Continuum Crowds

Adrien Treuille, Siggraph 2006

9557550王上文

Outline

Introduction Related work Approach

The Governing Equations Optimal Path Computation Speed & Density Dynamic Potential Field Approximation & approximation

Result & Demo Video Conclusion

Introduction

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

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.

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…

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.

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

Approach - Overview

Program flowchart

Approach – The Governing Equations Maximum Speed Field f

People move at the maximum speed possible.

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”.

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

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)

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

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.

Approach – Speed & Density

Speed is a density-dependent variable. A crowd density field Slow speed with high density High speed with low density

Approach – Speed

Speed is a density-dependent variable. Convert each person into an individual density

field. The average velocity field

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

Approach – Speed

High density Flow speed is average velocity field.

Approach – Speed

Medium density Interpolate between the topographical and flow sp

eeds.

Approach - Density

How to get density to compute the speed field? Splat the crowd particles onto a density grid

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.

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

Approach – Density & Speed

With the density field, we can compute maximum speed field f.

So we can calculate the unit cost field C

Approach – The Algorithm

Approach – Dynamic Potential Field Approximation Dynamic Potential Field Approximation

Solve Equation (5) to get potential field is expensive

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)

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

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

Result

Demo

Demo Video

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.

Conclusion

Advantages Can capture a number of emergent phenomena.

Lane formation Short lived vortices during turbulent congestion.

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.

Q&A

Any Question?