Talk:Level set method

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Could somebody add a section that explains how to derive the level set equation? In Osher's book it is just given, as if it were obvious. But it's not obvious to me.Singularitarian 08:53, 11 July 2007 (UTC)

I guess you mean the Hamilton-Jacobi equation for the level set function:
\phi_t = v|\nabla \phi|.
This is easiest to check when φ is a linear function in one variable, φ(x) = mx + b. For m > 0, moving the x-intercept to the right by vΔt is the same thing as moving the line down by mvΔt, which then easily implies the above equation. The general case is done in the same way for each point x on the zero level set. Let me know if you'd like more details. Oleg Alexandrov (talk) 14:46, 11 July 2007 (UTC)

Well, I hope this is what you want. Supose that

\Gamma(t=0) : \quad \phi(\vec{r},t=0) = 0

is the curve you have at t = 0. So, the level set method say that this identity holds for any time, so there's φ that satisfy

\Gamma(t) : \, \, \phi(\vec{r},t) = 0

for every t in your domain. Take its derivatives with respect to time,

\phi_t + \nabla \phi (\vec{r}) \cdot \dot{\vec{r}} (t) = 0 ,

and then you get your Level Set equation. Hamilton-Jacobi equation is a particularly case of this: if the \nabla \phi(\vec{r}) has no second order derivatives in space, so we have the the Hamilton-Jacobi equation. (And I hope my English don´t confuse you too much) Do you work with this method? Thschiavo 13:24, 27 July 2007 (UTC)

[edit] Sign convention?

Both of Osher's and Sethian's books (given in the references section) give level set function values as negative inside the curve and positive outside. Osher does so explicitly on page 5 (a circle shown with phi < 0 inside and phi > 0 outside) whereas Sethian's plots on page 8 do so implicitly. I would argue that their sign convention trumps whatever was used as the basis for this article, but I don't want to be the one who has to recreate all of the pretty graphs if a change is made. WokYai 17:56, 16 October 2007 (UTC)