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# Summary3 - strongest • We looked at 2D version in...

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12/1/2011 1 Summary After the second midterm … Calculus of Variation Integral functionals of the form 𝐽 𝑥 = 𝑓 𝑡, 𝑥 𝑡 , 𝑥 𝑡 𝑑𝑡 ? ? The input is a function 𝑥 = 𝑥(𝑡) from certain collection of “feasible” functions. For example, we may require 𝑥 ∈ 𝑓 𝑡 𝑓 ? = ?, 𝑓 ? = ?, 𝑓 ∈ 𝐶 2 [?, ?]}

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12/1/2011 2 Euler’s Equation To find the candidates for minimizing 𝐽(𝑥) among the feasible functions, CoV provides us Euler’s equation Idea: change it to a calculus problem (by an auxiliary variable to perturb the minimum function); integration by parts; applying the fundamental lemma of calculus of variation Fundamental Lemma Basic ideas: constructing “smoother” delta functions The three forms (from the weakest to the
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Unformatted text preview: strongest) • We looked at 2D version in homework/project • But only for rectangular regions 12/1/2011 3 Level Sets Method • Reviewed some basic differential geometry (tangent, normal, and curvature of curves in 2D) • Explicit representation using parametric equations (or vector-valued functions) • Implicit representation using the 0-level set of a surface (in a higher dimensional space) Curve Evolution • We used curve evolution for active contours method in image segmentation • Gave an alternative derivation to the evolution equation • Numerical implementation using central difference, upwind (when needed), … • Use signed distance function for the Level Set Function (LSF)...
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Summary3 - strongest • We looked at 2D version in...

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