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The Effect of the Ill-posed Problem on Quantitative Error Assessment in Digital Image Correlation

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Abstract

This work explores the effect of the ill-posed problem on uncertainty quantification for motion estimation using digital image correlation (DIC) (Sutton et al. [2009]). We develop a correction factor for standard uncertainty estimates based on the cosine of the angle between the true motion and the image gradients, in an integral sense over a subregion of the image. This correction factor accounts for variability in the DIC solution previously unaccounted for when considering only image noise, interpolation bias, contrast, and the software settings such as subset size and spacing.

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Notes

  1. Note, there are subtle differences between our definition of the cosine of the angle here and that used in the classical definition. Our modification is based on needing to represent only the acute angle between the two vectors. As we discuss in a subsequent section, if the motion points in the direction opposite the image gradient no loss of generality occurs.

  2. In the DIC literature, this function is often written as G(x + u,t) − F(x)where G and F are the deformed and reference images, respectively. There are many normalized varieties of this equation such as the sum-squared-differences, zero-normalized-sum-squared-differences, etc.

  3. We have intentionally presented the DIC problem in abstract form because the analysis that follows applies to both local and global formulations. A local, subset-based approach is used in the numerical examples to simplify the presentation. One can expect similar results for a global formulation, although this is not investigated here.

  4. Empirical evidence suggests that this angle is small. Were it not, DIC could not be used so successfully in so many application areas.

  5. We do not explicitly include the interpolation bias in the estimator above because the resulting error is small in comparison to the effects of noise and motion orientation.

  6. This directional pattern is far from acceptable for use in real DIC applications. We use it here simply to highlight the connection between directionality and error in the DIC solution.

  7. The specific details of the loading rate, specimen geometry, material type, and sample preparation are not of interest in this example and are therefore intentionally omitted.

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Authors and Affiliations

  1. Center for Computing Research, Sandia National Laboratories, P.O. Box 5800, MS 1320, Albuquerque, NM, 87185-1320, USA

    R. B. Lehoucq & D. Z. Turner

  2. Diagnostic Science & Engineering, Sandia National Laboratories, P.O. Box 5800, MS 0828, Albuquerque, NM, 87185-0828, USA

    P. L. Reu

Authors
  1. R. B. Lehoucq
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  2. P. L. Reu
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  3. D. Z. Turner
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Correspondence to D. Z. Turner.

Additional information

This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Appendix

Appendix

All symbols used in this work are defined in Table 5.

Table 5 Definition of all symbols used in this work
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Lehoucq, R.B., Reu, P.L. & Turner, D.Z. The Effect of the Ill-posed Problem on Quantitative Error Assessment in Digital Image Correlation. Exp Mech 61, 609–621 (2021). https://doi.org/10.1007/s11340-017-0360-5

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  • DOI: https://doi.org/10.1007/s11340-017-0360-5

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