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In-Situ Systematic Error Correction for Digital Volume Correlation Using a Reference Sample

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Abstract

The self-heating effect of a laboratory X-ray computed tomography (CT) scanner causes slight change in its imaging geometry, which induces translation and dilatation (i.e., artificial displacement and strain) in reconstructed volume images recorded at different times. To realize high-accuracy internal full-field deformation measurements using digital volume correlation (DVC), these artificial displacements and strains associated with unstable CT imaging must be eliminated. In this work, an effective and easily implemented reference sample compensation (RSC) method is proposed for in-situ systematic error correction in DVC. The proposed method utilizes a stationary reference sample, which is placed beside the test sample to record the artificial displacement fields caused by the self-heating effect of CT scanners. The detected displacement fields are then fitted by a parametric polynomial model, which is used to remove the unwanted artificial deformations in the test sample. Rescan tests of a stationary sample and real uniaxial compression tests performed on copper foam specimens demonstrate the accuracy, efficacy, and practicality of the presented RSC method.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant nos. 11427802, and 11632010), the Aeronautical Science Foundation of China (2016ZD51034), the Beijing Nova Program (xx2014B034), and the Academic Excellence Foundation of BUAA for PhD Students. We also thank King Abdullah University of Science and Technology (KAUST) for its support.

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Correspondence to B. Pan or G. Lubineau.

Appendix

Appendix

Here we theoretically analyze the relationship between rotation of X-ray optical axis and the induced artificial deformations in reconstructed volume images. By comparing with the ideal case, the change in imaging geometry due to rotation of X-ray optical axis can be decomposed into three aspects: (1) motion of emission point in Fig. 10(a), (2) rotation of sample in Fig. 10(b), (3) eccentric rotation of detector in Fig. 10(c).

Fig. 10
figure 10

Decomposition of the changes in imaging geometry of a lab-CT due to rotation of X-ray beam: (a) motion of emission point, (b) rotation of sample, (c) eccentric rotation of detector

First, the motion of emission point along y direction shown in Fig. 10(a) causes rigid-body translation v 2 in the volume image as discussed above.

$$ {v}_2=-\left(\frac{D_1}{d_1}-1\right)\cdot {d}_0\tan \theta =-\left(\frac{D_0}{d_0}-1\right)\cdot {d}_0\tan \theta $$
(7)

Second, since the direction of X-ray beam deviates from ideal case, X-ray beam passes through the sample along different paths. As such, the difference from ideal beam path can be equivalent to a rotation of the sample with an angle of θ, thus finally causing a rigid-body rotation θ of volume image. Third, the rotation of detector around point O′ first includes a deviation of the rotation center, thus a rigid-body translation v 3 from O to O′ may be present in the reconstructed volume image.

$$ {v}_3=\left(\frac{D_0-{d}_0}{d_0}\right)\cdot {d}_0\tan \theta $$
(8)

From the above two expressions, v 2 and v 3 are equal but in opposite directions, thus rotation of X-ray beam causes no rigid-body translation in the volume image. Also, since the detector deviates from the ideal direction (i.e., perpendicular to the X-ray beam), the upper part of the projection image may expand due to the increased amplification, while the lower part shrinks. As a result, a complex non-uniform deformation may occur within the reconstructed volume image due to the rotation of detector according to the commonly-used Feldkamp algorithm in a lab cone-beam CT system. The corresponding theoretical analysis is detailed as follows.

As illustrated in Fig. 11(a), assume the perpendicular distance from rotation center to the X-ray cone-beam is Y, the projection image P Φ(Y) can be denoted as a function of Y and rotation angle Φ of the sample. In ideal case, the detector should be perpendicular to the X-ray beam, then we can get the projected position y as follows

$$ y=\frac{Y}{d_1}\cdot {D}_1 $$
(9)
Fig. 11
figure 11

(a) Equivalent imaging geometry due to the rotation of X-raydetector, and (b) the resulting image deformation

While the projected position y’ on rotated detector can be expressed as

$$ {y}^{\prime }=\frac{D_1Y}{d_1-Y\tan \theta } $$
(10)

Then, the corresponding position Y′ should be written as

$$ {Y}^{\prime }=\frac{y^{\prime }}{D_1}\cdot {d}_1=\frac{Yd_1}{d_1-Y\tan \theta } $$
(11)

From the above expression, it is seen that the distance Y shared by the points along X-ray beam may change to Y′ due to the rotation of detector. In this regard, the effect of X-ray detector rotation can be equivalently described as: as for each rotation angle Φ of the test sample, the upper part of the sample (above optical axis) expand, while the lower part (below optical axis) shrink as exhibited in Fig. 11(b). As such, the final intensity or position changes in volume images highly depends on specific algorithm employed during volume image reconstruction. In most cases, however, these non-uniform deformations are limited to far less than 1 voxel and present as random intensity changes rather than a global bias, thus can be considered as random errors rather than systematic errors.

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Wang, B., Pan, B. & Lubineau, G. In-Situ Systematic Error Correction for Digital Volume Correlation Using a Reference Sample. Exp Mech 58, 427–436 (2018). https://doi.org/10.1007/s11340-017-0356-1

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