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On Error Assessment in Stereo-based Deformation Measurements

Part I: Theoretical Developments for Quantitative Estimates

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Abstract

Using the basic equations for stereo-vision with established procedures for camera calibration, the error propagation equations for determining both bias and variability in a general 3D position are provided. The results use recent theoretical developments that quantified the bias and variance in image plane positions introduced during image plane correspondence identification for a common 3D point (e.g., pattern matching during measurement process) as a basis for preliminary application of the developments for estimation of 3D position bias and variability. Extensive numerical simulations and theoretical analyses have been performed for selected stereo system configurations amenable to closed-form solution. Results clearly demonstrate that the general formulae provide a robust framework for quantifying the effect of various stereo-vision parameters and image-plane matching procedures on both the bias and variance in an estimated 3D object position.

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Notes

  1. 1.

    When x is n-dimensional, the covariance matrix (also called the variance-covariance matrix) is usually denoted Var(x) and has the expression:

    $$ Var({\mathbf{x}}) = E(({\mathbf{x}} - E({\mathbf{x}})){({\mathbf{x}} - E({\mathbf{x}}))^T}) = \left[ {\begin{array}{*{20}{c}} {{\sigma_{{x_1}}}^2} & {{\rm cov} ({x_1},{x_2})} & . & . & {{\rm cov} ({x_1},{x_n})} \\{{\rm cov} ({x_2},{x_1})} & {{\sigma_{{x_2}}}^2} & . & . & . \\. & . & . & . & . \\. & . & . & . & . \\{{\rm cov} ({x_n},{x_1})} & . & . & . & {{\sigma_{{x_n}}}^2} \\\end{array} } \right] $$
  2. 2.

    The cost function is usually obtained from bundle adjustment or other similar optimization methods.

  3. 3.

    Equation (10) in reference [8] is incorrect; the IM should be IN

  4. 4.

    It is assumed that all images have been corrected for distortions. Non-parametric methods [14] provide one approach that can be used to remove distortions so that the pinhole mode

  5. 5.

    The initial world system orientation relative to the camera 1 system is commonly used as the reference for all 3D measurements. For this reason, it is separated from the remaining Nviews extrinsic parameter variables.

  6. 6.

    During calibration, it is common to arrange the initial world system to coincide with the planar grid/dot pattern configuration and thereby simplify the analysis.

  7. 7.

    It is assumed that the radial distortion model provides an exact representation for the deviations relative to the pinhole model.

  8. 8.

    Figure 1 assumes ideal pinhole model projection to simplify the presentation.

  9. 9.

    The large subset size was selected to be consistent with use of previously published experimental data [13]. The large subset size was used to obtain sufficient pattern density and local contrast for accurate matching, since the pattern was quite coarse at the high magnification employed by the authors to achieve pixel translations in increments of ≈ 0.07 pixels.

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Acknowledgements

The financial support of Sandia National Laboratory and the technical advice and support of Dr. Timothy Miller and Dr. Phillip Reu through Sandia Contract PO#551836, the support of Dr. Bruce Lamattina through ARO# W911NF-06-1-0216 and the support provided by Dr. Stephen Smith through NASA NNX07AB46A are gratefully acknowledged. In addition, the research support provided by the Department of Mechanical Engineering at the University of South Carolina is also gratefully acknowledged.

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Correspondence to M. A. Sutton.

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Wang, Y., Sutton, M.A., Ke, X. et al. On Error Assessment in Stereo-based Deformation Measurements. Exp Mech 51, 405–422 (2011). https://doi.org/10.1007/s11340-010-9449-9

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Keywords

  • Probabilistic theoretical analysis
  • Stereovision
  • Camera parameters
  • Variance and expectation for 3D position