Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images

ARTICLE IN PRESS Pattern Recognition 43 (2010) 1917–1931

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Anisotropic diffusion with generalized diffusion coefficient function for defect detection in low-contrast surface images Shin-Min Chao, Du-Ming Tsai  Department of Industrial Engineering and Management, Yuan-Ze University, 135 Yuan-Tung Road, Nei-Li, Tao-Yuan, Taiwan, ROC

a r t i c l e in fo

abstract

Article history: Received 16 May 2009 Received in revised form 10 November 2009 Accepted 5 December 2009

In this paper, an anisotropic diffusion model with a generalized diffusion coefficient function is presented for defect detection in low-contrast surface images and, especially, aims at material surfaces found in liquid crystal display (LCD) manufacturing. A defect embedded in a low-contrast surface image is extremely difficult to detect, because the intensity difference between the unevenly illuminated background and the defective region is hardly observable and no clear edges are present between the defect and its surroundings. The proposed anisotropic diffusion model provides a generalized diffusion mechanism that can flexibly change the curve of the diffusion coefficient function. It adaptively carries out a smoothing process for faultless areas and performs a sharpening process for defect areas in an image. An entropy criterion is proposed as the performance measure of the diffused image and then a stochastic evolutionary computation algorithm, particle swarm optimization (PSO), is applied to automatically determine the best parameter values of the generalized diffusion coefficient function. Experimental results have shown that the proposed method can effectively and efficiently detect small defects in various low-contrast surface images. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Defect detection Surface inspection Anisotropic diffusion Entropy criterion Particle swarm optimization

1. Introduction Digital image processing has become a key technology in the area of automated visual inspection (AVI). The manual activity of inspection can be subjective and highly dependent on the experience of human inspectors. Intelligent AVI systems provide significant advantages over traditional human inspection with high process repeatability, accuracy and speed. In this paper, we consider the task of AVI in low-contrast surfaces, and especially focus on the key components used for thin film transistor-liquid crystal displays (TFT-LCDs). The inspection of defects in LCD panel surfaces ensures the display quality and improves the yield in LCD manufacturing. In a low-contrast surface image, the gray levels of a defect and the background are hardly distinguishable and there are no clear edges between the defect and its surroundings. For demonstration purposes, Fig. 1 presents two low-contrast surface images of backlight panels used for LCD. Fig. 1(a1) shows a faultless image, and Fig. 1(b1) shows a defective version of the panel. It can be seen from Fig. 1(b1) that the defect is difficult to be found in the low-contrast background. In order to visualize the subtle defect, the gray values of the panel images are linearly stretched

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E-mail addresses: [email protected], [email protected] (D.-M. Tsai).

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between 0 and 255 for an 8-bit display. Fig. 1(a2) and (b2) shows the contrast-stretched images of Fig. 1(a1) and (b1), respectively. The enhanced panel image in Fig. 1(b2) shows the defect clearly, but it also intensifies the background texture and non-uniform illumination. It increases the difficulty of defect detection if the inspection task is carried out in such an enhanced image. Fig. 1(a3) and (b3) illustrate the gradient images of Fig. 1(a1) and (b1), respectively. These resulting images reveal that the characteristic of a low-contrast surface image under uneven illumination invalidates the use of gradient magnitude or gray-level thresholding to identify local anomalies. There are a few machine vision techniques developed in recent years for defect detection in low-contrast images. Saitoh [1] presented a machine vision scheme for the inspection of brightness unevenness in LCD panel surfaces. An edge detection algorithm was used to identify discontinuous points first. A genetic algorithm was then applied to extract the boundary of anomalous brightness region. Kim et al. [2] studied the detection of spot-type defects in LCD panel surfaces. An adaptive multiplelevel thresholding method based on the statistical characteristics of the local area is applied to segment the defect from the background surface. Jiang et al. [3] used a luminance meter as the sensing device for detecting non-uniform brightness in LCD panels. Analysis of variance (ANOVA) and exponential moving average techniques were applied to detect the presence of nonuniform brightness in the panel surface.

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Fig. 1. Low-contrast sample images of backlight panels: (a1) faultless surface image; (b1) defective surface image; (a2), (b2) respective contrast–stretched images and (a3), (b3) respective gradient images.

Lee and Yoo [4] proposed a surface fitting approach to identify uneven-brightness blemish in LCD panels. A bivariate polynomial model was used to estimate the background surface. The graylevel of each pixel in the image was subtracted from the corresponding gray-level of the estimated background surface. A pixel with gray-level difference larger than some predetermined threshold was identified as a defective one. Wang and Ma [5] also proposed a recursive polynomial surface fitting method for brightness blemish detection in LCDs. Since the inspection surfaces involve low-contrast defects in an unevenly illuminated background, the currently available image smoothing or sharpening techniques may either oversmooth the small defects while removing the uneven illumination in the background, or over-enhance the textured pattern and uneven-brightness of the background while intensifying the contrast of the defect with respect to its surroundings. In this paper, an anisotropic diffusion scheme is proposed to detect the subtle defects in various low-contrast surface images. The proposed anisotropic diffusion method can simultaneously smooth the noisy and unevenly illuminated background and enhance the low-contrast defect in the inspection image. Anisotropic diffusion model was first introduced by Perona and Malik [6] in image processing for scale-space description and edge detection. It has been widely used as an adaptive edgepreserving smoothing technique for edge detection [7,8], image restoration [9–11], image smoothing [12,13], image segmentation [14,15] and texture segmentation [16]. The anisotropic diffusion approach is basically a modification of the linear diffusion (or heat equation), and the continuous anisotropic diffusion is given by @I t ðx; yÞ ¼ div½c t ðx; yÞ  rI t ðx; yÞ @t

ð1Þ

where I t ðx; yÞ refers to the image at time t, div the divergence operator, rI t ðx; yÞ the gradient of the image and c t ðx; yÞ the diffusion coefficient. The idea of anisotropic diffusion is to

adaptively choose c t such that intra-regions become smooth while edges of inter-regions are preserved. The diffusion coefficient c t must be a nonnegative function of gradient magnitude so that small variations in intensity such as noise or shading can be well smoothed, and edges with large intensity transition are retained. It is generally given by an exponential function or an inverse quadratic function, and determined by the gradient magnitude with respect to a predetermined edge strength threshold. You et al. [17] gave an in-depth analysis of the behavior of the Perona–Malik anisotropic diffusion model by considering the anisotropic diffusion as the steepest descent method for solving an energy minimization problem. Barash [18] addressed the fundamental relationship between anisotropic diffusion and adaptive smoothing. He showed that an iteration of adaptive smoothing PP I t þ 1 ðx; yÞ ¼

i

j I t ðx þi; y þ jÞwt ðx þ i; yþ jÞ

PP i

j wt ðx þ i; yþ jÞ

ð2Þ

is an implementation of the discrete version of the anisotropic diffusion equation if the weight wt in Eq. (2) is taken as the same of the diffusion coefficient c t in Eq. (1). Gilboa et al. [19] proposed a forward and backward (FAB) adaptive diffusion process to enhance edge and smooth noise in the image. The FAB diffusion model involves four ad hoc parameters, of which two critical threshold values of gradient must be manually and carefully chosen for the success of the diffusion result. The smaller threshold value determines the use of a forward function, while the larger threshold value determines the use of a backward function. A discontinuous diffusion function is, therefore, applied since nothing is done for the gradient magnitude between the two threshold values. Wang et al. [20] also proposed an improved FAB diffusion algorithm for edge enhancement and noise reduction. The conventional diffusion model can effectively perform adaptive smoothing for intra-regions in an image. However, it

ARTICLE IN PRESS S.-M. Chao, D.-M. Tsai / Pattern Recognition 43 (2010) 1917–1931

can only passively stop the diffusion process to preserve original gray values of edges in inter-regions. For defect detection in a low-contrast image, the conventional diffusion model can only smooth the faultless background, but cannot enhance the lowcontrast defects. The diffused result may still be a low-contrast image. Recently, Chao and Tsai [21] presented an anisotropic diffusion scheme to detect defects in low-contrast surface images, in which the diffusion function is given by an inverse quadratic function. Their approach unifies both smoothing and sharpening processes in diffusion equation. It carries out the smoothing process in faultless areas to make the background uniform and performs the sharpening operation in the defective area to enhance anomalies. In their model, the selection of parameter values has the critical effect for the success of the detection result. However, the parameter values must be manually fine-tuned by try-and-error for different inspection targets. In order to achieve better diffusion results, a large number of diffusion iterations were required in their model to smooth the background and intensify the low-contrast defect. A larger iteration number means a larger amount of computation time is required, which deters the method for on-line, real-time implementation in manufacturing. It cannot be compensated with a small iteration number by varying the main parameter value of the edge strength threshold in the conventional diffusion coefficient function. In this paper, we propose an anisotropic diffusion model with a generalized diffusion coefficient function that aims to enhance the gray-level difference between local anomalies and the background to detect defects in low-contrast surface images. The FAB diffusion model [19] and the Chao–Tsai diffusion model [21] can be considered as special cases of the proposed diffusion model in this paper. The proposed method can flexibly change the function curve by controlling the curve parameter values of the diffusion coefficient function. It activates a smoothing process in faultless regions to make the background uniform, and performs a sharpening process in defective regions to enhance anomalies. The proposed method can also automatically choose appropriate parameter values of the generalized diffusion coefficient function. In order to automatically find the best parameter values, an entropy criterion is proposed as the performance measure of a diffused image, and a particle swarm optimization (PSO) procedure is developed to search for the best parameter values of the generalized diffusion coefficient function under a given small number of diffusion iterations. It is therefore both effective and efficient for defect detection applications. The proposed method can distinctly enhance low-contrast defects and uniformly smooth the background without intensifying textured patterns and uneven illumination so that a simple binary thresholding can be effectively applied to segment defects in the diffused image. This paper is organized as follows. In Section 2, we first review the Perona–Malik anisotropic diffusion equation, and then discuss the generalized diffusion coefficient function that adaptively performs the smoothing and sharpening operations. The automatic parameters search procedure using PSO is presented in Section 3. Section 4 presents the experimental results from a variety of backlight panel, LCD glass substrate and brightness enhancement film surfaces. The performance comparison with the Chao–Tasi diffusion model is also discussed. Finally, Section 5 gives the conclusion of our research.

2. Adaptive anisotropic diffusion model for defect detection 2.1. Perona–Malik anisotropic diffusion model Let I t ðx; yÞ be the gray-level at coordinates ðx; yÞ of a digital image at iteration t, and I 0 ðx; yÞ the original input image. The

1919

continuous anisotropic diffusion in Eq. (1) can be discretely implemented by using four nearest neighbors and the Laplacian operator [6] I t þ 1 ðx; yÞ ¼ I t ðx; yÞ þ

4 1X ½ci ðx; yÞ  rI it ðx; yÞ 4i¼1 t

ð3Þ

where rI it ðx; yÞ, i= 1, 2, 3 and 4, represent the gradients of four neighbors in the north, south, east and west directions, respectively, i.e.

rI 1t ðx; yÞ ¼ I t ðx; y1ÞI t ðx; yÞ; rI 3t ðx; yÞ ¼ I t ðx þ1; yÞI t ðx; yÞ;

rI 2t ðx; yÞ ¼ I t ðx; y þ1ÞI t ðx; yÞ rI 4t ðx; yÞ ¼ I t ðx1; yÞI t ðx; yÞ

where cit ðx; yÞ is the diffusion coefficient associated with rI it ðx; yÞ, and is considered as a function of the gradient rI it ðx; yÞ in the Perona–Malik model, i.e. c it ðx; yÞ ¼ gðrI it ðx; yÞÞ For the sake of simplicity, rI it ðx; yÞ is subsequently denoted by rI. The function gðrIÞ has to be a nonnegative, monotonically decreasing function with gð0Þ ¼ 1 and limjrIj-1 gðrIÞ ¼ 0. The function gðrIÞ should result in low coefficient values at highgradient edges to preserve the gray levels of edges, and high coefficient values for low-gradient pixels within an image region so that the region can be uniformly smoothed. In the Perona– Malik anisotropic diffusion model, a possible diffusion coefficient function is given by gðrIÞ ¼ 1=½1 þ ðjrIj=KÞ2 

ð4Þ

where the parameter K is a constant, and acts as the edge strength threshold. Parameter K in the diffusion coefficient function must be finetuned for a particular application. If the K value is too large, the diffusion process will over-smooth and result in a blurred image. In contrast, if the K value is too small, the diffusion process will stop smoothing in early iterations and yield a restored image similar to the original one. Let fðrIÞ be a flux function [6] defined by

fðrIÞ ¼ gðrIÞ  rI

ð5Þ

A large flux value indicates a strong effect of smoothness. Figs. 2 and 3 depict the diffusion coefficient function and the flux function in Eqs. (4) and (5), respectively. For a given K value, it can be seen from Fig. 2 that the diffusion coefficient function in Eq. (4) drops dramatically and approximates to zero when the gradient magnitude jrIj is larger than 4K. That is, the diffusion stops as soon as jrIj 4 4K. The maximum smoothness occurs at jrIj ¼ 1K, as shown in the corresponding flux function in Fig. 3. The classical Perona–Malik model can effectively smooth intra-regions in an image. However, it can only stop the diffusion process to preserve the original gray values of edges in inter-regions. In a lowcontrast image, the Perona–Malik model can smooth the faultless background but cannot distinctly enhance subtle defects. Therefore, the diffusion result may still be a low-contrast image and defects cannot be reliably identified in the diffused image. It is not an acceptable result for low-contrast surface inspection. 2.2. Generalized diffusion coefficient function In an anisotropic diffusion equation, the diffusion coefficient function gðrIÞ plays an important role as a ‘‘fuzzy’’ edge detector to control the diffusion strength. It implies that the anisotropic diffusion with different diffusion coefficients will generate different diffusion results in the same image. In this paper, we propose a generalized diffusion coefficient function based on a linear-logarithmic function. It controls the

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1.0

g (∇I)

0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5

∇I/K Fig. 2. Graph of the diffusion coefficient function: gðrIÞ ¼ 1=½1 þ ðjrIj=KÞ2 .

0.5

 (∇I)

0.4 0.3 0.2 0.1 0.0 0

1

2

3

4

5

∇I/K Fig. 3. Graph of the flux function: fðrIÞ ¼ f1=½1þ ðjrIj=KÞ2 g  rI.

function curve by changing the parameter values of the linearlogarithmic function so that different function curves can be flexibly used for different diffusion applications. The proposed generalized diffusion coefficient function gðrIÞ is given by gðrIÞ ¼ a  lnðjrIjþ bÞ þ g

ð6Þ

where a, b and g are curve parameters. Parameter a is the slope of the linear-logarithmic function, which determines the increment (a 40) or decrement (a o0) of the function. A decreasing diffusion coefficient provides both smoothing and sharpening effects for the image, and an increasing function gives only a smoothing effect to the image. Since the generalized diffusion coefficient function is used for sharpening defects and smoothing the background in a low-contrast surface image, parameter value of a is always negative in this study. A larger value of |a| will result in a larger sharpening effect. Parameter g is the intercept of the linear-logarithmic function, which gives the starting diffusion coefficient value of gðrIÞ at rI ¼ 0. A larger value of g will provide a strong smoothing effect, whereas a smaller value of g will give a weak smoothing effect. Offset b controls the curvature of the function curve, which can fine-tune the speed of the diffusion process. As b value is increased, the function curvature is decreased and, therefore, the diffusion coefficient value will be slowly decreased. Note that jrIj þ b must be a positive real number. If gðrIÞ in Eq. (6) is a monotonically decreasing function (a o0), the function value drops as rI increases and then crosses over zero. The zero-crossing point, denoted by rI 0 , is given by

rI 0 ¼ ðeðg=aÞ bÞ

ð7Þ

The zero-crossing point determines the execution of a smoothing or a sharpening process in the diffusion model. If the magnitude of the gradient jrIjo rI 0 , the diffusion model will perform the smoothing process. In contrast, the diffusion model will perform the sharpening process for jrIj4 rI 0 .

Different combinations of the parameter values for a, b and g will generate different curves of the diffusion coefficient functions and, therefore, perform different diffusion effects. Fig. 4 depicts three diffusion coefficient curves of Eq. (6) with different combinations of (a, b, g)=(  0.3, 0.1, 0.4), (  0.8, 1, 0.9) and (  0.4, 0.5, 0.2). The corresponding zero-crossing points occur at jjjj ¼ 3:69; 2:08 and 1:15, respectively. If the curve of the diffusion coefficient function has a large starting value of gðrIÞ at rI ¼ 0 and drops slowly (e.g., (a, b, g)=(  0.3, 0.1, 0.4)), then the proposed diffusion model will perform the sharpening operation too late and the subtle defect in the low-contrast image cannot be well enhanced. Conversely, if the curve of the diffusion coefficient function has a small starting value of gðrIÞ at rI ¼ 0 and drops rapidly (e.g., (a, b, g) =( 0.4, 0.5, 0.2)), then the proposed diffusion model will carry out the sharpening process in early diffusion iterations and enhance both anomalies and the details in the background region. Therefore, a proper curve parameter setting is critical for defect detection in the low-contrast image. Fig. 5 illustrates the flux functions for the above three combinations of curve parameter values. The positive flux value indicates a smoothing process, whereas the negative flux value gives a sharpening process. The processing strength is proportional to the magnitude of the flux function. As a demonstration test image, Fig. 1(b1) displays the image of a defective backlight panel. Fig. 1(b2) is the enhanced version of the test image. Fig. 6 shows the diffusion results of the test image using the above three combinations of the parameters a, b and g. The number of iterations is set to 20 for the test image. With curve parameters (a, b, g) =( 0.3, 0.1, 0.4), the low-contrast defect cannot be enhanced in the resulting image, as seen in Fig. 6(b). It indicates the combination of parameters (a, b, g) performs a poor sharpening effect and over-smoothes the whole image. With (a, b, g)=(  0.4, 0.5, 0.2), Fig. 6(d) shows that the proposed diffusion model over-sharpens the image. It generates many falsely sharpened objects and noise. By using (a, b, g)=(  0.8, 1, 0.9),

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1921

1.0 (, , ) = (-0.3, 0.1, 0.4) (, , ) = (-0.8, 1, 0.9) (, , ) = (-0.4, 0.5, 0.2)

g (∇I)

0.5

0.0

-0.5

-1.0 0

1

2

3

4

5

6

∇I Fig. 4. Plots of diffusion coefficient function gðrIÞ with three different combinations of curve parameters.

1.0 (, , ) = (-0.3, 0.1, 0.4) (, , ) = (-0.8, 1, 0.9) (, , ) = (-0.4, 0.5, 0.2)

 (∇I)

0.5

0.0

-0.5

-1.0 0

1

2

3

4

5

6

∇I Fig. 5. Plots of flux function fðrIÞ ¼ gðrIÞ  rI with three different combinations of curve parameters.

the diffusion model presents a good diffusion result that enhances the low-contrast defect and removes noise in the resulting image, as seen in Fig. 6(c). The experiment on the test image reveals that the success of the diffusion process for defect detection in low-contrast images critically relies on the proper choice of the parameter values of the generalized diffusion coefficient function.

3. Automatic parameters search The generalized diffusion coefficient function provides the flexibility to design a best diffusion model for specific images and applications. The proposed diffusion coefficient function of a linear-logarithmic form involves up to three curve parameters a, b and g. In the diffusion model, there is actually one additional parameter, the number of diffusion iterations T. The value of T

determines the processing time of the whole diffusion procedure. The larger the value of T, the more computation time it takes. For defect detection applications, the processing time should be as small as possible so that it can be implemented for on-line inspection in manufacturing. In this study, the number of iterations T is predetermined, and a small T value is initially given. The best parameter values of a, b and g are then determined accordingly. The initial setup of T value will be increased only if no proper combinations of a, b and g exist for the success of defect detection in the training image. To automatically choose the proper combination of the parameter values of the linear-logarithmic function, we must first design a performance metric to evaluate the diffusion quality of an image, and then develop a search algorithm that finds the best parameter values by optimizing the performance metric. In this study, we propose an entropy criterion to evaluate the diffusion quality and apply particle swarm optimization (PSO)

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H = 8.97

H = 8.53

H = 5.80

H = 6.83

Fig. 6. Diffusion results of the defective backlight panel image using different combinations of curve parameters: (a) original defective image; (b)–(d) diffusion results from (a, b, g) = ( 0.3, 0.1, 0.4), ( 0.8, 1, 0.9) and ( 0.4, 0.5, 0.2) and (e)–(h) normalized Laplacian images of (a)–(d), respectively. (The number of diffusion iterations T= 20 for the test image.)

algorithm to search for the best fit. They are individually described in the following subsections.

3.1. Entropy criterion In a low-contrast surface image, a local anomaly has smooth changes of brightness from its surrounding region and, therefore, the defect area and its surrounding background area are hardly distinguishable. Observing the previous diffusion results in Fig. 6(b)–(d), one can find if the proposed diffusion model performs only the smoothing process with minor sharpening effect, then the gray levels between defect and background areas are still close to each other. As seen in Fig. 6(b), the defect still cannot be visually observed in the diffused image. If the proposed diffusion model carries out excessive sharpening, then both the defect and background noise are enhanced. Fig. 6(d) demonstrates the effect of over-sharpening. It is unable to verify that sharpened objects are true anomalies in the diffused image. However, the diffusion model with proper parameter values of the linearlogarithmic diffusion coefficient function can well perform the required smoothing and sharpening operations so that the defect area is distinctly intensified and the background region is uniformly smoothed, as seen in Fig. 6(c). To automatically determine the parameter values of a, b and g in the linear-logarithmic function, a performance measure must be given. It should generate an optimal (either max. or min.) objective value for the best combination of a, b and g that can properly sharpen defects and smooth the background. In this study, an entropy criterion is proposed as the performance measure of a diffused image. In information theory, entropy is a measure of the uncertainty associated with a random variable. A large value of entropy indicates high degree of uncertainty and high complexity about an event. Since the gray-level variation is small in the original low-contrast image, the second-order derivative of the gray-level function is used for measuring the complexity in the diffused image. It is desired that both oversmoothed and over-sharpened images result in higher entropy

values, and the best diffused image generates a minimum entropy value. In this study, the entropy criterion H of an image I t at diffusion iteration t is given by XX Lt ðx; yÞ  ln Lt ðx; yÞ ð8Þ HðI t Þ ¼  x

y

where jr2 I t ðx; yÞj Lt ðx; yÞ ¼ P P 2 u v jr I t ðu; vÞj

ð9Þ

where Lt ðx; yÞ is the normalized Laplacian of the image I t ðx; yÞ at 2 coordinates ðx; yÞ. The discrete Laplacian r I t ðx; yÞ in Eq. (9) is obtained by

r2 I t ðx; yÞ ¼ ½I t ðx þ 1; yÞ þ I t ðx1; yÞ þ I t ðx; y þ1Þ þI t ðx; y1Þ4I t ðx; yÞ: Since the magnitude of Laplacian for the original low-contrast image and over-smoothed image are generally very small, the Laplacian magnitude is normalized in Eq. (9) to ensure the resulting entropy values are distinctly large. The objective function that finds the best combination of parameter values of a, b and g for the linear-logarithmic diffusion coefficient function is therefore modeled as XX LT ðx; yÞ  ln LT ðx; yÞ Minimize HðIT Þ ¼  ða;b;gÞ

subject to

x

a o0;

y

b 4 0 and g 4 0

ð10Þ

where LT ðx; yÞ is the normalized Laplacian of the diffused image at a given number of diffusion iterations T. Note that the number of iterations of T can also be a decision variable in the optimization model. Since the computation time of the diffusion process for a given image size is completely determined by T, the upper bound of T can be set at the maximum inspection time allowed in manufacturing. As demonstration sample images, Fig. 6(e)–(h) shows the normalized Laplacian images of Fig. 6(a)–(d), respectively. The normalized Laplacian images of the original gray-level image

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(Fig. 6(a)) and the over-smoothed diffusion image (Fig. 6(b)) present complicated image contents and, therefore, both have high entropy values of 8.97 and 8.53, respectively. The normalized Laplacian image of the over-sharpened image in Fig. 6(h) also displays a complicated diffusion result with many false defects and results in a high entropy value of 6.83. The diffused image shown in Fig. 6(c) apparently has the best detection result. Its corresponding normalized Laplacian image is less complicated than those of the original gray-level image and the oversmoothed and over-sharpened images, and yields a minimum entropy value of 5.80. A particle swam optimization technique is developed in this study to search for the best combination of parameters (a, b, g) based on the objective function of Eq. (10). It is described in detail in the following subsection. 3.2. Particle swarm optimization search algorithm Particle swarm optimization (PSO) is an evolutionary computation technique originally proposed by Kennedy and Eberhart [22]. PSO resembles the social interaction from a group of flying birds. Individuals in a flock of flying birds are evolved by cooperation and competition among the individuals themselves through generations [23]. Each individual, named as a ‘‘particle’’, adjusts its flying according to the flying experience from itself and companions. Each particle with its current position represents a potential solution to the problem. In this study, each particle is treated as a point in a three dimensions space since the proposed generalized diffusion coefficient function has three unknown parameters a, b and g. In PSO, a number of particles, which simulate a group of flying birds, are simultaneously used to find the best fitness in the search space. At each iteration, every particle keeps track of its personal best position by dynamically adjusting its flying velocity. The new velocity is evaluated by its current flying velocity and the distances of its current position with respect to its previous best local position and the global best position. After a sufficient number of iterations, the particles will eventually cluster around the neighborhood of the fittest solution. Let the ith particle be represented as wi ¼ ½wi1; wi2 ; wi3  ¼ ½ai ; bi ; gi , which gives a possible combination of the three parameter values a, b and g in the linear-logarithmic diffusion coefficient function. The best previous position, i.e. the position giving the best fitness value, of particle i is recorded and represented as pi ¼ ½pi1; pi2 ; pi3 . Assume there are a total of P particles in the PSO. Let pg ¼ ½pg1 ; pg2 ; pg3  denote the best particle that gives the current optimal fitness value among all the particles in the population. The rate of the position change, i.e. the velocity, for particle i is represented by vi ¼ ½vi1 ; vi2 ; vi3 . Each particle moves over the search space with a velocity dynamically adjusted according to its historical behavior and its companions. The velocity and position of a particle are updated according to the following Eqs. [22,24]: ¼ vij þ c1  rand1  ðpij wij Þ þ c2  rand2  ðpgj wij Þ vnew ij

ð11Þ

¼ wij þ vnew wnew ij ij ;

ð12Þ

8i ¼ 1; 2; . . . ; P; j ¼ 1; 2; 3

where c1 and c2 are two positive constants, rand1 and rand2 are two random numbers in the range 0 and 1. The second term in the right-hand side of the velocity equation in Eq. (11) is interpreted as the ‘‘cognition’’ part, which represents the private thinking of the particle itself. The third term in Eq. (11) is the ‘‘social’’ part, which represents the collaboration among the particles [23,24]. The two positive constants c1 and c2 are the weights used to regulate the acceleration of self-cognition (a local best) and social interaction (a global best). Kennedy and Eberhart

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[22] and Shi and Eberhart [23] suggested c1 = c2 =2 since they make the weights for ‘‘cognition’’ and ‘‘social’’ parts to be 1 on average. The same values are also adopted in this study. The velocity equation in Eq. (11) calculates the particle’s new velocity in each dimension according to its previous velocity and the distances of its current position from its own best experience and the group’s best experience. Then the particle moves toward a new position according to the position update equation in Eq. (12). The performance of each particle is measured by the fitness value, i.e. the objective function value in Eq. (8). Since the proposed generalized diffusion coefficient function involves bounded ranges of parameter values, only the positions in the feasible space are recorded when calculating the local best pi in Eqs. (11) and (12) in the PSO search procedure. The PSO search procedure can now be formally presented for the proposed diffusion model with generalized diffusion coefficient function under the entropy criterion. The following symbols are the notation used in the PSO search algorithm. Notation: C = the number of search iterations in PSO. T= the number of diffusion iterations. P= the number of particles in the swarm. I t ðx; yÞ: the input image of size M  N at diffusion iteration t, x= 0, 1, 2, y, M  1 and y=0, 1, 2, y, N 1. I 0 ðx; yÞ is the original gray-level image and I T ðx; yÞ is the resulting diffusion image at iteration T. wi ¼ ½wi1 ; wi2 ; wi3  ¼ ½ai ; bi ; gi  =position of particle i, for i= 1, 2, y, P, which are the curve parameters, i.e. the unknown parameters to be estimated. HðI T jwi Þ = the fitness value of an image, i.e. the entropy value of the diffused image I T ðx; yÞ generated from particle i at position wi . The procedure: Step 1:

Initialize the parameter values: Randomly generate the initial position and velocity of each particle i: wi ¼ ½wi1; wi2 ; wi3  and vi ¼ ½vi1; vi2 ; vi3  for i ¼ 1; 2; . . . ; P. In order to prevent overshoot of the target position, initial values of wi and vi are selected in the ranges of 2 rwi1 r 0:1, 0:1 r wi2 r2, 0:1 rwi3 r 1 and 0 o vij o 0:1 for all j ¼ 1; 2 and 3. Compute the fitness value HðI T jwi Þ of each particle i, for i ¼ 1; 2; . . . ; P, using Eqs. (8) and (9). Determine the local best position of particle i and the global best position: pi ¼ ½pi1 ; pi2 ; pi3  ¼ wi for i ¼ 1; 2; . . . ; P, pg ¼ ½pg1 ; pg2 ; pg3  ¼ arg minHðI T jpi Þ

Step 2:

and position wnew of each Update the velocity vnew ij ij

i

particle i using Eqs. (11) and (12), for i ¼ 1; 2; . . . ; P and j =1, 2, 3. violates the range constraint for some j, If wnew ij and wnew by generating new then recalculate vnew ij ij random numbers rand1 and rand2 . new new ¼ ½wnew Let wnew i i1 ; wi2 ; wi3  and new new new new vi ¼ ½vi1 ; vi2 ; vi3 . Swap the new and old velocities and positions by setting vi ¼ vnew and i wi ¼ wnew . i For a given new particle position wnew , the input i image I 0 ðx; yÞ is diffused T times with the linearlogarithmic diffusion coefficient function of

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Step 3:

S.-M. Chao, D.-M. Tsai / Pattern Recognition 43 (2010) 1917–1931

parameter values new new a ¼ wnew i1 ; b ¼ wi2 and g ¼ wi3 . Evaluate the new fitness values, and update local best positions pi and the global best position pg : Þ o HðI T jpi Þ, then let pi ¼ wnew , If HðI T jwnew i i else retain the current value of pi , for i ¼ 1; 2; . . . ; P. Let pg ¼ arg minHðI T jpi Þ i

Step 4:

Step 5:

Check for the stopping criterion: If the maximum number of search iterations C is reached, then go to Step 5. Otherwise, repeat Steps 2–3 until the stopping criterion is met. Deliver the solution: The final global best pg is the best solution obtained, and is used as diffusion parameters, i.e. a ¼ pg1 ; b ¼ pg2 and g ¼ pg3 .

PSO is an extremely simple and easily implemented algorithm that can dynamically adjust the current local and global positions with a large number of particles in the swarm to simultaneously find the solution from different positions in the search space. It is well suited for automatic parameters search of the proposed diffusion model that involves three unknown parameters. The stability and convergence of PSO were theoretically analyzed in detail by Clerc and Kennedy [25]. Jiang et al. [26] gave the parameter selection guides of the PSO algorithm. Based on the study of Liu et al. [27], the PSO algorithm is stable when ðc1  rand1 þc2  rand2 Þ defined in Eq. (11) is less than 4. In this study, the two weights c1 and c2 are set to 2 and the range of two

T = 5, H = 6.81

random numbers is between 0 and 1. It meets the stability condition of the PSO algorithm. To verify the stability and convergence of the PSO algorithm on the proposed entropy criterion, we have conducted two experiments: (1) run the PSO algorithm 20 times with the same initial solution, and (2) run the PSO algorithm 20 times, each with a different initial solution. The surface image in Fig. 1(b1) is used for training. It has an original entropy value of 8.97. The number of particles P= 20, and the maximum number of search iterations C= 50 are used in the experiments. In the first experiment, the initial solution ða; b; gÞ is given by (  0.1, 0.1, 0.1) and the corresponding entropy value is 8.55. In the 20 replications with the same initial solution, they all converges to the final solution of (  1, 1, 1) with an entropy value of 5.23. It is apparent from the first experiment that the PSO algorithm can stably converge and find good parameter values with a given initial solution. In the second experiment, we replicated the PSO algorithm 20 times and randomly generated different initial solutions. The entropy mean and standard deviation of the 20 initial solutions are 8.83 and 0.87, respectively. They are relatively high and deviated widely. The resulting entropy mean and standard deviation of the 20 final solutions are 5.34 and 0.10. Although the 20 final solutions are not consistently the same, they all converge to similar entropy values between 5.23 and 5.47. All these 20 final solutions can well intensify the low-contrast defect without showing noisy points. When the number of search iterations is increased, the convergence and stability of the PSO algorithm can be further improved. To further evaluate the effect of the number of diffusion iterations T, Fig. 7(a)–(d) presents the diffusion results of the defective backlight panel image shown in Fig. 1(b) using the automatic parameters search procedure at four different diffusion

T = 10, H = 5.23

T = 15, H = 5.28

T = 20, H = 5.25

Fig. 7. Diffusion results using the automatic parameters search scheme at varying number of diffusion iterations T: (a) 5; (b) 10; (c) 15 and (d) 20. (The original defective backlight panel image is shown in Fig. 1(b1).)

10

Entropy H

8 6 4 2 0 5

10 T

15

20

Fig. 8. Plot of entropy values as a function of the number of diffusion iterations (T= 1–20) using the automatic parameters search scheme for the test image in Fig. 1(b1).

ARTICLE IN PRESS S.-M. Chao, D.-M. Tsai / Pattern Recognition 43 (2010) 1917–1931

iterations T= 5, 10, 15 and 20, respectively. In the PSO algorithm, the number of particles P= 20, and the number of search iterations C =50 are used for the test image. The corresponding combinations of parameter values are (a, b, g)=(  1.4, 1, 1), (  1, 1, 1), ( 0.5, 0.4, 0.4) and (  0.8, 1.3, 0.9) for T=5, 10, 15 and 20, respectively. The resulting images show that the proposed automatic parameters search procedure can find a suitable parameter combination for defect detection in a low-contrast image at a sufficient number of diffusion iterations. An

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insufficient number of iterations, such as T= 5 in Fig. 7(a), cannot completely sharpen the defective region. When T values are sufficiently large, the defect shape can be well intensified in the diffused images, and the diffusion results are not sensitive to the increment of T value, as seen in Fig. 7(b)–(d). Fig. 8 shows the plot of entropy values with T= 1–20 using the automatic parameters search scheme. It can be seen from the results that the entropy value becomes stable when the number of diffusion iterations T is larger than 10.

H = 9.51

H = 8.32

H = 9.41

H = 5.40

H = 8.96

H = 5.64

H = 8.99

H = 5.20

H = 9.14

H = 5.96

Fig. 9. Diffusion results of backlight panel surfaces: (a1)–(e1) a faultless and four defective test images; (a2)–(e2) contrast-stretched images of (a1)–(e1), respectively; (a3)–(e3) respective diffusion results with (a, b, g) = (  0.9, 1, 1) for all samples (number of diffusion iterations T= 10) and (a4)–(e4) thresholding results using the 3-sigma control limits. (H is the entropy value defined in Eq. (8).)

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In the automatic parameters search procedure, the training image should be a defective one and contain only one single defect in the whole image. This is because the minimize entropy value will occur for the diffused image that results in one single distinct defect area in the uniform background. A training image that contains multiple defects may fail to find the best combination of parameter values because the minimize entropy value is obtained when only one of the defects is enhanced in the diffused image. To avoid the effect of speckle noise spots contained in a

training image, a smoothing process using a 3  3 median filter can be applied first to the original image to eliminate the interference of noise in the PSO search process.

4. Experimental results In this section, three types of low-contrast surface images are evaluated. The detection performance between the Chao–Tsai

H = 9.09

H = 8.37

H = 9.43

H = 6.42

H = 9.40

H = 6.98

H = 9.38

H = 6.57

H = 9.19

H = 5.94

Fig. 10. Diffusion results of LCD glass substrates: (a1)–(e1) a faultless and four defective test images; (a2)–(e2) contrast-stretched images of (a1)–(e1), respectively; (a3)– (e3) respective diffusion results with (a, b, g)= (  0.6, 1, 0.8) for all samples (number of diffusion iterations T= 10) and (a4)–(e4) thresholding results using the 3-sigma control limits. (H is the entropy value defined in Eq. (8).)

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diffusion model [21] and the proposed diffusion model with generalized diffusion coefficient function is then compared in the experiments.

4.1. Detection results of low-contrast surface images This subsection presents experimental results of three different types of low-contrast surface images including backlight panels, glass substrates and brightness enhancement films to

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evaluate the performance of the proposed diffusion model. All test images are 200  200 pixels wide with 8-bit gray levels. The experiments were conducted on an Intel Core 2 Duo 3.0 GHz personal computer. The number of diffusion iterations T was set at 10 for all test images in the experiments. In the PSO search algorithm, the required parameters were set up as follows: the population size (the number of particles) P =20, the weights c1 = c2 = 2, and the maximum number of search iterations (stopping criterion) C= 50. The training time of the PSO search procedure is 220 s on average for the three test image sets.

H = 9.66

H = 8.68

H = 9.82

H = 6.97

H = 9.73

H = 6.95

H = 9.83

H = 6.85

H = 9.81

H = 7.23

Fig. 11. Diffusion results of brightness enhancement film surfaces: (a1)–(e1) a faultless and four defective test images; (a2)–(e2) contrast-stretched images of (a1)–(e1), respectively; (a3)–(e3) respective diffusion results with (a, b, g) =(  0.7, 0.8, 1) for all samples (number of diffusion iterations T= 10) and (a4)–(e4) thresholding results using the 3-sigma control limits. (H is the entropy value defined in Eq. (8).)

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Fig. 9(a1)–(e1) demonstrates one faultless and four defect images of backlight panel surfaces. Fig. 9(a2)–(e2) presents, respectively, the enhanced images of Fig. 9(a1)–(e1) so that the defect locations and shapes can be visually observed. The values of curve parameters a, b and g were automatically set at 0.9, 1 and 1, respectively, using Fig. 9(b1) as the training image in the PSO search procedure, and are applied to all test backlight panel surface images. The results from the proposed diffusion model are presented in Fig. 9(a3)–(e3), which show that all subtle defects are well enhanced in the diffused images. In order to segment defects in the diffused image, the simple statistical control limits are used to set up the thresholds. The upper and lower control limits for intensity variation in the diffused image are

given by

md 7S  sd

ð13Þ

where md and sd are the mean and standard deviation of gray values in the whole diffused image, and S is the control constant. In the diffused image, if the gray-level of a pixel falls within the control limits, the pixel is classified as a faultless point. Otherwise, it is classified as a defective one. In this study, the control constant S = 3 is used for all test samples to follow the 3-sigma standard deviation. Fig. 9(a4)–(e4) illustrates the thresholding results of the diffused images in Fig. 9(a3)–(e3). In the faultless surface image of Fig. 9(a1), the resulting binary image is uniformly white and no defect is claimed. In the defective

Fig. 12. Comparison of the diffusion results of backlight panel surfaces: (a1)–(e1) contrast-stretched images from Fig. 9(a1)–(e1), respectively; (a2)–(e2) diffusion results of the Chao–Tsai diffusion model; (a3)–(e3) thresholding results from (a2)–(e2); (a4)–(e4) diffusion results of the proposed diffusion model and (a5)–(e5) thresholding results from (a4)–(e4). (Control constant S= 3 for both models.)

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images of Fig. 9(b1)–(e1), all hardly visible defects are well detected. Fig. 10 shows the detection results of five LCD glass substrate images that contain horizontal structure patterns on the surfaces. The defects in the images also present low-contrast intensities from their surroundings. The parameter values (a, b, g) =( 0.6, 1, 0.8) obtained from the PSO search procedure is based on the training image in Fig. 10(b1), and are applied to all test images. Fig. 10(a1) is a clear glass substrate image, and Fig. 10(b1)–(e1) are four defective glass substrate images. The enhanced images show the structure textures and uneven lighting on the glass substrate surfaces, as seen in Fig. 10(a2)–(e2). The results from the proposed diffusion model are illustrated in Fig. 10(a3)–(e3),

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which show that the defects are well enhanced and the structural background patterns are smoothed. Fig. 10(a4)–(e4) demonstrates the detection results as binary images using the statistical control limits with control constant S= 3. The thresholding results also reveal that all local defects embedded in the low-contrast surface images can be effectively detected, and the binarization result of the defect-free surface image is approximately a uniform white image. Finally, Fig. 11 demonstrates the third test sample set, which involves five brightness enhancement film images that contain small low-contrast defects in the rough surfaces. Fig. 11(a1) shows a faultless brightness enhancement film image. Fig. 11(b1)–(e1) illustrates four defective surface images of the

Fig. 13. Comparison of the diffusion results of LCD glass substrates: (a1)–(e1) contrast-stretched images from Fig. 10(a1)–(e1), respectively; (a2)–(e2) diffusion results of the Chao–Tsai diffusion model; (a3)–(e3) thresholding results from (a2)–(e2); (a4)–(e4) diffusion results of the proposed diffusion model and (a5)–(e5) thresholding results from (a4)–(e4). (Control constant S= 3 for both models.)

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1 0.8 0.6 Generalized diffusion model

g (∇I)

0.4

Chao-Tsai diffusion model

0.2 0 -0.2

0

1

2

3

4

5

∇I

-0.4 -0.6 -0.8 -1

Fig. 14. Plots of diffusion coefficient function gðrIÞ of the Chao–Tsai diffusion model with ðZ; KÞ ¼ ð0:2; 1Þ and the generalized diffusion model with ða; b; gÞ ¼ ð0:9; 1; 1Þ for the backlight panel images in Fig. 12.

1 0.8

Generalized diffusion model

0.6

Chao-Tsai diffusion model

g (∇I)

0.4 0.2 0 -0.2

0

1

2

-0.4

3

4

5

∇I

-0.6 -0.8 -1 Fig. 15. Plots of diffusion coefficient function gðrIÞ of the Chao–Tsai diffusion model with ðZ; KÞ ¼ ð0:2; 2Þ and the generalized diffusion model with ða; b; gÞ ¼ ð0:6; 1; 0:8Þ for the LCD glass substrates images in Fig. 13.

brightness enhancement film. Fig. 11(a2)–(e2) are the respective enhanced images of Fig. 11(a1)–(e1) to visualize the locations and shapes of defects. Fig. 11(b1) is used as the training image in the PSO search procedure. Fig. 11(a3)–(e3) shows the diffusion results using diffusion parameters (a, b, g)=(  0.7, 0.8, 1). The subtle defects are well highlighted in the diffused images. Fig. 11(a4)– (e4) displays the thresholding results of the diffused images in Fig. 11(a3)–(e3). The defects in defective surfaces are correctly segmented in the binarized images, and the defect-free surface image is uniformly white in the segmented image. 4.2. Performance comparison The Chao–Tsai diffusion model is also used in the experiment for defect detection in low-contrast surface images. In order to show the superiority of the proposed model with a generalized diffusion coefficient function, the detection results of backlight panels and LCD glass substrates are compared with those of the Chao–Tsai diffusion model. Fig. 12 presents the diffusion results of backlight panel surfaces (the original gray-level images are in Fig. 9) from the Chao–Tsai diffusion model and the proposed diffusion model for visual comparison of detection effectiveness. It can be seen from Fig. 12 that all defects can be effectively enhanced in both diffusion models. However, the proposed diffusion method outperforms the Chao–Tsai diffusion model in the detection of tiny defects, as seen in the segmented images of Fig. 12(b3) and (b5). Fig. 13 further demonstrates the diffusion results of LCD glass substrate surfaces (the original gray-level images are in Fig. 10) from the Chao–Tsai diffusion model and the proposed diffusion model, respectively. Fig. 13(c3), (e3) and (c5),

(e5) shows that the defect shapes are better extracted in the binary images when the proposed diffusion model is used for image diffusion. The computation time of the Chao–Tsai diffusion model was 0.16 s with the number of diffusion iterations T=30. However, the proposed diffusion model took only 0.06 s with T= 10 for an image of size 200  200. The Chao–Tsai diffusion model requires a large number of diffusion iterations since the diffusion coefficient function uses only one parameter K and the function curve cannot achieve a fast background smoothing and defect sharpening in early diffusion iterations. It failed to detect the defects when T was set at 10. The best K values for the Chao–Tsai model were set at 1 and 2 for backlight panels and LCD glass substrates, respectively. The best K values were manually chosen by tryand-error in the Chao–Tsai model. Conversely, the proposed diffusion model with generalized diffusion coefficient function can flexibly select a desired function curve and makes the diffusion process converge fast with a very small number of diffusion iterations. Figs. 14 and 15 shows the plots of the diffusion coefficient functions gðrIÞ of the Chao–Tsai diffusion model and the generalized diffusion model for defect detection in backlight panel surfaces and LCD glass substrates, respectively. The diffusion coefficient function curves of the generalized diffusion model drop faster than those of the Chao–Tsai diffusion model in both test image sets. They indicate that the proposed generalized diffusion model performs a stronger sharpening process when the diffusion coefficient function crosses the zero point rI 0 . The zerocrossing point of the generalized diffusion model also occurs earlier than that of the Chao–Tsai diffusion model. It indicates

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that the generalized diffusion model can carry out the sharpening process in early diffusion iterations, and thus requires a smaller number of diffusion iterations. The proposed generalized diffusion model can, therefore, well enhance tiny defects and converge faster than the Chao–Tsai diffusion model.

5. Conclusion In this paper, an anisotropic diffusion model with a generalized diffusion coefficient function has been proposed to detect subtle defects in low-contrast, unevenly illuminated images. The proposed model is flexible to implement for defect detection in various low-contrast surface images by controlling the curve parameter values of the diffusion coefficient function. The merits of this paper are the proposition of: (1) a generalized diffusion coefficient function that provides varying smoothing/sharpening process for different inspection tasks with a small number of diffusion iterations, (2) the entropy criterion that can effectively measure the diffusion quality of an image in the application of surface inspection and (3) the PSO search procedure that can automatically determine the best parameter values of the generalized diffusion coefficient function without human intervention. Experimental results have shown that the proposed diffusion model can be well applied to low-contrast images of backlight panels, LCD glass substrates and brightness enhancement films. Performance evaluation shows that the proposed diffusion model for defect detection is more effective and efficient than the Chao– Tsai diffusion model. The proposed diffusion method can be best applied to defect detection in non-textured surfaces. It is worth further investigation of the diffusion method for defect detection in textured surfaces. References [1] F. Saitoh, Boundary extraction of brightness unevenness on LCD display using genetic algorithm based on perceptive grouping factors. in: Proceedings of the International Conference on Image Processing, Kobe, Japan, October 1999, pp. 308–312. [2] W.S. Kim, D.M. Kwak, Y.C. Song, D.H. Choi, K.H. Park, Detection of spot-type defects on liquid crystal display modules, Key Engineering Materials 270–273 (2004) 808–813. [3] B.C. Jiang, C.C. Wang, H.C. Liu, Liquid crystal display surface uniformity defect inspection using analysis of variance and exponentially weighted moving average techniques, International Journal of Production Research 43 (1) (2005) 67–80. [4] J.Y. Lee, S.I. Yoo, Automatic detection of region-mura defect in TFT-LCD, IEICE Transactions on Information and Systems E 87-D (2004) 2371–2378. [5] Z. Wang,. Ma, Implementation of region-mura detection based on recursive polynomial-surface fitting algorithm, Second Asia International Symposium on Mechatronics, Hong Kong, China, December 2006, pp. 1–6. [6] P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence 12 (7) (1990) 629–639.

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