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


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.

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  1. 1.

    The proposed calculation is adapted from

  2. 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:

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    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 \({\varvec{\phi }}\).


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Correspondence to Frédéric Sur.

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

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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).

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  • Image registration
  • Digital image correlation
  • Fattening effect
  • Savitzky–Golay filtering
  • Experimental mechanics
  • Stereo-imaging
  • Optical flow