# Defect detection for patterned fabric images based on GHOG and low-rank
decomposition

Chunlei Li(1), Guangshuai Gao(1), Zhoufeng Liu(1), Di Huang(2), Sheng Liu(1),
Miao Yu(1)  (1) School of Electronic and Information Engineering, Zhongyuan
University of Technology, ZhengZhou 450007, China
(2) School of Computer Science and Engineering, Beihang University, BeiJing,
100191, China

###### Abstract

In order to accurately detect defects in patterned fabric images, a novel
detection algorithm based on Gabor-HOG (GHOG) and low-rank decomposition is
proposed in this paper. Defect-free pattern fabric images have the specified
direction, while defects damage their regularity of direction. Therefore, a
direction-aware descriptor is designed, denoted as GHOG, a combination of
Gabor and HOG, which is extremely valuable for localizing the defect region.
Upon devising a powerful directional descriptor, an efficient low-rank
decomposition model is constructed to divide the matrix generated by the
directional feature extracted from image blocks into a low-rank matrix
(background information) and a sparse matrix (defect information). A nonconvex
log det($\cdot$) as a smooth surrogate function for the rank instead of the
nuclear norm is also exploited to improve the efficiency of the low-rank
model. Moreover, the computational efficiency is further improved by utilizing
the alternative direction method of multipliers (ADMM). Thereafter, the
saliency map generated by the sparse matrix is segmented via the optimal
threshold algorithm to locate the defect regions. Experimental results show
that the proposed method can effectively detect patterned fabric defects and
outperform the state-of-the-art methods.

###### Index Terms:

patterned fabric, defect detection, GHOG, low-rank decomposition, ADMM.

##  I Introduction

Fabric defect detection always plays a key role in the quality control of
textile industry. Currently, it is mainly performed visually by skilled
workers. However, its reliability is restricted by eye fatigue and human
errors. An automated detection system based on machine vision can provide a
promising solution that not only minimizes labor costs, but will also improve
accuracy and efficiency. Moreover, an automated system is better equipped to
deal with different kinds of fabric patterns, from the non-motif pattern
(plain and twill fabrics, as shown in Fig.1 (a)) to the motif pattern (star-,
box-, and dot-patterned fabrics, as shown in Fig.1 (b-d)).

Most existing defect detection methods focus on simple plain and twill
fabrics, which can be classified into four categories: statistical method [1],
frequency analysis method [2], model method [3], and dictionary learning
method [4].

The aforementioned defect detection methods achieve high detection accuracy.
However, because of the complexity and sophisticated design on patterned
fabric, these proposed methods cannot be extended to detect patterned fabric
defects. Moreover, few studies have been conducted on patterned fabric so far.
In this paper, our research is focused on defect detection in patterned
fabric.

Figure 1: (a) Plain and twill. (b) Star-patterned fabric.(c) Box-patterned
fabric. (d) Dot-patterned fabric.

The patterned fabrics are defined as fabrics with repetitive patterned units
in their designs. Even within the same class of ’patterned’ fabrics, there are
still many categories, and the pattern sizes are different. Therefore,
patterned fabric defect detection is a challenging task. Traditional methods
cannot efficiently recognize the normal pattern; furthermore, they fail to
localize the defect region. Some methods devised for pattern fabric defect
detection, such as the ELO rating (ER) method [31], and wavelet-preprocessing
golden image subtraction (WGIS) [21], et al.,which are performed in a
supervised approach, require non-defective samples; moreover, the detection
accuracy of these methods depends on precise alignment and choosing a suitable
template.

Visual attention mechanism can enable machine and biological vision systems to
quickly find the most salient regions or objects from a scene [5]. Generally,
pattern fabric images always exhibit a high periodic texture among sub-
patterns. However, the defect will disrupt their periodicity (regularity),
resulting in the defects outstanding from homogeneous background. Therefore,
visual saliency models provide a promising method for pattern fabric defect
detection.

Many saliency models have been proposed to detect salient objects [5]. A
representative series of papers are based on the low-rank matrix decomposition
(LR) theory [6,7]. The background usually lies in a low-dimensional subspace,
while the objects that are different from the background can be considered as
noises or errors. Therefore, these methods divide an image into a low-rank
matrix plus a sparse matrix in a learned feature space, where the low-rank
matrix represents the background regions, and the sparse matrix indicates the
salient object regions.

For different kinds of patterned fabrics, the background of the fabric has a
visually homogeneous texture, and any defects stand out from the background.
Compared with object detection in a natural scene, patterned fabric defect
detection better fits the low-rank decomposition model.

However, using low-rank decomposition directly in a color space or in some
other feature space of images does not deal with the task of pattern fabric
defect detection with any sort of efficiency. This is because sub-parts have
different colors or other features locally, but globally they belong to the
normal texture. Therefore, a new powerful descriptor should be proposed to
efficiently characterize the fabric texture, and should be required to possess
the following attributes: (1) defect regions should have a totally different
feature descriptor compared with the background; (2) the background should
have a similar feature descriptor (that can be easily regarded as the low-rank
part).

Defect-free pattern fabric images have a specified direction, while defects
damage their regularity of direction. Therefore, a direction-aware descriptor
can better represent the fabric feature, which is extremely valuable for
separating a salient defect from a non-salient background.

Therefore, in this paper, a direction-aware descriptor, denoted as GHOG, is
designed to be a combination of Gabor and HOG. Upon devising a powerful
directional descriptor, an efficient low-rank decomposition model is
constructed to divide the matrix generated by the directional feature
extracted from image blocks into a low-rank matrix (background information)
and a sparse matrix (defect information). Moreover, we also propose the use of
a non-convex log det( ) as a smooth surrogate function for the rank as opposed
to the nuclear norm in order to improve the model’s efficiency. Finally, the
saliency map generated by the sparse matrix is segmented to locate the defect
regions.

The reminder of this paper is organized as follows. Section 2 introduces the
related works of fabric defect detection. In Section 3, we focus on the
proposed algorithm and its specific procedures. Section 4 evaluates the
performance of the proposed algorithm and compares it with other state-of-the-
art methods. Finally, we conclude the paper in Section 5.

##  II Related works

Defect detection plays an important role in the fabric quality control
process. Many different fabric defect detection methods have been proposed to
solve this problem, and are generally used to detect defects in plain and
twill fabrics. These methods are divided into the following categories:
spatial statistical analysis, spectral analysis, model-based methods, and
dictionary learning. Spatial statistical methods detect defects by calculating
gray values contrasted with their surroundings, including histogram character
analysis [8], morphology [9], local contrast enhancement [10], and the fractal
method [11]. The detection results of these methods depend largely on the size
of a selected window and its discrimination threshold; it is difficult to
detect smaller sizes defects for them. Additionally, these methods cannot
effectively exploit the image’s global information, and are always influenced
by noise.

Spectral analysis methods transform the image to the spectral domain by
choosing a suitable orthogonal basis, which can make better use of the image’s
global information detect defects. These methods include the Fourier transform
(FT) [12], the Gabor transform [13] and the orthogonal wavelet transform [14].
However, these methods always have high computational complexity and poor
detection performance for the complex texture fabric images.

Model-based methods first extract image texture features through modeling and
parameter estimation techniques. Defect detection is realized by
discriminating whether the test image conforms with the normal texture model.
Existing methods include the Gaussian-Markov random field (GMRF) [15], and the
Gaussian mixture model (GMM) [16, 17]. These methods have obtained
satisfactory detection performance; however, they usually share a high
computational complexity, and these methods cannot efficiently detect smaller
size defects.

Dictionary learning-based methods learn a dictionary of training images or
test images, and then reconstruct the defect-free fabric image; thereafter,
defect detection can be realized by subtracting the recovered image from the
test image [18, 19]. In a different way, dictionary learning based methods
also reduce the dimension of an image block by projecting the image block into
a dictionary learning from reference image, then the SVDD is adopted to
discriminate whether an image block is a defect block [20]. However, these
methods are unable to achieve ideal detection performance because the
reconstructed image by the dictionary learning from themselves may exist some
defects, or the self-adaptability of these methods are reduced if the
dictionary learns from the reference images.

Regarding complicated patterned fabrics, several methods have been recently
published, such as the wavelet-preprocessing golden image subtraction method
(WGIS) [21], the Bollinger band method (BB) [22], the regular band method (RB)
[23], template matching for discrepancy measures (TMPM) [24], the pattern
matching and subtraction approach [25-28], the Hash function method [29], and
the regularity and local orientation (RLO) method [30].

The WGIS method utilized a golden image to perform a moving subtraction of
each pixel along each row of every wavelet-pre-processed tested image. The BB
and RB methods, designed by different combination of moving averages and
standard deviations, utilized the regularity property of a patterned texture
to carry out on dot-, box-and star-patterned fabrics. The TMPM method used a
golden image-like approach to exploit a discrepancy measure as a fitness
function to detect defectson patterned textures. The patterned matching and
subtraction method performs a point-to-point comparison, which is inherently
sensitive to image noise, misalignment and distortion. The hash function
method utilizes the offset properties between defect-free and regular patterns
to detect the defects; it is fast but is also sensitive to noise and unable to
show the shape of any defects after segmentation. The ELO rating (ER) method
[31], is a method in which the detection of fabric defects is similar to
carrying out fair matches in the spirit of good sportsmanship. However, this
method depends on partition size, the number of randomly located partitions,
$w$-variable and constant $K$.

The above pattern fabric defect detection methods adopt traditional approaches
to characterize the fabric texture, such as wavelet transform, Gabor
transform, average value, standard deviation and regular bands. They devised
the feature descriptors, but did not consider the characteristics of the
pattern fabric image. On the other hand, most of the complicated pattern
fabric defect detection methods adopted template-matching technology to
localize the defect; they are performed in a supervised approach. The
detection accuracy depends on precise alignment and choosing a suitable
template. Therefore, in this paper, a powerful direction-aware descriptor was
designed, denoted as GHOG. The descriptor considers the characteristic of the
patterned fabric image, and a low-rank decomposition model that can better fit
the defect detection problem is adopted to separate the salient defect from
the non-salient background.

##  III The proposed algorithm

Defect-free pattern fabric images have a specified direction, while defects
damage their regularity of direction. Moreover, compared with the object
detection in a natural scene, patterned fabric defect detection can better fit
a low-rank decomposition model. Therefore, our paper proposes a novel method
of patterned fabric defect detection, based on GHOG and low-rank
decomposition, and it includes the following four steps: 1) GHOG feature
extraction; 2) low-rank decomposition model construction; 3) optimal solution
of the model; 4) the generation and segmentation of the saliency map.

Figure 2: Construction process of the proposed GHOG descriptor

###  III-A GHOG feature extraction

Highly effective fabric defect detection algorithms should resort to
meaningful and powerful feature descriptors to facilitate uniqueness
measurements and warrant sufficiently discriminative capabilities. Defect-free
patterned fabric images have a specified direction, while defects damage their
regularity of direction. Therefore, a direction-aware descriptor can better
represent the fabric feature.

Due to the outstanding mathematical properties of the Gabor filter and its
analogy to human visual mechanisms, the two-dimensional (2D) Gabor filter has
been widely applied in the texture analysis. In this paper, a bank of Gabor
directional filters have been adopted to extract directional information, and
generate the directional Gabor filtered maps.

In addition, the gradient space of an image provides a measure of change in
intensity over the pixels, as opposed to the absolute values of the pixels’
intensity values. Although the existence of defects destroys the regularity of
a patterned texture, it will have a more enhanced effect in gradient space
domain than in image space domain. As a consequence, features extracted from
gradient space domain of a defective periodic block will differ significantly
to those extracted from a defective-free block. Considering that the Histogram
of Oriented Gradients (HOG) features are extracted from generated Gabor
filtered maps in quantized directions, a novel Gabor-HOG (GHOG) feature
description is proposed to greatly reduce the feature dimension and to capture
the directional feature. The construction process of the proposed GHOG
descriptor is shown in Figure.2, and the specific procedures are described as
follows:

1) Gabor directional filtered map generation. The complex function expression
of a 2-D Gabor filter is described as follows [34]:

$g\left({x,y;\lambda,\psi,\sigma,\gamma}\right)=\exp\left({-\frac{{x^{\prime
2}+\gamma^{2}y^{\prime
2}}}{{2\sigma^{2}}}}\right)\exp\left({i\left({2\pi\frac{{x^{\prime}}}{\lambda}+\psi}\right)}\right)$
(1)

Its real part can be formulated as:

$g\left({x,y;\lambda,\psi,\sigma,\gamma}\right)=\exp\left({-\frac{{x^{\prime
2}+\gamma^{2}y^{\prime
2}}}{{2\sigma^{2}}}}\right)\cos\left({2\pi\frac{{x^{\prime}}}{\lambda}+\psi}\right)$
(2)

And its imaginary part is described as:

$g\left({x,y;\lambda,\psi,\sigma,\gamma}\right)=\exp\left({-\frac{{x^{\prime
2}+\gamma^{2}y^{\prime
2}}}{{2\sigma^{2}}}}\right)\sin\left({2\pi\frac{{x^{\prime}}}{\lambda}+\psi}\right)$
(3)

where $x^{\prime}=x\cos\theta+y\sin\theta$,
$y^{\prime}=-x\sin\theta+y\cos\theta$, $\lambda$ is wave length, which is
always greater than or equal to 2, but not more than 1/5 of the size of the
input image; $\theta$ represents directions, their values ranging from 0 to
$2\pi$; $\psi$ indicates phase shift, whose range is from $-\pi$ to $\pi$, 0
and $\pi$ correspond to center-on and center-off functions, respectively,
while $-\pi/2$ and $\pi/2$ correspond to anti-symmetric function; $\gamma$ is
the length-width ratio, also known as the ratio of the vertical and the
horizontal, which determines the ellipticity of the Gabor function; if
$\gamma=1$, the shape is round, if $\gamma<1$ , shape stretches along the
direction of parallel stripes, in this paper, $\gamma$ is set to 0.5; $\sigma$
is the Gaussian factor standard deviation of the Gabor function.The value of
$\sigma$ cannot be preset directly, its change just depends on the variation
of the bandwidth $b$. $b$ must be a positive constant, which is related to the
ratio of $\sigma/\lambda$, we usually set it as 1. Then the relationship of
$\sigma$ and $\lambda$ is $\sigma=0.56\lambda$. In this paper, empirically, we
choose eight orientations with one scale to filter the patterned fabric image,
and accordingly generate eight directional filtered maps $I_{G_{o}}$
($o$=1,2,…,_N_ , _N_ =8 here) that capture the directional features.

2) Uniformly sampling for the Gabor filtered maps. Once these directional
filtered maps of all quantized directions are obtained, they are exploited as
the inputs for the next computing histogram of orientated gradients features
over the same image region. For each generated Gabor filtered map , the size
of which is the same as the given original image, is equally decomposed into
segments $\left\\{{I_{G_{o}}^{i}}\right\\}_{i=1,2...K}$ with sizes of
$N_{\rm{b}}\times N_{\rm{b}}$ ; where $o$ indicts the orientations, and
$o$=1,2,…8, $K$ is the number of segments, $N_{b}$ equals 16 in this paper.

3)  HOG feature extraction. For each segment $I_{G_{o}}^{i}$, the HOG is
similar to the method described in [35]. The detail procedures are as follows:

First of all, normalize the input image block $I_{G_{\rm{i}}}^{j}$ obtained by
gamma rectification, and it is described as follows.

$\emph{H}_{o}(x,y)=(\emph{I}_{G_{o}}^{i})^{gamma}$ (4)

In this paper, gamma is set to 1/2.

Calculate the gradient of image pixels along the horizontal and vertical
directions:

$mag_{o}(x,y)=\sqrt{\left({\frac{{\partial H_{o}(x,y)}}{{\partial
x}}}\right)^{2}+\left({\frac{{\partial H_{o}(x,y)}}{{\partial
y}}}\right)^{2}}$ (5)

$\theta_{o}(x,y)=\arctan(\frac{{\partial H_{o}(x,y)}}{{\partial
y}}/\frac{{\partial H_{o}(x,y)}}{{\partial x}})$ (6)

$\frac{{\partial H_{o}\left({x,y}\right)}}{{\partial
x}}=H_{o}\left({x+1,y}\right)-H_{o}\left({x-1,y}\right)$ (7)

$\frac{{\partial H_{o}\left({x,y}\right)}}{{\partial
y}}=H_{o}\left({x,y+1}\right)-H_{o}\left({x,y-1}\right)$ (8)

Then each orientation is mapped to the range of $[0,2\pi]$ from that of
$[-\pi/2,\pi/2]$, which keeps consistent with the number of the Gabor filter
maps. After quantization, the entry $n_{o}$ of each orientation $\theta_{o}$
is computed as follows:

$n_{o}(x,y)=\bmod(\left\lfloor{\frac{{\theta_{o}(x,y)}}{{2\pi/N}}+\frac{1}{2}}\right\rfloor,N),{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}o=1,2,...,N$ (9)

The histogram of orientation gradients features, $h_{oi}$, is constructed as
(10) by accumulating the gradient magnitude $mag_{o}$ of all the pixels with
the same quantized orientation entry $n_{o}$.

$h_{oi}(j)=\sum{f(n_{o}(x,y)=j)}*mag_{o}(x,y)$ (10)

where _j_ =0, 1,…, _N_ -1; _o_ =1,2,…,_N_ ; _i_ =1,2,…,_K_.

$f(x)=\left\\{\begin{array}[]{l}1,{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}if{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}x{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}is{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}true\\\ 0,{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}otherwise\\\ \end{array}\right.$ (11)

Then, for each Gabor filtered maps $I_{G_{o}}$, its HOG features $h_{o}$ is
generated by concatenating all the histograms of all the segments:

$h_{o}=[h_{o1},h_{o2},h_{o3},...,h_{oK}]^{T}$ (12)

The final GHOG descriptor is generated by concatenating all _N_ histogram of
HOG features of all segments as (13). Each histogram $h_{o}$ is normalized to
a unit norm vector $\mathop{h_{o_{K}}}\limits^{\wedge}$ before the
concatenation.

$GHOG_{K}=\left[{\mathop{h_{1K}}\limits^{\wedge},\mathop{h_{2K}}\limits^{\wedge},...,\mathop{h_{NK}}\limits^{\wedge}}\right]^{T}$
(13)

We define a feature matrix F as the final GHOG descriptors to represent the
information of the entire image.

$\textbf{F}=[GHOG_{1},GHOG_{2},...,GHOG_{K}]$ (14)

###  III-B Low-rank decomposition model

Low-rank decomposition and subspace recovery have been widely used for
detection and recognition. This is because the background of an image always
lies in a low-dimensional subspace, while the objects to be detected are
different from the background can be considered as either noise or errors.
Therefore, these methods divide an image into a low-rank matrix, plus a sparse
matrix in a learned feature space, where the low-rank matrix represents the
background regions, and the sparse matrix indicates the salient object
regions. For different kinds of patterns, the fabric’s background has a
visually homogeneous texture, and the defects stand out. Compared with object
detection in a natural scene, the patterned fabric defect detection better
fits the low-rank decomposition model. Therefore, in this paper, the low-rank
decomposition model is proposed to detect the defect region.

For the aforementioned generated feature matrix $F$, a low-rank decomposition
model is constructed as follows:

$\begin{split}\left({L^{*},S^{*}}\right)=\arg{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\mathop{\min}\limits_{\left({L,S}\right)}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}\left({rank{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\left(L\right)+\lambda\left\|S\right\|_{0}{\kern 1.0pt}}\right){\kern
1.0pt}{\kern 1.0pt}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}s.t.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}F=L+S\\\
\end{split}$ (15)

where $L$ is a low-rank matrix, indicating the background, i.e., repeated
patterns. $S$ is a sparse matrix, and it represents the defective objects.

Since the above problem is NP-hard, it is difficult to approximate the optimal
solution of (7), so the convex surrogate is adopted as follows:

$\left({L^{*},S^{*}}\right)=\arg{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\mathop{\min}\limits_{\left({L,S}\right)}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}\left({\left\|L\right\|_{*}+\lambda\left\|S\right\|_{1}}\right){\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}s.t.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}=L+S$ (16)

where $\left\|L\right\|_{*}$ is the nuclear norm of $L$, $\left\|{{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}}\right\|_{1}$ indicates the $l_{1}$ norm, $\lambda$ is a weighted
factor that controls the low-rank and sparsity degree.

Using the nuclear norm as a convex surrogate in the first term of Eq. (8),
i.e., rank minimization problem is correct, while non-convex optimization
toward the rank minimization problem could lead to better recovery results.

In this paper, a smooth but non-convex surrogate of the rank is adopted
instead of the nuclear norm. For a given matrix with a symmetric positive
semi-definite $X\in R^{n\times n}$, the rank minimization problem can be
approximately surrogate by minimizing the following equation [34]:

$E(X,\xi)=\log\det(X+\xi I)$ (17)

where $\xi$ is a positive scalar. $E\left({X,\xi}\right)$ approximates the sum
of the logarithm of singular values, thus it is smooth and non-convex. The log
det as a non-convex surrogate of the rank has also been more carefully proofed
from an information-theoretic perspective. As shown in Figure 3, the surrogate
function $E\left({X,\xi}\right)$ can better approximate the rank than the
nuclear norm.

Figure 3: Comparison of $L\left({x,\xi}\right)$, $rank\left(x\right)$,
$\left\|x\right\|_{*}$ in the case of a scalar.

For the low-rank matrix $L$, Eq.(9) is rewritten as follows:

$\begin{split}L(L,\xi)=\log\det((LL^{T})^{1/2}+\xi I)\\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}=\log\det(U\Sigma^{1/2}U^{-1}+\xi I)\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}=\log\det(\Sigma^{1/2}+\xi I)\\\
\end{split}$ (18)

where $\Sigma$ is the diagonal matrix whose diagonal elements are eigenvalues
of matrix $LL^{T}$, i.e., $LL^{T}=U\Sigma U^{-1}$, meanwhile, $\Sigma^{1/2}$
is also the diagonal matrix whose diagonal elements are the singular values of
the matrix $L$. Hence, $L\left({L,\xi}\right)$ is a log det( ) surrogate
function of $rank\left(L\right)$ obtained by setting
$X=\left({LL^{T}}\right)^{1/2}$. Finally, Eq.(8) can be rewritten as follows:

$(L^{*},S^{*})=\arg{\kern 1.0pt}{\kern
1.0pt}\mathop{\min}\limits_{(L,S)}{\kern 1.0pt}{\kern
1.0pt}(L(L,\xi)+\lambda\left\|S\right\|_{1}){\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}s.t.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}F=L+S$ (19)

###  III-C Optimal solution of the model

The alternative direction method of multipliers (ADMM) demonstrates a good
balance between efficiency and accuracy in solving optimization problems. In
this paper, ADMM has been adopted to solve Eq. (11).

The Augmented Lagrange Multiplier function of Eq.(11) is as follows:

$\begin{split}L(L,S,Z)=L(L,\xi)+\lambda\left\|S\right\|_{1}+\frac{\beta}{2}\left\|{L+S-F}\right\|_{F}^{2}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}-\left\langle{Z,L+S-F}\right\rangle\\\ \end{split}$ (20)

where $Z\in R^{m\times n}$ is the multiplier of the linear constraint,
$\beta>0$ is the penalty parameter for the violation of the linear constraint,
$\left\langle\cdot\right\rangle$ is the inner product and
$\left\|\cdot\right\|_{F}$ is the induced Frobenius norm. The proposed
objective function can be solved by alternatively minimizing the objective
function with respect to the $L$, $S$ and the multiplier $Z$. It can be
described as solving the following three sub-problems:

$\left\\{\begin{array}[]{l}L_{k+1}=\arg\min_{L}L(L,S_{k},Z_{k};\beta)\\\
S_{k+1}=\arg\min_{S}L(L_{k+1},S,Z_{k};\beta)\\\
Z_{k+1}=Z_{k}-\beta\left({L_{k+1}+S_{k+1}-F}\right)\\\ \end{array}\right.$
(21)

For the first sub-problem in (13) which solves for $L$ at fixed $S$ and $Z$,
it can be explicitly represented as the following form:

$\begin{split}L^{*}=\mathop{\arg\min}\limits_{L}\sum\nolimits_{j=1}^{n_{0}}{\log(\sigma_{j}\left(L\right)+\xi)}+\frac{\beta}{2}\left\|{L+S-F}\right\|_{F}^{2}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}-\left\langle{Z,L+S-F}\right\rangle\\\
\end{split}$ (22)

where $n_{0}=\min\left\\{{m,n}\right\\}$, and $\sigma_{j}\left(L\right)$
indicates the $j$-th singular value of $L$. For simplicity, we use
$\sigma_{j}$ to denote $\sigma_{j}\left(L\right)$ . Even though
$\sum\nolimits_{j=1}^{n}{{\rm{log}}\left({\sigma_{j}+\xi}\right)}$ is non-
convex, it can be solved by utilizing a local minimization approach. We define
the equality
$f\left(\sigma\right)=\sum\nolimits_{j=1}^{n}{{\rm{log}}\left({\sigma_{j}+\xi}\right)}$.
Then $f\left(\sigma\right)$ can be approximated by using its first-order
Taylor expansion, as follows:

$f\left(\sigma\right)=f\left({\sigma^{(k)}}\right)+\left\langle{\nabla
f\left({\sigma^{(k)}}\right),\sigma-\sigma^{(k)}}\right\rangle$ (23)

where $\sigma^{(k)}$ is the solution obtained in the $k$-th iteration.
Therefore, Eq.(14) can be solved by iteratively solving:

$\begin{split}L^{(k+1)}=\mathop{\arg\min}\limits_{L}\frac{\beta}{2}\left\|{L+S-F-\frac{Z}{\beta}}\right\|_{F}^{2}\\\
{\kern 1.0pt}{\kern
1.0pt}+\sum\nolimits_{j=1}^{n}{\frac{{\sigma_{j}}}{{\sigma_{j}^{(k)}+\xi}}}\\\
\end{split}$ (24)

where we use the fact that $\nabla
f(\sigma^{(k)})=\sum\nolimits_{j=1}^{n_{0}}{\frac{1}{{\sigma_{j}^{(k)}+\xi}}}$
and ignore the constants in Eq.(14). For convenience, we rewrite Eq.(16) as:

$L^{(k+1)}=\mathop{\arg\min}\limits_{L}\frac{1}{2}\left\|{L+S-F-\frac{Z}{\beta}}\right\|_{F}^{2}+\tau\varphi\left({L,\omega^{(k)}}\right)$
(25)

where $\tau=1/\beta$.
$\varphi(L,\omega)=\sum\nolimits_{j}^{n_{0}}{\omega_{j}^{(k)}}\sigma_{j}$
indicates a weighted nuclear norm whose weights
$\omega_{j}^{(k)}=1/\left({\sigma_{j}^{(k)}+\xi}\right)$. Note that the
weights are ascending, since the singular values $\sigma_{j}$ are ordered in a
descending order.

In general, for a real matrix, the weighted nuclear norm is a convex function
only if the weights are descending, and the optimal solution to Eq. (17) is
obtained by a weighted singular value thresholding operator, referred to as
the proximal operator. In this paper, the weights are ascending, thus Eq. (17)
is non-convex. It is therefore difficult to find its global minimizer.
Nevertheless, we could find that the weighted singular value thresholding
gives one minimizer to Eq. (17) via Theorem 1 (Proximal Operator of Weighted
Nuclear Norm) [35]. According to this Theorem, we can obtain the low-rank
matrix at the $(k+1)$-th iteration by

$L^{\left({k+1}\right)}=U\left({\Sigma-\tau
diag\left({\omega^{\left(k\right)}}\right)}\right)_{+}V^{T}$ (26)

where $U\Sigma V^{T}$ is the SVD of the feature matrix $F$, and
$\omega_{j}^{(k)}=1/\left({\sigma_{j}^{(k)}+\xi}\right)$. Even though the
weighted thresholding is only a local minimizer, it always leads to a decrease
in the objective function value. In this paper, the initial value
$\omega^{\left(0\right)}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}$ is set to
$\left[{1,1,...1}\right]^{T}$.

After solving the low-rank matrix L, the sparse matrix S can be solved by
fixing L and Z. Indeed, we can easily obtain the solution using the widely-
used shrinkage problem [36]:

$S^{k+1}=\frac{1}{\beta}Z^{k}-L^{k}+F-P_{\Omega_{\infty}^{\gamma/\beta}}\left[{\frac{1}{\beta}Z^{k}-L^{k}+F}\right]$
(27)

where $P_{\Omega_{\infty}^{\gamma/\beta}}$ indicts the Euclidean projection
onto:

$\Omega_{\infty}^{\gamma/\beta}:=\left\\{{X\in R^{n\times
n}\left|{-\gamma/\beta\leq X_{ij}\leq\gamma/\beta}\right.}\right\\}$ (28)

Then the multipliers $Z$ can be updated as follows:

$Z^{k+1}=Z^{k}-\beta(L^{k+1}+S^{k+1}-F)$ (29)

###  III-D The generation and segmentation of the saliency map

Saliency map generation. After decomposing the feature matrix $F$ of the given
patterned fabric image into the low-rank matrix $L$, which corresponds to the
patterned fabric background, and a sparse matrix $S$, which corresponds to the
defective regions by the aforementioned methods. A saliency map $M$ is
generated by the $l_{1}$-norm of each column $S_{i}$ in $S$, and it is
described as follows:

$M(I_{i})=\left\|{S_{i}}\right\|_{1}$ (30)

The larger value of $M\left({I_{i}}\right)$ indicates that the block $I_{i}$
has higher probability of being defective.

The segmentation of saliency map.

1). Denoise the saliency map $M$ to generate a new saliency map
$\mathop{M}\limits^{\wedge}$

$\mathop{M}\limits^{\wedge}=g*(M\circ M)$ (31)

where $g$ is the radius of the circular smoothing filter and ”$\circ$” denotes
the Hadamard inner product operator and ”*” is the convolution operator.

2). Transform the saliency map $\mathop{M}\limits^{\wedge}$ into a gray-scale
image $G$

$G=\frac{{\mathop{M}\limits^{\wedge}-\min(\mathop{M}\limits^{\wedge})}}{{\max(\mathop{M}\limits^{\wedge})-\min(\mathop{M}\limits^{\wedge})}}\times
255$ (32)

3). Segment $G$ by using improved adaptive threshold algorithm [37] to locate
the defect regions.

##  IV Experimental results and analysis

In order to verify the effectiveness of the proposed algorithm, we chose
images from the dot-, box-, and star-patterned fabric databases for
performance evaluation. The size of the fabric image is 512 pixels $\times$
512 pixels. The dot-patterned fabric database contains 30 defect-free and 30
defective images, the star-patterned fabric database contains 25 defect-free
and 25 defective images, and the box-patterned fabric database contains 30
defect-free and 26 defective images. All the defective images have a
corresponding ground-truth that shows the defect-free regions as black and
defective regions as white. All experiments in this paper were implemented in
an Inter(R) Core(TM) i3-2120 3.3GHZCPU environment, using software MATLAB
2011a.

_A. Qualitative Results_

The test patterned fabric image matrix was divided into the superposition of
two parts, i.e., a low-rank matrix and a sparse matrix, and the visual
saliency map was generated from the sparse matrix and outstands the defective
regions. Thus a simple threshold method can easily localize the defect region.

The feature extraction method and detection model are equally important for
fabric defect detection. In order to validate the effectiveness of our method,
we firstly compared the saliency maps generated by different descriptors,
including Gabor [32], HOG [33], and different detection models, such as the
template matching model (TMM) [38] and the context analysis model (CAM) [39]
with our method, as shown in Figure 4. The first column consists of the
original images, the second column consists of saliency maps generated by the
Gabor feature [32] and the LR detection model, the third column consists of
the saliency maps generated by the HOG feature [33] and the LR detection
model, the fourth column consists of the saliency maps generated by the GHOG
feature and the TMM [38] detection model, the fifth column consists of the
saliency maps generated by the GHOG feature and the CAM [39] detection model,
and the last column consists of the saliency maps generated by our method
(GHOG feature and the LR detection model). From the second column and the
third column in Figure 4, we can conclude that the detection result based on
the Gabor or HOG features cannot outstand the defect region for star-patterned
fabric and box-patterned fabric images, but they can outstand the detect
region for dot-patterned fabric images. On the other hand, from the fourth
column and the fifth column in Figure 4, the saliency map generated by the
GHOG feature and the TMM or CAM detection models cannot outstand the defect
region. In the last column we can see that the saliency map generated by our
method can efficiently outstand the defect region for all the three types of
images. And the performance of these methods can be concluded as Table1.

TABLE I: Comparisons with other methods

Methods$/$ fabric image |  Gabor+LR |  HOG+LR |  GHOG+TMM |  GHOG+CAM |  GHOG+LR
---|---|---|---|---|---
star-patterned |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  $\surd$
box-patterned |  $\times$ |  $\times$ |  $\times$ |  $\times$ |  $\surd$
dot-patterned |  $\surd$ |  $\surd$ |  $\times$ |  $\times$ |  $\surd$

(a) Saliency map for star-patterned fabric image (b) Saliency map for box-
patterned fabric imagec (c) Saliency map for dot-patterned fabric image

Figure 4: The saliency maps generated by different features and detection
model. The first column shows the original images, the second column shows
saliency maps generated by Gabor [32] with the LR model (Gabor + LR), the
third column shows the saliency maps generated by HOG [33] with LR model (HOG
+ LR), the fourth column shows the saliency maps generated by TDVSM [38](GHOG
+ TDVSM), the fifth column shows the saliency maps generated by LSF-GSA
[39](GHOG + LSF-GSA), and the last column shows the saliency maps generated by
our method(GHOG + LR).

From Figure 4 and Table 1, we can see that the proposed defection method based
on GHOG and LR is suitable for pattern fabric defect detection.

In addition, we compared our method with some state of the art visual saliency
models, including the wavelet transform (WT) method [40], the prior guided
least squares regression method (PGLSR) [41], the textural differential visual
saliency model (TDVSM) [38] and the local statistic features and global
saliency analysis model (LSF-GSA) [39].

As is shown in Figure 5, WT [40] first transformed the image into a frequency
domain, then generated the saliency map by analyzing the wavelet coefficient.
However, even in a normal background with a complicated pattern, its wavelet
coefficients are larger, which will lead to incorrect detection results. The
PGLSR method [41] could effectively detect defects in the patterned fabric,
but similarities in texture between the background and the defect lead to
inaccurate shape descriptions of the defects. In Li et al. [38], saliency was
calculated by comparing their textural features with the average texture
features, it obtains a successful performance for fabric images with a
stochastic texture, while the method fails to detect defects in a patterned
fabric image, especially in box-patterned fabrics, as shown in Figure.5(b).
Liu et al. [39] detected fabric defects by using the local statistic feature
and global saliency analysis, while the detection results have a large amount
of noise. Our method generates a saliency map by combining the GHOG feature
descriptor with a low-rank decomposition model, and it effectively highlights
the defect regions. Subsequently, the detection results are obtained by
segmenting the generated saliency map.

(a) Saliency map for star-patterned fabric image (b) Saliency map for box-
patterned fabric image (c) Saliency map for dot-patterned fabric image

Figure 5: The saliency maps of our method compared with other state-of-the-art
saliency models. The first column shows the original images, the second column
shows saliency maps generated by WT [40], the third column shows the saliency
maps generated by PGLSR [41], the fourth column shows the saliency maps
generated by TDVSM [38], the sixth column shows the saliency maps generated by
LSF-GSA [39], and the last column shows the saliency maps generated by our
method.

In order to further verify the effectiveness of our proposed method, we
compared our detection method with the existing fabric detection algorithm.
Because most of the traditional detection methods are invalid for detecting
patterned fabric defect, in this paper, we only compared the proposed method
with the other two valid detection methods for detecting patterned fabrics.
The experimental results are described in Figure 6, where the white pixels
represent the defect regions, and the black pixels represent the background.
The first row represents the detection results generated by PGLSR [41], the
second row represents the detection results generated by LSF-GSA [39], the
third row indicates the detection results generated by our method and the last
row indicates the ground-truth images. In Figure 6, we can see that the other
two methods (as shown in the first two rows) can almost localize the defect
region, but the detected shape of the defect is different from the ground
truth. Our detection results (as shown in the third row) are similar to the
ground truth (as shown in the last row), can efficiently localize the defect
regions, and the detected shape of the defect is similar to the ground truth.
This demonstrates our proposed method is superior to the other two methods.

(a) Detection result for star-patterned fabric image (b) Detection result for
box-patterned fabric image (c) Detection result for dot-patterned fabric image

Figure 6: Detection results of our method compared with other fabric defect
detection methods. The first row shows the detection results of PGLSR method,
the second row shows the detection results of LSF-GSA method, the third row
shows the detection results of our method, the last row shows the ground-truth
images.

_B. Quantitative Evaluations_

In order to further evaluate the performance of our proposed method, the
receiver operating characteristic curve (ROC) is adopted, as shown in Figure
7. Figure.7(a) shows the ROC curves of star-patterned fabric images,
Figure.7(b) shows the ROC curves of box-patterned fabric images, and
Figure.7(c) shows the ROC curves of dot-patterned fabric images. From Figure
7(a) and (b), we can see that our method outperforms the other three methods
for star-patterned and box-patterned fabrics. This demonstrates the
effectiveness of our proposed method. However, our proposed method is
unsatisfactory regarding dot-patterned fabrics, as shown in Figure.7(c). The
proposed GHOG feature is orientation-aware, while the orientation features of
the dot-patterned fabric are indistinct; therefore, GHOG is not enough to
characterize the dot-patterned fabric. In fact, for most of the fabric
samples, they are weaved using warp and weft, and have distinct directions.

(a) Star-patterned ROC curves (b) Box-patterned ROC curves (c) Dot-patterned
ROC curves

Figure 7: ROC curves of patterned fabric images compared with other detection
methods. (a) ROC curves of the Star-patterned fabric images. (b) ROC curves of
the Box-patterned fabric images. (c) ROC curves of the Dot-patterned fabric
images.

In addition, some other statistical parameters, such as true positive (TP),
true negative (TN), false positive (FP), false negative (FN), are adopted in
this paper, originally proposed by Ng.et.al [42]. Based on these parameters,
some measure metrics, such as precision, are calculated as:

$precision=\frac{{TP}}{{TP+FP}}$ (33)

and recall is calculated as:

$recall=\frac{{TP}}{{TP+FN}}$ (34)

As shown in Figure 8, we can see that the proposed approach not only provided
the highest rate in recall but also provided a balanced performance with
respect to precision. Figure 9 takes into account both recall and precision,
and adopts the criteria of the F-measure. Additionally, it also demonstrates
the effectiveness of our proposed method.

$F=2\frac{{precision\cdot recall}}{{precision+recall}}$ (35)

(a) Star-patterned PR curves (b) Box-patterned PR curves (c) Dot-patterned PR
curves

Figure 8: ROC curves of patterned fabric images compared with other detection
methods. (a) ROC curves of star-patterned fabric images. (b) ROC curves of
box-patterned fabric images. (c) ROC curves of dot-patterned fabric images.
Figure 9: The F-measure of our method compared with other methods.

##  V Conclusion

In this paper, we proposed a novel patterned fabric defect algorithm based on
GHOG and low-rank decomposition. GHOG descriptor is adopted to extract the
features of the patterned fabric image, and the proposed low-rank model is
adopted to decompose the fabric image into normal background and defect,
respectively. Experimental results demonstrate that the proposed method can
efficiently and correctly locate the defect region, and show that our approach
outperforms other state-of-the-art methods. In the future, additional
theoretical development will be beneficial for defect detection in the glass
surface, rail surface, etc.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China
(No.61202499, No.61379113), Science and technology innovation talent project
of Education Department of Henan Province(17HASTIT019), Science and technology
leader project of Zhengzhou City(131PLJRC643). In addition, the authors would
like to thank Dr. Henry Y.T. Ngan, Industrial Automation Research Laboratory,
Dept. of Electrical and Electronic Engineering, The University of Hong Kong,
for providing the database of patterned fabric images.

## References

  * [1] Selver, M. A., Avsar, V., $\&$ $\ddot{O}$zdemir, H. (2014). Textural fabric defect detection using statistical texture transformations and gradient search. Journal of the Textile Institute, 105(9), 998-1007.
  * [2] Jing, J. (2015). Automatic defect detection of patterned fabric via combining the optimal gabor filter and golden image subtraction. Journal of Fiber Bioengineering $\&$ Informatics, 8, 229-239.
  * [3] Allili, M. S., Baaziz, N., $\&$ Mejri, M. (2014). Texture modeling using contourlets and finite mixtures of generalized gaussian distributions and applications. Multimedia IEEE Transactions on, 16(3), 772-784.
  * [4] Zhou, J., Semenovich, D., Sowmya, A., $\&$ Wang, J. (2014). Dictionary learning framework for fabric defect detection. Journal of the Textile Institute, 105(3), 223-234.
  * [5] Borji, A., $\&$ Itti, L. State-of-the-art in visual attention modeling. Pattern Analysis and Machine Intelligence, IEEE Transactions on,2013, 35(1), 185-207.
  * [6] Lang C, Liu G, Yu J, et al. Saliency detection by multitask sparsity pursuit [J]. Image Processing, IEEE Transactions on, 2012, 21(3): 1327-1338.
  * [7] Chen C, Li S, Qin H, et al. Structure-Sensitive Saliency Detection via Multilevel Rank Analysis in Intrinsic Feature Space[J]. Image Processing, IEEE Transactions on, 2015, 24(8): 2303-2316.
  * [8] Selver, M. A., Avsar, V., $\&$ $\ddot{O}$zdemir, H. Textural fabric defect detection using statistical texture transformations and gradient search [J]. Journal of Thoracic Oncology Official Publication of the International Association for the Study of Lung Cancer, 2014, 6(105):2017-2023.
  * [9] Hu J, He Z, Weng G, et al. Detection of chemical fabric defects on the basis of morphological processing [J]. Journal of the Textile Institute, 2015:1-9.
  * [10] Guan S, Shi H, Zhao T. Image segmentation of fabric defect based on object rarity feature [J].Textile Research Journal, 2015,36(11): 45-50.
  * [11] Hanmandlu M, Choudhury D, Dash S. Detection of defects in fabrics using topothesy fractal dimension features [J]. Signal, Image and Video Processing, 2015, 9(7): 1521-1530.
  * [12] Sakhare K, Kulkarni A, Kumbhakarn M, et al. Spectral and spatial domain approach for fabric defect detection and classification [C]//Industrial Instrumentation and Control (ICIC), 2015 International Conference on. IEEE, 2015: 640-644.
  * [13] Jing J, Chen S, Li P. Automatic Defect Detection of Patterned Fabric via Combining the Optimal Gabor Filter and Golden Image Subtraction [J]. Journal of Fiber Bioengineering and Informatics, 2015, 8(2): 229-239.
  * [14] Wen Z, Cao J, Liu X, et al. Fabric defects detection using adaptive wavelets [J]. International Journal of Clothing Science and Technology, 2014, 26(3): 202-211.
  * [15] Xiaobo Y. Fabric defect detection of statistic aberration feature based on GMRF model [J]. Journal of Textile Research, 2013, 4: 026.
  * [16] Li M, Cui S, Xie Z. Application of Gaussian mixture model on defect detection of print fabric [J]. Journal of Textile Research, 2015, 8: 019.
  * [17] Allili M S, Baaziz N, Mejri M. Texture modeling using contourlets and finite mixtures of generalized Gaussian distributions and applications [J]. Multimedia, IEEE Transactions on, 2014, 16(3): 772-784.
  * [18] Zhou J, Wang J. Fabric defect detection using adaptive dictionaries [J]. Textile Research Journal, 2013.
  * [19] Qu T, Zou L, Zhang Q, et al. Defect detection on the fabric with complex texture via dual-scale over-complete dictionary [J]. The Journal of The Textile Institute, 2015: 1-14.
  * [20] Zhou J, Semenovich D, Sowmya A, et al. Dictionary learning framework for fabric defect detection [J]. The Journal of The Textile Institute, 2014, 105(3): 223-234.
  * [21] Ngan H Y T, Pang G K H, Yung S P, et al. Wavelet based methods on patterned fabric defect detection [J]. Pattern recognition, 2005, 38(4): 559-576.
  * [22] Ngan H Y T, Pang G K H. Novel method for patterned fabric inspection using Bollinger bands [J]. Optical Engineering, 2006, 45(8): 087202-087202-15.
  * [23] Ngan H Y T, Pang G K H. Regularity analysis for patterned texture inspection [J]. Automation Science and Engineering, IEEE Transactions on, 2009, 6(1): 131-144.
  * [24] G. Stubl, J.-L. Bouchot, P. Haslinger, B. Moser, Discrepancy norm as fitness function for defect detection on regularly textured surfaces, LNCS 7476 (2012) 428-437.
  * [25] Costa M F M, Rodrigues F, Guedes J, et al. Automated evaluation of patterned fabrics for defect detection [C]//Education and Training in Optics and Photonics (ETOP’99). International Society for Optics and Photonics, 2000: 403-407.
  * [26] Sanby C, Norton-Wayne L, Harwood R. The automated inspection of lace using machine vision [J]. Mechatronics, 1995, 5(2): 215-231.
  * [27] Farooq U, King T, Gaskell P H, et al. Machine vision using image data feedback for fault detection in complex deformable webs [J]. Transactions of the Institute of Measurement and Control, 2004, 26(2): 119-137.
  * [28] Cano J, Perez J C, Valiente J M, et al. Textile inspection with a parallel computer cluster [C]//5th Int. Conf. Quality Control by Artificial Vision (QCAV-2001), Le Creusot, France. 2001.
  * [29] Baykal I C, Muscedere R, Jullien G A. On the use of hash functions for defect detection in textures for in-camera web inspection systems [C]//Circuits and Systems, 2002. ISCAS 2002. IEEE International Symposium on. IEEE, 2002, 5: V-665-V-668 vol. 5.
  * [30] Chetverikov D, Hanbury A. Finding defects in texture using regularity and local orientation [J]. Pattern Recognition, 2002, 35(10): 2165-2180.
  * [31] Tsang C S C, Ngan H Y T, Pang G K H. Fabric inspection based on the Elo rating method [J]. Pattern Recognition, 2016, 51: 378-394.
  * [32] Kruizinga P, Petkov N, Grigorescu S E. Comparison of Texture Features Based on Gabor Filters [J]. IEEE Transactions on Image Processing, 1999, 11(10):142-147.
  * [33] Dalal N, Triggs B. Triggs, B.: Histograms of Oriented Gradients for Human Detection. In: CVPR [J]. Proceedings / CVPR, IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 1(12):886-893 vol. 1.
  * [34] Fazel M, Hindi H, Boyd S P. Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices [C]//American Control Conference, 2003. Proceedings of the 2003. IEEE, 2003, 3: 2156-2162.
  * [35] Dong W, Shi G, Li X, et al. Compressive sensing via nonlocal low-rank regularization [J]. Image Processing, IEEE Transactions on, 2014, 23(8): 3618-3632.
  * [36] Tibshirani R. Regression shrinkage and selection via the lasso [J]. Journal of the Royal Statistical Society. Series B (Methodological), 1996: 267-288.
  * [37] Zhoufeng L, Jiuge W, Quanjun Z, et al. Research on fabric defect detection algorithm based on improved adaptive threshold [J]. Microcomputer $\&$ Its Applications, 2013, 10: 016.
  * [38] LI C, ZHANG Z, LIU Z, et al. A novel fabric defect detection algorithm based on textural differential visual saliency model [J]. Journal of Shandong University (Engineering Science), 2014, 44(4): 1-9.
  * [39] LIU Z, ZHAO Q, LI C, et al. Fabric defect detection algorithm using local statistic features and global saliency analysis [J]. Journal of Textile Research, 2014, 11: 013.
  * [40] Imamo glu N, Lin W, Fang Y. A saliency detection model using low-level features based on wavelet transform [J]. Multimedia, IEEE Transactions on, 2013, 15(1): 96-105.
  * [41] Cao J, Zhang J, Wen Z, et al. Fabric defect inspection using prior knowledge guided least squares regression [J]. Multimedia Tools and Applications, 2015: 1-17.
  * [42] Ng M K, Ngan H Y T, Yuan X, et al. Patterned Fabric Inspection and Visualization by the Method of Image Decomposition[J]. IEEE Transactions on Automation Science $\&$ Engineering, 2014, 11(3):943-947.