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2014 | OriginalPaper | Chapter

2. Background Preliminaries

Authors : Amar Mitiche, J.K Aggarwal

Published in: Computer Vision Analysis of Image Motion by Variational Methods

Publisher: Springer International Publishing

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Abstract

In this preliminary chapter we will give definitions, descriptions, and formulas, concerning curvature, Euler-Lagrange equations, unconstrained descent optimization, and level sets, all fundamental topics and tools underlying the variational methods of motion analysis described in the subsequent chapters.

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previous chapter Image Motion Processing in Visual Function

next chapter Optical Flow Estimation

Note that we could have defined the unit normal of the implicit curve as \(\mathbf{n }=-\nabla \phi {/}\Vert \nabla \phi \Vert \) instead of \(\mathbf{n }=\nabla \phi {/}\Vert \nabla \phi \Vert \) as in Eq. (2.9), in which case curvature would change sign, i.e., it would have the expression in Eq. (2.20) but without the minus sign. At the same time it would be written \(\kappa = \mathrm{div} \left( \nabla \phi {/}\Vert \nabla \phi \Vert \right) \) rather than with the minus sign as in Eq. (2.21).

The discussions apply as well to the maximization of similar functionals.

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Title: Background Preliminaries
Authors: Amar Mitiche
J.K Aggarwal
Publisher: Springer International Publishing
Book: Computer Vision Analysis of Image Motion by Variational Methods
Print ISBN: 978-3-319-00710-6

Electronic ISBN: 978-3-319-00711-3

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-00711-3_2