Automatic ridgelet image enhancement algorithm for road crack image based on fuzzy entropy and fuzzy divergence

ARTICLE IN PRESS Optics and Lasers in Engineering 47 (2009) 1216–1225

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Automatic ridgelet image enhancement algorithm for road crack image based on fuzzy entropy and fuzzy divergence Daqi Zhang , Shiru Qu, Li He, Shuang Shi Department of Automatic Control, Northwestern Polytechnical University, 127 Youyi Xilu, Xi’an, Shaanxi 710072, China

a r t i c l e in f o

a b s t r a c t

Article history: Received 23 October 2008 Received in revised form 21 May 2009 Accepted 24 May 2009 Available online 24 June 2009

True estimation of the boundary of a road crack and its size is a major task for its automatic detection. The improvement of visual effects of a road image is necessary for such a task. Therefore, we propose an automatic ridgelet image enhancement algorithm. A nonlinear function plays an important role in the enhancement algorithm in the ridgelet domain of an image. However, it is difficult to adjust the parameters of the nonlinear function adaptively with the variation of the road crack image input. Based on the fuzzy entropy criterion, we introduce two fuzzy divergences and two supplementary linear combinations between the fuzzy entropy and two fuzzy divergences as new measurements to solve the threshold segmentation problem in the ridgelet domain. According to the distribution histogram of magnitudes of the ridgelet high-frequency coefficients, we obtain the optimal segmentation thresholds that act as the parameters of the nonlinear function by using the maximum or minimum measurements of fuzzy entropy and fuzzy divergence, respectively. The self-adaptive nonlinear function makes it possible to realize the automatic enhancement of a road crack image. Experimental results show that our image enhancement algorithm can effectively enhance the global and local contrastive effects on road crack images. & 2009 Elsevier Ltd. All rights reserved.

Keyword: Road crack detection Image enhancement Ridgelet transform Fuzzy entropy Fuzzy divergence

1. Introduction Roads are important public facilities, and therefore accurate and real-time information on their conditions is very necessary for road authorities to effectively manage them. The extent and type of crack, two of the most important road-quality indicators, were measured only by visual inspection in the past. However, human visual inspection has certain limitations. With the CCD camera-based image-processing device, we can carry out the automatic detection of the image of a road crack, extract crack information and characterize the crack in terms of crack length and width. However, imaging techniques tend to result in images with poor contrast and relatively high noise level. Image enhancement is crucial because it can help to improve the quantity and quality of information on road crack. The conventional image enhancement methods such as histogram equalization tend to amplify the noise and at the same time enhance the visibility of object characterization. Considerable success has been achieved in the development of wavelet-transform image enhancement algorithms with noise suppression [1–16]. Wavelets perform very well for objects with point singularities but their performance is not good for representing 1D singularities. As

 Corresponding author.

E-mail address: [email protected] (D. Zhang). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.05.014

extension of the wavelet multi-scale analysis framework, ridgelet and curvelet can effectively deal with such linear singularities in 2D signals [7,9]. Therefore, many image enhancement algorithms used in ridgelet or curvelet domains develop rapidly and achieve better enhancement results [11–15]. The wavelet-transform image enhancement algorithm employs one transform to one image and then applies a nonlinear function to subband coefficients in the transform domain. Finally, the enhanced image is reconstructed when the inverse transform is performed. The example shown in Fig. 1 is a typical nonlinear function [11] for image enhancement. When the absolute value of the transform-domain coefficients is smaller than threshold T1 they will be suppressed. The coefficients whose absolute values are between T1 and T2 can get a uniform gain. For coefficients whose absolute value exceeds T2, their gain decreases with the increasing value. The purposes of the nonlinear function are not only to enhance image details but also to prevent the amplification of various kinds of image noise. The threshold T1 in Fig. 1 usually corresponds to the noise level in source image. Moreover, frequency-domain transforms can reduce information to a relatively small number of independent transform coefficients that capture important features such as point and edge. In doing so, the nonlinear function can carry out good image enhancement. The fuzzy set was introduced into ridgelet domain and the parameterized fuzzy transform function was used to enhance crack road images. The parameters of the fuzzy set were decided

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1 0.75 0.5 0.25 0 T1

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-0.75

-0.5

-0.25

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singularities in 2D signals. However, its approximation ability is not satisfactory when there are straight singularities. Worst of all, the decay rate of m-terms nonlinear wavelet approximation is O(n1) only, which is worse than the Fourier analysis. Candes et al. initiated ridgelet transform [7,8] to represent linear singularities in images. At first, it encountered such difficulties as overcompleteness and perfect reconstruction. Different from the conventional discretization approaches, Do and Vetterli devised a discrete orthogonal transform defined in the finite field, which is called finite ridgelet transform (FRIT) [10]. After the finite radon transform (FRAT) maps line singularities into image domain with point singularities in FRAT domain, a series of orthogonal transforms are applied to FRAT coefficients column wise to accomplish the FRIT. The FRIT transform, thus, obtained is invertible, nonredundant and leads to a family of directional orthogonal bases for digital images. Moreover, compared with the wavelet transform, it shows better characters such as nonlinear approximation and denoising for images with straight features [16–19].

Fig. 1. A typical nonlinear function for wavelet-transform image enhancement.

2.1. Ridgelet transform The ridgelet transform effectively represents line singularities of 2D signals. It maps line singularities into point singularities in the radon domain by employing the embedded radon transform. Therefore, the wavelet transform can be effectively applied to discovering the point singularities in this new domain. Having the ability to approximate singularities along a line, several terms with common ridge lines can effectively be superposed by the ridgelet transform. The bivariate ridgelet transform in R2 is defined by Ra;b;y ðx1 ; x2 Þ ¼ a1=2 cððx1 cos y þ x2 sin y  bÞ=aÞ

(1)

where c(x) is a univariate wavelet function. a40, b and y parameters of scale, location and orientation, respectively. Ridgelets are constant along ridgelet lines x1 cos y+x2 sin y and are equal to the wavelets in the orthogonal direction. In Fig. 2, the Morlet ridgelet is plotted and it is easy to see its characteristics. A bivariate function f(x) in R2 can be decomposed into the following ridgelet coefficients: Z Ra;b;y ðxÞf ðxÞ dx (2) Rf ða; b; yÞ ¼ and it can be reconstructed by Z 2p Z þ1 Z þ1 da dy f ðxÞ ¼ Rf ða; b; yÞRa;b;y ðxÞ 3 db 4p a 1 0 0

(3)

1 0.5 R (x,y)

by the global fuzzy entropy maximum method in Ref. [12]. Many researchers made efforts to implement adaptive enhancement algorithms for different images. Many artificial intelligence methods were used to select the optimal parameters of the nonlinear function under certain evaluation measurements. Based on ridgelet transform and driven by the immune clonal algorithm (ICA), an automatic image enhancement algorithm for a large category of images was proposed [13], where ICA was used to search for the optimal parameters of each special image. Satellite cloud images were enhanced by fuzzy wavelet neural network (FWNN) and genetic algorithm (GA) in curvelet domain. The GA takes information entropy as fitness function to determine optimal gray transform parameters, and FWNN is used to approximate the in-complete beta transform (IBT) in the coarse scale [14]. To restrain the noise and emphasize the hand-vein linear pattern in an image, we conduct the multi-scale and selfadaptive enhancement transform of the captured hand-vein image based on ridgelet transform [15]. The goal of enhancing a road crack image is to suppress noise and give prominence to linear crack information. Therefore, the feature representation capability of frequency-domain transform is crucial for the quality of the enhanced image and the nonlinear function for transform-domain image enhancement influences the enhancement results in the same way. We prefer the ridgelet transform to other transforms for road crack image enhancement because it effectively provides the sparse representation of smooth functions and line singularities of the image such as linear profile in road crack images. That is to say, this technique can accurately represent both smooth functions and line singularities with a few nonzero coefficients and makes fewer mean square errors (MSE) than the wavelet transform. In addition, we choose a nonlinear function with shape-control parameters in the ridgelet-transform domain to enhance the image. Two parameters of the nonlinear function can be flexibly adjusted to achieve the best quality of the enhanced image. Different images should have different optimal combinations of function parameters based on the ridgelet transform. Therefore, depending on different images, we establish several objective assessment criteria to automatically determine the optimal parameters of the nonlinear function.

0 -0.5 -1 -2 -1

2. Image enhancement using ridgelet transform Wavelet transform can achieve optimal nonlinear approximation for function classes that are smooth away from point

0 x

1 2 -2

-1

Fig. 2. The Morlet ridgelet.

1

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Like the Fourier and wavelet transforms, any arbitrary functions can be represented by continuous superposition of ridgelets. Because the 2D ridgelet transform can be regarded as a 1D wavelet transform in the radon domain, the ridgelet coefficients of function f(x) can be defined as Z Rt ðy; tÞa1=2 wððt  bÞ=aÞ dt (4) Rf ða; b; yÞ ¼ where Rt(y,t) is the radon transform of function f(x) and given by the Dirac distribution d(*) as Z f ðxÞdðx1 cos y þ x2 sin y  tÞdx1 dx2 (5) Rt ðy; tÞ ¼

2.2. Nonlinear enhancement function in ridgelet domain

M [ N [ uij i¼1 j¼1

dij

ðuij 2 ½1; 1Þ

(7)

where uij denotes the degree of membership of the subband coefficient dij to set A. The initial values {uij} are calculated by the magnitude normalization of the subband coefficients using the following formula: t ¼ signðwÞnðjwj  jwjmin Þ=ðjwjmax  jwjmin Þ;

1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1

To highlight the linear pattern in a road crack image, we conduct the ridgelet transform of the road crack image and enhance such a crack feature by modifying the high-frequency coefficients in the ridgelet domain. It is generally accepted that the high-frequency coefficients follow the general Gaussian distribution and its probability density function is   b jxj b exp  (6) Pðx; a; bÞ ¼ 2a  Gð1=bÞ a R where G(t) ¼ +N exp(u)ut1 du, a and b are the standard 0 variance and the shape parameters, respectively [12]. These subband coefficients are almost symmetrically distributed at the two sides of a zero and this phenomenon is obvious from Fig. 4(a). So we introduce the general fuzzy set A¼

Adjusted Values by Enhancement Function

1218

t 2 ½1; 1

(8)

Here, |w|max and |w|min are the magnitude maximum and the minimum, respectively, of the subband coefficients. To suppress the noise effectively and magnify the relatively large coefficients, we add the side-lobe effect of the histogram of high-frequency coefficients. The enhancement gain also continuously increases with increasing the magnitude of the highfrequency coefficients. The smoothness of the enhancement function (the continuity of enhancement gain) enables the nonlinear enhancement function adaptively modify the highfrequency coefficients in ridgelet domain. The discontinuity and the hard enhancement gain of the nonlinear enhancement function are distinctly shown in Fig. 1. Considering the continuity of enhancement gain and information entropy magnification of the distribution histogram of ridgelet high-frequency coefficients, we use the following nonlinear ridgelet enhancement function [12,15], which is also called generalization fuzzy membership function, to adjust the multi-scale ridgelet decomposition coefficients. Here the enhancement function is SðtÞ ¼ a½lðcðt  bÞÞ  lðcðt þ bÞÞ

-0.75

-0.5

-0.25 0 0.25 0.5 Degree of Membership

0.75

1

Fig. 3. The curve of ridgelet image enhancement function when b ¼ 0.25 and c ¼ 40.

interrelated to the image noise, while the larger ones interrelated to the contour outline of a source image. The nonlinear function is to be used in our ridgelet enhancement algorithm for road crack image. In Fig. 4(a) and (b), there is a visible contrast between the distribution of the enhanced ridgelet high-frequency coefficients at the third decomposition level of a road crack image and one of these coefficients is not enhanced. We can see clearly that the distribution of original ridgelet coefficients observes the generalized Gaussian distribution from Fig. 4(a), while the distribution of these enhanced coefficients does not from Fig. 4(b); the horizontal-axis range of ridgelet coefficients by nonlinear enhancement adequately extends, and a new peak rises at about the value of 90 in the right histogram graph and the lower peak at about 25 in the left histogram graph disappears. Thanks to the nonlinear enhancement, we have to find the peak which lies at the position where the linear features focus on the ridgelet domain of the original road crack image. Moreover, the peak enlarges the amount of information gain in the ridgelet domain.

3. Automatic computation of nonlinear enhancement function’s parameter As is well-known, the entropy of a system as defined by Shannon is a measure of uncertainty about its actual structure. Shannon’s entropy function is based on the concept that the information gain from an event is inversely related to the probability of the event. Consequently, it can be used to quantify the complexity of the magnitude-level histogram of ridgelettransform coefficients. Here, we introduce fuzzy entropy and fuzzy divergence functions to obtain the optimal segmentation threshold to be used for the parameter b of the nonlinear enhancement function (9). So we intend to carry out the fuzzy initialization of the magnitude set I of ridgelet high-frequency coefficients at a certain decomposition level for threshold segmentation in the ridgelet domain.

(9) t

where a ¼ 1/[l(c(1b))l(c(1+b))],0obo1, l(t) ¼ 1/(1+e ) and tA[1,1]. The parameter b is the fuzzy enhancement threshold which determines the enhancement range and c the curve shape factor which controls the enhancement intensity. It is easily seen in Fig. 3 that the enhancement function magnifies the coefficients with relatively larger absolute values. The smaller coefficients are

3.1. Fuzzy initialization of ridgelet high-frequency coefficients Supposing that the magnitude set I of the ridgelet highfrequency coefficients is treated as a fuzzy set, it can be divided into two classes by using the threshold T(TAI): the featuredominant class A and the nonfeature-dominant class B. We

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0 -100

-75

-50 -25 0 25 50 75 Ridgelet High-frequency Coefficients

100

0

-90 -70 -50 -30 -10 10 30 50 70 90 Ridgelet High-frequency Coefficients by Fuzzy Enhancement

Fig. 4. The histogram distribution of ridgelet high-frequency coefficients and that of enhanced coefficients.

characterize the fuzzy set A and B by the following fuzzy membership functions: ( 1  jðw  w ¯ A ðTÞÞ=Cja =2; 0  jwj  T (10) mA ðwÞ ¼ Tojwj  1 jðw  w ¯ B ðTÞÞ=Cja =2;

mB ðwÞ ¼ 1  mA ðwÞ

(11)

where a is a parameter to adjust the compactness, and C ¼ |w|max|w|min which yields mA(w) is normalized to the range ¯ A(T) [0,1]. The mean value of the magnitudes of the two classes W ¯ B(T) can be estimated by the formulas and W , T T X X w wPðwÞ PðwÞ (12) ¯ A ðTÞ ¼ w¼0

w ¯ B ðTÞ ¼

1 X w¼T

w¼0

, wPðwÞ

3.2. The measure of fuzzy entropy (determination of the optimal threshold t) Assuming that the magnitude histogram of all ridgelet highfrequency coefficients does completely satisfy the information gain of a certain decomposition level, we introduce the measure of fuzzy entropy Z T Z 1 Z 1 PðwÞHðwÞ dw ¼ PðwÞHðwÞ dw þ PðwÞHðwÞ dw EðTÞ ¼ 0

0

T

(14) where H(w) is the fuzzy entropy, i.e. HðwÞ ¼ ½mA ðwÞ logðmA ðwÞÞ þ ð1  mA ðwÞÞ logð1  mA ðwÞÞ ¼ ½mB ðwÞ logðmB ðwÞÞ þ ð1  mB ðwÞÞ logð1  mB ðwÞÞ

(15)

Formula (14) has the discrete form 1 X

PðwÞ

(13)

w¼T

EðTÞ ¼

L X

Pi  H

i¼1

where P(w) is the histogram of magnitude of the high-frequency coefficients. The character function is considered as a weighting coefficient which reflects the ambiguity (fuzziness) in a set. Because the fuzziness of A decreases when the values of w ¯ B(T), the character function has certain ¯ A(T) or W approach W degrees of cluster centralization, and therefore we can adjust the value of threshold T so as to segment one set into two subsets in ridgelet domain. Searching for the optimal parameter of the nonlinear enhancement function is a challenging job for the ridgelet image enhancement algorithm because the distribution of coefficients in ridgelet domain is uncertain. It is hard to estimate the distribution of ridgelet high-frequency coefficients because of its complexity and additive noise. To avoid the assumptive estimation of the distribution of these high-frequency coefficients in ridgelet domain, we establish the histogram of magnitude level of high-frequency coefficients by dividing uniformly the magnitude range into L magnitude levels. The number L of magnitude levels is determined by the multiples of the magnitude range, i.e. L ¼ l  ceil(|w|max|w|min), where the multiple l is an integer from 2 to 5 and the function ceil(x) rounds x to the nearest integer greater than or equal to x. Consequently, the discrete magnitude level set IDis ¼ {Ti|i ¼ 1, 2 ,y, L} of ridgelet high-frequency coefficients at a certain decomposition level can be generated by recording all values at the center of every magnitude level.

  Ti N ; Pi ¼ i L N

(16)

where the threshold variable T is in {Tj/L|j ¼ 1, 2 ,y, L;TjAIDis}, Pi the probability of high-frequency coefficients deposited at magnitude level, Ni the number of high-frequency coefficients at magnitude level i, and N the sum of Ni. After acquiring the fuzzy set of entropy, the minimization criterion of fuzzy entropy can be used to calculate the segmentation threshold TE, i.e. T E ¼ minfEðTÞjT ¼ T j =L; 1  j  Lg

(17)

Through many tests on a large sample of road crack images, the image enhancement results are not very satisfactory. In addition to the fuzzy entropy, we introduce two fuzzy divergences for better feature segmentation in ridgelet domain. 3.3. The measure of fuzzy divergence The fuzzy divergence [20,21] between two classes A and B is defined as Z T PðwÞ  ½HA=B ðwÞ þ HB=A ðwÞ dw DAB ðTÞ ¼ 0

Z

1

PðwÞ  ½HA=B ðwÞ þ HB=A ðwÞ dw

þ T

where HA/B(w) and HB/A(w) are two fuzzy cross-entropies, i.e.

(18)

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HA=B ðwÞ ¼ ½1 þ mA ðwÞ log½ð1 þ mA ðwÞÞ=ð1 þ mB ðwÞÞ þ ½2  mA ðwÞ log½ð2  mA ðwÞÞ=ð2  mB ðwÞÞ

(19)

HB=A ðwÞ ¼ ½1 þ mB ðwÞ log½ð1 þ mB ðwÞÞ=ð1 þ mA ðwÞÞ þ ½2  mB ðwÞ log½ð2  mB ðwÞÞ=ð2  mA ðwÞÞ

T EDðABÞ ¼ min fEDAB ðTÞjT ¼ T j =Lg (20)

It is easy to prove that HA/B(w) is equal to HB/A(w). Formula (18) has the discrete form DAB ðTÞ ¼

1jL

(30)

1jL

P i  ½HA=B ðT i =LÞ þ HB=A ðT i =LÞ

(21)

i¼1

¼ min fl1 ½EðT j =LÞ  T E 2 þ l2 ½DAA0 ðT j =LÞ  T DAA0 2 g 1jL

To introduce the fuzzy divergence between the same classes, we suppose that the class A0 and B0 obtained by threshold segmentation be characterized by the flowing function CA0 (w) and CB0 (w): ( 1; 0  jwj  T C A0 ðwÞ ¼ (22) 0; Tojwj  1 C B0 ðwÞ ¼ 1  C A0 ðwÞ

(23)

It is rational to think that A0 and A are two fuzzy sets defined on the same universal set and A0 is a certain set of A. Then, this fuzzy divergence of A0 and A is defined as Z T DAA0 ðTÞ ¼ PðwÞ  ½HA=A0 ðwÞ þ HA0 =A ðwÞ dw Z

1

þ T

PðwÞ  ½HA=A0 ðwÞ þ HA0 =A ðwÞ dw

(24)

where HA/A0 (w) and HA0 /A(w) are two fuzzy cross-entropies, i.e. HA=A0 ðwÞ ¼ ½1 þ mA ðwÞ  log½ð1 þ mA ðwÞÞ=ð1 þ C A0 ðwÞÞ þ ½2  mA ðwÞ  log½ð2  mA ðwÞÞ=ð2  C A0 ðwÞÞ

(25)

HA0 =A ðwÞ ¼ ½1 þ C A0 ðwÞ  log½ð1 þ C A0 ðwÞÞ=ð1 þ mA ðwÞÞ þ ½2  C A0 ðwÞ  log½ð2  C A0 ðwÞÞ=ð2  mA ðwÞÞ

(26)

Formula (24) has the discrete form DAA0 ðTÞ ¼

1jL

¼ min fl1 ½EðT j =LÞ  T E 2 þ l2 ½DAB ðT j =LÞ  T DAB 2 g T EDðAA0 Þ ¼ min fEDAA0 ðTÞjT ¼ T j =Lg

L X

0

measurements by constructing two linear combinations and minimizing any one of the following two criterion functions to determine the optimal threshold:

L X

pðT i Þ  ½HA=A0 ðT i =LÞ þ HA0 =A ðT i =LÞ

(27)

i¼1

Based on the maximum criterion of fuzzy divergence between two classes and the minimum criterion of fuzzy divergence in the same class, we generate another two segmentation thresholds TD(AB) and TD(AA0 ) T DðABÞ ¼ maxfDAB ðTÞjT ¼ T j =L; 1  j  Lg

(28)

T DðAA0 Þ ¼ minfDAA0 ðTÞjT ¼ T j =L; 1  j  Lg

(29)

When the disparity between the feature-dominant class A and the nonfeature-dominant class B in the ridgelet domain of a road crack image is tremendous, the maximum criterion of fuzzy divergence between two classes has a better segmentation threshold. However, linear-feature information is always blended together with noise, so the feature-dominant class A segmented by the threshold TD(AB) includes not only linear-feature information but also noise information. As a result, the linear-feature information included in the nonfeature-dominant class B is greatly or even fully weakened by the nonlinear enhancement function. The fuzzy divergence of the same class A0 and A globally considers the similarity of feature information before and after threshold segmentation in ridgelet domain other than the disparity between the feature-dominant class A and the nonfeature-dominant class B. It is obvious that the feature-dominant class A includes noise information. To keep balance between linear-feature information and noise information, we make a compromise on these fuzzy

(31)

where l1 and l2 are two weighting factors which satisfy l1+l2 ¼ 1(l1,l2)Z0). 3.4. Estimating the fuzzy measures for image enhancement The aim of image enhancement is to restore the distortion or loss of information. But it is complicated to judge the quality of restoration through image enhancement. To have objective quality criterion, the following assumptions are made [11]: between two edge enhancement methods, the better one should produce the best results for standard image-processing tasks such as segmentation or edge detection. Here, we examine edge detection results using the Canny edge detector to evaluate five fuzzy measurements for computing the segmentation threshold in ridgelet domain. Fig. 5(a) is a synthetic image containing six gray bars with increasing intensity and Fig. 5(b) is its ideal bar edge image. The synthetic image with the normal random noise with the mean being 0 and the standard deviation being 10 is shown in Fig. 6(a). We acquire its five ridgelet enhancement images using the nonlinear function (9) whose parameter b is equal to the fuzzy measurement TE, TD(AA0 ), TD(AB), TED(AA0 ) or TED(AB) and c ¼ 15, respectively. Using the Canny edge detector of the five enhanced gray bar images, we detect edge pixels shown in Fig. 6(b–f), respectively, and the edge detection accuracy is 72.75%, 87.1%, 52.28%, 75.09% and 76.33%, respectively. In Fig. 7, the accuracy of detected edge pixels versus the edge SNR using the Canny edge detector on the ridgelet enhanced image using the nonlinear function enhancement where the parameter b is, respectively, equal to: (i) TE, represented by the dark curve; (ii) TD(AA0 ), represented by the red curve; (iii) TD(AB), represented by the blue curve; (iv) TED(AA0 ), represented by the cyan curve; (v) TED(AB), represented by the magenta curve. Here, we think the segmentation thresholds in ridgelet domain produced by the two fuzzy divergences outgo the ones produced by the fuzzy entropy when they are used as the parameter b of the nonlinear enhancement function (9) for the ridgelet image enhancement. Besides, the complexity of automatically obtaining the optimal parameter b should also be considered. If the same

Fig. 5. The synthetic bar image and its ideal bar edge.

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Fig. 6. The noised synthetic bar edge image (a) and detected edge images using the Canny edge detector on the ridgelet enhanced images using the nonlinear function enhancement, where the parameters b is, respectively, equal to: TE (b; detection accuracy%, 72.75), TD(AA0 ) (c; 87.1), TD(AB) (d; 52.28), TED(AA0 ) (e; 75.09) or TED(AB) (f; 76.33) and c ¼ 15.

Accuracy of Detected Edge Pixels (%)

100 90 80 b=TE b=TD(AA') b=TD(AB) b=TED(AA') b=TED(AB)

70 60 50 40

0

1

2

3

4 5 Input Edge SNR

6

7

8

9

Fig. 7. Accuracy of detected edge pixels versus the edge SNR using the Canny edge detector.

search algorithm is used to look for the extreme points of five fuzzy measurements, the time spent on the acquisition of the segmentation threshold T in ridgelet domain is also the same because all the time and complexity of computing five fuzzy measurements are O(L2).

4. The procedures of the image enhancement algorithm Our enhancement algorithm for road crack image follows the primary procedures of the general transform-domain image enhancement scheme mentioned in the introduction section and its procedures are as follows:

Step 1. Input a low-quality road crack image and conduct Nlevel ridgelet decomposition of the image and extract all the subband high-frequency coefficients. Step 2. Calculate the maximum |wi|max, the minimum |wi|min and the mean |wi|mean of the ridgelet high-frequency coefficients at decomposition level i(i ¼ 1, 2 ,y, N). Step 3. Divide uniformly the magnitudes of high-frequency coefficients at decomposition level i into Li, Li ¼ l  ceil(|wi|max|wi|min) and construct the discrete magnitude level set Ii ¼ {Tji} by recording all values Tji(j ¼ 1, 2 ,y, Li) at the center of every magnitude level. Then, establish the histogram {hi(w)|wAIi} of magnitude level of high-frequency coefficients at decomposition level i.

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Step 4. Let parameters a be in the interval[1,3], Ci ¼ |wi|max|wi|min in the fuzzy membership functions (10) and (11). Then, apply the two functions to transforming the elements of the discrete magnitude level set {Tji}. Step 5. Compute the five segmentation thresholds TEi, TD(AAi0 ), TD(AB)i , TED(AAi0 ) and TED(AB)i by using the distribution histogram of the magnitude level set Ii according to formulas (17) and

(28)–(31). Two blending parameters are simply set l1 ¼ l2 ¼ 0.5 in the combination formulas (30) and (31). Step 6. Perform the magnitude normalization of the ridgelet high-frequency coefficients level by level according to formula (8). Step 7. Let parameter b be, respectively, equal to the magnified thresholds TEi, TD(AAi0 ), TD(AB)i , TED(AAi0 ) or TED(AB)i , and cA[1/ (|wi|meanK), 1/(|wi|mean+K)] (K is adjusting between 10 and 15)

Fig. 8. Results of contrast enhancement of three road crack images using different methods: (a–c) original images; (d–f) enhancement results by fuzzy entropy measurement; (g–i) enhancement results by the measurement of fuzzy divergence of one sort; (j–l) enhancement results by the measurement of fuzzy divergence of two sorts; (m–o) enhancement results by the measurement of the combination of fuzzy entropy and fuzzy divergence of one sort; (p–r) enhancement results by the measurement of the combination of fuzzy entropy and fuzzy divergence of two sorts; (s–u) enhancement results by histogram equalization; (v–x) enhancement results by wavelet transform.

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in the nonlinear enhancement function (9). Then transform the ridgelet high-frequency coefficients at decomposition level i by the nonlinear function. Step 8. Perform the inverse of the normalization formula (8) on the transformed high-frequency coefficients at decomposition level i. Then, execute the inverse ridgelet transform with the modified ridgelet coefficients to reconstruct the enhanced road crack image.

5. Experimental results To verify the effectiveness and feasibility of our image enhancement algorithm for the automatic detection of road cracks, we implement the image enhancement algorithm on a large number of road crack images, which not only record the polished concrete pavements but also the asphalt concrete ones. We compare our algorithm with other methods such as the histogram equalization and the wavelet-transform enhancement. Here, it is necessary to present the two parameters referred to in the enhancement algorithm at the beginning of the paper. The compactness factor a of the fuzzy membership function (10) and (11) are set as a ¼ 1 and the curve shape factor c of the nonlinear enhancement function (9) is in the interval [10, 40] (the values of the two parameters have been testified on a large number of road crack images for contrast enhancement). After preparing all parameter settings, we execute the enhancement algorithm according to the procedures described in Section 4, and the original road crack images shown in Fig. 8(a–c) where the 3-level ridgelet decomposition is performed. The visual effects of the enhanced road crack images are shown in: (i) Fig. 8(d–f), using our ridgelet enhancement algorithm where the parameter b is equal to TE in the nonlinear function (9); (ii) Fig. 8(g–i), using our ridgelet enhancement algorithm where the parameter b is equal to TD(AA0 ); (iii) Fig. 8(j–l), using our ridgelet enhancement algorithm where

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Fig. 9. Fuzzy entropy E (red curve), two fuzzy divergences DAA0 (dark curve), DAB (blue curve) and the linear combinations EDAA0 (pink curve), EDAB (green curve) versus the threshold T (magnitude level of normalized high-frequency coefficients) in the ridgelet domain. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the parameter b is equal to TD(AB); (iv) Fig. 8(m–o), using our ridgelet enhancement where the parameter b is equal to TED(AA0 ); (v) Fig. 8(p–r), using our ridgelet enhancement where the parameter b is equal to TED(AB); (vi) Fig. 8(s–u), using the histogram equalization enhancement; (vii) Fig. 8(v–x), using the wavelet-transform enhancement. In Fig. 9, the fuzzy entropy E(T), two fuzzy divergences DAA0 (T), DAB(T), and two supplementary linear combinations EDAA0 (T), EDAB(T), which are all defined on the ridgelet high-frequency coefficients at the first decomposition level, are all plotted in the same coordinate system after their respective normalization. According to visual effects, it is obvious from Fig. 8 that our method performs better than the histogram equalization en-

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Fig. 10. Marginal density of the original image shown in Fig. 8(b) and its enhanced image: (a) marginal density of original image; (b) marginal density of enhanced image using our method; (c) marginal density of enhanced image using the histogram equalization method; (d) marginal density of enhanced image by using wavelet-transform enhancement.

hancement and wavelet-transform enhancement because the latter two neither give prominence to the crack information that appears in the background nor noises. It is also obvious from Fig. 8(x) that some linear or zigzag crack information in road crack images is ruined by wavelet-transform enhancement. Although the noises are enhanced by using our enhancement method, the gap between the intensity of crack pixels and that of background pixels is widened so greatly that it meets the requirement of automatic detection of road cracks. The extreme points of five fuzzy measurements are revealed in Fig. 9, from which we also find that the blending threshold strategy for computing the parameter b in the nonlinear function (9) is moderate for the ridgelet enhancement of road crack images, because they are the trade-off between image contrast enhancement and image noise reduction as shown in Fig. 8(m–r). The amount of information gain or image restoration after image enhancement can be learned from the comparison of the marginal density between the enhanced images. Fig. 10 shows, respectively, the marginal densities of the original road crack image in Fig. 8(b) and of the enhanced ones in Fig. 8(h, t, w). It is apparent that the histogram equalization essentially destroys information on crack extraction through marginal density fitting,

and the gray level of the image enhanced by our method is greatly raised.

6. Conclusions The paper proposes an automatic image enhancement algorithm in ridgelet domain by using the fuzzy entropy and two fuzzy divergences for road crack detection. A self-adaptive nonlinear function is essential for transforming the subband high-frequency coefficients in our image enhancement algorithm. As is well-known, it is difficult to acquire the optimal parameter of the nonlinear function for the automatic enhancement of an inhomogeneous image. For this reason, we put forward some criteria based on the measurements of fuzzy entropy and those of fuzzy divergence to determine the ridgelet-domain segmentation threshold, which is helpful for automatically searching for the respective optimal parameters of the nonlinear function. Using the self-adaptive nonlinear function to transform the high-frequency coefficients, we obtain the desired enhanced image by carrying out the inverse ridgelet transform of it. The results of comparative experiments on the histogram distributions and visual effects of a large number of enhanced road crack images show preliminarily that our image enhancement algorithm

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