Euler lagrange equations for a curve integral

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I'm reading through this paper, where a variational framework to some computer vision feature detection techniques is given. I'm familiar with variational techniques, though I'm not an expert. There's the following integral (formula (9) on the paper)



$$
E(vecn) = oint_0^L sign(langle v, nabla I rangle) langle nabla I, vecn rangle ds
$$



It is claimed that the corresponding Euler-Lagrange equation is given by



$$
sign(langle v, nabla I rangle) Delta I = 0
$$



Where $Delta $ is the laplacian operator.
In the above integral $I$ is a 2D image, $vecn$ is the normal to some curve $gamma$ that we want to find, $v$ is vector orthogonal to the level set.



I'm quite confused how the Euler lagrange equations might be derived, and I don't actually know where I can start. Either an explanation or just a clue would be useful,



thank you.




integration optimization calculus-of-variations computer-vision




share|cite|improve this question



















  • Did you read the paper? It is clear and sound in the derivation of the EL equations! What the step you miss?
    – Cesareo
    Jul 28 at 10:30











  • I might be blind, but where exactly is the derivation of the equation above?
    – user8469759
    Jul 29 at 10:43










  • Unfortunately in the paper, the equations are not numbered. The above equations are a consequence of considering $rho(alpha) = |alpha|$ (half left column, third page)
    – Cesareo
    Jul 29 at 11:00











  • So the derivation is in section II ?
    – user8469759
    Jul 29 at 11:40










  • Yes. It is in section II
    – Cesareo
    Jul 29 at 11:54














up vote
1
down vote

favorite
2












I'm reading through this paper, where a variational framework to some computer vision feature detection techniques is given. I'm familiar with variational techniques, though I'm not an expert. There's the following integral (formula (9) on the paper)



$$
E(vecn) = oint_0^L sign(langle v, nabla I rangle) langle nabla I, vecn rangle ds
$$



It is claimed that the corresponding Euler-Lagrange equation is given by



$$
sign(langle v, nabla I rangle) Delta I = 0
$$



Where $Delta $ is the laplacian operator.
In the above integral $I$ is a 2D image, $vecn$ is the normal to some curve $gamma$ that we want to find, $v$ is vector orthogonal to the level set.



I'm quite confused how the Euler lagrange equations might be derived, and I don't actually know where I can start. Either an explanation or just a clue would be useful,



thank you.




integration optimization calculus-of-variations computer-vision




share|cite|improve this question



















  • Did you read the paper? It is clear and sound in the derivation of the EL equations! What the step you miss?
    – Cesareo
    Jul 28 at 10:30











  • I might be blind, but where exactly is the derivation of the equation above?
    – user8469759
    Jul 29 at 10:43










  • Unfortunately in the paper, the equations are not numbered. The above equations are a consequence of considering $rho(alpha) = |alpha|$ (half left column, third page)
    – Cesareo
    Jul 29 at 11:00











  • So the derivation is in section II ?
    – user8469759
    Jul 29 at 11:40










  • Yes. It is in section II
    – Cesareo
    Jul 29 at 11:54












up vote
1
down vote

favorite
2









up vote
1
down vote

favorite
2






2





I'm reading through this paper, where a variational framework to some computer vision feature detection techniques is given. I'm familiar with variational techniques, though I'm not an expert. There's the following integral (formula (9) on the paper)



$$
E(vecn) = oint_0^L sign(langle v, nabla I rangle) langle nabla I, vecn rangle ds
$$



It is claimed that the corresponding Euler-Lagrange equation is given by



$$
sign(langle v, nabla I rangle) Delta I = 0
$$



Where $Delta $ is the laplacian operator.
In the above integral $I$ is a 2D image, $vecn$ is the normal to some curve $gamma$ that we want to find, $v$ is vector orthogonal to the level set.



I'm quite confused how the Euler lagrange equations might be derived, and I don't actually know where I can start. Either an explanation or just a clue would be useful,



thank you.




integration optimization calculus-of-variations computer-vision




share|cite|improve this question











I'm reading through this paper, where a variational framework to some computer vision feature detection techniques is given. I'm familiar with variational techniques, though I'm not an expert. There's the following integral (formula (9) on the paper)



$$
E(vecn) = oint_0^L sign(langle v, nabla I rangle) langle nabla I, vecn rangle ds
$$



It is claimed that the corresponding Euler-Lagrange equation is given by



$$
sign(langle v, nabla I rangle) Delta I = 0
$$



Where $Delta $ is the laplacian operator.
In the above integral $I$ is a 2D image, $vecn$ is the normal to some curve $gamma$ that we want to find, $v$ is vector orthogonal to the level set.



I'm quite confused how the Euler lagrange equations might be derived, and I don't actually know where I can start. Either an explanation or just a clue would be useful,



thank you.





integration optimization calculus-of-variations computer-vision





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 26 at 10:54









user8469759

1,4271513




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  • Did you read the paper? It is clear and sound in the derivation of the EL equations! What the step you miss?
    – Cesareo
    Jul 28 at 10:30











  • I might be blind, but where exactly is the derivation of the equation above?
    – user8469759
    Jul 29 at 10:43










  • Unfortunately in the paper, the equations are not numbered. The above equations are a consequence of considering $rho(alpha) = |alpha|$ (half left column, third page)
    – Cesareo
    Jul 29 at 11:00











  • So the derivation is in section II ?
    – user8469759
    Jul 29 at 11:40










  • Yes. It is in section II
    – Cesareo
    Jul 29 at 11:54
















  • Did you read the paper? It is clear and sound in the derivation of the EL equations! What the step you miss?
    – Cesareo
    Jul 28 at 10:30











  • I might be blind, but where exactly is the derivation of the equation above?
    – user8469759
    Jul 29 at 10:43










  • Unfortunately in the paper, the equations are not numbered. The above equations are a consequence of considering $rho(alpha) = |alpha|$ (half left column, third page)
    – Cesareo
    Jul 29 at 11:00











  • So the derivation is in section II ?
    – user8469759
    Jul 29 at 11:40










  • Yes. It is in section II
    – Cesareo
    Jul 29 at 11:54















Did you read the paper? It is clear and sound in the derivation of the EL equations! What the step you miss?
– Cesareo
Jul 28 at 10:30





Did you read the paper? It is clear and sound in the derivation of the EL equations! What the step you miss?
– Cesareo
Jul 28 at 10:30













I might be blind, but where exactly is the derivation of the equation above?
– user8469759
Jul 29 at 10:43




I might be blind, but where exactly is the derivation of the equation above?
– user8469759
Jul 29 at 10:43












Unfortunately in the paper, the equations are not numbered. The above equations are a consequence of considering $rho(alpha) = |alpha|$ (half left column, third page)
– Cesareo
Jul 29 at 11:00





Unfortunately in the paper, the equations are not numbered. The above equations are a consequence of considering $rho(alpha) = |alpha|$ (half left column, third page)
– Cesareo
Jul 29 at 11:00













So the derivation is in section II ?
– user8469759
Jul 29 at 11:40




So the derivation is in section II ?
– user8469759
Jul 29 at 11:40












Yes. It is in section II
– Cesareo
Jul 29 at 11:54




Yes. It is in section II
– Cesareo
Jul 29 at 11:54















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