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Unformatted text preview: Variational Method for Image Analysis Derivation of the Euler Equations (11/2010) We derive the necessary condition for the minimization of an integral functional of the form: I [ y ] := Z b a F ( x,y,y ) dx, y ( a ) = y y ( b ) = y 1 Recall that F is a given function of three variables and y is the unknown function to be determined so that the functional I [ y ] is minimized. 1 First Variation We want to use some ideas from Calculus to find a necessary condition on a function y min such that y min ( a ) = y , y min ( b ) = y 1 , and I [ y min ] I [ y ] for all y satisfying y ( a ) = y and y ( b ) = y 1 . Let us rephrase this extremal property of y min : For any and any y satisfying y ( a ) = y and y ( b ) = y 1 , I [ y min ] I [ y min + ( y- y min )] . Now, let y min and y be fixed and only consider as the independent variable. That is, consider the function of one variable f ( ) := I [ y min + ( y- y min )] . Then f has a minimum when = 0. So, it is necessary 1 that d d f ( ) fl fl fl fl =0 = 0 . Definition . We call I [ y min ] := d d f ( ) fl fl fl fl =0 the first variation of I at y min . Theorem . A necessary condition for y min to yield the minimum of I is: I [ y min ] = 0 . (1) 1 Of course, we need to assume that f ( ) is differentiable in a neighborhood of 0 1 2 Euler Equation We need to make the necessary condition more useful, that is, we need to work out the condition so that it gives us an equation for y min to satisfy. We have d d f ( ) = d d Z b a F ( x,y min + ( y- y min ) ,y min + ( y- y min ) ) dx = Z b a d d F ( x,y min + ( y- y min ) ,y min + ( y- y min ) ) dx = Z b a F y ( x,y min + ( y- y min ) ,y min...
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- Fall '11