{"id":779,"date":"2015-10-21T08:13:46","date_gmt":"2015-10-21T07:13:46","guid":{"rendered":"http:\/\/wp.cs.ucl.ac.uk\/gift-surg-intranet\/?p=779"},"modified":"2015-10-21T08:13:46","modified_gmt":"2015-10-21T07:13:46","slug":"phd-meeting-minute-sebastiano-ferraris-20151020","status":"publish","type":"post","link":"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/2015\/10\/21\/phd-meeting-minute-sebastiano-ferraris-20151020\/","title":{"rendered":"PhD Meeting Minute \u2014 Sebastiano Ferraris. \u2014 20151020"},"content":{"rendered":"<p class=\"p1\" style=\"text-align: center\"><span class=\"s1\"><b>Meeting Minutes<\/b><\/span><\/p>\n<p class=\"p1\" style=\"text-align: center\"><span class=\"s1\">October 20, 2015<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\"><b>1. Attendees:<\/b><\/span><\/p>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Tom Vercauteren<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Sebastiano Ferraris<\/span><\/li>\n<li class=\"li2\">Marc Modat<\/li>\n<\/ul>\n<p class=\"p1\"><span class=\"s1\"><b>2. Discussed Topics:<\/b><\/span><\/p>\n<ul class=\"ul1\">\n<li class=\"li2\">Numerical methods for the computation of the exponential of an SVF.<br \/>\n(Details below)<\/li>\n<li class=\"li2\">Next steps.<\/li>\n<\/ul>\n<p class=\"p1\">\n<p class=\"p1\"><span class=\"s1\">Given an SVF over a domain \\Omega (in this case matrix coordinates):<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/0_vector_field.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-804\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/0_vector_field-300x109.png\" alt=\"_0_vector_field\" width=\"300\" height=\"109\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/0_vector_field-300x109.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/0_vector_field-250x90.png 250w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/0_vector_field.png 362w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">the goal is to find the diffeomorphism \\varphi that generated this vector field as its tangent vector field (or velocity field). <\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">This enable to reformulate the problem in terms of solving the differential equations that relates the diffeomorphism and the SVF. Since by definition the field involved is invariant respect to the time, the corresponding differential equation that defines the velocity field is a stationary (or homogeneous) one.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">If g is the one parameter subgroup that defines the transformation, the problem of finding g^{1} is equivalent to find the solution to the differential equation:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/1_ode.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-806\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/1_ode-300x59.png\" alt=\"_1_ode\" width=\"300\" height=\"59\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">The operation of integrating the previous differential equation, is in Lie group theory called Lie exponentiation. Therefore<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/2_def_exp.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-805\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/2_def_exp-300x63.png\" alt=\"_2_def_exp\" width=\"300\" height=\"63\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/2_def_exp-300x63.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/2_def_exp-250x52.png 250w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/2_def_exp.png 425w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">and any numerical method to solve the stationary ODE, become a method to compute the Lie exponential.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">The diffeomorphism that defines the solution is computed and represented as a displacement field: the vector field defined at each point of the domain as the difference between the diffeomorphism and the identity computed at this point:<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">disp(x) = \\varphi(x) &#8211; x \u00a0\\qquad \\forall x \\in \\Omega<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">and when the vector field is small enough, the diffeomorphism can be approximated by the SVF itself, when written ad displacement field, so as<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">\\varphi(x) = \\exp(v(x)) = v(x) + Id<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">The poliaffine scaling and squaring is based on this approximation, and on the fact that at point wise the SVF can be approximated by a linear transformation.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">Scaling and squaring:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/3_poliaffine_eq.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-807 size-full\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/3_poliaffine_eq.png\" alt=\"_3_poliaffine_eq\" width=\"852\" height=\"112\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">Linear approximation when v is small:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/4_polyaffine_matrix.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-808\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/4_polyaffine_matrix-300x88.png\" alt=\"_4_polyaffine_matrix\" width=\"300\" height=\"88\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">Other methods considered in these experiments are the following:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-801 size-large\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods-1024x443.png\" alt=\"_5_methods\" width=\"700\" height=\"303\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods-1024x443.png 1024w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods-300x130.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods-768x332.png 768w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods-250x108.png 250w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods-1568x678.png 1568w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/5_methods.png 1700w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">By a theoretical point of view, these approximation makes sense in the case of diffeomorphism, even if the sum of diffeomorphisms is not defined, because we rely on the displacement field generated by a diffeomorphism.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">(Displacement fields are vector fields, whose set is an infinite dimensional space that posses sum, scalar product and norm, otherwise not defined on the group of diffeomorphisms).<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">Results are considered for an initial set of 200 svn generated by matrices of SE(2), group of rigid body transformations, where the integration can be computed with a closed formula (as the vector field generated by the exponential of the matrix \u2014 subtracting the id!).<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">An example of these vector fields are in the following picture <\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">RED: SVF input value.<br \/>\n<\/span><span class=\"s1\">BLUE: ground truth of the displacement field generated by the diffeomorphisms, our goal.<br \/>\n<\/span><span class=\"s1\">MAGENTA: solution computed with one of the approximation methods.<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/matrix_random_generated.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-787 size-full\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/matrix_random_generated.png\" alt=\"matrix_random_generated\" width=\"1008\" height=\"396\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/matrix_random_generated.png 1008w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/matrix_random_generated-300x118.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/matrix_random_generated-768x302.png 768w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/matrix_random_generated-250x98.png 250w\" sizes=\"auto, (max-width: 1008px) 100vw, 1008px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">The norm of the error normalised by the square root of the size of the domain are:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/6_num_norm_matrix.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-802\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/6_num_norm_matrix-300x253.png\" alt=\"_6_num_norm_matrix\" width=\"300\" height=\"253\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/6_num_norm_matrix-300x253.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/6_num_norm_matrix-250x211.png 250w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/6_num_norm_matrix.png 585w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">For the dataset of 200 random we have the following box plot, where the error is computed having a ground truth:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_200_random_generated_svf_from_matrix.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-789 size-full\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_200_random_generated_svf_from_matrix.png\" alt=\"boxplot_200_random_generated_svf_from_matrix\" width=\"1008\" height=\"504\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_200_random_generated_svf_from_matrix.png 1008w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_200_random_generated_svf_from_matrix-300x150.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_200_random_generated_svf_from_matrix-768x384.png 768w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_200_random_generated_svf_from_matrix-250x125.png 250w\" sizes=\"auto, (max-width: 1008px) 100vw, 1008px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">(note outliers and y axis values).<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">In a second set of experiments we used instead a gaussian smoothed random generated SVF, as the one plotted below:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/svf_randomgen.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-790 size-full\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/svf_randomgen.png\" alt=\"svf_randomgen\" width=\"1008\" height=\"396\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/svf_randomgen.png 1008w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/svf_randomgen-300x118.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/svf_randomgen-768x302.png 768w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/svf_randomgen-250x98.png 250w\" sizes=\"auto, (max-width: 1008px) 100vw, 1008px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">with the following norm of the error for this particular case, where the ground truth has been considered the Euler Method:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/7_num_norm_random_svf.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-803\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/7_num_norm_random_svf-300x237.png\" alt=\"_7_num_norm_random_svf\" width=\"300\" height=\"237\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/7_num_norm_random_svf-300x237.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/7_num_norm_random_svf-250x198.png 250w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/7_num_norm_random_svf.png 637w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">(Midpoint method is very similar to the Euler one, this is why so small error).<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">For a dataset of 100 random generated SVF, considering different possible ground truth (not only the Euler method but any method) we have:<\/span><\/p>\n<p class=\"p1\"><a href=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-792 size-large\" src=\"http:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF-1024x477.png\" alt=\"boxplot_100_random_SVF\" width=\"700\" height=\"326\" srcset=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF-1024x477.png 1024w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF-300x140.png 300w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF-768x358.png 768w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF-250x116.png 250w, https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/wp-content\/uploads\/sites\/14\/2015\/10\/boxplot_100_random_SVF.png 1280w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/a><\/p>\n<p class=\"p1\"><span class=\"s1\">Now, the next steps are:<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">1) correct the value of the norm in the graphic the matrix computation (still debugging!)<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">2) understand the origin of the outliers, that appears consistently, with almost every time the same ratio over the number of tests (3 every 100).<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">3) go in deep of each method, theoretically.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">4) Add other methods as the Implicit Runge Kutta.<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">5) Test for different resampling strategies and see what are the improvements that can be made<\/span><\/p>\n<p class=\"p1\"><span class=\"s1\">6) Collect some example of SVF that possesses an analytic integration. Use the analytic diffeomorphisms to have a ground truth in these particular cases.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Meeting Minutes October 20, 2015 1. Attendees: Tom Vercauteren Sebastiano Ferraris Marc Modat 2. Discussed Topics: Numerical methods for the computation of the exponential of an SVF. (Details below) Next steps. Given an SVF over a domain \\Omega (in this<\/p>\n","protected":false},"author":48,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-779","post","type-post","status-publish","format-standard","hentry","category-electronic-journal"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>PhD Meeting Minute \u2014 Sebastiano Ferraris. \u2014 20151020 - GIFT-Surg Intranet<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/wp1.cs.ucl.ac.uk\/gift-surg-intranet\/2015\/10\/21\/phd-meeting-minute-sebastiano-ferraris-20151020\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"PhD Meeting Minute \u2014 Sebastiano Ferraris. \u2014 20151020 - GIFT-Surg Intranet\" \/>\n<meta property=\"og:description\" content=\"Meeting Minutes October 20, 2015 1. 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