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Say I have a set of sample measurements of a one dimensional curve in the image plane $f(x)$. What is the purpose of minimizing the following functional $$E(S) = \int \left[\lambda(S''(x))^2 + (f(x) - S(x))^2 \sum_k \delta (x - x_k)\right]dx$$

I believe it has something to do with energy minimization and snakes.

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    $\begingroup$ Perhaps you might like to edit your question to include where you ran across this expression? $\endgroup$ – D.W. Oct 13 '14 at 8:36
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I would agree - energy minimization for snakes (active contours).

It minimizes second derivative - bending of the curve.

Minimizes the difference between the image (probably it's edges) and contour (at discreet contour points - the reason for having Dirac impulse in the expression).

Usually one could expect also to have minimization of the first derivative - stretching of the snake.

More details and references on wiki - active contour

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