 Research
 Open Access
A novel segmentation method for uneven lighting image with noise injection based on nonlocal spatial information and intuitionistic fuzzy entropy
 Haiyan Yu^{1, 2}Email author and
 Jiulun Fan^{2}
https://doi.org/10.1186/s1363401705095
© The Author(s). 2017
 Received: 18 September 2016
 Accepted: 12 October 2017
 Published: 27 October 2017
Abstract
Local thresholding methods for uneven lighting image segmentation always have the limitations that they are very sensitive to noise injection and that the performance relies largely upon the choice of the initial window size. This paper proposes a novel algorithm for segmenting uneven lighting images with strong noise injection based on nonlocal spatial information and intuitionistic fuzzy theory. We regard an image as a gray wave in threedimensional space, which is composed of many peaks and troughs, and these peaks and troughs can divide the image into many local subregions in different directions. Our algorithm computes the relative characteristic of each pixel located in the corresponding subregion based on fuzzy membership function and uses it to replace its absolute characteristic (its gray level) to reduce the influence of uneven light on image segmentation. At the same time, the nonlocal adaptive spatial constraints of pixels are introduced to avoid noise interference with the search of local subregions and the computation of local characteristics. Moreover, edge information is also taken into account to avoid false peak and trough labeling. Finally, a global method based on intuitionistic fuzzy entropy is employed on the wave transformation image to obtain the segmented result. Experiments on several test images show that the proposed method has excellent capability of decreasing the influence of uneven illumination on images and noise injection and behaves more robustly than several classical global and local thresholding methods.
Keywords
 Image segmentation
 Local thresholding
 Non local spatial constraint
 Intuitionistic fuzzy theory
1 Introduction
Image segmentation, which is the extraction of an object from the background in an image, is one of the essential techniques in areas of image processing and computer vision [1, 2]. However, in some cases, some undesired disturbances in the thresholding segmentation process may generate a false segmentation result. Uneven lighting is one of the leading disturbance sources that can affect the segmentation result, which often is produced in the capturing of an image. The primary causes for the disturbance of uneven illumination are (a) the scene cannot be isolated from the shadows of other objects optically, (b) the light may be unstable in some cases, and (c) the object is very large, and thus it creates an uneven light distribution [3].
Thresholding is a direct and effective technique for image segmentation. The thresholding techniques performed on graylevel images can be divided into two categories, namely, bilevel and multilevel thresholding. In bilevel thresholding, pixels are classified into two different brightness regions as background and object. Multilevel thresholding is applied to more complex images, which contain several classes with different graylevel ranges.
Moreover, the current bilevel thresholding (binarization) techniques are usually divided into two classes, global and local thresholding. The global algorithms generally compute a threshold for an image. Most of the global methods originated in the twentieth century, i.e., the 1970s, which can be classified into several main categories. The first category is based on the shape of the histogram, such as valleyseeking method [4] and histogram approximation method [5]. The second category is based on clustering algorithm, such as Otsu’s method and the fuzzy clustering method. The Otsu’s method [6] is one of the most classical clustering methods, which segments an image by maximizing the betweenclass variance of the thresholded image. The fuzzy clustering [7] method is another classical global method that computes the fuzzy membership between the pixel and the mean value of two classes and finds groups by applying cluster analysis. Entropybased methods are the third category of global methods, such as the Shannon entropybased method [8], the Tsallis entropybased method [9], the Renyi entropybased method [10, 11], and the fuzzy entropybased method [12]. To improve the robustness to noise, the spatial information is taken into account and many modified versions of the Otsu method [13], the fuzzy entropybased method [14, 15], the fuzzy clustering method [16–18], and the Renyi entropybased method [19, 20] have emerged.
Meanwhile, a local method usually computes a different threshold for the neighbor of each pixel or for each appointed block in the image. Local thresholding algorithms are superior to global ones for segmenting uneven lighting images because they can select adaptive threshold values according to the local area information [21]. Neighborbased and blockbased methods are two major styles of local adaption methods. The neighborbased methods compute a threshold for each pixel based on the statistics of the arrangement, i.e., the variance of its neighborhood region. For example, Bernsen [22] selects the threshold by a function of the highest and lowest grayscale values. In Niblack’s method [23], a pixelwise threshold is calculated based on the standard deviation and the local mean of all pixels in the moving window over the gray image. Sauvola et al. [24, 25] first classify each window by content into text, picture, and background and then apply different segmentation rules to the various types of window. Kim [26] modifies Sauvola’s algorithm by introducing more than one window size for the type of text. Moreover, in order to improve the above approaches for the determination of the local threshold, several special features that are extracted in the pixel neighborhood are also taken into account, such as character stroke width [27] and gradient information [28]. In addition, Bradley and Roth [29] introduce spatial variations in lighting and propose a realtime adaptive thresholding technique which has strong robustness to lighting changes in the image by using the integral image of the input. Kim et al. [30] introduce a water flow model for document image binarization. In this model, an image surface is considered as a threedimensional terrain which is composed of valleys and mountains. Then, they find the local characteristic of the original terrain by pouring some water onto the terrain and computing the filled water. Lastly, they apply a global thresholding algorithm to find the text regions. Moreover, M. Valizadeh and E. Kabir in [31] improve the water flow model and propose an adaptive method to segment degraded document images.
Blockbased methods are another category of local thresholding algorithms, which divide the image into different subblocks. The subblocks are regarded as separate images and segmented by some principles. For example, Taxt [32] obtains the local threshold for each subblock based on EM algorithm. Eikvil et al. [33] propose a fast text binarization method by segmenting the subblocks based on the Otsu method. Park et al. [34] improve Eikvil’s method by segmenting the object subblocks with the Otsu method and the background subblocks based on their mean value. Huang et al. [3] propose a method that adaptively selects the block size. Chou et al. [35] discriminate the classification of the subblocks based on support vector machine (SVM) and segment the different subblock types with different strategies.
However, there are still several problems in these local thresholding methods. First, the segmentation accuracies of these window merging methods greatly depend on the reasonable selection of the initial window size. Second, partitioning the image into several subblocks usually leads to incoherent segmentation results between adjacent subblocks. Lastly, the existence of high noise level in the image may cause adjacent pixels of a pixel to contain abnormal features, thus leading to unsatisfactory segmentation results.
A wave transformation model, which is introduced by Wei et al., is a prospective idea for uneven lighting image segmentation [36]. They consider an image surface as a threedimensional terrain that is composed of mountains and valleys, corresponding to peaks and troughs, respectively, and partition the subregions with the local peaks and troughs in multidirections. Then, a wave transformation is performed on the grayscale waves in the local subregion, and a matrix of multidimensional vectors is obtained. Lastly, the vectors are compressed to one dimension using the principal component analysis (PCA) method, and an Otsu global method is employed to find an optimal wave threshold for segmenting the matrix. This algorithm does not require image partitioning and can yield good segmentation results for uneven light images. However, there are two serious drawbacks of the method. First, it is very sensitive to noise since it does not take into consideration the spatial information in the wave transformation. Second, when the variation of light intensity in the background is too large, it may lead to misclassification of some pixels.
On the other hand, since Zadeh [37] introduced the fuzzy set (FS) theory, it has been used to solve image segmentation problems regarding vague images. Pal and King [38] first introduce the fuzzy membership function and apply it in grayscale image processing. Then, many image segmentation algorithms based on the fuzzy theory are widely studied and are considered as efficient ways because they can describe the fuzzy uncertainty of images excellently [39]. Atanassov [40] proposes a novel concept of higher order FSs, i.e., intuitionistic fuzzy sets (IFSs), which provides a flexible mathematical frame to address the hesitancy derived from imprecise information. He describes the IFSs by two characteristic functions that express the degree of membership and the degree of nonmembership, representing the degree of belongingness and nonbelongingness, respectively, of elements to the IFS.
In this paper, we propose a novel local thresholding algorithm for segmenting uneven lighting images with noise injection. In particular, we introduce the idea of the wave transformation in Wei’s method and partition the image into subregions based on the local peaks and troughs in many straight lines extracted by rows and columns. Then, we perform the transformation of grayscale waves using fuzzy membership so that the relative characteristic (the local membership value) of each pixel substitutes its absolute characteristic (its gray level) to reduce the influence of uneven background light. Simultaneously, nonlocal spatial constraint and edge information obtained by the Sobel operator [41] are taken into account in order to avoid false peak and trough labeling caused by noise injection and large variation of light intensity. Lastly, we model the wave transformation image with the intuitionistic fuzzy theory and use a global intuitionistic fuzzy measure to segment the transformed image.
The rest of this paper is organized as follows. Section 2 introduces the wave transformation for images and intuitionistic fuzzy set theory. Section 3 describes our segmentation method. Section 4 presents the experimental results and comparison with several wellknown segmentation algorithms. Section 5 gives the conclusions.
2 Preliminaries
2.1 Wave transformation for computing the local characteristics of an image
In this section, we give a brief introduction to the wave transformation model proposed by Wei et al. [36], which is used to reduce the influence of uneven light on the segmentation of images.
Intuitively speaking, the image can be viewed as waves of the gray pixels. For an individual pixel, its local characteristic is immediately concerned with the location in the corresponding wave. The pixel located at the peak of the wave has a relatively higher level, while the pixel located at the trough of the wave has a relatively lower level. The location of the pixel in the local wave, namely, the wave transformation value that reflects the relative characteristic of a pixel in the local subregion, can be used to replace its original gray level for segmenting images [36].
Therefore, the pixels located at the two peaks have the same wave transformation values, namely, \( {w}_{p_1}={w}_{p_2} \), no matter how large their original gray levels are. Likewise, \( {w}_{t_1}={w}_{t_2} \) for the pixels at the two troughs. Suppose two pixels are located at k _{1}, k _{2} with g(k _{1}) < g(k _{2}) in local subregions ϕ _{1}, ϕ _{2}, i.e., the yellow shadows in Fig. 2, respectively. If H(k _{1}; T _{1}, P _{1}) > H(k _{2}; T _{2}, P _{2}), their transformation values satisfy u(H(k _{1}; T _{1}, P _{1})) > u(H(k _{2}; T _{2}, P _{2})) because u is a monotonous increasing function, which indicates that the wave transformation values, located in the green shadows in Fig. 2, depend on the relative characteristics of the pixels rather than their absolute gray levels, thereby reducing the impact of uneven lighting [36].
2.2 Intuitionistic fuzzy sets and intuitionistic fuzzy entropy
In this section, we present the basic elements of intuitionistic fuzzy set theory and two fuzzy membership functions, which will be used in the wave transformation model and image segmentation.
For fuzzy sets \( \tilde{A} \) and \( \tilde{B} \), ∀x ∈ X, the membership functions of \( \tilde{A}\cap \tilde{B} \) and \( \tilde{A}\cup \tilde{B} \) are defined as \( {\mu}_{\tilde{A}\cap \tilde{B}}(x)=\min \left({\mu}_{\tilde{A}}(x),{\mu}_{\tilde{B}}(x)\right) \) and \( {\mu}_{\tilde{A}\cup \tilde{B}}(x)=\max \left({\mu}_{\tilde{A}}(x),{\mu}_{\tilde{B}}(x)\right) \). \( {\tilde{A}}^c \) is used to express the complement of \( \tilde{A} \), that is, \( {\mu}_{{\tilde{A}}^c}(x)=c\left({\mu}_{\tilde{A}}(x)\right) \) and ∀x ∈ X, where c is a complementary function.
Moreover, an FS \( \tilde{A} \) defined on X can also be represented using the notation of IFSs as follows: \( \tilde{A}=\left\{<x,{\mu}_A(x),1{\mu}_A(x)>x\in X\right\} \) with π _{ A }(x) = 0 for all x ∈ X.
In addition, an axiom definition of intuitionistic fuzzy entropy measures is also introduced by Burillo and Bustince [41] to measure the fuzziness of an intuitionistic fuzzy set.
Definition 5. Intuitionistic fuzzy entropy is a function E : F(X) → R ^{+}(R ^{+} = [0, +∞)) and satisfies the following conditions: IFS1: E(A) = 0 iff A is an FS. IFS2: E(A) = Cardinal(X) = n iff μ _{ A }(x _{ i }) ≥ v _{ A }(x _{ i }) = 0 for all x _{ i } ∈ X. IFS3: E(A) ≥ E(B) iff A≼B, i.e., μ _{ A }(x _{ i }) ≤ μ _{ B }(x _{ i }) and v _{ A }(x _{ i }) ≤ v _{ B }(x _{ i }), for all x _{ i } ∈ X. IFS4: E(A) = E(A ^{ c }).
3 The proposed method
3.1 Overview of the approach
As mentioned in Section 2.1, the wave transformation in Wei’s method can reduce the bad influence of uneven light on the image segmentation by obtaining the local characteristic of each pixel. It is accomplished by dividing the image into a number of local subregions and computing the local characteristic value of each pixel based on its location in its corresponding region. Specially, the subregions are obtained by searching for local peaks and troughs based on the grayscale levels of pixels within a straight line extracted from the image in a given direction.

Step 1. We first obtain the nonlocal meanfiltered image of a graylevel image in order to improve the noise resistance ability.

Step 2. Then, we apply the wave transformation on the filtered image to eliminate uneven light of the image, namely, computing the local membership value of each pixel by a fuzzy membership function. This procedure is completed by mainly four steps. (1) First, we need to find the local subregions that each pixel is located in. More concretely, all straight lines of the image in two given directions (namely, the horizontal and the vertical directions) are extracted. The significant local peaks and troughs in each line are searched for, and two neighboring troughs and peaks constitute a local subregion. (2) Then, the local membership degrees of each pixel located in the two corresponding subregions (respectively in the horizontal and vertical directions) are computed by a fuzzy membership function. (3) Moreover, the local membership degree of each pixel is further revised by combining with its edge information, in order to avoid false peak and trough labeling caused by the large variation of light intensity. (4) Lastly, the local membership degrees in the horizontal and vertical directions for each pixel are integrated with its nonlocal weight matrix, thus obtaining the final local membership values of all pixels and constituting the wave transformation matrix for the image.

Step 3. The final membership matrix is used to replace the grayscale matrix of the image; then, it is modeled and segmented with the intuitionistic fuzzy theory.
3.2 Wave transformation of an image using nonlocal spatial information and fuzzy membership
3.2.1 Nonlocal filter of the image
Here, h is the filtering degree parameter, N _{ k } is a z × z square neighborhood centered at the pixel Q(x _{ k }, y _{ k }), a(a > 0) is the standard deviation of the Gaussian kernel, and \( Z(k)=\sum \limits_{j\in {V}_k^r}{e}^{\left({\left\Vert {N}_k{N}_j\right\Vert}_{2,a}^2\right)/{h}^2} \) is a normalized constant [45]. The weight v(k, j) depends on the similarity between the neighborhood configurations of the pixel (x _{ k }, y _{ k }) and the pixel (x _{ j }, y _{ j }), which satisfies 0 ≤ v(k, j) ≤ 1 and \( \sum \limits_{j\in {V}_k^r}v\left(k,j\right)=1 \). For the pixel (x _{ k }, y _{ k }), the spatial information v(k, j) will be used for the integration of fuzzy memberships in the next section.
Theorem 1. Suppose the gray level of a pixel (x, y) in the uneven lighting image f _{ δ } is constituted by f _{ δ }(x, y) = f(x, y) + δ(x, y), where f(x, y) is the original intensity of (x, y) in the image with even light and δ(x, y) is the intensity of the uneven light in (x, y). Given that δ(x, y) remains approximately constant in the local region, the estimated value \( {\overline{Q}}_{\delta}\left(x,y\right) \) of each pixel (x, y) in the uneven lighting image f _{ δ } by the nonlocal filter is equal to the estimated value \( \overline{Q}\left(x,y\right) \) of (x, y) in the original image f plus the uneven light intensity of (x, y), namely, \( {\overline{Q}}_{\delta}\left(x,y\right)=\overline{Q}\left(x,y\right)+\delta \left(x,y\right) \) .
Theorem 1 indicates that the nonlocal filter does not change the light intensities of an image and removes the noise under the premise that the light intensity δ(x, y) remains approximately unchanged in the local region, which is prepared for the followup process, i.e., wave transformation.
3.2.2 Wave transformation with fuzzy membership theory and nonlocal spatial information
Divide the filtered image into local subregions by searching for local peaks and troughs in straight lines
Compute the wave transformation value (local membership value) of each pixel in the horizontal and vertical directions using fuzzy membership
After finding all the subregions constituted by peaks {P _{1}, P _{2}, …, P _{ R }} and troughs {T _{1}, T _{2}, …, T _{ S }} in the gray wave curve g(k), the wave transformation can be applied on the pixels in g(k). Suppose there are several local subregions ϕ _{1}, ϕ _{2}, ⋯, ϕ _{ s }, ⋯ in g(k). Let a local subregion ϕ _{ s } = [T _{ s }, T _{ s + 1}] consist of a peak P _{ s } and two troughs T _{ s } and T _{ s + 1}, where T _{ s } < P _{ s } < T _{ s + 1}. Let \( {\phi}_{s_1}=\left[{t}_s,{p}_s\right] \) be the rising edge interval and \( {\phi}_{s_2}=\left[{P}_s,{T}_{s+1}\right] \) be the training edge interval. The local membership degree G(k) of each pixel k in the region \( {\phi}_{s_1}=\left[{T}_s,{P}_s\right] \) can be determined as follows.
Theorem 2. Suppose the gray level of a pixel (x, y) in the uneven lighting image f _{ δ } is constituted by f _{ δ }(x, y) = f(x, y) + δ(x, y), where f(x, y) is the original intensity of (x, y) in the image with even light and δ(x, y) is the intensity of the uneven light in (x, y). Given that δ(x, y) remains approximately constant in a local subregion, the wave transformation matrix by the Sfunction Ψ(f) of the original image f is approximately equal to the wave transformation matrix Ψ(f _{ δ }) of the uneven lighting image f _{ δ } .
Theorem 2 indicates that the wave transformation of an image using the fuzzy membership function (Sfunction) can reduce the light intensity difference between neighborhood subregions, thus markedly decreasing the influence of uneven light on the image segmentation.
Revise the wave transformation values of each pixel using its edge information
However, when the premise that the intensity of uneven light remains approximately unchanged in local subregions cannot be satisfied, the local membership by the wave transformation should be modified. That is, when the variation of the light intensity in a pure background is so large that it is bigger than the threshold α in Eq. (13), a false subregion composed by a peak and a trough must be extracted, thus easily leading to misclassification. This is because in a subregion, the pixels close to the peak correspond to the object, and the pixels close to the trough correspond to the background according to Wave transformation for computing the local characteristics of an image. Therefore, the pixels in the pure background will be classified into two classes, i.e., the object and the background, by the segmentation of the wave transformation matrix, which actually leads to the misclassification of some pixels to object.
According to Theorem 2, it holds under the premise that the intensity of the uneven light δ(x, y) remains approximately constant in each local subregion. For the pixel k ∈ [T _{ s }, P _{ s }], the condition ∃j ∈ [T _{ s }, P _{ s }], E(x _{ j }, y _{ j }) ≥ T indicates that there is edge information, and the trough T _{ s } and the peak P _{ s } are searched for due to the radical change of the gray level in the subregion \( {\phi}_{s_1}=\left[{T}_s,{P}_s\right] \). That is, there are two different classes in the subregion. Moreover, the intensity of the uneven light δ(x, y) can be regarded as approximately constant in \( {\phi}_{s_1}=\left[{T}_s,{P}_s\right] \). Then, Theorem 2 holds, and the wave transformation value can remain unchanged, namely, \( {G}_{d_H}^{\prime }(k)={G}_{d_H}(k) \). If the pixel k ∈ [T _{ s }, P _{ s }] does not satisfy the condition ∃j ∈ [T _{ s }, P _{ s }], E(x _{ j }, y _{ j }) ≥ T, this indicates that there is no edge information. That is, there is only one class in the subregion, and the trough T _{ s } and the peak P _{ s } are searched for due to the gradual change of uneven light intensity in the subregion \( {\phi}_{s_1}=\left[{T}_s,{P}_s\right] \). Therefore, the variation of uneven light intensity is too large, and its value cannot be regarded as approximately constant in \( {\phi}_{s_1}=\left[{T}_s,{P}_s\right] \). Then, Theorem 2 does not hold, and the wave transformation value \( {G}_{d_H}^{\prime }(k) \) should be modified to the intensity of the background.
Integrate the horizontal and vertical transformation values of each pixel with its nonlocal information
Theorem 3. Suppose the gray level of a pixel (x, y) in the uneven lighting image f _{ δ } is constituted by f _{ δ }(x, y) = f(x, y) + δ(x, y), where f(x, y) is the original intensity of (x, y) in the image with even light, and δ(x, y) is the intensity of the uneven light in (x, y). Given that δ(x, y) remains approximately constant in a local subregion, the integration of the horizontal and vertical transformation values of each pixel in the original image f with its nonlocal information is approximately equal to that of each pixel in the uneven lighting image f _{ δ } , namely, \( {\varPsi}_{\delta}^{\prime}\left({x}_k,{y}_k\right)\approx {\varPsi}^{\prime}\left({x}_k,{y}_k\right) \) .
Theorem 3 indicates that the addition of the nonlocal space information does not change the uneven light intensity, and the integration of the horizontal and vertical wave transformation values with the nonlocal space information can reduce the influence of the uneven light on the image segmentation.
After calculating all the local membership values of pixels in the image according to the abovementioned method, we obtain the final 2D wave transformation matrix. In the matrix, the characteristic of a pixel is represented by the relative vector (the local membership value) only related to its subregion in order to reduce the lighting difference between two neighborhood subregions. At the same time, the nonlocal information is incorporated to overcome the influence of the local highfrequency signal on the establishment of the membership matrix. That is, although the membership degrees of pixels to the local peaks substitute the gray levels as a new expression of pixels, they do not separate with the original gray level and space information completely. Last, the membership matrix Ψ ^{′} will be classified using IFS entropy to obtain a final segmented image.
3.3 Segmentation of transformed image using intuitionistic fuzzy set
We apply L level quantization on the membership matrix Ψ ^{′}(x, y) of size M × N to obtain a new matrix I ^{′}(x, y), which is also called wave transformation image. Then, the image is modeled based on intuitionistic fuzzy set [47]. Suppose the image I ^{′}(x, y) has L gray levels G _{ x } = {0, 1, ⋯, L − 1}, and its histogram is H = {h _{0}, h _{1}, …, h _{ L − 1}}. Let the 1D sample space X = G _{ x } = {0, 1, ⋯, L − 1}, and p is the probability of a gray level, i.e., p({i}) = h _{ i }, i = 0, 1, ⋯, L − 1, where i is the quantization level.
4 Results and discussion
4.1 Performance of wave transformation using the fuzzy membership function
Then, the proposed method is applied on the test image, with α = 60 (see in Eq. (13)) for the search of peaks and troughs in S4. Several peaks and the troughs of the 45th column are searched for in Fig. 6e of the rice image, where the symbol “○” represents the peaks and the symbol “△” represents the troughs. The local membership values (wave transformation values) of the 45th column are obtained in Fig. 6f by the Sfunction. It is obvious that all of the peaks and troughs are respectively located in the same horizontal lines. The local characteristics of other pixels are represented by their locations in the subregions.
Figure 6h shows the wave transformation result of 45th–50th columns. Figure 6b shows the transformation image of the rice image. It is obvious that the dark region in the lower part of the rice image is appropriately lightened to the same as the upper part of the rice image. Thus, a threshold of 167 can be easily found by the IFS entropy in S, and all of the “rice” objects are extracted by the IFS entropybased method, as shown in Fig. 6d. Moreover, the threshold of 167 corresponds to a membership degree of \( 167\cdot \frac{1}{L}=0.6523 \) because the transformation image is obtained by applying L = 256 level quantization on the membership matrix.
4.2 Performance of the revision of the wave transformation values using edge information
Figure 7c_{2} shows the gray wave transformation of the 160th row when the edge information is taken into account by Eq. (19), with T = 800. For the pixels in the pure background subregions without edge information, i.e., k ∈ [T _{1}, P _{1}] or k ∈ [P _{2}, T _{3}], the transformation values are set to 1, namely, the value of the background \( {G}_{\mathrm{Row}\_{160}^{\mathrm{th}}}^{\prime }(k)=\mathrm{background}=1 \). For the pixels in the subregions k ∈ [P _{1}, T _{2}] or k ∈ [T _{2}, P _{2}] with edge information, the transformation values are unchanged, i.e., \( {G}_{\mathrm{Row}\_{160}^{\mathrm{th}}}^{\prime }(k)={G}_{\mathrm{Row}\_{160}^{\mathrm{th}}}(k) \). Figure 7b_{1}, c_{1} shows the corresponding gray wave transformation images of Fig. 7b_{2}, c_{2}. It is obvious that the transformation image in Fig. 7c_{1} agrees with the actual requirement more than that in Fig. 7b_{1}. Figure 7b_{3}, c_{3} shows the segmented images before and after adding edge information. Apparently, when the edge information is taken into account, the proposed method can obtain a better segmented result.
4.3 Performance of segmentation of uneven lighting images with noise injection
To prove the effectiveness of our method for uneven lighting images with strong noise injection, experimental tests are implemented on six uneven illumination images corrupted by the Gaussian noise. The intensity value of the pixel in these images varies from 0 to 255, namely, L = 256. In the experiments, several classical local methods, Bernsen’s method [22], Niblack’s method [23], and Sauvola’s method [24]; related works for the local methods, Bradley’s method [29], Chou’ s method [35], and Valizadeh’s method [31]; based on an improved water flow modelbased method, Wei’s method [36]; and several related global twodimensional methods with space information, the twodimensional Otsu method (2DOtsu) [13] proposed by Liu et al., the fuzzy cmeans clustering algorithm with nonlocal spatial information (FCM_NLS) [16] proposed by Zhao et al., and the twodimensional weak fuzzy partition entropybased method (2DWFPE) [14] proposed by Yu et al. are implemented on the test images, and their results are compared with our method.
The parameter settings for these methods are as follows. Bernsen’s method uses a 93 × 93 neighborhood. Niblack’s method uses a 50 × 50 neighborhood with k = − 0.2. Sauvola’s method uses a 50 × 50 neighborhood with k = 0.2. Bradley’s method uses a 30 × 30 neighborhood. Valizadeh’s method uses the parameter W = 2. Chou’s method uses a mean threshold of 128 and a variance threshold of 10 with a block size 3 × 3. Wei’s method uses the parameter α = 60. FCM_NLS uses the parameter β = 10. Specifically, for the parameters of the nonlocal filter in our method and FCM_NLS, we set r = 5, a = 2, and h = 15 for the fingerprint image and r = 5, a = 2, and h = 30 for the other images.
Missclassfication rate (ME) obtained by the eight methods on images corrupted by addictive noise
ME  Gaussian noise  Bernsen’s method  Niblack’s method  Sauvola’s method  Bradley’s method  Chou’s method  Valizadeh’s method  Wei’s method  2D Otsu  FCM_NLS  2D WFPE  Our method 

Rice  0.010  0.1718  0.3497  0.2225  0.3537  0.3502  0.2064  0.3307  0.0884  0.0462  0.0686  0.0284 
Fingerprint  0.002  0.1386  0.1707  0.1758  0.1417  0.1751  0.1394  0.1023  0.2359  0.2091  0.3645  0.0644 
Mouse  0.010  0.2897  0.2818  0.2341  0.2421  0.2852  0.2929  0.4277  0.5326  0.5184  0.5982  0.0471 
Block  0.015  0.3246  0.5262  0.4734  0.4685  0.5254  0.5007  0.4637  0.1497  0.0815  0.1708  0.0089 
Licenseplate  0.015  0.1348  0.2552  0.2264  0.2817  0.2549  0.2526  0.2042  0.0523  0.0487  0.0414  0.0258 
Coin  0.010  0.2229  0.3704  0.3164  0.3540  0.3154  0.2813  0.3096  0.1925  0.2000  0.2017  0.1167 
4.4 Influence of the parameters α and T
Actually, the value of α seriously depends on the gray difference between the objects and the background in the dark regions. In most cases, the larger the gray level difference of the objects and the background in the dark regions, the larger the reasonable value of α and the larger the range in which α can take value in, and vice versa. That is, the smaller the degree of uneven lighting, the larger the range in which α can take value in. Under the condition of the Gaussian noise (0,0.010), if we want to obtain a ME value smaller than 0.05, α can take value in [22, 120] for the rice image as shown in Fig. 14a, and α can take value in [20, 220] for the licenseplate image as shown in Fig. 15b. However, for the mouse image, α only can take value in [40, 80] for the relatively satisfactory results in Fig. 15a. Moreover, the ME value also depends on many factors, such as the proportion of the dark regions in the whole image and the intensity of local illumination variation.
Noise injection also has an important influence on the segmentation performance especially on the dark region with a small gray difference between the objects and the background. To reduce the influence of noise injection on the search of local peaks and troughs and differentiate the objects and the background in the dark regions, the reasonable range of α is narrowed with increasing noise strength. Taking the rice image for example, in order to obtain a ME value smaller than 0.05, the reasonable range of the parameter α under the condition of the Gaussian noise (0,0.020) is [42, 125], which is narrower than the range [22, 125] of α under the condition of the Gaussian noise (0,0.010), as shown in Fig. 14a, b.
The segmentation results of our method also depend on the edge detection with the parameter T in Eq. (19). For the rice image, three curves with the parameter T respectively equal to 600, 800, and 1000 tend to change similarly with the parameter α. However, a reasonable parameter T is crucial to the segmentation performance for the mouse image that has relatively more serious uneven light. As shown in Fig. 15a, the ME value with T = 600 is bigger than 0.05, since incomplete edge information leads to some incorrect computation of local memberships and consequently bad segmentation results.
In conclusion, the parameter selection of our method is affected by multiple factors, such as the noise strength, the degree of uneven lighting, and the gray difference between objects and the background. However, the above experimental results show that the intervals [60, 80] and [800, 1000] may be two reasonable ranges respectively for α and T to take value in. Moreover, when the light intensity of the subregions becomes darker, the value of the two parameters should decrease properly.
5 Conclusions
In this paper, we presented a novel algorithm for the segmentation of uneven background lighting images with strong noise injection. We first treated the image as a gray wave in threedimensional space and extracted grayscale wave curves in the horizontal and vertical directions. Then, we applied wave transformation on the curves using fuzzy membership to obtain the relative characteristic of each pixel in order to reduce the influence of the uneven background lighting. Simultaneously, the nonlocal spatial weight matrix and edge information were also taken into account in the transformation in order to improve the robustness of the transformation to noise injection and avoid false peak and trough labeling. Finally, we segmented the wave transformation image using intuitionistic fuzzy theory. In different experiments, our algorithm demonstrated superior performance against some wellknown algorithms on several uneven background lighting images.
Although the proposed algorithm for uneven lighting image segmentation has some advantages, there are still two problems requiring further study. The first critical problem is the selection of the parameter α. The parameter α is set manually based on experience in this paper. Therefore, further research on the automatic determination of α with consideration of both the uneven lighting background and noise injection is necessary. The second problem is the detection of edge information for a noisy image. The edge information has an important impact on the wave transformation of pixels in an uneven lighting image. In this paper, we used two 5 × 5 Sobel models for the edge detection. However, when we used a given global threshold T to extract the edge information, there were still small noise edges being detected in some cases. Therefore, how to effectively detect large edges in the dark regions and ignore locally fluctuated edges caused by noise will also be part of our future research.
Declarations
Acknowledgements
This work was supported in by the National Natural Science Foundation of China (Nos. 61102095, 61671377, 61571361, 61340040, 61601362), Natural Science Basic Research Plan in Shaanxi Province of China (No. 2012JQ8045), and Special Research Project of Shaanxi Department of Education (No. 2013JK1131).
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 SB Chaabane, M Sayadi, F Fnaiech, E Brassart, Colour image segmentation using homogeneity method and data fusion techniques. EURASIP J. Adv. Signal. Process. 2010(367297), 11 (2010)Google Scholar
 S Kasaei, M Hasanzadeh, Fuzzy image segmentation using membership connectedness. EURASIP J. Adv. Signal. Process. 2008(417293), 13 (2008)MATHGoogle Scholar
 Q Huang, W Gao, W Cai, Thresholding technique with adaptive window selection for uneven lighting image. Pattern Recogn. Lett. 26, 801–808 (2005)View ArticleGoogle Scholar
 C Sahasrabudhes, SD Gupta, A valleyseeking threshold selection technique. Comput. Vis. Image. Underst. 56, 55–65 (1992)Google Scholar
 N Ramesh, H Yoo, IK Sethi, Thresholding based on histogram approximation. IEEE Proc. Vis. Image. Signal. Process. 142(5), 271–279 (1995)View ArticleGoogle Scholar
 N Otsu, A thresholding selection method from grayscale histogram. IEEE Trans. Syst. Man. Cybern. 9, 62–66 (1979)View ArticleGoogle Scholar
 JC Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms (Plenum, NewYork, 1981)View ArticleMATHGoogle Scholar
 TW Ridler, S Calvard, Picture thresholding using an iterative selection method. IEEE Trans. Syst. Man. Cybern. 8, 630–632 (1978)View ArticleGoogle Scholar
 G Johannsen, J Bille, A threshold selection method using information measures, Proceedings of the 6th International Conference on Pattern Recognition, Munich (1982), pp. 140–143Google Scholar
 PK Sahoo, C Wilkins, J Yeage, Threshold selection using Renyi’s entropy. Pattern Recogn. 30(1), 71–84 (1997)View ArticleMATHGoogle Scholar
 FY Nie, PF Zhang, JQ Li, DH Ding, A novel generalized entropy and its application in image thresholding. Signal Process. 134, 23–34 (2017)View ArticleGoogle Scholar
 AKC Wong, PK Sahoo, A graylevel threshold selection method based on maximum entropy principle. IEEE Trans. Syst. Man. Cybern. 19, 886–871 (1989)View ArticleGoogle Scholar
 JZ Liu, WQ Li, The automatic threshold of graylevel pictures via twodimensional Otsu method. Acta Automat. Sin. 19(1), 101–105 (1993) (in Chinese)Google Scholar
 HY Yu, XB Zhi, JL Fan, Image segmentation based on weak fuzzy partition entropy. Neurocomputing 168, 994–1010 (2015)View ArticleGoogle Scholar
 JH Lan, YL Zeng, Multithreshold image segmentation using maximum fuzzy entropy based on a new 2D histogram. Optik. Int. J. Light. Electron. Opt. 124(18), 3756–3760 (2013)View ArticleGoogle Scholar
 F Zhao, LC Jiao, HQ Liu, XB Gao, A novel fuzzy clustering algorithm with non local adaptive spatial constraint for image segmentation. Signal Process. 91, 988–999 (2011)View ArticleMATHGoogle Scholar
 F Zhao, HQ Liu, JL Fan, A multiobjective spatial fuzzy clustering algorithm for image segmentation. Appl. Soft Comput. 30, 48–57 (2015)View ArticleGoogle Scholar
 LH Son, TM Tuan, Dental segmentation from Xray images using semisupervised fuzzy clustering with spatial constraints. Eng. Appl. Artif. Intell. 59, 186–195 (2017)View ArticleGoogle Scholar
 PK Sahoo, G Arora, A thresholding method based on two dimensional Renyi’s entropy. Pattern Recogn. 37, 1149–1161 (2004)View ArticleMATHGoogle Scholar
 AB Ishak, Choosing parameters for Rényi and Tsallis entropies within a twodimensional multilevel image segmentation framework. Physica A: Statistical Mechanics and its Applications. 466, 521536 (2017).Google Scholar
 Y Liu, SN Srihari, Document image binarization based on texture features. IEEE Trans. Pattern Anal. Mach. Intell. 19, 540–544 (1997)View ArticleGoogle Scholar
 J Bernsen, Dynamic Thresholding for GrayLevel Images, Proceedings of the Eighth International Conference Pattern Recognition, Paris (1986), pp. 1251–1255Google Scholar
 W Niblack, An Introduction to Digital Image Processing (Prentice Hall, Englewood Cliffs, 1986), pp. 115–116Google Scholar
 J Sauvola, M Pietikainen, Adaptive document image binarization. Pattern Recogn. 33, 225–236 (2000)View ArticleGoogle Scholar
 J Sauvola, T Seppanen, S Haapakoski, M Pietikainen, Adaptive document binarization, Proceedings of the ICDAR (1997), pp. 147–152Google Scholar
 IJ Kim, in Ninth International Workshop on Frontiers in Hand Writing Recognition. Multiwindow binarization of camera image for document recognition (2004), pp. 323–327Google Scholar
 Y Yang, H Yan, An adaptive logical method for binarization of degraded document images. Pattern Recogn. 33(5), 787–807 (2000)View ArticleGoogle Scholar
 ØD Trier, T Taxt, Improvement of ‘integrated function algorithm’ for binarization of document images. Pattern Recogn. Lett. 16(3), 277–283 (1995)View ArticleGoogle Scholar
 D Bradley, G Roth, Adaptive thresholding using the integral image. J. Graph. Gpu. Game. Tools. 12(2), 13–21 (2007)View ArticleGoogle Scholar
 IK Kim, DW Jung, RH Park, Document image binarization based on topographic analysis using a water flow model. Pattern Recogn. 35, 265–277 (2002)View ArticleMATHGoogle Scholar
 M Valizadeh, E Kabir, An adaptive water flow model for binarization of degraded document images. Int. J. Doc. Anal. Recognit. 16, 165–176 (2013)View ArticleGoogle Scholar
 T Taxt, PJ Flynn, AK Jain, Segmentation of document images. IEEE Trans. Pattern Anal. Mach. Intell. 11(12), 1322–1329 (1989)View ArticleGoogle Scholar
 L Eikvil, T Taxt, K Moen, in Proceedings of the International Conference on Document Analysis and Recognition. A fast adaptive method for binarization of document images (1991), pp. 435–443Google Scholar
 JH Park, IH Jang, NC Kim, in Proceedings of the IEEE Pacific Rim Conference on Communications, Computers and Signal Processing. Skew Correction of Business Card Images in PDA (2003), pp. 724–727Google Scholar
 CH Chou, WH Lin, F Chang, A binarization method with learningbuilt rules for document images produced by cameras. Pattern Recogn. 43, 1518–1530 (2010)View ArticleMATHGoogle Scholar
 W Wei, XJ Shen, QJ Qian, An adaptive thresholding algorithm based on grayscale wave transformation for industrial inspection images. Acta Automat. Sin. 37(8), 944–953 (2011) (in Chinese)Google Scholar
 LA Zadeh, Fuzzy sets. Inf. Control. 8(3), 338–353 (1965)View ArticleMATHGoogle Scholar
 SK Pal, RA King, Image enhancement using smoothing with fuzzy set. IEEE Trans. Syst. Man Cybern. 11(7), 494–501 (1981)View ArticleGoogle Scholar
 S Krinidis, V Chatzis, A robust fuzzy local information Cmeans clustering algorithm. IEEE Trans. Image Process. 19(5), 1328–1337 (2010)View ArticleMATHMathSciNetGoogle Scholar
 KT Atanassov, Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)View ArticleMATHGoogle Scholar
 P Burillo, H Bustince, Entropy on intuitionistic fuzzy sets and on intervalvalued fuzzy sets. Fuzzy Sets Syst. 78, 305–316 (1996)View ArticleMATHMathSciNetGoogle Scholar
 KT Atanassov, S Steova, Intuitionistic Fuzzy Sets, in: Proceedings of the Polish Symposium on Interval and Fuzzy Mathematics, Pozan (1993), pp. 23–26Google Scholar
 SK Pal, DKD Majumder, Fuzzy Mathematical Approach to Pattern Recognition (J Wiley, New York, 1986)MATHGoogle Scholar
 T Chaira, AK Ray, Segmentation using fuzzy divergence. Pattern Recogn. Lett. 24, 1837–1844 (2003)View ArticleGoogle Scholar
 A Buades, B Coll, JM Morel, in In Proceeding of IEEE Computer Society Conference on Computer Vision & Pattern Recognition. A nonlocal algorithm for image denoising, vol 2 (2005), pp. 60–65Google Scholar
 PF Jin, Improved algorithm for Sobel edge detection of image. J. Appl. Opt. 29(4), 625–628 (2008) in ChineseMathSciNetGoogle Scholar
 IK Vlachos, GD Sergiadis, Intuitionistic fuzzy information—applications to pattern recognition. Pattern Recogn. Lett. 28, 197–206 (2007)View ArticleGoogle Scholar
 M Sezgin, B Sankur, Survey over image thresholding techniques and quantitative performance evaluation. J. Electron. Imaging. 13(1), 146–168 (2004)View ArticleGoogle Scholar