Numerical techniques for the integral defined by a contour

In summary, the conversation discusses the doubts of the speaker on how to solve an integral and what numerical method to use for it. The integral is related to the position where the contour is equal to zero and obtaining the value of the control point at that position. The speaker also mentions a formula for integrating only where the contour of the level set is equal to zero. They then simplify the question to how to solve a discrete integral over an image and wonder if they can use any technique with this type of functions.
  • #1
alitas
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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
 
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  • #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?.
 

FAQ: Numerical techniques for the integral defined by a contour

What is the purpose of numerical techniques for the integral defined by a contour?

The purpose of numerical techniques for the integral defined by a contour is to approximate the value of a complex integral that cannot be solved using traditional methods. These techniques use numerical algorithms to evaluate the integral along a specified contour in the complex plane.

How do numerical techniques for the integral defined by a contour differ from traditional integration methods?

Numerical techniques for the integral defined by a contour involve evaluating the integral along a specific path in the complex plane, rather than using the fundamental theorem of calculus. This allows for the evaluation of complex integrals that cannot be solved analytically.

What are some common numerical techniques used for evaluating integrals defined by a contour?

Some common numerical techniques include the trapezoidal rule, Simpson's rule, and the Gaussian quadrature method. These methods use a combination of discrete points and weights to approximate the value of the integral along the contour.

How accurate are numerical techniques for the integral defined by a contour?

The accuracy of these techniques depends on the complexity of the contour and the chosen method. In general, the accuracy increases as the number of evaluation points along the contour increases. However, these techniques may still introduce some error, especially for highly oscillatory integrals.

Are there any limitations to using numerical techniques for the integral defined by a contour?

Yes, there are limitations to these techniques. They are most effective for integrals with smooth contours and well-behaved integrands. Highly oscillatory integrals or those with singularities may require specialized techniques for accurate evaluation.

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