EBSD and BKD
Pattern Solving - The Radon Transformation

 

 

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    The geometry of a Kikuchi pattern is specific for the structure and the orientation of the crystallite. For indexing, one only needs to know the location and the width of some of the bands in the pattern. Whereas an experienced user may recognize diffuse bands even on a strong background, automated band localization is not an easy task for the software. Because of the small electron wavelength as compared to the lattice spacings, Bragg angles are very small in electron diffraction so that the border lines of Kikuchi bands can very well be approximated by straight lines.

    Band localization is performed by applying a Radon transform on the Kikuchi patterns and analyzing peak profiles in Radon space. The general Radon transformation [1] is a functional to transform a manifold in an m-dimensional space into a manifold in an n-dimensional space. With the normal parametrization of a straight line as the functional kernel, a line in a plane is thus tranformed into a single point R(r, j) in 2-dimentional Radon space by

    f(x,y) stands for the intensity distribution along the line,  d denotes the Dirac function, r is the perpendicular distance of the line from the origin, j is the angle between the x axis and the normal from the origin to the line. Every point (r, j) in the Radon space thus corresponds to the sum of intensity values along a certain straight line in the image space.

    Reconstruction of the pattern by applying the inverse Radon transform

      

    literally means that a peak in Radon space is projected back into image space along its incoming path. By definition, the intensity I(r, j) of every single cell (ri, ji) in Radon space is attributed equally to all pixels (xl, yl) on the related straight line ri = x·cos ji + y·sin ji in the reconstructed Kikuchi pattern.  (N.B.: The Radon transformation and its Inverse are the basis of back-transformation in computer tomography.)

    The EBSD image (i.e., the Kikuchi pattern) is made up by a discrete array [x, y] of gray-tone image points (so-called pixels). So we consider a discrete Radon transform whereby the integrals are replaced by sums, and the Radon domain consists of an array of discrete cells on a Cartesian grid (r, j). A Kikuchi band can be imagined to be composed of a bundle of straight lines that may be slightly twisted and shifted against each other. By a "forward" Radon transformation, the intensity profile of a Kikuchi band in the image space is mapped to a "butterfly-like" shape in Radon domain with a bright center and dark cusps on the sides in r direction, rather than a single point as were the case of a single line.

    A "backward" Radon transformation from cells in Radon space to the pattern according to the inverse transformation (equation 2) has some conceptual and numeric advantages. As mentioned above, a cell in Radon space corresponds to a single straight line in the pattern. So the Radon space rather than the image is scanned point by point, and for each point R(r, j) the intensities along the corresponding straight line in the pattern are accumulated. However, the intensity profile can, as contrasted to a "forward" RT, in addition be interrogated along this line. Thus short fragments of the line outside a band can be excluded from transformation by applying a run length threshold, as well as artifacts of "hot" spots in the pattern can be skipped by defining intensity thresholds. The result is a clearer domain in Radon space. Noise can further be removed and contrast enhanced effectively in the Radon transformed pattern by applying a square intensity operator.

    The task of locating lines or bands in the diffraction pattern is so reduced to the simpler task of searching for isolated peaks. The "butterfly peak" is localized by a peak search with constraints or by evaluating some coefficients of a 1D FFT of this Radon domain (Schwarzer and Sukkau, 2003).
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    [1] J. Radon: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Naturw. Klasse 69 (1917) 262-267
    English translation by P.C. Parks: On the determination of functions from their integral values along certain manifolds. IEEE Trans. Medical Imaging  MI-5 (1986) 170-176

    [2] R.A. Schwarzer and J. Sukkau: Automated evaluation of Kikuchi patterns by means of Radon and Fast Fourier Transformations, and verification by an artificial neural network. Adv. Eng. Mat. 5 (2003) 601-606

  


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