Ur code is really helpful. May I know how the 3D SFM works?

Hi, thank you for sharing this code, it’s works for what I want to do. but it hard for me to understand all of the mathematical equations on your paper “Localized Region Based Active Contours”(code:Chan&Vese code); I wondering if you have any link where can I find the explication or more details?. if you have something, this is my @mail: kasmireda@gmail.com.

thanks and regards
Reda

Let’s check my claim on an extreme case, where your method should have a better advantage over mine. The new F would be zero for all Lz. I would iterate once over Ln and Lx clearing all cells value. Then I would iterate Lz again, and rediscover all the neighbors. It would mean that if we have N cells in the the active, I would have 2N accesses. You on the hand would have only N access.
Considering the other extreme case that the whole Lz moved, you would need to relocate all your items through Sx. So you’ll spend 3N, while I still spend 2N. Then we should consider the amortized average case, although if you normalize the force as you do, the level set should change quite often.
Well, I admit my examples were a bit sketchy, but I hope that you get the idea, that we access the memory, and move the list points quite in the same amount, and I don’t think it would matter much in terms of performance.

Why my solution is cleaner: The code is shorter and simpler, and it’s really easy to track the changes, and prove that the distance invariant is kept (cell in level i has it value in the allowed distance range), as opposed to the one example I gave you that your invariant can fail (I noticed it also in my experiments).

The approach you lay out may be faster in some situations (e.g., where there are few voxels in Lz). In the case where the number of voxels gets large (and only a sub-set of those voxels cross the zero-level-set on each iteration) the method presented here will be MUCH faster.

In terms of invariant distances, it’s important to ensure floating-point accuracy on the phi function to ensure sub-voxel accuracy of the surface.

1. Calculate F for Lz.
2. Go over all the other lists Ln[], Lp[], and delete them, while setting the cells they were pointing to to the highest label (+/-3 in your case).
3. Iterate Lz:
3.1 Update the cell value according to F.
3.2 If the value is still in Lz interval, add the neighbors with label > 0 to Lp1 with a value of the Lz cell + 1, and neighbors with label i
to Lp(i+1). Update the value of all neighbors with label i+1 to Lpi’s value + 1.

It would replace your 2,3 procedures, won’t need the intermediate Sx lists, would be faster, and it would be cleaner, in the sense that it would keep the distance invariant.

Which means according to Sethian that your method should be very stable (you also preserve the CFL condition by normalizing the force, and limiting it to 0.5), but at a price of precision.

Did you try a fourth order flow such as the Laplacian curvature of Willmore flows?

So it probably my mistake, since I’m not sure what is the invariant the algorithm keeps in each level. For example I assumed the algorithm tries to preserve for each level its data range. For example Lp2 should always have values between (1.5,2.5]. But according to line 7 at procedure 2, p would still have a Lp1 value and it would be transferred to Lp2. I find it hard to follow all the cases, and see if this is crucial or not, but currently I admit that the algorithm behaves well, and my previous impression was due to a bug in my implementation. Maybe the worst that could happen are orphan cells (as implied by [Whitaker98], or wrong approximation of the distance function at higher levels. I would appreciate your opinion on the matter.

Thanks,
Zohar