In 1996, James Sethian showed a remarkably clean idea. You can get shortest routes through a messy world by letting a wave expand once. No trial paths. No scanning beams. Just one growing front. You compute an arrival time field T(x,y). At each point, T means how long the front needs to reach that point. The rule is ‖∇T‖ = 1/F. If the medium is fast, meaning F is large, the front moves quickly and T grows slowly. If the medium is slow, meaning F is small, the front moves slowly and T grows quickly. Obstacles act like F is near zero, so the front cannot pass through and instead wraps around. Then you get the path without searching. Once T exists, you pick any start point and follow xdot proportional to minus grad T. You slide downhill on the time landscape and trace a fastest route back to the source. This is the Hamilton Jacobi and optimal control view that Tsitsiklis (1995) made precise. Compute the value or arrival time function first, and the optimal trajectories follow from it. #FastMarching #EikonalEquation #HamiltonJacobi #OptimalControl #ShortestPath #ComputationalGeometry

In 1996, James Sethian showed something almost unfair...you can find shortest routes through a messy world by letting a wave expand once...no trial paths, no search beams, just one growing front. Here’s how: we solve for an arrival-time field T(x,y) so that T literally means how long the wave needs to reach this point. The rule is ||∇ T|| = 1/F, where the medium is fast (F large) the front sprints, where it’s slow it trudges, and obstacles are speed ≈ 0, so the front wraps around them because that’s the only way forward. Then comes the satisfying part: once T exists, a path doesn’t need to search at all...drop a bead anywhere and let it follow ẋ ∝ -∇ T, it slides downhill on the time landscape and traces a globally fastest route back to the source. This “wave = optimal control” viewpoint is exactly what Tsitsiklis (1995) made precise from the Hamilton-Jacobi side...compute the value/arrival-time function and the optimal trajectories fall out from it. #FastMarching #EikonalEquation

Were they in a hurry to get home for the footie.... #fastmarching

Comvoissi le collègue André kil a xepté le masking😷 pour aller faire le fastmarching demain à Magne avec ManZa. Lechebottt. 😾