Numerical techniques for the integral defined by a contour

[alitas]

Homework Statement

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


Homework Equations

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

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

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

 

[alitas]

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?.