1. Continuum Crowds Adrien Treuille, Siggraph 2006 9557550 王上文

2. Outline Introduction Related work Approach  The Governing Equations  Optimal Path Computation  Speed & Density  Dynamic Potential Field Approximation & approximation Result & Demo Video Conclusion

3. Introduction What is Crowds?  Large groups of people.  Enormous complexity and subtlety.

4. Introduction Crowds’ difficulty  Computation Environmental constraints. Dynamic interactions between people. Intelligent path planning.  The characteristic of dense crowds  Real-time crowd simulation is difficult due to large computation.

5. Related work Most previous work has been “agent-based”  Motion is computed separately for each individual.  It can capture each person’s unique situation. Visibility Proximity of other pedestrians Other local factors  Different simulation parameters may be defined for each member.  But…

6. Related work (continue) The agent-based approach has some drawbacks.  Difficult to consistently produce realistic motion.  Global path planning for each agent expensive. Most models separate local collision avoidance from global path planning. Conflicts arise.

7. Approach - Overview A dynamic potential field model  Optimal Path Computation  Density & Speed Computation  The Governing Equations Maximum Speed Field Discomfort Field Unit Cost Field  Discretized grid structure Density conversion Unit cost computation Dynamic Potential Field Construction

8. Approach - Overview Program flowchart

9. Approach – The Governing Equations Maximum Speed Field f  People move at the maximum speed possible.

10. Approach – The Governing Equations Discomfort Field  People generally follow trodden paths when they exist.  People do not cross a street until they reach a crosswalk  Achieving these by assuming a “discomfort field”.

11. Approach – The Governing Equations Unit Cost Field  Choose paths as to minimize a linear combination of the following three terms. The length of the path The amount of time to the destination The discomfort felt, per unit time, along the path

12. Approach – The Governing Equations Unit Cost Field (Continued)  Equation (2) can be rewritten as Eq(3)  Then Eq(3) can be simplified to Eq(4)

13. Approach – Optimal Path Computation A Dynamic Potential Function  For any person, the optimal strategy is to move opposite the gradient of the this function   Else satifies the equation:  So every person moves with the scaled speed

14. Approach – Optimal Path Computation It need to calculate the potential function for the group only once  With the same identical speed field, discomfort, and goal.  Calculate potential function is the slowest aspect of simulation.  As few groups as possible.

15. Approach – Speed & Density Speed is a density-dependent variable.  A crowd density field  Slow speed with high density  High speed with low density

16. Approach – Speed Speed is a density-dependent variable.  Convert each person into an individual density field.  The average velocity field

17. Approach – Speed Low density  The terrain is bounded to lie within the minimum and maximum slopes &  is the slope of the height field h in direction  Topographical speed

18. Approach – Speed High density  Flow speed  is average velocity field.

19. Approach – Speed Medium density  Interpolate between the topographical and flow speeds.

20. Approach - Density How to get density to compute the speed field?  Splat the crowd particles onto a density grid

21. Approach - Density two requirements of the density conversion function  The density field must be continuous. Could be satisfied by any number of splatting technique, including Bilinear and Gaussian  Each person should contribute no less than to their own grid cell and no more than to any neighboring grid cell.  is a threshold.

22. Approach - Density In order to satisfy the second requirement  The density is then added to the grid as  The density exponent determines the speed of density falloff.  Then each person contributes at least to their grid cell, but no more than to neighboring cells, with

23. Approach – Density & Speed With the density field, we can compute maximum speed field f. So we can calculate the unit cost field C

24. Approach – The Algorithm

25. Approach – Dynamic Potential Field Approximation Dynamic Potential Field Approximation  Solve Equation (5) to get potential field is expensive

26. Approach – Dynamic Potential Field Approximation First find the less costly adjacent grid cell along the both x- and y-axes Then use these upwind directions to calculate a finite difference approximation to Equation (5)

27. Approach – Dynamic Potential Field Construction Algorithm 1. Assigning 0 inside the goal and marked as KNOWN. 2. Assigning all other cells and marked as UNKNOWN. 3. Those UNKNWON cells adjacent to KNOWN cells are included in the list of CANDIDATE cells and approximate by solving Eq. (11) 4. The CANDIDATE cell with the lowest potential is marked as KNOWN and its neighbors are marked as CANDIDATE and re-approximating the potential. 5. Repeat 4

28. Approach Then we can get each person’s position and speed.  Maximum speed field f From density field  Potential field From unit cost field C From maximum speed field f

29. Result

30. Demo Demo Video

31. Conclusion Advantages  The individuals do not face conflicting.  Smoother motion than previous methods.  It’s possible to integrate this model with agent models. The moving cars and the UFO in demo are all agents.

32. Conclusion Advantages  Can capture a number of emergent phenomena. Lane formation Short lived vortices during turbulent congestion.

33. Conclusion Disadvantage  Not feasible for real crowds in unknown environment. It assume people really know the dynamic properties of the environment.  It change direction without respect to inertia. Can be solved, but it would not be real-time.  Without the flexibility and individual variability of the full agent-based approach. Can be solved by adding some agents.

34. Q&A Any Question?