[Fwd: Re: [Insight-users] Level Set / Fast Marching explanations]

[Chris McIntosh]

Hello,

gatts gatsu wrote:
> Hi all,
>
> I would like to have some details on the way it works because I don't 
> fully understand it.

Best bet is to read a few key papers:

http://www.iua.upf.es/~vcaselles/papers_v/GAC_article.pdf
http://www.math.ucla.edu/~lvese/PAPERS/IEEEIP2001.pdf

>
> First, I was wondering what the "speed" of an image refers to. What do 
> you call the speed of an image?

The velocity at which the wave front (surface or contour) is moving at 
each grid location.

>
> On the figure 9.14 of the itksoftwareguide, numbers are written next 
> to the level set surface gamma. I guess that those are the distance 
> between gamma and the pixel next to it. However, how is it calculated? 
> For instance, if you pick the pixel at the bottom with the value 0.4 
> and the one at the left is -0.4. Why is the first one positive and the 
> second one negative ? They both seem to be outside the real gamma. I 
> don't figure out how it is computed.


It is a signed distance transform. Negative values are inside, positive 
values are outside (or vice versa).

Each pixel's intensity value represents its distance from the surface 
(the zero level set).

The surface is the zero level set. i.e. where the image you refer to is 
equal to all zeros.

Adding one to the entire image ( a speed image with all values equal to 
one) would change all zeros to one, and all -l's to zero.
So where there were -1's, there would now be a surface.


>
> I tried to look at this following url to understand better : 
> http://math.berkeley.edu/~sethian/2006/Explanations/level_set_explain.html 
>
> If you look at the xample with the cone-shaped surface and the circle. 
> The starting front is a circle. The fronts at the next steps are the 
> intersection between the surface of the cone-shaped surface and the 
> starting front plane.  It seems that in the next steps, the front will 
> still be a circle. Why is it keeping the same form if the algorithm 
> allows us to get random forms ?

Not random forms, forms driven by the image and internal smoothness forces.

>
> I am sorry to ask you those stupid questions but I would really 
> appreciate if someone could give some explanations.

No worries. I'd suggest googling around for power-point presentations, 
or checking out a good book (like those listed on Sethian's
website).

Cheers,

Chris

>
> Thanks a lot,
> Regards,
>
> Philippe
>
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