Both transformations have their advantages and disadvantages. <\/p>\n\n\n\n
Hough (or Radon) transformation<\/p>\n\n\n\n
Purely projection-technically the Hough transformation fits better to<\/strong> the gnomonic projection of the crystal lattice on the<\/strong> planar detector screen<\/strong>. The drawback, however, is that we do not see the lattice projection, but<\/strong> diffraction phenomena of the lattice produce two hyperbola-shaped<\/strong> intensity drops, which are taken as evidence of geometrically predicted band edges<\/strong>. The curvature<\/strong> of<\/strong> the hyperbolas<\/strong> resulting from the gnomonic projection of small circles (band edges) grows<\/strong>, the<\/strong> larger the angular distance<\/strong> to the pattern center as well as the Bragg angles<\/strong> become. <\/p>\n\n\n\n According to Bragg’s law, the Bragg angle changes<\/strong>: (1)<\/strong> inversely proportional with the<\/strong> squareroot of the acceleration voltage<\/strong> and\/or (2)<\/strong> proportional with the lattice parameters<\/strong> of the phase, (3)<\/strong> with the indexing<\/strong> of the lattice plane (hkl)<\/strong>, and (4)<\/strong> with the diffraction order n<\/strong>. <\/em> To reduce<\/strong> these curvature<\/strong>, first the electron energy<\/strong> can be increased<\/strong> which reduces, however, the spatial resolution of the technique, and PCz<\/sub><\/strong> can be increased<\/strong> which increases the necessary dwell time and so a possible charging. Since the applied technique works preferably for large angle Kikuchi patterns, an increase of PCz<\/sub> is from this point of view limited.<\/p>\n\n\n\n Please note:<\/span> For high-accurate techniques<\/strong> is valid: The Hough peak does not define the lattice plane trace<\/strong>, except it crosses the pattern center. With increasing angular distance to the pattern center the deviation increases. <\/p>\n\n\n\n The biggest advantage<\/strong> of the currently used Hough transform is that PC is not needed<\/strong>. This explains why the pattern quality<\/strong> of pattern is always available<\/strong>, even if PC or the diffracting phase are unknown.<\/p>\n\n\n\n One could also easily improve the Hough transformation considering PC. Then gnomonic distortions could be immediately corrected so that equivalent bands would have equivalent peak widths. However, lattice planes of a zone would then be aligned along sine-shaped curves which is certainly not easy to interprete as great circles in the Funk transformation. <\/em><\/p>\n\n\n\n Funk transformation<\/p>\n\n\n\n The main disadvantage<\/strong> of the Funk transformation is that the recorded Kikuchi pattern<\/strong> cannot be used immediately, but must first be<\/strong> correctly projected onto a spherical surface<\/strong> what requires PC<\/strong>. The Funk transformation<\/strong> converts<\/strong> great circles into points and points into great circles. More generally one could say that it transforms small circles with the opening angle \u03c7 into small circles with the opening angle 90\u00b0-\u03c7<\/strong>, because for a great circle \u03c7=90\u00b0 is valid.<\/p>\n\n\n\n All critical distortions<\/strong> mentioned above in Hough transformation<\/strong> are inherently compensated<\/strong>. It represents some kind of projection of the reciprocal lattice, where <\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" Both transformations have their advantages and disadvantages. Hough (or Radon) transformation Purely projection-technically the Hough transformation fits better to the gnomonic projection of the crystal lattice on the planar detector screen. The drawback, however, is that we do not see Read More …<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"parent":1410,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/1463"}],"collection":[{"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/comments?post=1463"}],"version-history":[{"count":19,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/1463\/revisions"}],"predecessor-version":[{"id":1494,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/1463\/revisions\/1494"}],"up":[{"embeddable":true,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/1410"}],"wp:attachment":[{"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/media?parent=1463"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
The Hough peaks<\/strong> become more blurred<\/strong> as<\/strong> the deviations between straight lines and hyperbola<\/strong>-shaped curvature<\/strong> becomes stronger<\/strong>.<\/p>\n\n\n\n
The angular distance<\/strong> to the projection center PC, on the other hand, grows the smaller<\/strong> the distance to the detection screen (<\/em>PCz<\/sub><\/strong>) becomes and<\/strong> the further the pattern center is shifted from the center of the image, i.e. the bigger<\/strong> <\/em>\u0394<\/strong> = (PCx<\/sub>-\u00bd)\u00b2 + (PCy<\/sub>-\u00bd)\u00b2.<\/em>
This is one reason for the later implemented, special treatment of TKD pattern indexing, especially if PCy<\/sub><0.<\/em><\/p>\n\n\n\n
By this spherical projection<\/strong> alone, lattice planes become great circles and diffraction cones become small circles, i.e., their shape is unambiguously determined, which considerably limits possible errors<\/strong> in analyses.<\/p>\n\n\n\n