## Abstract

Gabor wavelets are applied to develop an unsupervised novelty method for defect detection and segmentation that is fully automatic and free of any adjustable parameter. The algorithm combines the Gabor analysis of the sample image with a statistical analysis of the wavelet coefficients corresponding to each detail. The statistical distribution of the coefficients corresponding to the defect-free background texture is calculated from the coefficient’s distribution of the sample under inspection. Once the background texture features are estimated, a threshold is automatically fixed and applied to all the details, whose information is merged into a single binary output image in which the defect appears segmented from the background. The method is applicable to random, nonperiodic, and periodic textures. Since all the information to inspect a sample is obtained from the sample itself, the method is proof against heterogeneities between different samples of the material, in-plane positioning errors, scale variations, and lack of homogeneous illumination. Experimental results are presented. Some results are compared with other unsupervised methods designed for defect segmentation in periodic textures.

© 2009 Optical Society of America

## 1. INTRODUCTION

Surface defect detection is an important task in many fields of industrial inspection that have traditionally relied on trained human vision. The development of automated and objective inspection methods has played an important role in the industrial progress of the past decades. A large number of algorithms, devices, and systems for machine-vision-based inspection have been extensively reported and reviewed. Recently, for instance, Xie [1] has reviewed the techniques to detect local abnormalities in textures and has classified them into four categories, namely, statistical approaches, structural approaches, filter-based methods, and model-based approaches, with a comprehensive list of references to some recent works. Statistical and filter-based approaches have been very popular in the design of some of the key texture analysis methods applied to defect detection that are listed in the survey. Filter-based approaches can operate in the spatial domain, the frequency domain, or in the joint spatial/spatial-frequency domain.

Regarding the latter case, Gabor filters and wavelet transforms have been widely used (see, for instance, [2, 3, 4, 5, 6, 7, 8, 9]) and efficiently combined with multiresolution decomposition schemes to detect and segment flaws in textured surfaces. Many papers report a variety of applications to fabric inspection in the textile industry [2, 3, 5, 6, 7, 8, 9].

A fault or defect can be seen as a region where the homogeneity of the background texture is locally broken. Inspection may involve different levels of blob (defect) analysis which can be ordered as follows, according to the increasing amount of knowledge demanded: detection, localization, segmentation, and classification. Some defect detection algorithms are designed assuming a training stage of the machine vision system. During training, the system learns all possible object classes whose data are to be encountered when testing. But this is difficult to ensure in many industrial inspection processes either because abnormalities are rare or because there may be no complete data that describe all possible faults and only defect-free samples can be analyzed. For this reason, it is important for a vision system to be able to identify new or unknown data that was not available at the time of training, and in consequence, to discriminate between known and unknown object information during testing. This challenging task is called novelty detection [10].

Similarly, supervised defect detection algorithms rely on the fact that the characteristics of a defect-free region (background) are learned through the presentation of training defect-free templates to the machine vision system. As a result, any part of the test image that might look different from the background is classified as a defect. Supervised approaches provide very good performance when the defect-free background does not change from one image to another. However, when the defect-free background is liable to variability, supervised methods become unreliable, and the system may have to be retrained on a new set of images. As stated by Rohrmus in [11], p. 1547, “the key to inspection robustness is invariance against critical image distortions: rotation and translations (Euclidean motion), the scaling, and inhomogeneous illumination conditions,” as well as other possible changes in the imaging system occurring between the acquisition of the defect-free template and the test sample. In such situations, unsupervised defect detection algorithms that need minimal knowledge of both the defect and the defect-free background are clearly advantageous. Recently, some unsupervised methods have been proposed for defect detection in textured materials [8, 12, 13, 14, 15], mainly textile [8, 13, 14, 15]. Some methods use a statistical-based approach [8, 13, 15], while others use structural and filter-based approaches in the Fourier domain [12, 14]. All of them assume that the local defect occupies a relatively small area in the defect-free background and extract the information about both the background and the defect from the single image under inspection.

Abouelela *et al.* [13] analyze textured materials consisting of uniformly gray, plain fabric samples illuminated by an infrared source. Infrared imaging has been demonstrated to be useful to reduce variances due to dyed color patterns that could interfere with variances due to possible structural defects of the weave pattern [16]. In [8, 13], the input image is subdivided into blocks, assuming that the background fabric texture occupies the larger part of the image to be inspected and only a few blocks may correspond to defects. On the one hand, block size has to be small enough to be sensitive to the variations caused by possible defects. On the other hand, block size has to be large enough to be insensitive to the deterministic variations of the woven structure and minute unimportant fluctuations. The algorithms require fixing a number of parameters, such as the block sizes and thresholds, which limits their applicability. Kim and Kang [8] use Daubechies wavelet packet frames and a Gaussian mixture model, so the applicability of their method is advantageously extended to fabrics with nonuniform color patterns (e.g., warp and weft threads of different color) and defects that alter slightly the mean gray level of the region. These methods, however, would not be sensitive to small defects (in comparison with the subblock size) that could repeat regularly in other subblocks (see, for instance, Fig. 1 ).

The methods proposed in [13, 14] adopt a global approach based on bandpass filtering the Fourier transform of the sample image. Tsai and Huang deal with statistical textures with isotropic patterns in [12]. They apply a suppression mask in the frequency domain, an inverse Fourier transform, and a threshold for image binarization that work well in cases of low complexity. In our prior paper [14] we proposed a method for unsupervised defect detection in periodic textures. It was based on the structural feature extraction of the weave repeat from the Fourier transform of the sample image. These features were then used to define a set of multiresolution bandpass filters that operated in the frequency domain. The segmentation of the defective area was achieved after inverse Fourier transform, binarization, and merging of the information obtained at different scales. The method was applied to fabrics and demonstrated to be sensitive even to tiny defects (Fig. 1) and to other sorts of defects in a single image. However, it was specifically limited to the inspection of periodic textures.

In this work, we propose a new unsupervised novelty detection method for defect detection and segmentation in textures using Gabor filters. It uses the basic multiresolution Gabor filter scheme already described in a previous paper [2]. The method proposed here shows the following properties: it has no need of any defect-free references or a training stage, it has no adjustable parameters, and it is applicable to both random and periodic textures. In addition, the filter bank has been reduced to a half part since only the real part of the Gabor filters will be considered in the algorithm. From the analysis of every sample image under inspection, we extract a number of background texture features, and then we discriminate the irregular areas, provided any exist. The feature extraction is performed by a global analysis of the details (Gabor filtered images) obtained at different scales and orientations. In the analysis, we assume that the wavelet coefficients of pixels can be suitably fitted by Gaussian mixtures, more specifically, by combining two normal distributions, which provide information about the background texture and the defective area. The statistical procedures applied are robust against possible sample-to-sample variations or fluctuations. The efficacy of this statistical analysis was demonstrated in our previous paper [14], but for an unsupervised defect detection method limited to periodic patterns.

The method addresses superficial defects that can be imaged by a camera. It is primarily designed to handle grayscale digital images of texture samples; therefore, color defects would be detected only insofar as they were grayscale distinguishable. As we will show for fabric samples made of different color yarns, the performance of the method is not affected by the influence of color. In the case of patterned dyed fabrics and some fuzzy or velvet fabrics, the detection of structural defects can be improved by using near infrared (NIR) imaging [16, 17]. The woven structure of the sample appears enhanced in NIR images and flaw detection becomes less difficult.

The method’s performance is illustrated with a complete step-by-step example consisting of a flawed random texture obtained by producing a numerical local alteration (flaw) on the photographic image of a real sample of rough paper (random texture background). Afterwards, the method is applied to a variety of photographic images such as fabric and paper samples that exhibit either periodic or random texture. For periodic textures, we also compare the results obtained by the method proposed here with those obtained in [14].

The final use of this method would be its implementation for on-line texture inspection in an industrial setting. The method proposed in this paper is unsupervised, therefore it neither requires any training stage nor is dependent on environmental conditions. In the textile sector, for instance, there are several trade systems for automated defect detection based on image processing techniques: BarcoVision’s Cyclops, Elbit Vision System’s I-Tex, or Zellweger Uster’s Fabriscan, just to mention some of them. But all these fabric inspection systems are based on adaptive, neural networks. They use training sets of samples and, consequently, any variations in the environmental conditions may alter the system response.

## 2. BACKGROUND: MULTIRESOLUTION ANALYSIS USING GABOR FILTERS

Gabor functions have been extensively applied to texture analysis and surface inspection for defect detection, segmentation, and classification. Some detailed reviews of texture analysis techniques and wavelet analysis in industrial applications are recommended for further reading [1, 9, 18].

In the wavelet setting, images are decomposed into a low-resolution approximation and a set of details of different orientations and different resolution levels. In this work, a multiresolution analysis of digital images is carried out according to the scheme described in [2, 19] as a log-polar sampling of the frequency spectrum of the image. Details corresponding to four scales and four orientations result from the application of a set of dyadic $4\times 4$ Gabor filters. The Gabor functions can be represented by the continuous expressions

*p*value, the lower the frequency and, consequently, the resolution of the channel.

The Gabor functions are complex valued. They give rise to two families of real Gabor filters: the filters that take the even (real) part ${g}_{pq}^{e}$, and the filters that take the odd (imaginary) part ${g}_{pq}^{o}$. In the scheme used in [2], the complete set of 32 Gabor filters, including both even and odd parities, was considered. However, the results obtained using just one of the two parities frequently suffices for defect detection purposes (see, for instance, Fig. 2 ). When the Gabor filters are applied to a sample image $s(x,y)$ in the spatial domain, the details of the wavelet decomposition are obtained through convolution:

As described in [19], an efficient implementation of Gabor filtering in the spatial domain can be obtained using convolution masks and following a pyramidal algorithm. At the second and subsequent levels of the algorithm, the original image is decimated, having previously been convolved with a low-pass filter to remove the high-frequency terms and to avoid artifacts caused by subsampling (aliasing). In our method, Gabor filters have fixed size kernels of $11\times 11$ because they provide high fidelity approximating the original filters. Taking advantage of the Gabor filter separability, they can be sequentially applied, horizontally (by rows) and vertically (by columns) using kernels of $11\times 1$ on a one-dimensional base, thus maintaining relatively low computational cost [20]. To further simplify the method, only the imaginary part ${g}_{pq}^{o}$ of Gabor filters will be used in the following sections.

Taking into account the design of the filters, the texture sample should be imaged and digitized so that the grain size is $\sim 4\text{\hspace{0.17em} pixels}$. For instance, in the case of fabrics, samples are captured so that threads are approximately $4\text{\hspace{0.17em} pixels}$ wide.

## 3. ANALYSIS OF A FLAWED TEXTURE MODEL

Let us consider a digital image of a flawed sample $s(x,y)$. In order to find out how the defect propagates across scales and orientations in the details obtained from Gabor filters, we model this image as the sum (or superposition) of the background defect-free texture $t(x,y)$ and the intensity variations that account for the defective area $d(x,y)$ [14]. Figure 3 displays an example of such decomposition for a rough paper sample of random texture. The defect image $d(x,y)$ has positive and negative values in general. As it is commonly represented, gray areas correspond to values close to zero, bright areas to positive values, and dark areas to negative values. Assuming exclusively local defects in the sample image, the defect image $d(x,y)$ is equal to zero throughout except in the small area of the flaws. Sharp gradients appear at the edges of the defect area. The rapid variations at these edges give rise to high absolute values of the wavelet coefficients obtained in specific scales and orientations of the decomposition. See, for example in Fig. 4 , the wavelet coefficients ${s}_{33}^{o}$, ${t}_{33}^{o}$, and ${d}_{33}^{o}$ yielded by the Gabor wavelet decomposition of the images $s(x,y)$, $t(x,y)$, and $d(x,y)$ of Fig. 3. It is clear from this figure that the distribution of the coefficients over the flawed area is different from the distribution of the coefficients over the regular area corresponding to the background texture. The flawed area in ${s}_{33}^{o}$ contains both the darkest and the brightest pixels, thus accounting for the coefficients farthest from zero.

We will take advantage of this fact to separate the coefficients of the flawed area from the coefficients of the background texture to segment the defect from the background. If we look at the histograms of the wavelet coefficients of the details ${s}_{33}^{o}$ and ${t}_{33}^{o}$ of our example [Figs. 5a, 5b ] we see that they are similar. However, if we look at the magnified versions of those figures [truncated histograms of Figs. 5c, 5d] we note that ${s}_{33}^{o}$ ranges over a wider domain than ${t}_{33}^{o}$. The density distribution of the wavelet coefficients of ${s}_{33}^{o}$ is the convolution of the density distributions of ${t}_{33}^{o}$ and ${d}_{33}^{o}$. Those few pixels of ${s}_{33}^{o}$ with extreme coefficients basically correspond to the defect. They are encircled in the tail ends of the truncated ${s}_{33}^{o}$ histogram [Fig. 5c]. They account for the greater variability of the coefficient distribution in the flawed area than in the rest of the sample.

## 4. SEGMENTATION

After the Gabor analysis of the sample image, the segmentation of a possible defect can be accomplished in two steps. In the first step, the wavelet coefficients of each ${s}_{pq}$ (details of the decomposition) are classified and those corresponding to a defect area are identified. In the second step, the presumed defects found by this procedure in the details of all scales and orientations $(p,q=1,2,3,4)$ are merged.

In the first step, some statistical considerations concerning the probability distributions of wavelet coefficients are essential. Crouse *et al.* [21] developed a hidden Markov model for the wavelet transform that captures complex dependencies between the coefficients and non-Gaussian statistics, yet remains tractable so that they can be applied to real-world problems. To match the non-Gaussian nature of the wavelet coefficients, they modeled the marginal probability of each coefficient as a mixture density with a hidden state variable. To characterize the key dependencies between the wavelet coefficients, more specifically, their similarity due to location closeness or propagation through scales, they introduced Markovian dependencies between the hidden state variables. We consider the simple two-state model that provides satisfactory results in estimation and detection problems. For each detail, the mixture combines two zero-mean normal distributions, one associated with the coefficients of the flawed area with high standard deviation, including the coefficients at the ends (long tails) of the histogram, the other associated with the coefficients corresponding to the regular regions of the signal, with smaller standard deviation. Following the ideas presented in [8], Kim and Kang have recently used a Gaussian mixture model for texture classification and segmentation of some Brodatz texture images with success [22].

Based on these works, we model the probability density function of the wavelet coefficients corresponding to a particular scale and orientation $p,q$ as a Gaussian mixture of two zero-mean normal distributions,

A reasonably good estimate of the standard deviation ${\sigma}_{pq}^{t}$ of ${f\phantom{\rule{0.1em}{0ex}}}_{pq}^{t}\left(i\right)$ can be directly obtained from the coefficients of the sample under inspection ${s}_{pq}$ with $p,q=1,2,3,4$. However, the simple assumption that ${\sigma}_{pq}^{t}$ is equal to the standard deviation of the coefficients of ${s}_{pq}$ can be inaccurate. In the standard deviation, the distances from the mean are squared, so on average, large deviations (corresponding to potential defects) are weighted more heavily, and thus outliers in the coefficient set can heavily influence it. For example, it would be biased to higher values in case of inspecting noticeably flawed samples. We recall that no *a priori* information about the sample or the background texture or the presence or absence of flaws is available. Based on our former work [14], taking into account the spreading behavior of the defect across scales/orientations and assuming that the defect occupies a relatively small part of the image, it can be said that the vast majority of the coefficients of ${s}_{pq}$ and ${t}_{pq}$ are similar.

We will use the median absolute deviation $\left(\mathit{MAD}\right)$ [23] of ${s}_{pq}$ because it is a resistant statistical measure of the variability of a sample. $\mathit{MAD}$ first computes the residuals or deviations from the coefficient’s median and second, it takes the median of their absolute values, that is,

*i*and

*j*indicate points represented by their coordinates $(x,y)$ elsewhere. The $\mathit{MAD}$ is a robust statistic, being more resistant to outliers in the coefficient set than the standard deviation. In the $\mathit{MAD}$, the magnitude of the distances of a small number of outliers is irrelevant. Consequently, it can be assumed that $\mathit{MAD}\left\{{s}_{pq}\right\}=\mathit{MAD}\left\{{t}_{pq}\right\}$, and we call it simply ${\mathit{MAD}}_{pq}$. For a symmetric distribution about the mean, such as the normal distribution $N(0,{\sigma}_{pq}^{t})$, the ${\mathit{MAD}}_{pq}$ is the 3rd quartile, equivalently the 75th percentile $\left({p}_{pq,0.75}^{t}\right)$. It is known that the relationship between the 75th percentile of a normal distribution

*N*and its standard deviation

*σ*is ${p}_{0.75}\left(N\right)\approx 0.674492\sigma $. In our case,From Eqs. (4, 5) the standard deviation ${\sigma}_{pq}^{t}$ can be straightforwardly estimated from the histogram of the wavelet coefficients ${s}_{pq}$.

For every detail, we proceed now to the classification of pixels into two categories, those of the flawed area and those of the regular background area of the sample. The classification procedure is based on the magnitude of the wavelet coefficients. Since the manufactured materials to which the method is addressed have few flaws in general, we try to assure—as much as possible—that a pixel classified as belonging to the flawed area is actually part of the flaw. The opposite, that is, a pixel classified as belonging to the flawed area that is not actually part of any flaw, would be a false alarm at the pixel scale. Thus, we intend to keep the false alarm rate limited to a small value.

Once the normal distribution of the coefficients of the background texture $N(0,{\mathit{MAD}}_{pq}\u22150.674492)$ is determined, a double threshold is fixed at $3{\sigma}_{pq}^{t}\approx 4.45{\mathit{MAD}}_{pq}$ and applied to each ${s}_{pq}$, which has positive and negative values, according to the expression

The second step is the integration of the binarized images obtained for each scale and orientation. The information is combined by means of addition. Before adding, details of different scale must be resized so that the addition is applied to images of the same size. To do this, a pixel in a low scale image is properly replicated to produce a square with the appropriate number of identical pixels. To reduce misclassification errors, we consider that a pixel will eventually correspond to a defect if the aforementioned addition is greater than or equal to 2 for that pixel. This means that such pixel would be out of the double threshold in at least two wavelet coefficient distributions (details). With this constraint the final false alarm rate for pixels would be less than ${10}^{-3}$. Figure 7b shows the result obtained for the image $s(x,y)$ of our model (Fig. 3) after this second step.

Apart from the defect, it can be seen that some small points corresponding to the irregularities of the texture (rough paper) are detected and segmented by the algorithm. A simple morphological opening applied to the binary image [Fig. 7b] is useful to limit the possibly excessive sensitivity of the algorithm and to reduce noise [Fig. 7c].

## 5. RESULTS

The algorithm described in the previous sections has been applied to a variety of real samples (pieces of fabric or paper in Figs. 8, 9 ) containing different types of defects and background texture (periodic and random textures). We probe the algorithm on test samples that usually are out of the range in the vast majority of flaw detection systems because of their complexity. The test samples are difficult to analyze even by some unsupervised methods mentioned in the introduction [8, 12, 13, 14, 15]. Thus, for instance, the methods based on the definition of blocks generally fail when they are applied to samples such as those of Figs. 8a, 8b, 8c, 8i, 8j, 9b, 9c. In these samples, the defect is tiny (similar in size to the basic repeat or grain of the background structure), and it often appears repeated with some regularity. Sometimes there is more than one defect in the sample and the defects are not always of the same type. In such situations it is difficult to define a block with size small enough to be sensitive to the variations caused by possible defects and, at the same time, large enough to be insensitive to the deterministic variations of the texture. As for the Fourier-based methods, they are more limited when dealing with random and nonperiodic background textures [Figs. 9a, 9b] than with regular and periodic structures. We will see an example of this immediately.

We start comparing the results obtained using the algorithm described in this paper with those obtained by the algorithm specifically designed for periodic textures described in a previous paper [14]. Photographic images of 12 woven fabric samples were analyzed by both unsupervised methods, and they provided satisfactory results in the detection and segmentation of defects. Some of them are shown in Fig. 8. The comparison leads us to confirm a good agreement in the areas corresponding to flaws segmented by the two methods. Even for the difficult cases shown in Fig. 1 [now in Figs. 8a, 8b] and for the null case—a faultless actual fabric sample [Fig. 8f]—the results are highly satisfactory. In the case shown in Fig. 8e the detection of two defects occurs at different scales and the effects of resizing images before integration (Section 3) becomes observable in the result obtained by the unsupervised-Gabor-filtering-based method. Although the algorithm reported in [14] exhibits a neat performance and a more accurate segmentation in general, it can be said that the Gabor-filtering-based method proposed in this paper can also be applied to periodic textures with very good results.

Now we apply the unsupervised Gabor filtering based method to real samples with textures that are beyond the applicability of the method for periodic structures reported in [14], such as all samples of random texture [e.g., pieces of paper in Fig. 9a, 9b]. Moreover, some samples of periodic texture cannot be analyzed by that method either. For example, if the warp and weft lines of a fabric are excessively slanted [see Fig. 9c], the method will fail, because it is based on a precise determination of the periodic structure of the fabric sample and assumes the approximate alignment of the threads in the vertical and the horizontal directions [14]. There is also a test sample of actual fabric that is composed of a mixture of two periodic textures at close to 50% in area each [Fig. 9d]. This problem is too difficult for the method to test periodic structures [14]. We have tried it and proved that it fails to identify the main fabric structure and it cannot build the set of filters in the frequency domain. However, all these test samples are successfully analyzed by the unsupervised-Gabor-filtering-based method. Figure 9 shows on the right the segmentation of flaws obtained for each case.

## 6. CONCLUSIONS

We have applied Gabor wavelets to develop an unsupervised novelty method for defect detection and segmentation that is fully automatic and free of any adjustable parameter. As an additional very important property it is applicable to samples of random as well as periodic textures. Since all the information to inspect a sample is obtained from the sample itself and it does not require the comparison with any standard (faultless) sample or training defect set, the unsupervised novelty detection method proposed in this paper is a method that proof against heterogeneities between different samples of the material, in-plane positioning errors, scale variations, and lack of homogeneous illumination.

A detailed presentation of the Gabor filtering based algorithm has been illustrated step by step for a model sample. The algorithm combines the Gabor analysis of the sample image with a statistical analysis of the wavelet coefficients corresponding to each detail. The statistical distribution of the coefficients corresponding to the background texture is estimated from the coefficient’s distribution in the sample under inspection. As a result, the background texture features are identified. The following steps of the sequence consist of thresholding and merging all details, rescaling whenever necessary.

In the experiments, we have applied this new method to a variety of random and periodic textured test samples, more specifically, to photographic images of real pieces of materials of industrial interest such as paper and fabric. Although the test samples were difficult cases for unsupervised flaw detection, the results were satisfactory and the defect segmentation was correct in all cases. Not only is the sensitivity of the algorithm high, but the specificity is also good, since a negative result was obtained when analyzing a defect-free sample. We have compared this Gabor filtering based method with our unsupervised method specifically designed for periodic structures and limited to the analysis of materials with periodic texture [14]. The results show a very good agreement between both methods. Although the algorithm reported in [14] performs more accurate defect segmentation in well-aligned fabrics, the method proposed in this paper also obtains very good results for these samples. Occasionally, the decimation (decrease of the sample image resolution) associated with the implementation of the Gabor wavelets yields lower, but still acceptable, accuracy of the segmented area. On the other hand, the Gabor-filtering-based method proposed in this paper has a wider applicability. It is applicable to random textured samples and is resistant to in-plane rotation of the periodic textured sample, mixtures of periodic structures in the same sample, and other difficult problems that the method reported in [14] could not overcome.

## ACKNOWLEDGMENTS

The authors acknowledge the financial support of the Spanish Ministerio de Educación y Ciencia and the Fondo Europeo de Desarrollo Regional (FEDER) under project DPI2006-05479.

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