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Geometry in X-ray Testing

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Computer Vision for X-Ray Testing

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Abstract

Geometry is of basic importance for understanding in X-ray testing. In this chapter, we present a mathematical background of the monocular and multiple view geometry which is normally used in X-ray computer vision systems. The chapter describes an explicit model which relates the 3D coordinates of an object to the 2D coordinates of the digital X-ray image pixel, the calibration problem, the geometric and algebraic constraints between two, three, and more X-ray images taken at different projections of the object, and the problem of 3D reconstruction from n views.

Cover image: Average of X-ray images of a wheel in motion (series C0008 colored with “parula” colormap).

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Notes

  1. 1.

    Although industrial X-ray generators use standard tubes with larger focal size that blur the X-ray images slightly, the assumption that the X-ray source can be modeled as a point is valid for geometrical measurements. This is because the position of a point in the X-ray image can still be estimated as the center of the blurred point [5].

  2. 2.

    The reader should note that at this moment there are two retinal planes: \(\varPi \) for the central projection and \(\Phi \) for the image intensifier.

  3. 3.

    The reader should note that \(\Gamma \), the retinal plane of the CCD camera, is the third retinal plane of our model (see footnote 2).

  4. 4.

    Nevertheless, in Sect. 8.4.3 the reader can find an interesting X-ray testing application where the 3D model is estimated using a self-calibration method based on bundle adjustment.

  5. 5.

    Image intensifier developed by YXLON International Inc.

  6. 6.

    This is a gradient method that uses the Broyden–Fletcher–Goldfarb–Shanno formula for updating the approximation of the Hessian matrix iteratively, which reduces the computational cost of the minimization.

  7. 7.

    The word epipole comes from the Greek \(\grave{\epsilon } \pi \iota \) (epi): over and \(\pi \acute{o} \lambda o \varsigma \) (polos): attractor.

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Authors and Affiliations

  1. Pontificia Universidad Católica de Chile, Santiago, Chile

    Domingo Mery

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  1. Domingo Mery
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Correspondence to Domingo Mery .

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Mery, D. (2015). Geometry in X-ray Testing. In: Computer Vision for X-Ray Testing. Springer, Cham. https://doi.org/10.1007/978-3-319-20747-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-20747-6_3

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