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Numerical techniques for the integral defined by a contour

  1. Aug 14, 2012 #1
    1. The problem statement, all variables and given/known data
    I have a doubt about how to solve an integral and what numerical method I could use to solve it.

    Example

    A = [1,2;2,4]
    Contour = [0,0,1,1]
    r = I-mean;

    integral of the position where the contour is equal zero and obtaining from that positon the value Ai which is the value of the control point at that position


    ∫σ==0 Ai*r ds


    or the original that would be
    ∇E(λ) = ∫{Φλ=0} ϕi(s)r(s)/||∇Φλ(s)||ds

    where Φ is the contour and ϕ interpolated values
    

    2. Relevant equations

    ∫σ==0 interpolation in control points*r ds

    integrate only where the contour of the level set (active contour) is equal 0.

    3. The attempt at a solution

    summatory of the points of the image that are equal 0 following the formula but it does not work. So I wonder how to make an integral of a interpolation and if I can use any technique with this kind of functions.

    Thanks
     
    Last edited: Aug 14, 2012
  2. jcsd
  3. Aug 15, 2012 #2
    Let's simplify the question, I hope it is simple to understand. how would you solve a discrete integral over an image. So I have discrete points over the image and I am just interested in a subset. If it was a polynomial function it would be straightforward but how do you deal with the width between the points to use for example the trapezoidal method?.
     
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