{"id":1410,"date":"2021-12-01T11:22:43","date_gmt":"2021-12-01T11:22:43","guid":{"rendered":"http:\/\/www.calm-ebsd.com\/?page_id=1410"},"modified":"2021-12-03T11:38:41","modified_gmt":"2021-12-03T11:38:41","slug":"funk-transformation","status":"publish","type":"page","link":"https:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/","title":{"rendered":"Funk transformation"},"content":{"rendered":"\n<p>The application of another transformation than Hough or Radon commonly used for band detection goes back to [1]. There, this function is called SCRUFF transformation but seem to be similar to the Funk transformation used in CALM. <\/p>\n\n\n\n<p>The <strong>edge-filtered Funk transformation<\/strong> as shown below is exclusively <strong>used<\/strong> to indicate [hkl]*, i.e., <strong>to discover<\/strong> as many as possible <strong>Kikuchi bands<\/strong>. It is not and cannot be used for band profile extraction. <\/p>\n\n\n\n<p style=\"font-size:22px\">Phenomenological description assuming a complete signal<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-style-default\"><p><span class=\"has-inline-color has-black-color\">If the diffraction signal is projected onto a spherical surface, the <strong>intensity of <\/strong>any <strong>direction<\/strong> [x,y,z] <strong>in<\/strong> the <strong>Funk<\/strong> <strong>transformation<\/strong> is equal to the <strong>summed intensity along the great circle perpendicular<\/strong> to [x,y,z].<\/span><\/p><\/blockquote>\n\n\n\n<p>As well-known, each Kikuchi band can be interpreted by means of a very flat double cone with half opening angle of (90-\u03b8). In the stereographic projection (left image below) the band edges represent so-called small circles very close to the great circle describing the band-forming lattice plane trace. <br>After Funk transformation (right image) of the complete diffraction signal the band becomes again a double cone. <strong>The visible circle has<\/strong> now, however,<strong> a diameter of 2\u03b8<\/strong>. The <strong>cone axis indicates<\/strong> the lattice plane normal <strong>[hkl]*<\/strong>.   <\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-2 is-cropped\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1.png\" alt=\"\" data-id=\"1416\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1.png\" data-link=\"http:\/\/www.calm-ebsd.com\/al_20kv_001_4-049_stereographic-2\/\" class=\"wp-image-1416\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-1-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><figcaption class=\"blocks-gallery-item__caption\">sterographic projection of the master pattern<\/figcaption><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2.png\" alt=\"\" data-id=\"1433\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/al_20kv_001_4-049_funk-2-2\/\" class=\"wp-image-1433\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-2-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><figcaption class=\"blocks-gallery-item__caption\"><strong>edge corrected<\/strong> Funk transformation of the master pattern<\/figcaption><\/figure><\/li><\/ul><\/figure>\n\n\n\n<p>This perfect condition of <strong>complete and well-shaped circles<\/strong> displaying the band widths undistorted is <strong>only available for<\/strong> a <strong>complete simulation<\/strong> as used in a digital twin (&#8220;master pattern&#8221;).<\/p>\n\n\n\n<p style=\"font-size:22px\">The typical, incomplete signal (Kikuchi pattern)<\/p>\n\n\n\n<p>If only a <strong>part<\/strong> <strong>of the signal<\/strong> is available, e.g. a single cube plane (left image below), all <strong>circles<\/strong> in the edge-filtered Funk transformation (right) <strong>degenerate<\/strong> to <em>circle-like<\/em> features which are still well detectable.  <\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-2 is-cropped\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic.png\" alt=\"\" data-id=\"1415\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic.png\" data-link=\"http:\/\/www.calm-ebsd.com\/al_20kv_001_4-049_stereographic\/\" class=\"wp-image-1415\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk.png\" alt=\"\" data-id=\"1413\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk.png\" data-link=\"http:\/\/www.calm-ebsd.com\/al_20kv_001_4-049_funk\/\" class=\"wp-image-1413\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/figure><\/li><\/ul><\/figure>\n\n\n\n<p>If we reduce the captured segment further using the typical size and shape of a <strong>Kikuchi pattern<\/strong> (only between 12-15% of the entire signal) with a resolution of <strong>320&#215;230 pixels only<\/strong> and shift the projection center to <strong>PC = [0.5,0.25,0.667]<\/strong>, the Funk transformation becomes asymmetric. <strong>Reducing the sector size<\/strong> the <strong>empty region<\/strong> <strong>in<\/strong> the <strong>Funk<\/strong> transformation (right) <strong>grows<\/strong> and the <strong>degeneration of circles increases<\/strong> since the respective bands become shorter an shorter. The projection center is responsible for the shift of the empty area.<br>The medium <strong>image resolution<\/strong> <strong>has no <\/strong>considerable <strong>effect<\/strong> on the Funk transformation. <\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-2 is-cropped\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2.png\" alt=\"\" data-id=\"1423\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/al_20kv_001_4-049_stereographic-3\/\" class=\"wp-image-1423\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_stereographic-2-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1.png\" alt=\"\" data-id=\"1422\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/al_20kv_001_4-049_funk-2\/\" class=\"wp-image-1422\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/Al_20kV_001_4.049_Funk-1-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/figure><\/li><\/ul><\/figure>\n\n\n\n<p class=\"has-large-font-size\">Experimental pattern<\/p>\n\n\n\n<p>In comparison to simulated patterns an experimental Kikuchi pattern is slightly more blurred. In experiments, the <strong>wavelength<\/strong> is represented by a <strong>distribution<\/strong> and not a distinct value, but also the crystal is not as perfect as described in simulations. <br>Nevertheless, the <strong>Funk transformation<\/strong> <strong>is<\/strong> a <strong>great<\/strong> tool <strong>because<\/strong> of the strong<strong> constraints between [hkl]* and (uvw)*<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-2 is-cropped\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1.png\" alt=\"\" data-id=\"1431\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/ebsp_al_17mmdd-01_stereographic-1\/\" class=\"wp-image-1431\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_stereographic-1-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><figcaption class=\"blocks-gallery-item__caption\">experimental pattern from Al<\/figcaption><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1.png\" alt=\"\" data-id=\"1430\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/ebsp_al_17mmdd-01_funk-1\/\" class=\"wp-image-1430\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/EBSP_Al_17mmDD-01_Funk-1-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/figure><\/li><\/ul><\/figure>\n\n\n\n<p>Experimental influences like excess-deficiency will certainly affect the alignment of [hkl]* along great circles displaying (uvw)* and also the Bragg angles but the impact is small. In case of <strong>bigger deviations<\/strong>, either <strong>PC<\/strong> is <strong>incorrect<\/strong>, <strong>or<\/strong> considerable <strong>image distortions<\/strong> occur. <\/p>\n\n\n\n<p style=\"font-size:22px\">TKD<\/p>\n\n\n\n<p>Also TKD patterns can be processed in CALM. However, it should be clear that the captured sector is commonly smaller than for EBSD geometry. The following pattern has been aquired for a comparatively unusual PC=[0.521,-0.334,0.6].<\/p>\n\n\n\n<figure class=\"wp-block-gallery columns-2 is-cropped\"><ul class=\"blocks-gallery-grid\"><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic.png\" alt=\"\" data-id=\"1498\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/tkd-al_7_stereographic\/\" class=\"wp-image-1498\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_stereographic-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><figcaption class=\"blocks-gallery-item__caption\">Experimental TKD pattern from Al<\/figcaption><\/figure><\/li><li class=\"blocks-gallery-item\"><figure><img loading=\"lazy\" width=\"800\" height=\"800\" src=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk.png\" alt=\"\" data-id=\"1497\" data-full-url=\"http:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk.png\" data-link=\"http:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/tkd-al_7_funk\/\" class=\"wp-image-1497\" srcset=\"https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk.png 800w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk-300x300.png 300w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk-150x150.png 150w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk-768x768.png 768w, https:\/\/www.calm-ebsd.com\/wp-content\/uploads\/2021\/12\/TKD-Al_7_Funk-270x270.png 270w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/figure><\/li><\/ul><\/figure>\n\n\n\n<p>[1] Day, A. P.: Spherical EBSD, <em>J. Microscopy, <\/em><strong>2008<\/strong><em>, 230<\/em>, 472-486<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The application of another transformation than Hough or Radon commonly used for band detection goes back to [1]. There, this function is called SCRUFF transformation but seem to be similar to the Funk transformation used in CALM. The edge-filtered Funk <a href=\"https:\/\/www.calm-ebsd.com\/index.php\/useful-additional-information\/funk-transformation\/\" class=\"read-more\">Read More &#8230;<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"parent":389,"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\/1410"}],"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=1410"}],"version-history":[{"count":23,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/1410\/revisions"}],"predecessor-version":[{"id":1499,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/1410\/revisions\/1499"}],"up":[{"embeddable":true,"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/pages\/389"}],"wp:attachment":[{"href":"https:\/\/www.calm-ebsd.com\/index.php\/wp-json\/wp\/v2\/media?parent=1410"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}