Binarization of ESPI fringe patterns based on local entropy

Mingming Chen; Chen Tang; Min Xu; Zhenkun Lei

doi:10.1364/OE.27.032378

1. Introduction

Electronic speckle pattern interferometry (ESPI) is a well-known non-contact measurement means [1–3]. It has been continuously demonstrated for many investigations including displacement [4–6], deformation [7,8], strain [9–11], temperature [12], and so on. The key of the successful application of ESPI is the accurate extraction of phase terms from fringe patterns [13,14]. Many methods have been developed to estimate the phase terms. Among these methods, the fringe skeleton method is the most straightforward approach, and is often used in dynamic measurement [15]. In the fringe skeleton method, the phase terms can be obtained by interpolating the assigned skeletons [16]. Unfortunately, the fringe patterns obtained from ESPI contain massive inherent speckle noise which masks significant features [17,18]. To obtain fringe skeletons, a commonly used method is that the fringe patterns should be smoothed enough firstly, then binarization results of the smoothed fringe patterns are obtained with threshold methods, and finally fringe skeletons are extracted using thinning operation. The filtering is often a necessary step prior to binarization. Without filtering preprocessing, the traditional binarization method for ESPI fringe patterns can not give desired binarization results [19]. Therefore, the investigation of a simple method which can effectively binarize ESPI fringe patterns without filtering preprocessing is expected, especially in dynamic measurement.

Up to now, the fringe skeleton method has been improved from three aspects. In the first aspect, a majority of effort is to improve the denoising algorithms [20]. A lot of filtering methods for ESPI fringe patterns have been proposed. We refer to [21] for more details. If the filtered fringe pattern is smoothed enough, the skeletons can be obtained with a common binarization thinning method. In the second aspect, some efforts are put into improving the accuracy of interpolation algorithms. The existing interpolation algorithms mainly include the nearest interpolation method [22], the bilinear interpolation method [23], the C-spline interpolation method [24], back propagation neural networks (BPNN) method [25], and radial basis function (RBF) method [26]. It is worth mentioning that the RBF method can work well even with the disconnected skeletons. In the last aspect, some efforts are put into extracting the skeleton directly from fringe patterns [15]. Recently, the gradient vector fields (GVFs) method was proposed [27–29]. Some PDEs for calculating GVFs have been proposed [15,28,30]. Nevertheless, to the best of our knowledge, compared with aforementioned aspects, there are few efforts are concentrated on developing the binarization method for the skeletons extraction of ESPI fringe patterns. The existing method used for the binarization of fringe patterns is known as the threshold method. The threshold method includes global threshold and local threshold [31–33]. Both global threshold and local threshold lie in the determination of gray level threshold when used for the fringe patterns binarization. These methods can not give the desired binarization results of ESPI fringe patterns without filtering preprocessing, because it is difficult and sometimes impossible to precisely calculate the gray level threshold due to the presence of speckle noise. We will verify this point in the experiment section. In addition, recently, we proposed a binarization method for fringe patterns with intensity inhomogeneities based on modified FCM algorithm in [19]. This method can resist a certain amount of noise, but it can not work well on the ESPI fringe patterns which contain high levels of noise. Here, we focus our attention on improving the fringe skeleton method by proposing effective binarization method for ESPI fringe patterns.

Entropy is a statistical measure of randomness [34]. It has been widely used in image processing, such as image recovery, edge detection, image matching [35–38]. It can provide a good level of information to describe a given image. For instance, in a given gray image, if all pixels have the same grayscale, this image will exhibit the minimal entropy value; on the contrary, the image will present maximum entropy when each pixel in this image presents a specific grayscale [39]. The local entropy of the given image is related to the variance of grayscales in a local region [40,41]. In the ESPI fringe patterns, the intrinsic speckle noise creates a grainy appearance. Under the influence of speckle noises, the bright fringes and the dark fringes tend to result in different distribution of gray level. The distribution of gray level in bright fringes is more disordered than the one in dark fringes. Therefore, the local entropy values of bright fringes will be different from the ones in dark fringes. Thus, the local entropy can be taken as a feature which can characterize the bright and dark fringes in the ESPI fringe patterns. Based on this point, in this paper, we present a binarization method based on local entropy and fuzzy c-means (FCM) clustering algorithm for ESPI fringe patterns. In the proposed binarization method, different from the traditional method, the ESPI fringe pattern is clustered into white fringes and black fringes according to its local entropy map instead of the original intensity image. There is no need to perform the filtering preprocessing because the speckle noises are utilized as essentials. The skeletons can be extracted with simple thinning operation based on the proposed binarization method.

The main contributions of this paper are as follows. (1) The local entropy is introduced to the binarization algorithm of ESPI fringe pattern for the first time. (2) The proposed ESPI fringe pattern binarization method does not need filtering preprocessing.

The remainder of this paper is organized as follows: Section 2 details the basic principle and main steps of the proposed method. Section 3 gives experiments results and discussions. Finally, the conclusion is given in Section 4.

2. Description of the proposed method

In this section, we describe the proposed binarization method for ESPI fringe patterns in detail. As shown in Fig. 1, the local entropy of the ESPI fringe pattern is calculated first. Then the FCM is applied to the local entropy map. Finally, the ESPI fringe pattern is clustered into white fringes and black fringes according to its local entropy.

Fig. 1. The flowchart of the proposed method.

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2.1. Local entropy extracting

Different from the traditional binarization method for ESPI fringe patterns, taking into account the characteristic of intrinsic speckle noise, our binarization method is performed on the local entropy map of the given image instead of the original intensity image. Because entropy is a statistical measure of randomness, it can be used to describe the disorder of gray levels or intensity distribution of the pixels [42]. The local entropy values of bright fringes will be different from the ones in dark fringes, this provides a possible way to binarize the ESPI fringe pattern according to its local entropy.

Local entropy is derived by calculating the entropy in a local region of the given image. Let X be a given fringe pattern with the size of $m \times n$. The local entropy of pixel j can be defined as [39]:

(1)$$E(j )={-} \sum\limits_{g = 1}^N {P(g )} {log _2}P(g )$$

where $E(j )$ is the entropy of pixel $j$, $P(g )$ is the probability of gray level g in the neighborhood of pixel j, $N$ is the gray level of X. Let W be the size of neighbor window centered at pixel j.

From Eq. (1), we can see that the local entropy values are small for the regions with uniform distribution of gray levels, but large for the regions with uneven gray scale distribution. Therefore, the local entropy values in the regions of bright fringes are larger than those in the regions of dark fringe. The local entropy map can be obtained by moving the neighbor window pixel by pixel within the fringe pattern from left to right and top to bottom.

2.2. Applying FCM algorithm to the local entropy map

Because of the difference between the local entropy values in the regions of bright fringes and those of dark fringes, the local entropy map can be segmented by an appropriate threshold. Instead of threshold methods which are used for traditional ESPI fringe pattern binarization, we choose FCM clustering algorithm to segment the local entropy map.

Take the local entropy map E of ESPI fringe pattern as the dataset for FCM clustering. Then rearrange $E = {[{{e_1},{e_2}, \cdots ,{e_M}} ]^T}$, $M = m \times n$. FCM is an iterative optimization that minimizes the cost function [43]:

(2)$$J = \sum\limits_{j = 1}^M {\sum\limits_{i = 1}^c {u_{ij}^\beta } } {\left\Vert{E(j )- v(i )} \right\Vert^2}$$

where c is the number of cluster; $u_{ij}^{}$ represents the membership of the local entropy $E(j )$ in the $i$th cluster; $\beta \in [{1,\infty } ]$ is the cluster fuzziness, in our method, we set $\beta = 2$; $v(i )$ is the $i$th cluster center; $\left\Vert\cdot \right\Vert$ is the norm metric. The membership functions and clusters are updated as [44]:

(3)$${u_{ij}} = \frac{1}{{\sum\limits_{k = 1}^c {{{\left( {\frac{{\left\Vert{E(j )- v(i )} \right\Vert}}{{\left\Vert{E(j )- v(k )} \right\Vert}}} \right)}^{2/({\beta - 1} )}}} }}$$

and

(4)$${v_{ij}} = \frac{{\sum\limits_{j = 1}^M {u_{ij}^\beta E(j )} }}{{\sum\limits_{j = 1}^M {({u_{ij}^\beta } )} }}$$

After applying FCM clustering algorithm to local entropy map, a $c \times M$ partition matrix $U$ is obtained. $U$ can be expressed as:

(5)$$U = \left[ {\begin{array}{{cccc}} {{u_{11}}}&{u{}_{12}}& \cdots &{{u_{1M}}}\\ {{u_{21}}}&{{u_{22}}}& \cdots &{{u_{2M}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{u_{c1}}}&{{u_{c2}}}& \cdots &{{u_{cM}}} \end{array}} \right]$$

The rows of U denote clusters and the columns of U denote elements of the local entropy map. ${u_{ij}} \in [{0,1} ]$ is the membership degree of the local entropy $E(j )$ belonging to cluster i.

2.3. Binarization according to clustering results

Binarization of fringe pattern can be viewed as the classification of all the pixels of the given fringe pattern into two clusters: white fringes and dark fringes. Therefore, we set $c = 2$ in our method. The partition matrix

(6)$$U = \left[ {\begin{array}{{cccc}} {{u_{11}}}&{u{}_{12}}& \cdots &{{u_{1M}}}\\ {{u_{21}}}&{{u_{22}}}& \cdots &{{u_{2M}}} \end{array}} \right]$$

Furthermore, calculating the maximum value of each column in U, and these values constitute the matrix $MAXU \in {{\mathbb R}^{1 \times N}}$. Set A be zero matrixes with the same size as $MAXU$. If $MAXU(j )= {u_{ij}},({i = 1,2;j = 1,2, \cdots ,M} )$, the pixel j in X belongs to cluster i. Subsequently, set $A_j^{}$=0 (or 1). Finally, rearranging $A$ into the matrix of the size $m \times n$. The pixels of fringe pattern X are classified into 2 categories. One category is the white fringes and the other one is the dark fringes. Thus, the binarization of fringe pattern is realized.

3. Experiments and discussion

In this section, we demonstrate the performance of our method via application to six computer-simulated fringe patterns and ten real ESPI fringe patterns. Moreover, we compare our method with the typical global threshold method [31] and the global and local threshold combining method [33]. The parameters in all methods are chosen based on the better performance by trial. We set the values of the window sizes in the global and local threshold combining method [33] and our method to be 15×15, 15×15, 15×15, 13×13, 11×11, 15×15 for Figs. 2(a)–2(f); 35×35, 29×29, 29×29, 21×21, 19×19, 15×15 for Figs. 6(a)–6(f); 29×29, 27×27, 29×29, 13×13 for Figs. 9(a)–9(d), respectively; the parameters $\beta = 2$, $c = 2$, $\varepsilon = 0.001$ in all implementations for our method. Not that, the size of window should be adjusted according to the size of speckle. In general, the larger the size of speckle, the larger the size of window should be. All numerical simulations are carried out on the MATLAB R2014a platform with Processor Intel(R) Core(TM) i5-4590 CPU 3.30 GHz, Memory 4.00 GB RAM, and 64-bit Operating System.

Firstly, we validate the performance of our method by employing the computer-simulated fringe patterns. As shown in Fig. 2, the shape and density of fringes in each computer-simulated fringe pattern are different. The size of each test image is 512×512.

i. In order to confirm the feasibility of the proposed binarization method, we show the corresponding local entropy maps of Fig. 2 in Fig. 3. From Fig. 3, one can see that the local entropy values in the regions of bright fringes are larger than the ones in dark fringes. It is validated that it is feasible for the binarization of ESPI fringe pattern based on the local entropy.
ii. We compare our method with the methods of typical global threshold method [31] and global and local threshold combining method [33]. The comparison results are shown in Fig. 4. The Figs. 4(a-1), 4(b-1), 4(c-1), 4(d-1), 4(e-1), and 4(f-1) are the binarization results of the method of typical global threshold method [31]. The Figs. 4(a-2), 4(b-2), 4(c-2), 4(d-2), 4(e-2), and 4(f-2) are the binarization results of the global and local threshold combining method [33]. The Figs. 4(a-3), 4(b-3), 4(c-3), 4(d-3), 4(e-3), and 4(f-3) are the binarization results of our method. From Fig. 4, it is obviously seen that the methods of typical global threshold method [31] and the global and local threshold combining method [33] can not realize the binarization of the test images due to the influence of intrinsic speckle noise in the ESPI fringe patterns. On the contrary, our method can give a desired binarization result thanks to the speckle noise in the ESPI fringe patterns.
iii. We extract the black fringe skeletons of the binarization results of our method. In the fringe skeleton method, the purpose of ESPI fringe pattern binarization is to extract the skeletons. Note that good binarization will result in good skeletons with the same thinning method. Therefore, we extract the skeletons of the binarization results to confirm the performance of our method in further. Figure 5 shows the corresponding skeletons of black fringes of Figs. 4(a-3), 4(b-3), 4(c-3), 4(d-3), 4(e-3), and 4(f-3) which are obtained by using the Hilditch [45] thinning method and deburring processing. From Fig. 5, we can see that the binarization results of our method can give desired skeletons results.

Fig. 2. The computer-simulated fringe patterns.

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Fig. 3. The corresponding entropy maps of Fig. 2.

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Fig. 4. The binarization results of different methods.

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Fig. 5. The corresponding skeleton results of Fig. 4.

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Secondly, we validate the performance of our method by employing two groups of experimentally obtained ESPI fringe patterns. The first group of ESPI fringe patterns is shown in Fig. 6. It contains six real ESPI fringe patterns which were obtained from the dynamic measurement of thermal deformation of a plate under high intensity light. The size of each experimentally obtained ESPI fringe pattern in this group is $412 \times 513$. The second group of ESPI fringe patterns is shown in Fig. 9. It contains four real ESPI fringe patterns which have different shapes and were obtained from different experiments. The sizes of Figs. 9(a)–9(d) are $512 \times 512$, $611 \times 576$, $512 \times 512$, and $225 \times 226$, respectively.

i. We validate the feasibility of our method on the binarization of real ESPI fringe patterns. The corresponding binarization results of Fig. 6 and Fig. 9 are shown in Fig. 7 and Fig. 10, respectively. From Fig. 7 and Fig. 10, it can be seen that our method can realize the binarization of experimentally obtained ESPI fringe patterns.
ii. We extract the black fringe skeletons of the binarization results in Fig. 7, and the white fringe skeletons of the binarization results in Fig. 10 with the same thinning method as Fig. 5. From the skeleton results shown in Fig. 8 and Fig. 11, we can find that the binarization results of real ESPI fringe patterns of our method can give desired skeleton results.

Fig. 6. The first group of experimentally obtained ESPI fringe patterns.

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Fig. 7. The binarization results of Fig. 6 of our method.

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Fig. 8. The corresponding skeleton results of Fig. 7.

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Fig. 9. The second group of experimentally obtained ESPI fringe patterns.

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Fig. 10. The binarization results of Fig. 9 of our method.

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Fig. 11. The corresponding skeleton results of Fig. 10.

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From the above results of the computer-simulated and experimentally obtained ESPI fringe patterns, we can get the following conclusions: (1) The binarization of ESPI fringe patterns can be realized based on local entropy; (2) Without the denoising processing, the commonly used ESPI fringe pattern binarization methods which are proposed in [31] and [33] can not realize binarization, but our method can give desired binarization results; (3) The binarization results of our method can give desired skeleton results.

4. Conclusions

In this paper, we proposed a binarization method based on local entropy and FCM algorithm. There is no need to perform the filtering process before binarization, which is different from the traditional binarization methods for ESPI fringe patterns. Instead, our method takes the intrinsic speckle noise in ESPI fringe patterns as essentials and utilizes it by binarizing the ESPI fringe patterns according to their local entropy. The skeletons can be obtained with common thinning method based on the proposed binarization method. The experimental results of both computer-simulated fringe patterns and real fringe patterns demonstrate that our method can effectively binarize the ESPI fringe patterns, and the binarization results can give desired skeleton results. Our method can accelerate the development of the computer analysis of fringe patterns for dynamic measurement.

Funding

National Natural Science Foundation of China (11772081).

Acknowledgment

The authors thank the editor and anonymous reviewers for their comments and suggestions on the paper.

Disclosures

The authors declare no conflicts of interest.

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Binarization of ESPI fringe patterns based on local entropy

Abstract

1. Introduction

2. Description of the proposed method

2.1. Local entropy extracting

2.2. Applying FCM algorithm to the local entropy map

2.3. Binarization according to clustering results

3. Experiments and discussion

4. Conclusions

Funding

Acknowledgment

Disclosures

References

Cited By

Figures (11)

Equations (6)

Optics Express