To allow for the presence of outliers, slack variables ξ_i’s are introduced so that the problem of constructing an optimal separating hypersphere is converted to the following optimization problem:

(5)

where, ξ_i accounts for possible errors, v is a user-provided parameter specifying an upper bound on the fraction of outliers and controls the trade-off between the hypersphere volume and the errors and Φ is a map function which maps input data into higher dimensional space. The corresponding dual problem is:

(6)

where, α = {α₁, α₂,…, α_M} is called lagrange multiplier vector, K(-,-) is called kernel function such that K(x_i, x_j) = Φ (x_i)·Φ (x_j) and a most widely used kernel function is the Radial Basic Function (RBF):

(7)

The optimization problem in Eq. 6 can be solved using standard quadratic programming methods and an optimal solution for α can be obtained. All x_i corresponding to non-zero α_i are called Support Vectors (SV) which usually occupy a small quantity of the training data. Then the optimal solution for c is:

On testing, for a new sample x, it is subjected to the map function Φ and if the distance from the mapped data to the center of the optimal hypersphere is smaller or equal than the radius of the hypersphere, then it is accepted, otherwise it is rejected which can be formulated as:

(8)

Because optimal hypersphere is found by solving a quadratic programming problem, the SVDD do not suffer the problem of local minimum which means that SVDD training can always find a global minimum, thus has a good generalization capability.

DEFECT DETECTION ALGORITHM

Detection of detects is considered as a one-class classification problem. The defect detection algorithm is divided into two phases: Learning phase and classification phase. In the learning phase the fabric image is quantized and a set of GLCM features, as well as some new features are extracted from the quantized image to characterize the fabric texture. The extracted features are used as training data for SVDD training to generate a SVDD classifier. In classification phase the same features are extracted from the quantized testing image and subjected to the SVDD classifier to determine whether it is defective or not.

Feature extraction: Several features extracted from GLCM are used in the proposed detection algorithm. Generally, for GLCM, the displacement parameter (dx, dy), by which two highly related pixels are departed, is a good selection to characterize the fabric texture. A natural selection of displacement parameters for texture characterization is (0, 1) (1, 1) (1, 0) and (1, -1) which are also used for fabric texture characterization by Latif-Amet et al. (2000) and Lin (2010), since the neighboring pixels are considered highly related. These four displacement parameters correspond to 0, 45, 90 and 135°, respectively, where most defects are present. Chan and Pang (2000) find out from the frequency spectrum of the fabric texture image that texture periodicities are existing in the warp (0°) and fill (90°) direction in the fabric texture which can be calculated as the reciprocal of the first harmonic frequency f₀ and f₉₀ along warp and fill direction, respectively. Because of the highly periodicity of the fabric texture, two pixels departed by a texture periodicity are also considered highly related. Therefore, two additional displacement parameters are used in our algorithm: (0, T₀) (T₉₀, 0), where T₀ and T₉₀ denote the texture periodicities at 0 and 90°, respectively, that is T₀ = 1/f₀, T₉₀= 1/f₉₀. f₀ and f₉₀ can be obtained from the 1-D Fourier spectrum of fabric texture at 0 and 90°, respectively. Figure 4 shows a Fourier spectrum of a real fabric texture at 0°, the fabric image is made zero mean in advance to suppress the Direct Current (DC) component. Because T₀ and T₉₀ are usually floating-points while displacement parameters must be integers for the computation of GLCM, the first-order linear interpolation is used to estimate the values of I (x+dx, y+dy) where dx, dy are float-points. Haralick et al. (1973) proposed 14 features from GLCM for texture classification, in this study only two of them, namely contrast CON and inverse difference moment IDM, are used:

(9)

(10)

where, L is the column (or row) number of GLCM (i.e., the total quantized levels), p (I, j) refers to the normalized entry of the co-occurrence matrices. That is p (I, j) = P_d(I, j)/R, where P_d (I, j) is the GLCM with displacement parameter d and R is the total number of pixel pairs (I, j).

In addition, we propose 2 extra features for each displacement parameter (dx, dy) based on following conceptions. As discussed in earlier the pixel in the original image is quantized by its GCD and the pixel with larger GCD means more unlikely it belongs to the normal texture and its intensity tend to be much darker or lighter than the normal texture. It is found that defects, within their boundaries, tend to have more dark pixels (e.g., oil-stain, dirty-yarn, etc.) or light pixels (e.g., miss-pick, thin-place, etc.) than the normal texture. In turn large quantities of dark pixels or light pixels within a local region may indicate a defect in that region. Figure 5 shows a data fragment of quantized image of Fig. 2a with L = 8. All the lightness exceptional pixels whose quantized level is larger than a threshold UL which corresponds to the upper limit of 70% confidence interval of the second Gaussian, are marked by rectangles. There are mainly three reasons for appearance of lightness exceptional pixels: variation of normal texture, the noise and the real defects.

Fig. 4:	Magnitude of frequency spectrum of fabric image along horizontal direction

Fig. 5:	Data fragment of quantized image of Fig. 2a

Because of the random property of variation of normal texture and the noise, lightness exceptional pixels and darkness exceptional pixels tend to scattered within the fabric image. However, the real defects tend to be continuous and constitute a small portion of field. Thus it is obvious that the connected lightness exceptional pixels in the center of Fig. 5 indicate a defect corresponding to the mispick in Fig. 2a and other scattered lightness exceptional pixels are caused by the variation of normal texture or the noise. A feature is proposed to characterize this property of defect.

Given a displacement parameter (dx, dy), let Q be the quantized image, u_k = x+k·dx, v_k = y+k·dy, if for all k = 0, 1, …, C-1, Q (u_k, v_k) are larger than UL and for k = -1 and C, Q (u_k, v_k) are not larger than UL, then points in position (u_k, v_k) (k = 0, 1,…, C -1) are considered as a Lightness Exceptional Run (LER). C denotes the length of the LER. The feature Long Lightness Exceptional Run Emphasis (LLERE) is defined as:

(11)

where, S_l denotes a set containing all the LERs whose length is larger than l in quantized image matrix. C_s and s (k) denote the length and the k-th element of the LER s. The feature LLERE is similar to the feature long run emphasis of Gray Level Run Length Matrix (GLRLM) (Galloway, 1975) which has been widely used for texture characterization and classification. They are both built on the statistic of consecutive pixels with the same attribution. However, in Eq. 11 the LERs whose length is smaller than or equal to l which are probably caused by the variation of normal texture or the noise, are not involved in the computation so that LLERE can emphasize the presence of real defects. l is set to the 95th percentile of the order statistics obtained from non-defective image samples. Similarly, another feature called Long Darkness Exceptional Run Emphasis (LDERE) which is the counterpart of LLERE is also used as a feature:

(12)

where, Z_l denotes a set containing all the Darkness Exceptional Runs (DER) whose length is larger than l in quantized image matrix. The definition of DER is similar to LER except that the values of its elements are smaller than a threshold DL which corresponds to the lower limit of 70% confidence interval of the first Gaussian. C_z and z (k) denote the length and the kth element of the DER z. Compared to the features extracted from GLCM, feature LLERE and LDERE also characterize the relationship of pixels separated by a certain distance in a given direction but they put more emphasis on the continuousness of exceptional pixels along that direction which makes them suitable to detect tiny directional defects. In summary 6 displacement parameters are used and 4 features are extracted for each displacement parameter. In all 24 features are extracted to form a feature vector.

Learning phase: Images of non-defective texture are used in the learning phase. All these images are divided into non-overlapping subregions. The feature vectors V_m, m = 1, 2, …, M are extracted from subregions using the feature extraction method proposed earlier, where M denotes the total number of non-defective subregions. Then the feature vectors are normalized as:

(13)

Where:

(14)

(15)

r = 0, 1, 2, … is the feature index of the feature vector, NV_m denotes the normalized feature vector, η_r and β_r are called offset coefficient and scale factor of the normalization, respectively. The objective of the normalization is to set the values of all the features in the feature vector within interval [0, 1] and make sure that all the features in the feature vector have nearly the same weight. Then these feature vectors are subjected to the SVDD training. During the training, there are two parameters should be decided: the trade-off parameter v in Eq. 6 and the RBF width parameter σ in Eq. 7. A large v allows more outliers of the hypersphere in the training dataset which corresponds to larger rejection rate and larger fraction of support vectors. Tax and Duin (2004) find out that the fraction of support vectors relates to the false alarm rate, so the parameter v can be decided by the expected false alarm rate. To choose the optimal value of parameters σ, the cross validation method is used in the SVDD training.

Classification phase: All the fabric images under inspection are divided into non-overlapping regions of the same size as in learning phase. From each subregion, a feature vector E is extracted using the feature extraction method proposed earlier. Then the feature vector E is normalized using η_r and β_r which have been computed in the learning phase:

(16)

Different from the normalization in learning stage, the values of normalized features NE (r) can be less than zero or greater than one. The normalized feature vector NE is then subjected to the SVDD classifier and the final classification result can be acquired by Eq. 8. If its output is 1, then the subregion is considered as non-defective otherwise it is considered as defective.