6.2 And for 3D point clouds
6.2.2 Toward mean curvature flows for 3D point clouds
In this section, we test the evolution of a point cloud by the discrete mean curvature flow, xn+1k =xnk+dtH(xnk),
whereH(xnk) is the approximation of the mean curvature at the pointxnk given by formula (6.23) and dt is a prescribed time step. Of course, as this scheme is explicit, we expect that instabilities appear.
Let us begin with the flow of a ball of radius 1, with a large radius ε= 0.6 and a large enough time stepdt= 0.01. After 40 iterations, we obtain in Figure 6.10 a smaller ball of radius around 0.6 (which is coherent with the time step and the curvature of the ball).
As expected, we can see instabilities appearing after 40 iterations, and the point cloud is no longer a “ball” after 50 iterations. This corresponds to the time when the radius of the ball used for computing curvature is the same as the radius of the ball itself.
We now observe the effects of this mean curvature flow on the bunny of diameter around 7 constituted of N = 34835 points. We take a radius ε = 0.5 and a step time dt = 0.001. We can observe that after 120 iterations, the body of the bunny has been smoothed (see the back of the bunny which is wavy before the flow in Figure 6.11). We also understand why the flow crashes at that moment: the ears are thin and collapse after 120 iterations. The color corresponds to the intensity of the computed curvature.
Let us now end with an entertaining experiment illustrated in Figure 6.12: we let the bunny evolve in the same conditions, but by the reverse mean curvature flow,
xn+1k =xnk−dtH(xnk).
Let us conclude this chapter by mapping out some perspectives about the numerics related to the (direct) discrete mean curvature flow, whose stability issues have been mentioned above. A first aspect is to stabilize the approximation of the mean curvature itself, by changing our current approximation
(a)
(b) Details on the tail of the dragon
(c) Details on the head of the dragon
Figure 6.9: Intensity of the mean curvature of a dragon, the computations are done for ε= 0.02 for a dragon of diameter 1
(a) From 0 to 40 iterations (b) After 30 iterations
(c) After 40 iterations (d) After 50 iterations
Figure 6.10: Balls evolving by mean curvature flow,with radius ε= 0.6 and time step dt= 0.01.
(a) Time 0
(b) After 120 iterations
(c) Time 0 (d) After 120 iterations
(e) Time 0 (f) Time 50 (g) Time 100 (h) Time 120
Figure 6.11: Bunny evolving by mean curvature flow: global evolution and comparison after 120 152
Figure 6.12: Bunny after a reverse main curvature flow: After 340 iterations with a radius ε = 0.5 and a time step dt= 0.001
for the one proposed in (6.24) and by fixing the number of points used to the computation instead of fixing the radius of the ball used. Another aspect is the instability due to the explicit discretization in time: is it possible to design a reasonable implicit or semi-implicit scheme with our approximation of the mean curvature? This will be the purpose of future work.
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