{"version":1,"kind":"Article","sha256":"","slug":"756","location":"","dependencies":[],"doi":"10.54294/t5q4cl","frontmatter":{"title":"Itk::Transforms supporting spatial derivatives","abstract":"This document describes the use and implementation of first and second order spatial derivatives of coordinate transformations in the Insight Toolkit url{www.itk.org}). Spatial derivatives are useful for many types of regularising or penalty terms frequently used in image registration. These derivatives are dubbed 'SpatialJacobian' and 'SpatialHessian' to distinguish with the derivative to the transformation parameters themselves, which is called `Jacobian' in the ITK.\n\nIn addition to the spatial derivatives, we derived and implemented the derivatives to the registration/transform parameters of these spatial derivatives, required for gradient descent type optimisation routines. These derivatives are implemented in a sparse manner, reducing the computation time for transformations which have local support. All of these derivatives are implemented for the most common ITK coordinate transformation, such as the rigid, affine and B-spline transformation. In addition we derive formulae and code for arbitrary compositions of transformations. The spatial derivatives were subsequently exploited by implementing the bending energy penalty term.\n\nThis paper is accompanied with the source code, input data, parameters and output data that the authors used for validating the algorithm described in this paper. This adheres to the fundamental principle that scientific publications must facilitate reproducibility of the reported results.\n","license":"You are licensing your work to Kitware Inc. under the\nCreative Commons Attribution License Version 3.0.\n\nKitware Inc. agrees to the following:\n\nKitware is free\n * to copy, distribute, display, and perform the work\n * to make derivative works\n * to make commercial use of the work\n\nUnder the following conditions:\n\\\"by Attribution\\\" - Kitware must attribute the work in the manner specified by the author or licensor.\n\n * For any reuse or distribution, they must make clear to others the license terms of this work.\n * Any of these conditions can be waived if they get permission from the copyright holder.\n\nYour fair use and other rights are in no way affected by the above.\n\nThis is a human-readable summary of the Legal Code (the full license) available at\nhttp://creativecommons.org/licenses/by/3.0/legalcode","keywords":["image registration","regulizers","penalty terms","spatial derivatives"],"authors":[{"name":"Staring, Marius","email":"marius.staring@gmail.com","affiliations":["Leiden University Medical Center"],"corresponding":true},{"name":"Klein, Stefan","email":"s.klein@erasmusmc.nl","affiliations":[]}],"date_submitted":"2010-09-08 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