User:Ricktest

Level Sets

Rough idea: find the function phi, that's all you need to do

phi(x, y, t)

Imaging phi as a lumpy function, like hills on the x-y plane. (Elevation of phi.)

The lumpy function changes as a function of time.

phi(x, y, t) = 0 Where this statement is true defines a contour (or contours) that specify the contours that you are looking for.

There are infinitely many points on the expanding front, all of them in a set describe the contour or contours that form the front.

Choose any point on the front and call it x, y. The point moves as time passes so call it x(t), y(t). You can have a situation where lumps end up forming contours that merge or split but I'm going to ignore that.

phi(x(t), y(t), t) = 0 is true only at the expanding front and nowhere else

sethian then says: phi_t + F (phi_x^2 + phi_x^2)^(1/2) = 0 this would only be guaranteed to be true where the front is but, i think sethian is making an assumption that this is true everywhere

phi(x, y, t) is defined everywhere, at all x, y points for all time (time starts at zero and gets higher)

it might be ok to say you can pick a point anywhere and it will be on a front, as long as the time is chosen carefully

phi(parameterx, parametery, t) parameterx, parametery, and t can be anything if you have functions x(t) and y(t) you can use them to describe the motion of a particle on the front

when you talk about phi_parameterx, that's varying parameterx, not x(t)

when you talk about phi_parametery, that's varying parametery, not y(t)

when you talk about phi_t, that's varying t and you don't need to vary parameterx or parametery

imagine the bouy as a point dropped randomly onto the table. the table has a speed defined everywhere. if the table had a velocity that would be what i was thinking, so lets just say for now the table has a velocity defined everywhere. the little particle will move according to the velocity. if x(t) and y(t) give the location of the particle, then it does make sense to say that phi(x(t), y(t), t) = 0 for all t under consideration

the magic is in the chain rule. the chain rule takes things like x(t) and y(t) and ends up giving you partial derivatives. you probably need to look up the derivation of the chain rule to really see how this stuff works.

n is the outward normal of the expanding front

n = del phi / | del phi |

why is this true

probably it is true only because you claim that it is

this actually defines phi