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Rendering Deformed Speckle Images with a Boolean Model

Abstract

Rendering speckle images affected by a given deformation field is of primary importance to assess the metrological performance of displacement measurement methods used in experimental mechanics and based on digital image correlation (DIC). This article describes how to render deformed speckle images with a classic model of stochastic geometry, the Boolean model. The advantage of the proposed approach is that it does not depend on any interpolation scheme likely to bias the assessment process, and that it allows the user to render speckle images deformed with any deformation field given by an analytic formula. The proposed algorithm mimics the imaging chain of a digital camera, and its parameters are carefully discussed. A MATLAB software implementation and synthetic ground-truth datasets for assessing DIC software programs are publicly available.

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References

  1. 1.

    Standard 1288, standard for characterization of image sensors and cameras, release 3.0. Tech. rep., European Machine Vision Association (EMVA) (2010)

  2. 2.

    Amiot, F., Bornert, M., Doumalin, P., Dupré, J.C., Fazzini, M., Orteu, J.J., Poilâne, C., Robert, L., Rotinat, R., Toussaint, E., Wattrisse, B., Wienin, J.: Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark. Strain 49(6), 483–496 (2013)

    Article  Google Scholar 

  3. 3.

    Baddeley, A.: Spatial point processes and their applications. In: Weil, W. (ed.) Stochastic Geometry: Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 13–18, 2004, pp. 1–75 (2007)

  4. 4.

    Balcaen, R., Wittevrongel, L., Reu, P.L., Lava, P., Debruyne, D.: Stereo-DIC calibration and speckle image generator based on FE formulations. Exp. Mech. 57(5), 703–718 (2017)

  5. 5.

    Baldi, A.: Digital image correlation and color cameras. Exp. Mech. (2017). https://doi.org/10.1007/s11340-017-0347-2

  6. 6.

    Barranger, Y., Doumalin, P., Dupré, J.C., Germaneau, A.: Strain measurement by digital image correlation: influence of two types of speckle patterns made from rigid or deformable marks. Strain 48(5), 357–365 (2012)

    Article  Google Scholar 

  7. 7.

    Blaysat, B., Grédiac, M., Sur, F.: Effect of interpolation on noise propagation from images to DIC displacement maps. Int. J. Numer. Methods Eng. 108(3), 213–232 (2016)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Blaysat, B., Grédiac, M., Sur, F.: On the propagation of camera sensor noise to displacement maps obtained by DIC—an experimental study. Exp. Mech. 56(6), 919–944 (2016)

    Article  Google Scholar 

  9. 9.

    Blaysat, B., Grédiac, M., Sur, F.: Assessing the metrological performance of DIC applied on RGB images. In: Proceedings of the 2016 Annual Conference of the International Digital Imaging Correlation Society, Philadelphia (PA) USA (2017)

  10. 10.

    Bomarito, G., Hochhalter, J., Ruggles, T.: Development of optimal multiscale patterns for digital image correlation via local grayscale variation. Exp. Mech. (2017). https://doi.org/10.1007/s11340-017-0348-1

  11. 11.

    Bornert, M., Brémand, F., Doumalin, P., Dupré, J.C., Fazzini, M., Grédiac, M., Hild, F., Mistou, S., Molimard, J., Orteu, J.J., Robert, L., Surrel, Y., Vacher, P., Wattrisse, B.: Assessment of digital image correlation measurement errors: methodology and results. Exp. Mech. 49(3), 353–370 (2009)

    Article  Google Scholar 

  12. 12.

    Bornert, M., Doumalin, P., Dupré, J.C., Poilâne, C., Robert, L., Toussaint, E., Wattrisse, B.: Short remarks about synthetic image generation in the context of sub-pixel accuracy of digital image correlation. In: Proceedings of the 15th International Conference on Experimental Mechanics (ICEM15), Porto, Portugal (2012)

  13. 13.

    Bornert, M., Doumalin, P., Dupré, J.C., Poilâne, C., Robert, L., Toussaint, E., Wattrisse, B.: Assessment of digital image correlation measurement accuracy in the ultimate error regime: improved models of systematic and random errors. Exp. Mech. (2017). https://doi.org/10.1007/s11340-017-0328-5

  14. 14.

    Bornert, M., Doumalin, P., Dupré, J.C., Poilâne, C., Robert, L., Toussaint, E., Wattrisse, B.: Shortcut in DIC error assessment induced by image inerpolation used for subpixel shifting. Opt. Lasers Eng. 91, 124–133 (2017)

    Article  Google Scholar 

  15. 15.

    Briol, F.X., Oates, C., Girolami, M., Osborne, M., Sejdinovic, D.: Probabilistic integration: a role for statisticians in numerical analysis? Tech. Rep. arXiv:1512.00933, v5 (2016)

  16. 16.

    Cofaru, C., Philips, W., Paepegem, W.V.: Evaluation of digital image correlation techniques using realistic ground truth speckle images. Meas. Sci. Technol. 21(5), 055,102/1–17 (2010)

  17. 17.

    Corless, R., Fillion, N.: A Graduate Introduction to Numerical Methods. Springer, Berlin (2014)

    MATH  Google Scholar 

  18. 18.

    Doumalin, P., Bornert, M., Caldemaison, D.: Microextensometry by image correlation applied to micromechanical studies using the scanning electron microscopy. In: Proceedings of the International Conference on Advanced Technology in Experimental Mechanics, vol. I. The Japan Society of Mechanical Engineering, pp. 81–86 (1999)

  19. 19.

    Estrada, J., Franck, C.: Intuitive interface for the quantitative evaluation of speckle patterns for use in digital image and volume correlation techniques. J. Appl. Mech. 82(9), 095,001-1–095,005-5 (2015)

    Article  Google Scholar 

  20. 20.

    Galerne, B.: Computation of the perimeter of measurable sets via their covariogram. Applications to random sets. Image Anal. Stereol. 30(1), 39–51 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Galerne, B., Gousseau, Y.: The transparent dead leaves model. Adv. Appl. Probab. 44(1), 1–20 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Gallego, M.A., Ibanez, M.V., Simó, A.: Parameter estimation in non-homogeneous Boolean models: an application to plant defense response. Image Anal. Stereol. 34(1), 27–38 (2015)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23(1), 5–48 (1991)

    Article  Google Scholar 

  24. 24.

    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  25. 25.

    Healey, G., Kondepudy, R.: Radiometric CCD camera calibration and noise estimation. IEEE Trans. Pattern Anal. Mach. Intell. 16(3), 267–276 (1994)

    Article  Google Scholar 

  26. 26.

    Hua, T., Xie, H., Wang, S., Hu, Z., Chen, P., Zhang, Q.: Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation. Opt. Laser Technol. 43(1), 9–13 (2011)

    Article  Google Scholar 

  27. 27.

    Koljonen, J., Alander, J.: Deformation image generation for testing a strain measurement algorithm. Opt. Eng. 47(10), 107,202/1–13 (2008)

  28. 28.

    Lava, P., Cooreman, S., Coppieters, S., Strycker, M.D., Debruyne, D.: Assessment of measuring errors in DIC using deformation fields generated by plastic FEA. Opt. Lasers Eng. 47(7–8), 747–753 (2009)

    Article  Google Scholar 

  29. 29.

    Lecompte, D., Smits, A., Bossuyt, S., Sol, H., Vantomme, J., Hemelrijck, D.V., Habraken, A.: Quality assessment of speckle patterns for digital image correlation. Opt. Lasers Eng. 44(11), 1132–1145 (2006)

    Article  Google Scholar 

  30. 30.

    Lehoucq, R., Reu, P., Turner, D.: The effect of the ill-posed problem on quantitative error assessment in digital image correlation. Exp. Mech. (2017). https://doi.org/10.1007/s11340-017-0360-5

  31. 31.

    Mazzoleni, P., Matta, F., Zappa, E., Sutton, M., Cigada, A.: Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns. Opt. Lasers Eng. 66, 19–33 (2015)

    Article  Google Scholar 

  32. 32.

    Morel, J.M., Yu, G.: ASIFT: a new framework for fully affine invariant image comparison. SIAM J. Imaging Sci. 2(2), 438–469 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Murthagh, F., Starck, J., Bijaoui, A.: Image restoration with noise suppression using a multiresolution support. Astron. Astrophys. 112, 179–189 (1995)

    Google Scholar 

  34. 34.

    Newson, A., Delon, J., Galerne, B.: A stochastic film-grain model for resolution-independent rendering. Comput. Graph. Forum 36(8), 684–699 (2017)

  35. 35.

    Newson, A., Faraj, N., Delon, J., Galerne, B.: Analysis of a physically realistic film grain model, and a Gaussian film grain synthesis algorithm. In: Proceedings of the 6th Conference on Scale Space and Variational Methods in Computer Vision (SSVM), Kolding, Denmark (2017)

  36. 36.

    Newson, A., Faraj, N., Galerne, B., Delon, J.: Realistic film grain rendering. Image Process. Online (IPOL) 7, 165–183 (2017)

    Article  Google Scholar 

  37. 37.

    Orteu, J.J., Garcia, D., Robert, L., Bugarin, F.: A speckle texture image generator. Proc. SPIE 6341, 63,410H 1–6 (2006)

  38. 38.

    Pan, B., Xie, H.M., Xu, B.Q., Dai, F.L.: Performance of sub-pixel registration algorithms in digital image correlation. Meas. Sci. Technol 17(6), 1615–1621 (2006)

    Article  Google Scholar 

  39. 39.

    Perlin, K.: An image synthesizer. SIGGRAPH Comput. Graph. 19(3), 287–296 (1985)

    Article  Google Scholar 

  40. 40.

    Quenouille, M.: The evaluation of probabilities in a normal multivariate distribution, with special reference to the correlation ratio. Proc. Edinb. Math. Soc. 8(3), 95–100 (1949)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Reu, P.L.: Experimental and numerical methods for exact subpixel shifting. Exp. Mech. 51(4), 443–452 (2011)

    Article  Google Scholar 

  42. 42.

    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  43. 43.

    Schreier, H., Sutton, M.: Systematic errors in digital image correlation due to undermatched subset shape functions. Exp. Mech. 42(3), 303–310 (2002)

    Article  Google Scholar 

  44. 44.

    Serra, J.: The Boolean model and random sets. Comput. Graph. Image Process. 12(2), 99–126 (1980)

    Article  Google Scholar 

  45. 45.

    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  46. 46.

    Small, C.: Expansions and Asymptotics for Statistics. Monographs on Statistics and Applied Probability, vol. 115. CRC Press, Boca Raton (2010)

    Book  Google Scholar 

  47. 47.

    Society for Experimental Mechanics: DIC challenge. https://sem.org/dic-challenge/

  48. 48.

    Stoyan, D., Kendall, W., Mecke, J., Kendall, D.: Stochastic Geometry and Its Applications. Wiley, New York (1987)

    MATH  Google Scholar 

  49. 49.

    Su, Y., Zhang, Q., Gao, Z.: Statistical model for speckle pattern optimization. Opt. Express 25(24), 30259–30275 (2017)

    Article  Google Scholar 

  50. 50.

    Sutton, M., Orteu, J.J., Schreier, H.: Image Correlation for Shape, Motion and Deformation Measurements. Springer, Berlin (2009)

    Google Scholar 

  51. 51.

    Triconnet, K., Derrien, K., Hild, F., Baptiste, D.: Parameter choice for optimized digital image correlation. Opt. Lasers Eng. 47(6), 728–737 (2009)

    Article  Google Scholar 

  52. 52.

    Yu, L., Pan, B.: The errors in digital image correlation due to overmatched shape functions. Meas. Sci. Technol. 26(4), 045–202 (2015)

    Article  Google Scholar 

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Acknowledgements

F.S. is grateful to Antoine Fond (Magrit team, Loria) for his help with the NVidia Titan X GPU.

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Correspondence to Frédéric Sur.

Appendix A: Quantization Error for a Compound Probability Distribution

Appendix A: Quantization Error for a Compound Probability Distribution

We assume that each real value inside a quantization box [ab] (with \(a<b\)) is assigned to a quantized value. In Monte Carlo estimation, sample means are distributed according to a normal distribution of mean m and standard deviation s. A quantization error occurs when the sample mean is not assigned to the same quantized value as m. If s is small with respect to \(b-a\) and m is in the middle of the interval [ab], the probability of an error is small, while it is larger if m is close to a or b.

In order to estimate an overall error, we assume that the mean m is uniformly distributed over [ab]. In other words, we assume that the sample mean X is a compound random variable distributed according to a Gaussian distribution of standard deviation s and mean m distributed according to a uniform distribution on interval [ab]. We estimate the probability that X is not correctly quantized, i.e., that it falls outside the interval [ab]. In other words, we would like to estimate the probability \(\mathcal{E}(s,b-a)=1-\Pr (a\le X \le b)\).

We denote by g the standard normal probability distribution function (that is, \(g(x) = e^{-x^2/2}/\sqrt{2\pi }\)) and by G the associated cumulative distribution function. Marginalizing out m in X gives:

$$\begin{aligned} \Pr (a\le X \le b)&= \frac{1}{b-a} \int _a^b \Pr (a\le X \le b \,|\,\mu ) \;\text {d}\mu \end{aligned}$$
(29)
$$\begin{aligned}&= \frac{1}{b-a} \int _a^b \left( G\left( \frac{b-\mu }{s}\right) \right. \nonumber \\&\quad -\left. G\left( \frac{a-\mu }{s}\right) \right) \;\text {d}\mu . \end{aligned}$$
(30)

Now, for any two real values \(\alpha \) and \(\beta \ne 0\), an antiderivative of \(G(\alpha +\beta x)\) is given by:

$$\begin{aligned} \int G(\alpha +\beta x)= & {} \frac{1}{\beta }\left( (\alpha +\beta x)G(\alpha +\beta x) \right. \nonumber \\&\quad +\left. g(\alpha +\beta x) \right) . \end{aligned}$$
(31)

We calculate:

$$\begin{aligned} \begin{aligned} \frac{1}{b-a} \int _a^b G\left( \frac{b-\mu }{s}\right) \;\text {d}\mu = G\left( \frac{b-a}{s}\right) \\ + \frac{s}{b-a}g\left( \frac{b-a}{s}\right) -\frac{s}{\sqrt{2\pi }(b-a)} \end{aligned} \end{aligned}$$
(32)

and

$$\begin{aligned} \begin{aligned} \frac{1}{b-a} \int _a^b G\left( \frac{a-\mu }{s}\right) \;\text {d}\mu = G\left( \frac{a-b}{s}\right) \\ - \frac{s}{b-a}g\left( \frac{a-b}{s}\right) +\frac{s}{\sqrt{2\pi }(b-a)} \end{aligned} \end{aligned}$$
(33)

We eventually obtain, since \(G(-x)=1-G(x)\) and \(g(-x)=g(x)\):

$$\begin{aligned} \Pr (a\le X \le b)= & {} 2G\left( \frac{b-a}{s}\right) -1\nonumber \\&\quad +\frac{2s}{b-a}\left( g\left( \frac{b-a}{s}\right) - \frac{1}{\sqrt{2\pi }}\right) \end{aligned}$$
(34)

We are interested in the quantization error when the random fluctuation of the Monte Carlo estimation is small, that is, when \(s/(b-a)\) tends to 0.

Since \(G(x)=1- g(x)/x + \mathcal{O}( g(x)/x )\) when \(x\rightarrow +\infty \) (where \(\mathcal O\) is Landau’s “big-O”; this is a property of Mill’s ratio [46, chapter 2]), we have proved the following proposition.

Proposition 1

If a and b are two real numbers such that \(a<b\), and X is a random variable distributed according to a compound Gaussian distribution of variance \(s^2\) such that its mean is a random variable distributed according to a uniform distribution on [ab], the following asymptotic expansion holds:

$$\begin{aligned} \Pr (a\le X \le b) =_{s \rightarrow 0} 1 - \frac{\sqrt{2}s}{\sqrt{\pi }(b-a)} + \mathcal{O}\left( s e^{-(b-a)/(2s^2) }\right) \end{aligned}$$
(35)

If \(s/(b-a)<< 1\), the latter expression gives an accurate estimate of the probability \(\mathcal E\) of quantization error as:

$$\begin{aligned} \mathcal{E}(s,b-a) = \frac{\sqrt{2}s}{\sqrt{\pi }(b-a)}. \end{aligned}$$
(36)

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Sur, F., Blaysat, B. & Grédiac, M. Rendering Deformed Speckle Images with a Boolean Model. J Math Imaging Vis 60, 634–650 (2018). https://doi.org/10.1007/s10851-017-0779-4

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Keywords

  • Boolean model
  • Random speckle rendering
  • Digital image correlation (DIC)