{"id":48,"date":"2016-08-22T15:15:42","date_gmt":"2016-08-22T19:15:42","guid":{"rendered":"https:\/\/sites.bu.edu\/tron\/?p=48"},"modified":"2017-07-31T10:44:53","modified_gmt":"2017-07-31T14:44:53","slug":"the-space-of-essential-matrices-as-a-riemannian-manifold","status":"publish","type":"post","link":"https:\/\/sites.bu.edu\/tron\/2016\/08\/22\/the-space-of-essential-matrices-as-a-riemannian-manifold\/","title":{"rendered":"The space of essential matrices as a Riemannian manifold"},"content":{"rendered":"<p>The images of 3-D points in two views are related by the so-called _essential matrix_.<br \/>\nThere have been attempts to characterize the space of valid essential matrices as a Riemannian manifold. These approaches either put an unnatural emphasis on one of the two cameras, or do not accurately take into account the geometric meaning of the representation.<\/p>\n<p>We addressed these limitations[^1] by proposing a new parametrization which aligns the global reference frame with the baseline between the two cameras. This provides a symmetric, geometrically meaningful representation which can be naturally derived as a quotient manifold. This not only provides a principled way to define distances between essential matrices, but it also sheds new light on older results (such as the well-known twisted pair ambiguity).<\/p>\n<figure id=\"attachment37\" aria-describedby=\"caption-attachment37\" style=\"width: 531px\" class=\"wp-caption aligncenter\"><a href=\"\/tron\/files\/2016\/08\/essentialManifold.png\"><img loading=\"lazy\" src=\"\/tron\/files\/2016\/08\/essentialManifold.png\" alt=\"A graphical representation of the quotient representation\" class=\"size-full wp-image-37\" height=\"291\" width=\"521\" \/><\/a><figcaption id=\"caption-attachment37\" class=\"wp-caption-text\">A graphical representation of the quotient representation<\/figcaption><\/figure>\n<p>We provide an implementation of the basic function for working with the essential manifold integrated with the Matlab toolbox MANOPT. Download link: <a href=\"\/tron\/files\/2016\/08\/Manopt_1.0.6b_Essential.zip\">Manopt 1.06b with essential manifold<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The images of 3-D points in two views are related by the so-called _essential matrix_. There have been attempts to characterize the space of valid essential matrices as a Riemannian manifold. These approaches either put an unnatural emphasis on one of the two cameras, or do not accurately take into account the geometric meaning of [&hellip;]<\/p>\n","protected":false},"author":11510,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[7,5],"_links":{"self":[{"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/posts\/48"}],"collection":[{"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/users\/11510"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/comments?post=48"}],"version-history":[{"count":2,"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/posts\/48\/revisions"}],"predecessor-version":[{"id":81,"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/posts\/48\/revisions\/81"}],"wp:attachment":[{"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/media?parent=48"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/categories?post=48"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.bu.edu\/tron\/wp-json\/wp\/v2\/tags?post=48"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}