Abstract

The fringe skeleton method is the most straightforward analysis method for phase extraction and widely used in dynamic measurement. Binarization is often required in this method. In the traditional binarization methods, filtering is often a necessary step prior to binarization due to the influence of intrinsic speckle noises in ESPI fringe patterns. In this paper, we propose a binarization method based on local entropy and fuzzy c-means (FCM) clustering algorithm. In this method, the pixels in the given ESPI fringe pattern are clustered into white fringes and black fringes according to their local entropy instead of the original intensity information. There is no need to perform the filtering preprocessing, because the intrinsic speckle noises are utilized as essentials. We evaluate the performance of our method by applying it to the computer-simulated and real fringe patterns. Experimental results demonstrate that the proposed method can achieve the desired binarization results, and the binarization results can give desired skeleton results.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Electronic speckle pattern interferometry (ESPI) is a well-known non-contact measurement means [13]. It has been continuously demonstrated for many investigations including displacement [46], deformation [7,8], strain [911], 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 [2729]. 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 [3133]. 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 [3538]. 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.

Download Full Size | PPT Slide | PDF

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]:

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

$$J = \sum\limits_{j = 1}^M {\sum\limits_{i = 1}^c {u_{ij}^\beta } } {\left\Vert{E(j )- v(i )} \right\Vert^2}$$
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]:
$${u_{ij}} = \frac{1}{{\sum\limits_{k = 1}^c {{{\left( {\frac{{\left\Vert{E(j )- v(i )} \right\Vert}}{{\left\Vert{E(j )- v(k )} \right\Vert}}} \right)}^{2/({\beta - 1} )}}} }}$$
and
$${v_{ij}} = \frac{{\sum\limits_{j = 1}^M {u_{ij}^\beta E(j )} }}{{\sum\limits_{j = 1}^M {({u_{ij}^\beta } )} }}$$
After applying FCM clustering algorithm to local entropy map, a $c \times M$ partition matrix $U$ is obtained. $U$ can be expressed as:
$$U = \left[ {\begin{array}{{cccc}} {{u_{11}}}&{u{}_{12}}& \cdots &{{u_{1M}}}\\ {{u_{21}}}&{{u_{22}}}& \cdots &{{u_{2M}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{u_{c1}}}&{{u_{c2}}}& \cdots &{{u_{cM}}} \end{array}} \right]$$
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

$$U = \left[ {\begin{array}{{cccc}} {{u_{11}}}&{u{}_{12}}& \cdots &{{u_{1M}}}\\ {{u_{21}}}&{{u_{22}}}& \cdots &{{u_{2M}}} \end{array}} \right]$$
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 3.30GHz, 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.

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

 

Fig. 4. The binarization results of different methods.

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

 

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

Download Full Size | PPT Slide | PDF

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.

References

1. B. Sharp, “Electronic speckle pattern interferometry (ESPI),” Opt. Laser. Eng. 11(4), 241–255 (1989). [CrossRef]  

2. M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).

3. C. Dong, K. Li, Y. Jiang, D. Arola, and D. Zhang, “Evaluation of thermal expansion coefficient of carbon fiber reinforced composites using electronic speckle interferometry,” Opt. Express 26(1), 531–543 (2018). [CrossRef]  

4. S. Wang, M. Lu, L. M. Bilgeri, M. Jakobi, F. S. Bloise, and A. W. Koch, “Temporal electronic speckle pattern interferometry for real-time in-plane rotation analysis,” Opt. Express 26(7), 8744–8755 (2018). [CrossRef]  

5. P. P. Padghan and K. M. Alti, “Quantification of nanoscale deformations using electronic speckle pattern interferometer,” Opt. Laser Technol. 107, 72–79 (2018). [CrossRef]  

6. D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016). [CrossRef]  

7. P. D. Ruiz, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement by wavelength-scanning electronic speckle pattern interferometry,” Appl. Opt. 44(19), 3945–3953 (2005). [CrossRef]  

8. W. An and T. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Laser. Eng. 40(5-6), 529–541 (2003). [CrossRef]  

9. M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016). [CrossRef]  

10. G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016). [CrossRef]  

11. G. Tavera Ruiz, H. Manuel, M. Flores-Moreno, C. Frausto-Reyes, and M. Santoyo, “Cortical bone quality affectations and their strength impact analysis using holographic interferometry,” Biomed. Opt. Express 9(10), 4818–4833 (2018). [CrossRef]  

12. M. Kumar and S. Chandra, “Measurement of temperature and temperature distribution in gaseous flames by digital speckle pattern shearing interferometry using holographic optical element,” Opt. Laser. Eng. 73, 33–39 (2015). [CrossRef]  

13. M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019). [CrossRef]  

14. C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express 18(9), 8942–8947 (2010). [CrossRef]  

15. C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33(2), 183–185 (2008). [CrossRef]  

16. X. Zhu, Z. Chen, C. Tang, Q. Mi, and X. Yan, “Application of two oriented partial differential equation filtering models on speckle fringes with poor quality and their numerically fast algorithms,” Appl. Opt. 52(9), 1814–1823 (2013). [CrossRef]  

17. F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019). [CrossRef]  

18. Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018). [CrossRef]  

19. M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019). [CrossRef]  

20. M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019). [CrossRef]  

21. B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017). [CrossRef]  

22. H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018). [CrossRef]  

23. C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

24. C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the delta-mollification phase map method,” Appl. Opt. 45(28), 7392–7400 (2006). [CrossRef]  

25. C. Tang, W. Lu, S. Chen, Z. Zhang, B. Li, W. Wang, and L. Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46(30), 7475–7484 (2007). [CrossRef]  

26. G. Wang, Y. J. Li, and H. C. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50(19), 3110–3117 (2011). [CrossRef]  

27. X. Chen, C. Tang, B. Li, and Y. Su, “Gradient vector fields based on variational image decomposition for skeletonization of electronic speckle pattern interferometry fringe patterns with variable density and their applications,” Appl. Opt. 55(25), 6893–6902 (2016). [CrossRef]  

28. C. Tang, H. Ren, L. Wang, Z. Wang, L. Han, and T. Gao, “Oriented couple gradient vector fields for skeletonization of gray-scale optical fringe patterns with high density,” Appl. Opt. 49(16), 2979–2984 (2010). [CrossRef]  

29. B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019). [CrossRef]  

30. F. Zhang, D. Wang, Z. Xiao, L. Geng, J. Wu, Z. Xu, J. Sun, K. Wang, and J. Tao, “Skeleton extraction and phase interpolation for single ESPI fringe pattern based on the partial differential equations,” Opt. Express 23(23), 29625–29638 (2015). [CrossRef]  

31. N. Otsu, “A Threshold Selection Method from Gray-Level Histograms,” IEEE Trans. Syst., Man, Cybern. 9(1), 62–66 (1979). [CrossRef]  

32. J. Bernsen, “Dynamic thresholding of grey-level images,” In: 8th International Conference on Pattern Recognition, Paris France, 27-31 October, 1986, pp. 1251–1255.

33. L. Wang, G. Leedham, and S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recognit. 41(3), 920–929 (2008). [CrossRef]  

34. H. Zhang, J. E. Fritts, and S. A. Goldman, “Entropy-based objective evaluation method for image segmentation,” In: Proceedings of SPIE - International Society for Optics and Photonics, San Jose, California, United States, 18 December 2003, pp. 38–50.

35. C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999). [CrossRef]  

36. H. Deng, J. Liu, and Z. Chen, “Infrared small target detection based on modified local entropy and EMD,” Chin. Opt. Lett. 8(1), 24–28 (2010). [CrossRef]  

37. X. Wang and C. Chen, “Ship Detection for Complex Background SAR Images Based on a Multiscale Variance Weighted Image Entropy Method,” IEEE Geosci. Remote Sensing Lett. 14(2), 184–187 (2017). [CrossRef]  

38. W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018). [CrossRef]  

39. A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011). [CrossRef]  

40. J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).

41. C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003). [CrossRef]  

42. C. Qi, “Maximum entropy for image segmentation based on an adaptive particle swarm optimization,” Appl. Math. Inf. Sci. 8(6), 3129–3135 (2014). [CrossRef]  

43. C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984). [CrossRef]  

44. J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008). [CrossRef]  

45. http://cgm.cs.mcgill.ca/∼godfried/teaching/projects97/azar/skeleton.html.

References

  • View by:
  • |
  • |
  • |

  1. B. Sharp, “Electronic speckle pattern interferometry (ESPI),” Opt. Laser. Eng. 11(4), 241–255 (1989).
    [Crossref]
  2. M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).
  3. C. Dong, K. Li, Y. Jiang, D. Arola, and D. Zhang, “Evaluation of thermal expansion coefficient of carbon fiber reinforced composites using electronic speckle interferometry,” Opt. Express 26(1), 531–543 (2018).
    [Crossref]
  4. S. Wang, M. Lu, L. M. Bilgeri, M. Jakobi, F. S. Bloise, and A. W. Koch, “Temporal electronic speckle pattern interferometry for real-time in-plane rotation analysis,” Opt. Express 26(7), 8744–8755 (2018).
    [Crossref]
  5. P. P. Padghan and K. M. Alti, “Quantification of nanoscale deformations using electronic speckle pattern interferometer,” Opt. Laser Technol. 107, 72–79 (2018).
    [Crossref]
  6. D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
    [Crossref]
  7. P. D. Ruiz, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement by wavelength-scanning electronic speckle pattern interferometry,” Appl. Opt. 44(19), 3945–3953 (2005).
    [Crossref]
  8. W. An and T. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Laser. Eng. 40(5-6), 529–541 (2003).
    [Crossref]
  9. M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
    [Crossref]
  10. G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
    [Crossref]
  11. G. Tavera Ruiz, H. Manuel, M. Flores-Moreno, C. Frausto-Reyes, and M. Santoyo, “Cortical bone quality affectations and their strength impact analysis using holographic interferometry,” Biomed. Opt. Express 9(10), 4818–4833 (2018).
    [Crossref]
  12. M. Kumar and S. Chandra, “Measurement of temperature and temperature distribution in gaseous flames by digital speckle pattern shearing interferometry using holographic optical element,” Opt. Laser. Eng. 73, 33–39 (2015).
    [Crossref]
  13. M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
    [Crossref]
  14. C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express 18(9), 8942–8947 (2010).
    [Crossref]
  15. C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33(2), 183–185 (2008).
    [Crossref]
  16. X. Zhu, Z. Chen, C. Tang, Q. Mi, and X. Yan, “Application of two oriented partial differential equation filtering models on speckle fringes with poor quality and their numerically fast algorithms,” Appl. Opt. 52(9), 1814–1823 (2013).
    [Crossref]
  17. F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019).
    [Crossref]
  18. Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
    [Crossref]
  19. M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
    [Crossref]
  20. M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
    [Crossref]
  21. B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
    [Crossref]
  22. H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018).
    [Crossref]
  23. C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).
  24. C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the delta-mollification phase map method,” Appl. Opt. 45(28), 7392–7400 (2006).
    [Crossref]
  25. C. Tang, W. Lu, S. Chen, , Z. Zhang, B. Li, W. Wang, and L. Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46(30), 7475–7484 (2007).
    [Crossref]
  26. G. Wang, Y. J. Li, and H. C. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50(19), 3110–3117 (2011).
    [Crossref]
  27. X. Chen, C. Tang, B. Li, and Y. Su, “Gradient vector fields based on variational image decomposition for skeletonization of electronic speckle pattern interferometry fringe patterns with variable density and their applications,” Appl. Opt. 55(25), 6893–6902 (2016).
    [Crossref]
  28. C. Tang, H. Ren, L. Wang, Z. Wang, L. Han, and T. Gao, “Oriented couple gradient vector fields for skeletonization of gray-scale optical fringe patterns with high density,” Appl. Opt. 49(16), 2979–2984 (2010).
    [Crossref]
  29. B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
    [Crossref]
  30. F. Zhang, D. Wang, Z. Xiao, L. Geng, J. Wu, Z. Xu, J. Sun, K. Wang, and J. Tao, “Skeleton extraction and phase interpolation for single ESPI fringe pattern based on the partial differential equations,” Opt. Express 23(23), 29625–29638 (2015).
    [Crossref]
  31. N. Otsu, “A Threshold Selection Method from Gray-Level Histograms,” IEEE Trans. Syst., Man, Cybern. 9(1), 62–66 (1979).
    [Crossref]
  32. J. Bernsen, “Dynamic thresholding of grey-level images,” In: 8th International Conference on Pattern Recognition, Paris France, 27-31 October, 1986, pp. 1251–1255.
  33. L. Wang, G. Leedham, and S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recognit. 41(3), 920–929 (2008).
    [Crossref]
  34. H. Zhang, J. E. Fritts, and S. A. Goldman, “Entropy-based objective evaluation method for image segmentation,” In: Proceedings of SPIE - International Society for Optics and Photonics, San Jose, California, United States, 18 December 2003, pp. 38–50.
  35. C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999).
    [Crossref]
  36. H. Deng, J. Liu, and Z. Chen, “Infrared small target detection based on modified local entropy and EMD,” Chin. Opt. Lett. 8(1), 24–28 (2010).
    [Crossref]
  37. X. Wang and C. Chen, “Ship Detection for Complex Background SAR Images Based on a Multiscale Variance Weighted Image Entropy Method,” IEEE Geosci. Remote Sensing Lett. 14(2), 184–187 (2017).
    [Crossref]
  38. W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
    [Crossref]
  39. A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
    [Crossref]
  40. J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).
  41. C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003).
    [Crossref]
  42. C. Qi, “Maximum entropy for image segmentation based on an adaptive particle swarm optimization,” Appl. Math. Inf. Sci. 8(6), 3129–3135 (2014).
    [Crossref]
  43. C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984).
    [Crossref]
  44. J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
    [Crossref]
  45. http://cgm.cs.mcgill.ca/∼godfried/teaching/projects97/azar/skeleton.html .

2019 (5)

M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
[Crossref]

F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
[Crossref]

B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
[Crossref]

2018 (7)

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018).
[Crossref]

Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
[Crossref]

G. Tavera Ruiz, H. Manuel, M. Flores-Moreno, C. Frausto-Reyes, and M. Santoyo, “Cortical bone quality affectations and their strength impact analysis using holographic interferometry,” Biomed. Opt. Express 9(10), 4818–4833 (2018).
[Crossref]

C. Dong, K. Li, Y. Jiang, D. Arola, and D. Zhang, “Evaluation of thermal expansion coefficient of carbon fiber reinforced composites using electronic speckle interferometry,” Opt. Express 26(1), 531–543 (2018).
[Crossref]

S. Wang, M. Lu, L. M. Bilgeri, M. Jakobi, F. S. Bloise, and A. W. Koch, “Temporal electronic speckle pattern interferometry for real-time in-plane rotation analysis,” Opt. Express 26(7), 8744–8755 (2018).
[Crossref]

P. P. Padghan and K. M. Alti, “Quantification of nanoscale deformations using electronic speckle pattern interferometer,” Opt. Laser Technol. 107, 72–79 (2018).
[Crossref]

2017 (2)

B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
[Crossref]

X. Wang and C. Chen, “Ship Detection for Complex Background SAR Images Based on a Multiscale Variance Weighted Image Entropy Method,” IEEE Geosci. Remote Sensing Lett. 14(2), 184–187 (2017).
[Crossref]

2016 (4)

X. Chen, C. Tang, B. Li, and Y. Su, “Gradient vector fields based on variational image decomposition for skeletonization of electronic speckle pattern interferometry fringe patterns with variable density and their applications,” Appl. Opt. 55(25), 6893–6902 (2016).
[Crossref]

D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
[Crossref]

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

2015 (3)

M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).

M. Kumar and S. Chandra, “Measurement of temperature and temperature distribution in gaseous flames by digital speckle pattern shearing interferometry using holographic optical element,” Opt. Laser. Eng. 73, 33–39 (2015).
[Crossref]

F. Zhang, D. Wang, Z. Xiao, L. Geng, J. Wu, Z. Xu, J. Sun, K. Wang, and J. Tao, “Skeleton extraction and phase interpolation for single ESPI fringe pattern based on the partial differential equations,” Opt. Express 23(23), 29625–29638 (2015).
[Crossref]

2014 (1)

C. Qi, “Maximum entropy for image segmentation based on an adaptive particle swarm optimization,” Appl. Math. Inf. Sci. 8(6), 3129–3135 (2014).
[Crossref]

2013 (1)

2011 (2)

G. Wang, Y. J. Li, and H. C. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50(19), 3110–3117 (2011).
[Crossref]

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

2010 (3)

2008 (3)

C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33(2), 183–185 (2008).
[Crossref]

L. Wang, G. Leedham, and S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recognit. 41(3), 920–929 (2008).
[Crossref]

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (1)

2003 (3)

W. An and T. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Laser. Eng. 40(5-6), 529–541 (2003).
[Crossref]

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003).
[Crossref]

1999 (1)

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999).
[Crossref]

1989 (1)

B. Sharp, “Electronic speckle pattern interferometry (ESPI),” Opt. Laser. Eng. 11(4), 241–255 (1989).
[Crossref]

1984 (1)

C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984).
[Crossref]

1979 (1)

N. Otsu, “A Threshold Selection Method from Gray-Level Histograms,” IEEE Trans. Syst., Man, Cybern. 9(1), 62–66 (1979).
[Crossref]

Accadia, C.

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

Agarwal, R.

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

Alti, K. M.

P. P. Padghan and K. M. Alti, “Quantification of nanoscale deformations using electronic speckle pattern interferometer,” Opt. Laser Technol. 107, 72–79 (2018).
[Crossref]

An, W.

W. An and T. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Laser. Eng. 40(5-6), 529–541 (2003).
[Crossref]

Arola, D.

Arruda, G. F.

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

Barbieri, A. L.

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

Bernsen, J.

J. Bernsen, “Dynamic thresholding of grey-level images,” In: 8th International Conference on Pattern Recognition, Paris France, 27-31 October, 1986, pp. 1251–1255.

Bezdek, C.

C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984).
[Crossref]

Bhutani, R.

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

Bilgeri, L. M.

Bloise, F. S.

Bruno, O. M.

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

Cai, Y.

Carlsson, T.

W. An and T. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Laser. Eng. 40(5-6), 529–541 (2003).
[Crossref]

Casaioli, M.

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

Chandra, S.

M. Kumar and S. Chandra, “Measurement of temperature and temperature distribution in gaseous flames by digital speckle pattern shearing interferometry using holographic optical element,” Opt. Laser. Eng. 73, 33–39 (2015).
[Crossref]

Chen, C.

X. Wang and C. Chen, “Ship Detection for Complex Background SAR Images Based on a Multiscale Variance Weighted Image Entropy Method,” IEEE Geosci. Remote Sensing Lett. 14(2), 184–187 (2017).
[Crossref]

Chen, M.

M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
[Crossref]

B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
[Crossref]

Chen, S.

Chen, X.

Chen, Z.

Cho, S. Y.

L. Wang, G. Leedham, and S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recognit. 41(3), 920–929 (2008).
[Crossref]

Costa, L. D.

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

Deng, H.

Dong, C.

Ehrlich, R.

C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984).
[Crossref]

Flores-Moreno, J. M.

D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
[Crossref]

Flores-Moreno, M.

Frausto-Reyes, C.

Fritts, J. E.

H. Zhang, J. E. Fritts, and S. A. Goldman, “Entropy-based objective evaluation method for image segmentation,” In: Proceedings of SPIE - International Society for Optics and Photonics, San Jose, California, United States, 18 December 2003, pp. 38–50.

Full, W.

C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984).
[Crossref]

Gadow, R.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Gao, G.

Gao, T.

Gao, X.

H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018).
[Crossref]

Gaur, K.

M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).

Geng, L.

Goldman, S. A.

H. Zhang, J. E. Fritts, and S. A. Goldman, “Entropy-based objective evaluation method for image segmentation,” In: Proceedings of SPIE - International Society for Optics and Photonics, San Jose, California, United States, 18 December 2003, pp. 38–50.

Gu, Q.

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Guo, D.

J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).

Han, L.

Hao, F.

Hawkes, D. J.

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999).
[Crossref]

Hill, D. L. G.

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999).
[Crossref]

Huang, Y.

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Huntley, J. M.

Huo, W.

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Jakobi, M.

Jiang, Y.

Killinger, A.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Koch, A. W.

Kong, J.

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

Kumar, M.

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).

M. Kumar and S. Chandra, “Measurement of temperature and temperature distribution in gaseous flames by digital speckle pattern shearing interferometry using holographic optical element,” Opt. Laser. Eng. 73, 33–39 (2015).
[Crossref]

Lavagnini, A.

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

Leedham, G.

L. Wang, G. Leedham, and S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recognit. 41(3), 920–929 (2008).
[Crossref]

Lei, Z.

B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
[Crossref]

F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019).
[Crossref]

Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
[Crossref]

B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
[Crossref]

Li, B.

B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
[Crossref]

Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
[Crossref]

B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
[Crossref]

X. Chen, C. Tang, B. Li, and Y. Su, “Gradient vector fields based on variational image decomposition for skeletonization of electronic speckle pattern interferometry fringe patterns with variable density and their applications,” Appl. Opt. 55(25), 6893–6902 (2016).
[Crossref]

C. Tang, W. Lu, S. Chen, , Z. Zhang, B. Li, W. Wang, and L. Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46(30), 7475–7484 (2007).
[Crossref]

C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the delta-mollification phase map method,” Appl. Opt. 45(28), 7392–7400 (2006).
[Crossref]

Li, J.

H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018).
[Crossref]

Li, K.

Li, W.

J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).

Li, Y. J.

Liu, J.

Lu, M.

Lu, W.

Lu, Y.

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

Manuel, D.

D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
[Crossref]

Manuel, H.

Mariani, S.

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

Martínez-García, V.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Mi, Q.

Montes, H.

D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
[Crossref]

Osten, W.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Otsu, N.

N. Otsu, “A Threshold Selection Method from Gray-Level Histograms,” IEEE Trans. Syst., Man, Cybern. 9(1), 62–66 (1979).
[Crossref]

Padghan, P. P.

P. P. Padghan and K. M. Alti, “Quantification of nanoscale deformations using electronic speckle pattern interferometer,” Opt. Laser Technol. 107, 72–79 (2018).
[Crossref]

Pedrini, G.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Pei, J.

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Qi, C.

C. Qi, “Maximum entropy for image segmentation based on an adaptive particle swarm optimization,” Appl. Math. Inf. Sci. 8(6), 3129–3135 (2014).
[Crossref]

Qi, M.

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

Qiu, T.

J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).

Ren, H.

H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018).
[Crossref]

C. Tang, H. Ren, L. Wang, Z. Wang, L. Han, and T. Gao, “Oriented couple gradient vector fields for skeletonization of gray-scale optical fringe patterns with high density,” Appl. Opt. 49(16), 2979–2984 (2010).
[Crossref]

Rodrigues, F. A.

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

Ruiz, P. D.

Sang, N.

C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003).
[Crossref]

Santoyo, F. M.

D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
[Crossref]

Santoyo, M.

Schmauder, S.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Shakher, C.

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).

Sharp, B.

B. Sharp, “Electronic speckle pattern interferometry (ESPI),” Opt. Laser. Eng. 11(4), 241–255 (1989).
[Crossref]

Speranza, A.

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

Studholme, C.

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999).
[Crossref]

Su, Y.

Sun, J.

Tang, C.

B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
[Crossref]

F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019).
[Crossref]

Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
[Crossref]

B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
[Crossref]

X. Chen, C. Tang, B. Li, and Y. Su, “Gradient vector fields based on variational image decomposition for skeletonization of electronic speckle pattern interferometry fringe patterns with variable density and their applications,” Appl. Opt. 55(25), 6893–6902 (2016).
[Crossref]

X. Zhu, Z. Chen, C. Tang, Q. Mi, and X. Yan, “Application of two oriented partial differential equation filtering models on speckle fringes with poor quality and their numerically fast algorithms,” Appl. Opt. 52(9), 1814–1823 (2013).
[Crossref]

C. Tang, T. Gao, S. Yan, L. Wang, and J. Wu, “The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns,” Opt. Express 18(9), 8942–8947 (2010).
[Crossref]

C. Tang, H. Ren, L. Wang, Z. Wang, L. Han, and T. Gao, “Oriented couple gradient vector fields for skeletonization of gray-scale optical fringe patterns with high density,” Appl. Opt. 49(16), 2979–2984 (2010).
[Crossref]

C. Tang, W. Lu, Y. Cai, L. Han, and G. Wang, “Nearly preprocessing-free method for skeletonization of gray-scale electronic speckle pattern interferometry fringe patterns via partial differential equations,” Opt. Lett. 33(2), 183–185 (2008).
[Crossref]

C. Tang, W. Lu, S. Chen, , Z. Zhang, B. Li, W. Wang, and L. Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46(30), 7475–7484 (2007).
[Crossref]

C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the delta-mollification phase map method,” Appl. Opt. 45(28), 7392–7400 (2006).
[Crossref]

Tang, S.

Tao, J.

Tavera Ruiz, G.

Wang, D.

Wang, G.

Wang, J.

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

Wang, K.

Wang, L.

Wang, S.

Wang, W.

Wang, X.

X. Wang and C. Chen, “Ship Detection for Complex Background SAR Images Based on a Multiscale Variance Weighted Image Entropy Method,” IEEE Geosci. Remote Sensing Lett. 14(2), 184–187 (2017).
[Crossref]

Wang, Z.

Weber, U.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Weidmann, P.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Wenzdburger, M.

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

Wildman, R. D.

Wu, J.

Xiao, Z.

Xu, M.

M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
[Crossref]

F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
[Crossref]

Xu, Z.

Yan, C.

C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003).
[Crossref]

Yan, H.

Yan, S.

Yan, X.

Yang, J.

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Zhang, B.

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

Zhang, D.

Zhang, F.

Zhang, H.

H. Zhang, J. E. Fritts, and S. A. Goldman, “Entropy-based objective evaluation method for image segmentation,” In: Proceedings of SPIE - International Society for Optics and Photonics, San Jose, California, United States, 18 December 2003, pp. 38–50.

Zhang, Q.

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Zhang, T.

C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003).
[Crossref]

Zhang, Z.

Zheng, T.

B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
[Crossref]

Zhou, H. C.

Zhou, Q.

Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
[Crossref]

Zhu, X.

Zong, J.

J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).

Appl. Math. Inf. Sci. (1)

C. Qi, “Maximum entropy for image segmentation based on an adaptive particle swarm optimization,” Appl. Math. Inf. Sci. 8(6), 3129–3135 (2014).
[Crossref]

Appl. Opt. (9)

P. D. Ruiz, J. M. Huntley, and R. D. Wildman, “Depth-resolved whole-field displacement measurement by wavelength-scanning electronic speckle pattern interferometry,” Appl. Opt. 44(19), 3945–3953 (2005).
[Crossref]

X. Zhu, Z. Chen, C. Tang, Q. Mi, and X. Yan, “Application of two oriented partial differential equation filtering models on speckle fringes with poor quality and their numerically fast algorithms,” Appl. Opt. 52(9), 1814–1823 (2013).
[Crossref]

F. Hao, C. Tang, M. Xu, and Z. Lei, “Batch denoising of ESPI fringe patterns based on convolutional neural network,” Appl. Opt. 58(13), 3338–3346 (2019).
[Crossref]

B. Li, C. Tang, G. Gao, M. Chen, S. Tang, and Z. Lei, “General filtering method for electronic speckle pattern interferometry fringe images with various densities based on variational image decomposition,” Appl. Opt. 56(16), 4843–4853 (2017).
[Crossref]

C. Tang, F. Zhang, B. Li, and H. Yan, “Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the delta-mollification phase map method,” Appl. Opt. 45(28), 7392–7400 (2006).
[Crossref]

C. Tang, W. Lu, S. Chen, , Z. Zhang, B. Li, W. Wang, and L. Han, “Denoising by coupled partial differential equations and extracting phase by backpropagation neural networks for electronic speckle pattern interferometry,” Appl. Opt. 46(30), 7475–7484 (2007).
[Crossref]

G. Wang, Y. J. Li, and H. C. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. 50(19), 3110–3117 (2011).
[Crossref]

X. Chen, C. Tang, B. Li, and Y. Su, “Gradient vector fields based on variational image decomposition for skeletonization of electronic speckle pattern interferometry fringe patterns with variable density and their applications,” Appl. Opt. 55(25), 6893–6902 (2016).
[Crossref]

C. Tang, H. Ren, L. Wang, Z. Wang, L. Han, and T. Gao, “Oriented couple gradient vector fields for skeletonization of gray-scale optical fringe patterns with high density,” Appl. Opt. 49(16), 2979–2984 (2010).
[Crossref]

Appl. Phys. B: Lasers Opt. (1)

M. Chen, C. Tang, M. Xu, and Z. Lei, “The oriented bilateral filtering method for removal of speckle noise in electronic speckle pattern interferometry fringes,” Appl. Phys. B: Lasers Opt. 125(7), 121 (2019).
[Crossref]

Auk (1)

C. Accadia, S. Mariani, M. Casaioli, A. Lavagnini, and A. Speranza, “Speranza. Sensitivity of Precipitation Forecast Skill Scores to Bilinear Interpolation and a Simple Nearest-Neighbor Average Method on High-Resolution Verification Grids,” Auk 133(2), 129–130 (2003).

Biomed. Opt. Express (1)

Chin. Opt. Lett. (1)

Comput. Geosci. (1)

C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,” Comput. Geosci. 10(2-3), 191–203 (1984).
[Crossref]

Comput. Med. Imag. Grap. (1)

J. Wang, J. Kong, Y. Lu, M. Qi, and B. Zhang, “A modified FCM algorithm for MRI brain image segmentation using both local and non-local spatial constraints,” Comput. Med. Imag. Grap. 32(8), 685–698 (2008).
[Crossref]

Exp. Mech. (1)

G. Pedrini, V. Martínez-García, P. Weidmann, M. Wenzdburger, A. Killinger, U. Weber, S. Schmauder, R. Gadow, and W. Osten, “Residual Stress Analysis of Ceramic Coating by Laser Ablation and Digital Holography,” Exp. Mech. 56(5), 683–701 (2016).
[Crossref]

IEEE Geosci. Remote Sensing Lett. (1)

X. Wang and C. Chen, “Ship Detection for Complex Background SAR Images Based on a Multiscale Variance Weighted Image Entropy Method,” IEEE Geosci. Remote Sensing Lett. 14(2), 184–187 (2017).
[Crossref]

IEEE Trans. Syst., Man, Cybern. (1)

N. Otsu, “A Threshold Selection Method from Gray-Level Histograms,” IEEE Trans. Syst., Man, Cybern. 9(1), 62–66 (1979).
[Crossref]

J. JSEM (1)

M. Kumar, K. Gaur, and C. Shakher, “Measurement of Material Constants (Young's Modulus and Poisson's Ratio) of Polypropylene Using Digital Speckle Pattern Interferometry (DSPI),” J. JSEM 15(special), 87–91 (2015).

Opt. Eng. (2)

M. Kumar, R. Agarwal, R. Bhutani, R. Bhutani, and C. Shakher, “Measurement of strain distribution in cortical bone around miniscrew implants used for orthodontic anchorage using digital speckle pattern interferometry,” Opt. Eng. 55(5), 054101 (2016).
[Crossref]

B. Li, C. Tang, T. Zheng, and Z. Lei, “Fully automated extraction of the fringe skeletons in dynamic electronic speckle pattern interferometry using a U-Net convolutional neural network,” Opt. Eng. 58(2), 023105 (2019).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (1)

P. P. Padghan and K. M. Alti, “Quantification of nanoscale deformations using electronic speckle pattern interferometer,” Opt. Laser Technol. 107, 72–79 (2018).
[Crossref]

Opt. Laser. Eng. (7)

D. Manuel, H. Montes, J. M. Flores-Moreno, and F. M. Santoyo, “Laser speckle based digital optical methods in structural mechanics: A review,” Opt. Laser. Eng. 87, 32–58 (2016).
[Crossref]

B. Sharp, “Electronic speckle pattern interferometry (ESPI),” Opt. Laser. Eng. 11(4), 241–255 (1989).
[Crossref]

W. An and T. Carlsson, “Speckle interferometry for measurement of continuous deformations,” Opt. Laser. Eng. 40(5-6), 529–541 (2003).
[Crossref]

M. Kumar and S. Chandra, “Measurement of temperature and temperature distribution in gaseous flames by digital speckle pattern shearing interferometry using holographic optical element,” Opt. Laser. Eng. 73, 33–39 (2015).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “A clustering framework based on FCM and texture features for denoising ESPI fringe patterns with variable density,” Opt. Laser. Eng. 119, 77–86 (2019).
[Crossref]

Q. Zhou, C. Tang, B. Li, and Z. Lei, “Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns,” Opt. Laser. Eng. 100, 111–117 (2018).
[Crossref]

M. Chen, C. Tang, M. Xu, and Z. Lei, “Binarization of optical fringe patterns with intensity inhomogeneities based on modified FCM algorithm,” Opt. Laser. Eng. 123, 14–19 (2019).
[Crossref]

Opt. Lett. (1)

Optik (1)

H. Ren, J. Li, and X. Gao, “3-D shape measurement of rail achieved by a novel phase measurement profilometry based on virtual reference fringe generated by image interpolation,” Optik 161, 348–359 (2018).
[Crossref]

Pattern Recognit. (2)

L. Wang, G. Leedham, and S. Y. Cho, “Minutiae feature analysis for infrared hand vein pattern biometrics,” Pattern Recognit. 41(3), 920–929 (2008).
[Crossref]

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit. 32(1), 71–86 (1999).
[Crossref]

Pattern Recognit. Lett. (1)

C. Yan, N. Sang, and T. Zhang, “Local entropy-based transition region extraction and thresholding,” Pattern Recognit. Lett. 24(16), 2935–2941 (2003).
[Crossref]

Phys. A (Amsterdam, Neth.) (1)

A. L. Barbieri, G. F. Arruda, F. A. Rodrigues, O. M. Bruno, and L. D. Costa, “An entropy-based approach to automatic image segmentation of satellite images,” Phys. A (Amsterdam, Neth.) 390(3), 512–518 (2011).
[Crossref]

Sensors (1)

W. Huo, Y. Huang, J. Pei, Q. Zhang, Q. Gu, and J. Yang, “Ship Detection from Ocean SAR Image Based on Local Contrast Variance Weighted Information Entropy,” Sensors 18(4), 1196 (2018).
[Crossref]

Other (4)

H. Zhang, J. E. Fritts, and S. A. Goldman, “Entropy-based objective evaluation method for image segmentation,” In: Proceedings of SPIE - International Society for Optics and Photonics, San Jose, California, United States, 18 December 2003, pp. 38–50.

J. Bernsen, “Dynamic thresholding of grey-level images,” In: 8th International Conference on Pattern Recognition, Paris France, 27-31 October, 1986, pp. 1251–1255.

J. Zong, T. Qiu, W. Li, and D. Guo, “Automatic ultrasound image segmentation based on local entropy and active contour model,” Comput. Math. Appl., (to be published).

http://cgm.cs.mcgill.ca/∼godfried/teaching/projects97/azar/skeleton.html .

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1. The flowchart of the proposed method.
Fig. 2.
Fig. 2. The computer-simulated fringe patterns.
Fig. 3.
Fig. 3. The corresponding entropy maps of Fig. 2.
Fig. 4.
Fig. 4. The binarization results of different methods.
Fig. 5.
Fig. 5. The corresponding skeleton results of Fig. 4.
Fig. 6.
Fig. 6. The first group of experimentally obtained ESPI fringe patterns.
Fig. 7.
Fig. 7. The binarization results of Fig. 6 of our method.
Fig. 8.
Fig. 8. The corresponding skeleton results of Fig. 7.
Fig. 9.
Fig. 9. The second group of experimentally obtained ESPI fringe patterns.
Fig. 10.
Fig. 10. The binarization results of Fig. 9 of our method.
Fig. 11.
Fig. 11. The corresponding skeleton results of Fig. 10.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

E ( j ) = g = 1 N P ( g ) l o g 2 P ( g )
J = j = 1 M i = 1 c u i j β E ( j ) v ( i ) 2
u i j = 1 k = 1 c ( E ( j ) v ( i ) E ( j ) v ( k ) ) 2 / ( β 1 )
v i j = j = 1 M u i j β E ( j ) j = 1 M ( u i j β )
U = [ u 11 u 12 u 1 M u 21 u 22 u 2 M u c 1 u c 2 u c M ]
U = [ u 11 u 12 u 1 M u 21 u 22 u 2 M ]

Metrics