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Keywords:

  • digital image correlation/matching;
  • error assessment;
  • expectation and variance for image motion;
  • intensity interpolation;
  • intensity pattern noise;
  • probabilistic formulation

Abstract:  Basic concepts in probability are employed to develop analytic formulae for both the expectation (bias) and variance for image motions obtained during subset-based pattern matching. Specifically, the expectation and variance in image motions in the presence of uncorrelated Gaussian intensity noise for each pixel location are obtained by optimising a least squares intensity matching metric. Results for both 1D and 2D image analyses clearly quantify both the bias and the covariance matrix for image motion estimates as a function of: (a) interpolation method, (b) sub-pixel motion, (c) intensity noise, (d) contrast, (e) level of uniaxial normal strain and (f) subset size. For 1D translations, excellent agreement is demonstrated between simulations, theoretical predictions and experimental measurements. The level of agreement confirms that the analytical formulae can be used to provide a priori estimates for the ‘quality’ of local, subset-based measurements achievable with a given pattern. For 1D strain with linear interpolation, theoretical predictions are provided for the expectation and co-variance matrix for the local displacement and strain parameters. For 2D translations with bi-linear interpolation, theoretical predictions are provided for both the expectation and the co-variance matrix for both displacement components. Theoretical results in both cases show that the expectations for the local parameters are biased and a function of: (a) the interpolation difference between the translated and reference images, (b) magnitude of white noise, (c) decimal part of the motion and (d) intensity pattern gradients. For 1D strain, the biases and the covariance matrix for both parameters are directly affected by the strain parameter p1 as the deformed image is stretched by (1 + p1). For 2D rigid body motion case, the covariance matrix for measured motions is shown to have coupling between the motions, demonstrating that the directions of maximum and minimum variability do not generally coincide with the x and y directions.