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Abstract
The measurement accuracy of digital image correlation (DIC) is influenced by the quality of the speckle pattern. Although various models for generating random speckle patterns have been well discussed, obtaining appropriate speckle images with isotropic quality and performance could be a challenging issue in DIC. In this paper, we propose a novel (to our knowledge) method for generating speckle patterns based on modified Conway’s game of life (GoL). By sequentially assembling the speckle patterns generated from the modified GoL, we produced the GoL speckle image. Then, verification and comparison experiments were conducted through pure in-plane translations. The results show that the generated speckle image which was resized with ${k_s} = 6\& {k_r} = 2$ processing and subsequently fuzzified using a Gaussian filter, produces the best accuracy for DIC measurement. Furthermore, based on the rigid body in-plane rotation displacement tests in the physical experimental results of three different speckle images, the GoL speckle generated from our proposed method shows the smallest measurement error. This indicates that the proposed speckle patterns generating method could provide a new type of speckle pattern with better quality and accuracy.
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1. Introduction
Digital image correlation (DIC), as an effective method to calculate full-field displacement, has been applied in various technological domains [1,2]. This technique is conducted by comparing and correlating a series of images photographed from the surface of object during loading, then determining surface displacement fields at each load step. Since being proposed, it has been well discussed that the measurement accuracy of DIC is obviously influenced by the quality of speckle pattern [3–6]. Generally, a good speckle pattern on the test sample surface should have characteristics include high contrast, randomness, isotropy and stability [7].
The speckle pattern can be fabricated easily by randomly spraying black paints over a white base to ensure high quality. However, different fabrication method by different operators may have various qualities, which may lead to unexpected performance in DIC test [8]. Therefore, researchers tend to apply random speckle patterns generated by computer to provide well-controlled displacement and deformation computation. Circle speckle and Gaussian speckle had become well-received speckle patterns in numerical simulations and practical measurements. Orteu et al. [9] provided a software program to generate speckle patterns using Perlin noise. Gu et al. [10] compared two different speckle patterns simulation models through in-plane translation and rotation performance in DIC. Baldi [11] described a generating synthetic color speckles method by spraying the sum of several bell-shaped functions over the surface. Sur et al. [12] proposed a deformed speckle patterns rendering method based on Boolean model in stochastic geometry without any interpolation algorithm. Wang et al. [13] discussed the influence of aperture and the speckle generation parameters, then determined the optimal speckle size and density.
Due to the correlation between the displacement measurement accuracy of DIC and the quality of random speckle patterns, it is worthy to study generating method of speckle patterns with optimal intensity distribution. Moreover, generating morphologically repeatable speckle patterns could be still a challenging issue in DIC [14], and the controllability of gray intensity in speckle patterns still need to be enhanced. Especially for the object with a lager surface, generating speckle patterns with similar quality and performance on the whole surface relate to the consistency of measurement directly. As summarized by Bomarito et al. [15], the speckle quality of the subject of the image are related to the measurement accuracy of DIC. For the purpose to imitate the artificial speckle which are spraying ink onto the surface of the tested specimen, the current generated speckles often performs specified patterning such as circle, polygon or bell-shaped, which may lead to similarity and correlation among different subsets increased, and cause the measurement errors increased. As one of the most common model of cell automata created by Conway [16–18], the Conway’s Game of Life (GoL) could powerfully minimize the correlation between adjacent pixels or subsets [19], which may have potential to optimize the measurement accuracy of generated speckle patterns. In this paper, a novel method to generate synthetic speckle images from a certain speckle pattern was described based on the modified GoL. Then, numerical verification and comparison experiments were implemented to validate the accuracy and consistency of the proposed method. This study will provide a partial solution for generating speckle patterns with optimal quality in DIC test.
2. Modified Conway’s Game of Life
The classical GoL model starts with a random W × H binary matrix, in which the value of each cell (x) represents two possible state: dead when x = 0, or alive when x = 1. The state of each cell at the next generation is determined by its eight neighbours. At each generation t = 1, 2, …, n, the evolution rule in GoL is defined as follows:
- (1) Birth: Any dead cell with no more than 3 alive neighbours, switches to alive at the next generation (t + 1).
- (2) Survival: Any alive cell with no more than 3 alive neighbours, keeps survival at the next generation (t + 1).
- (3) Death: Any alive cell with more than 3 alive neighbours, switches to dead at the next generation (t + 1).
- (4) Otherwise, any dead cell with more than 3 alive neighbours, keeps dead at the next generation (t + 1).
Since being established, the GoL model has been introduced into several scientific fields and applications, especially in digital image generation, scrambling and encryption [19–21]. Aguilera-Venegas et al. [22] also developed this model using probabilistic rules for life or death of the cells at the next generation. In this work, the probabilistic evolutional model for generating speckle images is proposed through the modification of GoL. Each pixel $x ({i,j} )$ in a specific binarized speckle pattern I0 (size: W × H)can be regarded as an element of binary matrix, and the evolution of the pixels accord with the aforementioned rule in GoL. The neighbour life density of each pixels :
where ML is the number of alive neighbours, M is the total number of neighbours. For the purpose to obtain a non-deterministic speckle patterns, a probability rule was introduced based on the PCAGOL model proposed by Aguilera-Venegas et al [22]. Before each step of this process was implemented, a random vale hp (preset 0 ≤ hp ≤ 1) is assigned to each pixel as the probability of being alive or dead at the next generation. The basic idea is that the pixel with a higher hp value is more difficult to keep alive with a medium neighbour life density, while it is opposite with an extreme high or low neighbour life density. The main processes of the proposed approach were given as follows:Case 1: For pixel with ${\rho _L} = 1$, it keeps dead.
Case 2: For pixel with $0.75 \le {\rho _L} < 0$, it keeps alive when hp ≥ 0.75, otherwise becomes dead.
Case 3: For pixel with $0.5 \le {\rho _L} < 0.75$, it keeps alive when hp ≥ 0.5, otherwise becomes dead.
Case 4: For pixel with $0.25 \le {\rho _L} < 0.5$, it keeps alive when hp ≥ 0.25, otherwise becomes dead.
Case 5: For pixel with $0 < {\rho _L} < 0.25$, it keeps alive when hp ≥ 0.5, otherwise becomes dead.
Case 6: For pixel with ${\rho _L} = 0$, it keeps alive when hp ≥ 0.75, otherwise becomes dead.
Using the speckle patterns generated from the process list above, the GoL speckle image is obtained by assembling speckle patterns in sequence. If there are m × n speckle assembled, the size of GoL speckle image is (W × m) ×(H × n).
3. Studied example
An implementation of this proposed model was developed using Matlab. To analyze the effectiveness of the proposed method, a high contrast image in DIC Challenge image sets is employed to perform numerical experiment simulation. A section with the size of 100 × 100 pixels was randomly selected from the DIC Challenge image and binarized as the original pattern I0, see Fig. 1.
Fig. 1. Sections are randomly selected from the DIC Challenge image, and binarized as the original pattern.
By applying the above roles in this model, nine simulated patterns Ik (k = 1,2,…) were produced from the original speckle pattern I0. These generated patterns and their mean intensity gradient (MIG) were plotted in Fig. 2. The MIG value is defined by eq.(2):
Then, the studied speckle image was assembled by these nine generated patterns in sequence, see Fig. 3. The size of the studied GoL speckle image is 300 × 300 pixels.
4. Numerical simulations
4.1 Verification experiments
To verify the stability of speckle pattern generating model proposed above, the numerical experiments of pure in-plane translations tests were conducted on x direction using the original studied speckle image in Fig. 3. A series of translated images were obtained by applying the appropriate shift in the Fourier domain. The subpixel displacements ranged from 0 to 1 pixel, corresponding to a shift of 0.05 pixel between two successive images. The displacement of each translated image was computed using Ncorr in Matlab. The radius of the subsets was set to 29 pixel with a spacing of 7 pixel.
Considering about the significant aliasing effect when the speckle size is less than 1.3 pixel [23,24], five different post-processing methods, including fuzzified using mean filter (method I, Fig. 4(a)), disk filter (method II, Fig. 4(b)) or Gaussian filter (method III, Fig. 4(c)), resizing process (method IV, Fig. 4(d)), resizing and subsequently fuzzified using Gaussian filter (method V, Fig. 4(e)), were applied on the original assembled speckle image, see Fig. 4. The resizing process was defined as magnifying the speckle image ${k_s}$ times and then minifying it ${k_r}$ times, which was implemented using the function of ‘imresize’ in Matlab. This process could optimize intensity distribution of the binarized speckle image. The resizing ratio k in resizing process was defined as:
Fig. 4. Post-processed speckle images and their histogram. Fuzzified using (a) mean filter, (b) disk filter and (c) Gaussian filter. (d) Resizing process with ${k_s} = {k_r} = 3$; (e) resizing process with ${k_s} = {k_r} = 3$ and subsequently fuzzified using Gaussian filter.
To quantitatively evaluate the measured displacements of each translated images and compare them with the imposed ones, the measurement errors of displacements are decomposed into three components: mean bias error, standard error and generalization error. The mean bias error of the measured displacements is defined as:
where ${u_{\textrm{mean}}}$ is the mean value of measured displacement, ${u_{\textrm{imp}}}$ is the imposed displacement. The standard error is defined as:The generalization error was introduced to evaluate the performance of generated speckle images as follows:
where $\overline {|{{u_e}} |}$ is average mean bias error of sub-pixel displacement in x direction, $\overline {{\sigma _u}}$ is average standard error of sub-pixel displacement.The average values of mean bias errors and standard errors of the sub-pixel displacements experiments are listed in Table 1. The reconstructed displacement of the original assembled speckle image departs from the actual displacement markedly due to its too small speckle size [23,24], even it has the highest MIG value. Figure 5 shows the mean bias errors and standard errors of the reconstructed displacement using different post-processing methods. It is observed from Table 1 and Fig. 5 that values of MIG are decreased after post-processed, while the speckle size and measurement errors are optimized noticeably. The mean bias errors of method I, II, III and V are nearly sinusoidal distribution with a period of 1 pixel, but it of method IV, or resizing process, produces an unusual curve. Comparing these post-processing methods, the speckle image after resizing and subsequently fuzzified using Gaussian filter seems to produce the smallest mean bias error and generalization error, which would be the optimal post-processing method for the proposed model of generating speckle patterns.
Fig. 5. (a) Mean bias errors and (b) standard errors in sub-pixel displacement experiments for the five post-processed speckle images.
Table 1. Values of MIG, speckle size and measurement errors of sub-pixel displacements in the numerical experiments.a
4.2 Comparison experiments
To further evaluate the measurement performance of this proposed speckle patterns generation method, a series of comparison experiments were conducted on six GoL speckle images with various parameters, see Fig. 6. Then, six different speckle images, including circle speckle (Fig. 6 g), two Gaussian speckle (speckle size = 4 pixels for Fig. 6 h and 5 pixels for Fig. 6(i)), Fresnel-Kirchhoff speckle [10] (Fig. 6(j)), Perlin’s coherent noise speckle [9] (Fig. 6 k) and Poison speckle [12] (Fig. 6 m) were introduced as comparative images. All the speckle images were of the same size 450 × 450 pixels. GoL speckle a, c and e were generated using the proposed model by selecting a section randomly from the image presented in Fig. 1 with the size of 150 × 150 pixels, 75 × 75 pixels and 50 × 50 pixels, then resizing processed using parameters of ${k_s} = 3\& {k_r} = 3$ (Fig. 6(a)), ${k_s} = 6\& {k_r} = 3$ (Fig. 6(c)) and ${k_s} = 6\& {k_r} = 2$ (Fig. 6(e)) respectively. Speckle b, d and f were obtained through Gaussian filter fuzzified on speckle a, c and e respectively.
Fig. 6. Speckle images used in the comparison experiments. GoL speckle images: (a) ${k_s} = 3\& {k_r} = 3$, no Gaussian filter fuzzified; (b) ${k_s} = 3\& {k_r} = 3$, Gaussian filter fuzzified; (c) ${k_s} = 6\& {k_r} = 3$, no Gaussian filter fuzzified; (d) ${k_s} = 6\& {k_r} = 3$, Gaussian filter fuzzified; (e) ${k_s} = 6\& {k_r} = 2$, no Gaussian filter fuzzified; (f) ${k_s} = 6\& {k_r} = 2$, Gaussian filter fuzzified. (g) Circle speckle image with the speckle size of 4 pixels; (h) Gaussian speckle image with the size of 5 pixels; (i) Gaussian speckle image with the speckle size of 5 pixels; (j) Fresnel-Kirchhoff speckle (Ref. [10]); (k) Perlin’s coherent noise speckle (Ref. [9]); (m) Poison speckle (Ref. [12]).
Using the speckle images mentioned above, the comparison experiments of pure in-plane translations tests were conducted on x direction. The experimental settings were the same with verification experiments.
Table 2 presents the speckle quality and average values of measurement errors in the comparison experiments. Due to interpolation bias of subpixel registration algorithm, the measurement errors in the y direction are deviated from zero. As shown in Table 2, the speckle size increases with the rise of resizing ratio k, and measurement errors of speckle images after resizing process also decreases apparently. Speckle images after fuzzified using Gaussian filter leads to further optimized speckle size and measurement errors. Comparing the results of all speckle images, even the smallest mean bias error in the y direction appears on the Perlin speckle (speckle k) and Poison speckle (speckle m), the speckle image f produces the smallest generalization errors in both x and y directions, which indicates that the proposed model after resizing processed with ${k_s} = 6\& {k_r} = 2$ and subsequently fuzzified using Gaussian filter could generate a higher quality and more accurate speckle pattern.
Table 2. Values of MIG, average mean bias errors and standard errors of sub-pixel displacements in comparison experiments.a
5. Physical experiments
A rigid body in-plane rotation displacement test are following to validate the performance of proposed speckle generating method in the physical experiment. Three different speckle images, including a Gol speckle being resized processed with ${k_s} = 6\& {k_r} = 2$ and subsequently fuzzified using Gaussian filter (Fig. 7(a)), a Gaussian speckle with the speckle size of 5 pixels (Fig. 7(b)), and an artificial speckle by spraying white and black paints (Fig. 7(c)), were printed on an A4 paper sheet and then glued on the surface of a 30 mm thick wood block. These blocks were installed on a horizontal rotation stage with a positioning accuracy of 1’ (RS60-L, Kexin, China). The printed speckle image was captured using a 16 mm lens mounted on a CMOS industrial camera (MV-HS510GC2, Microvision, China; pixel size 3.45µm × 3.45µm), which was installed about 300 mm above the test surface, see Fig. 8. The applied rotation increment was 10°, and 10 images was acquired for each speckle image.
Fig. 7. Speckle images used in the physical experiments. (a) GoL speckle; (b) Gaussian speckle; (c) artificial speckle.
Table 3 showed the speckle quality and average values of measured rotation angle and its measurement errors. It can be seen from Table 3 that due to the high degree of visual contrast of white and black paints under natural light, the artificial speckle shows the highest MIG value among these three speckle images, but the measurement errors in the rigid body in-plane rotation displacement test are greater than GoL speckle and Gaussian speckle. The speckle size of GoL and Gaussian speckle are nearly same, while GoL speckle performed higher MIG value than Gaussian speckle. Among these three speckle images, the GoL speckle provided an optimal measurement accuracy, which are lower than the positioning accuracy of the horizontal rotation stage. This validates that the proposed GoL speckle could achieve an acceptable level of precision in real-world displacement test using DIC.
Table 3. Mean values and measurement errors in physical rotation experiments
6. Conclusion
In this paper, a novel speckle patterns generating model based on modified Conway’s game of life was proposed and evaluated for DIC. Through assembling the generated speckle patterns in sequence, the performances of this proposed model corresponding to various post-processing methods were discussed in sub-pixel displacement experiments by DIC. Comparing the original speckle image with or without post-processing, speckle image which was resizing processed and subsequently fuzzified using Gaussian filter produces the best measurement accuracy in the rigid body in-plane translation because of its smallest generalization error.
Then, the measurement performance of this proposed model using various resizing parameters were discussed through comparison experiments. Comparing the GoL speckle, circle speckle and Gaussian speckle, the GoL speckle with resizing parameters of ${k_s} = 6\& {k_r} = 2$ and subsequently fuzzified using Gaussian filter shows the smallest generation error in the pure in-plane translations tests. Through the rigid body in-plane rotation displacement test in the physical experiment, the GoL provided an optimal measurement accuracy, which indicates that it could produce an acceptable level of precision in DIC test. This study concluded that the proposed speckle patterns generating model with this post-processing method could provide a new type speckle pattern with better quality and accuracy.
Funding
Guizhou Provincial Science and Technology Projects (ZK[2022]-117); Guizhou Provincial Science and Technology Support Plan ([2023]-205); Introduction of Talent Research Fund of Guizhou University (2020-026); Science and Technology Program of Guizhou Province ([2023]010).
Disclosures
The authors declare no conflicts of interest.
Data availability
The authors confirm that the data supporting the findings of this study are available within the article, and also available from the corresponding author upon reasonable request.
References
1. B. Pan, “Digital image correlation for surface deformation measurement: historical developments, recent advances and future goals,” Meas. Sci. Technol. 29(8), 082001 (2018). [CrossRef]
2. J. Blaber, B. Adair, and A. Antoniou, “Ncorr: Open-Source 2D Digital Image Correlation Matlab Software,” Exp. Mech. 55(6), 1105–1122 (2015). [CrossRef]
3. B. Pan, Z. Lu, and H. Xie, “Mean intensity gradient: An effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48(4), 469–477 (2010). [CrossRef]
4. X.-Y. Liu, R.-L. Li, H.-W. Zhao, et al., “Quality assessment of speckle patterns for digital image correlation by Shannon entropy,” Optik 126(23), 4206–4211 (2015). [CrossRef]
5. X. Zhao, Y. Wen, J. Zhao, et al., “Study of the quality of wood texture patterns in digital image correlation,” Optik 171, 370–376 (2018). [CrossRef]
6. J. Park, S. Yoon, T.-H. Kwon, et al., “Assessment of speckle-pattern quality in digital image correlation based on gray intensity and speckle morphology,” Opt. Lasers Eng. 91, 62–72 (2017). [CrossRef]
7. Y. L. Dong and B. Pan, “A Review of Speckle Pattern Fabrication and Assessment for Digital Image Correlation,” Exp. Mech. 57(8), 1161–1181 (2017). [CrossRef]
8. B. Pan, H. Xie, Z. Wang, et al., “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037 (2008). [CrossRef]
9. J.-J. Orteu, D. Garcia, L. Robert, et al., eds. (2006), p. 63410 H.
10. G. Gu, “A comparative study of random speckle pattern simulation models in digital image correlation,” Optik 126(23), 3713–3716 (2015). [CrossRef]
11. A. Baldi, “Digital Image Correlation and Color Cameras,” Exp. Mech. 58(2), 315–333 (2018). [CrossRef]
12. F. Sur, B. Blaysat, and M. Grédiac, “Rendering Deformed Speckle Images with a Boolean Model,” J. Math. Imaging Vis. 60(5), 634–650 (2018). [CrossRef]
13. Y. Wang, Y. Gao, Y. Liu, et al., “Optimal Aperture and Digital Speckle Optimization in Digital Image Correlation,” Exp. Mech. 61(4), 677–684 (2021). [CrossRef]
14. J. Zhang, A. Sweedy, F. Gitzhofer, et al., “A novel method for repeatedly generating speckle patterns used in digital image correlation,” Opt. Lasers Eng. 100, 259–266 (2018). [CrossRef]
15. G. F. Bomarito, J. D. Hochhalter, T. J. Ruggles, et al., “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017). [CrossRef]
16. J. E. Contreras-Reyes, “Lerch distribution based on maximum nonsymmetric entropy principle: Application to Conway’s game of life cellular automaton,” Chaos, Solitons Fractals 151, 111272 (2021). [CrossRef]
17. K. Sakata, Y. Tanaka, and D. Takahashi, “Max-Plus Generalization of Conway’s Game of Life,” (2019).
18. M. D. Niemiec, “Object Synthesis in Conway’s Game of Life and Other Cellular Automata,” in Game of Life Cellular Automata, A. Adamatzky, ed. (Springer London, 2010), pp. 115–134.
19. G. Dursun, F. Özer, and U. Özkaya, “A new and secure digital image scrambling algorithm based on 2D cellular automata,” Turk. J. Electr. Eng. Comput. Sci. 25, 3515–3527 (2017). [CrossRef]
20. F. Musanna, D. Dangwal, and S. Kumar, “Novel image encryption algorithm using fractional chaos and cellular neural network,” J. Ambient Intell. Humaniz. Comput. 13(4), 2205–2226 (2022). [CrossRef]
21. L. P. Yaroslavsky, “The amazing dynamics of stochastic pattern formation and growth models inspired by the Conway’s Game of Life,” (n.d.).
22. G. Aguilera-Venegas, J. L. Galán-García, R. Egea-Guerrero, et al., “A probabilistic extension to Conway’s Game of Life,” Adv. Comput. Math. 45(4), 2111–2121 (2019). [CrossRef]
23. Y. Su, Q. Zhang, and Z. Gao, “Statistical model for speckle pattern optimization,” Opt. Express 25(24), 30259 (2017). [CrossRef]
24. Y. Su, Z. Gao, Z. Fang, et al., “Theoretical analysis on performance of digital speckle pattern: uniqueness, accuracy, precision, and spatial resolution,” Opt. Express 27(16), 22439 (2019). [CrossRef]
25. M. Bornert, F. Brémand, P. Doumalin, et al., “Assessment of Digital Image Correlation Measurement Errors: Methodology and Results,” Exp. Mech. 49(3), 353–370 (2009). [CrossRef]
