Continuum Crowds

1
Continuum Crowds

• Adrien Treuille, Siggraph 2006
• 9557550???

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 persons 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
• Assigning 0 inside the goal and marked as KNOWN.
• Assigning all other cells and marked as
  UNKNOWN.
• Those UNKNWON cells adjacent to KNOWN cells are
  included in the list of CANDIDATE cells and
  approximate by solving Eq. (11)
• The CANDIDATE cell with the lowest potential is
  marked as KNOWN and its neighbors are marked as
  CANDIDATE and re-approximating the potential.
• Repeat 4

28
Approach

• Then we can get each persons 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.
• Its 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
QA

• Any Question?