Skip to main content
SpringerLink
Log in

On Biases in Displacement Estimation for Image Registration, with a Focus on Photomechanics

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Image registration under small displacements is the keystone of several image analysis tasks such as optical flow estimation, stereoscopic imaging, or full-field displacement estimation in photomechanics. A popular approach consists in locally modeling the displacement field between two images by a parametric transformation and performing least-squares estimation afterward. This procedure is known as “digital image correlation” in several domains as in photomechanics. The present article is part of this approach. First, the estimated displacement is shown to be impaired by biases related to the interpolation scheme needed to reach subpixel accuracy, the image gradient distribution, as well as the difference between the hypothesized parametric transformation and the true displacement. A quantitative estimation of the difference between the estimated value and the actual one is of importance in application domains such as stereoscopy or photomechanics, which have metrological concerns. Second, we question the extent to which these biases could be eliminated or reduced. We also present numerical assessments of our predictive formula in the context of photomechanics. Software codes are freely available to reproduce our results. Although this paper is focused on a particular application field, namely photomechanics, it is relevant to various scientific areas concerned by image registration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

Notes

  1. The proposed calculation is adapted from https://math.stackexchange.com/questions/245327/.

  2. We use BSpeckleRender_b which renders speckle images with patterns of intensity at pixel \({{\mathbf {x}}}\) varying as \(\exp (-4|{{\mathbf {x}}}-{{\mathbf {x}}}_0|^2/R^2) \) with a center \({{\mathbf {x}}}_0\) given by a Poisson point process and a random radius R, instead of random black disks over a white background, so that the image gradient is a smooth function. Matlab software code is available at the following URL: https://members.loria.fr/FSur/software/BSpeckleRender/.

  3. It should be noted that these four components are the derivatives of the local displacement field \({\varvec{\phi }}^{{\mathbf {x}}}\) estimated over \(\varOmega _{{\mathbf {x}}}\). In photomechanics, the derivatives of the displacement field, which are related to strain components, are rather computed from the derivatives of the global displacement \({\varvec{\phi }}\).

References

  1. Blanchet, G., Buades, A., Coll, B., Morel, J.M., Rougé, B.: Fattening free block matching. J. Math. Imaging Vis. 41(1), 109–121 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. 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 

  4. Bomarito, G., Hochhalter, J., Ruggles, T., Cannon, A.: Increasing accuracy and precision of digital image correlation through pattern optimization. Opt. Lasers Eng. 91, 73–85 (2017)

    Article  Google Scholar 

  5. 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 

  6. 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 

  7. Bouthemy, P., Toledo-Acosta, B., Delyon, B.: Robust model selection in 2D parametric motion estimation. J. Math. Imaging Vis. 61(7), 1022–1036 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cacoullos, T.: Exercices in Probability. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  9. Delon, J., Rougé, B.: Small baseline stereovision. J. Math. Imaging Vis. 28(3), 209–223 (2007)

    Article  MathSciNet  Google Scholar 

  10. Fayad, S., Seidl, D., Reu, P.: Spatial DIC errors due to pattern-induced bias and grey level discretization. Exp. Mech. 60, 249–263 (2020)

    Article  Google Scholar 

  11. Fleet, D., Weiss, Y.: Optical flow estimation. In: Handbook of Mathematical Models in Computer Vision. Springer, pp. 237–257 (2006)

  12. Fortun, D., Bouthemy, P., Kervrann, C.: Optical flow modeling and computation: a survey. Comput. Vis. Image Underst. 134, 1–21 (2015)

    Article  MATH  Google Scholar 

  13. Getreuer, P.: Linear methods for image interpolation. Image Process. On Line 1, 238–259 (2011)

    Article  MATH  Google Scholar 

  14. Grédiac, M., Blaysat, B., Sur, F.: A critical comparison of some metrological parameters characterizing local digital image correlation and grid method. Exp. Mech. 57(6), 871–903 (2017)

    Article  Google Scholar 

  15. Grédiac, M., Blaysat, B., Sur, F.: Extracting displacement and strain fields from checkerboard images with the localized spectrum analysis. Exp. Mech. 59(2), 207–218 (2019)

    Article  Google Scholar 

  16. Grédiac, M., Blaysat, B., Sur, F.: A robust-to-noise deconvolution algorithm to enhance displacement and strain maps obtained with local DIC and LSA. Exp. Mech. 59(2), 219–243 (2019)

    Article  Google Scholar 

  17. Grédiac, M., Blaysat, B., Sur, F.: Comparing several spectral methods used to extract displacement fields from checkerboard images. Opt. Lasers Eng. 127, 105984 (2020)

    Article  Google Scholar 

  18. Grédiac, M., Blaysat, B., Sur, F.: On the optimal pattern for displacement field measurement: Random speckle and DIC, or checkerboard and LSA? Exp. Mech. 60(4), 509–534 (2020)

    Article  Google Scholar 

  19. Grédiac, M., Hild, F. (eds.): Full-Field Measurements and Identification in Solid Mechanics. Wiley, Hoboken (2012)

    Google Scholar 

  20. Hild, F., Roux, S.: Digital image correlation: from displacement measurement to identification of elastic properties: a review. Strain 42(2), 69–80 (2006)

    Article  Google Scholar 

  21. Horn, B., Schunck, B.: Determining optical flow. Artif. Intell. 17(1–3), 185–203 (1981)

    Article  Google Scholar 

  22. Lavatelli, A., Balcaen, R., Zappa, E., Debruyne, D.: Closed-loop optimization of DIC speckle patterns based on simulated experiments. IEEE Trans. Instrum. Meas. 68(11), 4376–4386 (2019)

    Article  Google Scholar 

  23. Lehmann, T., Gonner, C., Spitzer, K.: Survey: interpolation methods in medical image processing. IEEE Trans. Med. Imaging 18(11), 1049–1075 (1999)

    Article  Google Scholar 

  24. Lehoucq, R., Reu, P., Turner, D.: The effect of the ill-posed problem on quantitative error assessment in digital image correlation. Exp. Mech. (2017)

  25. Lucas, B., Kanade, T.: An iterative image registration technique with an application to stereo vision. In: Proceedings of the 7th International Joint Conference on Artificial Intelligence (IJCAI), pp. 674–679. Vancouver (BC, Canada) (1981)

  26. Luo, J., Ying, K., He, P., Bai, J.: Properties of Savitzky–Golay digital differentiators. Digit. Signal Process. 15(2), 122–136 (2005)

    Article  Google Scholar 

  27. Pan, B., Xie, H., Wang, Z.: Equivalence of digital image correlation criteria for pattern matching. Appl. Opt. 49(28), 5501–5509 (2010)

    Article  Google Scholar 

  28. Passieux, J.C., Bouclier, R.: Classic and inverse compositional Gauss–Newton in global DIC. Int. J. Numer. Methods Eng. 119(6), 453–468 (2019)

    Article  MathSciNet  Google Scholar 

  29. Reu, P.: All about speckles: aliasing. Exp. Tech. 38(5), 1–3 (2014)

    Article  Google Scholar 

  30. Réthoré, J., Besnard, G., Vivier, G., Hild, F., Roux, S.: Experimental investigation of localized phenomena using digital image correlation. Philos. Mag. 88(28–29), 3339–3355 (2008)

    Article  Google Scholar 

  31. Sabater, N., Morel, J.M., Almansa, A.: How accurate can block matches be in stereo vision? SIAM J. Imaging Sci. 4(1), 472–500 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Savitzky, A., Golay, M.: Smoothing and differentiation of data by simplified least-squares procedures. Anal. Chem. 36(3), 1627–1639 (1964)

    Article  Google Scholar 

  33. Scharstein, D., Hirschmüller, H., Kitajima, Y., Krathwohl, G., Nesic, N., Wang, X., Westling, P.: High-resolution stereo datasets with subpixel-accurate ground truth. In: Proceedings of the 36th German Conference on Pattern Recognition (GCPR), pp. 31–42. Münster (Germany) (2014)

  34. Scharstein, D., Szeliski, R.: A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. Int. J. Comput. Vis. 47(1), 7–42 (2002)

    Article  MATH  Google Scholar 

  35. Schreier, H., Braasch, J., Sutton, M.: Systematic errors in digital image correlation caused by intensity interpolation. Opt. Eng. 39(11), 2915–2921 (2000)

    Article  Google Scholar 

  36. 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 

  37. Su, Y., Gao, Z., Fang, Z., Liu, Y., Wang, Y., Zhang, Q., Wu, S.: Theoretical analysis on performance of digital speckle pattern: uniqueness, accuracy, precision, and spatial resolution. Opt. Express 27(16), 22439–22474 (2019)

    Article  Google Scholar 

  38. Su, Y., Zhang, Q., Fang, Z., Wang, Y., Liu, Y., Wu, S.: Elimination of systematic error in digital image correlation caused by intensity interpolation by introducing position randomness to subset points. Opt. Lasers Eng. 114, 60–75 (2019)

    Article  Google Scholar 

  39. Su, Y., Zhang, Q., Xu, X., Gao, Z.: Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors. Opt. Lasers Eng. 86, 132–142 (2016)

    Article  Google Scholar 

  40. Sur, F., Blaysat, B., Grédiac, M.: Determining displacement and strain maps immune from aliasing effect with the grid method. Opt. Lasers Eng. 86, 317–328 (2016)

    Article  Google Scholar 

  41. Sur, F., Blaysat, B., Grédiac, M.: Rendering deformed speckle images with a Boolean model. J. Math. Imaging Vis. 60(5), 634–650 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sur, F., Blaysat, B., Grédiac, M.: On biases in displacement estimation for image registration, with a focus on photomechanics—Extended version. Technical Report hal-02862808, HAL (2020). https://hal.archives-ouvertes.fr/hal-02862808

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

    Google Scholar 

  44. Szeliski, R., Scharstein, D.: Sampling the disparity space image. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 419–425 (2004)

    Article  Google Scholar 

  45. Wang, Y., Lava, P., Reu, P., Debruyne, D.: Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements. Strain 52(2), 110–128 (2016)

    Article  Google Scholar 

  46. Xu, X., Su, Y., Zhang, Q.: Theoretical estimation of systematic errors in local deformation measurements using digital image correlation. Opt. Lasers Eng. 88, 265–279 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LORIA, CNRS, INRIA, Université de Lorraine, 54000, Nancy, France

    Frédéric Sur

  2. CNRS, Clermont Auvergne INP, Institut Pascal, Université Clermont Auvergne, 63000, Clermont-Ferrand, France

    Benoît Blaysat & Michel Grédiac

Authors
  1. Frédéric Sur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benoît Blaysat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michel Grédiac
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frédéric Sur.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by the French National Research Agency (ANR) under Grant ANR-18-CE08-0028-01 (ICAReS project).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sur, F., Blaysat, B. & Grédiac, M. On Biases in Displacement Estimation for Image Registration, with a Focus on Photomechanics. J Math Imaging Vis 63, 777–806 (2021). https://doi.org/10.1007/s10851-021-01032-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-021-01032-4

Keywords

Search

Navigation