Continuum Crowds

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.

High density

Flow speed is average velocity field.
Approach – Speed

Medium density

Interpolate between the topographical and flow speeds.
Approach - Density


Result & Demo Video Conclusion
Introduction

What is Crowds?

Large groups of people. Enormous complexity and subtlety.
Introduction

Crowds’ difficulty
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
Approach – Dynamic Potential Field Construction

1. 2. 3.
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
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
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”.

A Dynamic Potential Function

For any person, the optimal strategy is to move opposite the gradient of the this function Else satifies the equation:


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.
4.
5.
Approach

Then we can get each person’s position and speed.
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

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)

With the same identical speed field, discomfort, and goal. Calculate potential function is the slowest aspect of simulation. As few groups as possible.

So every person moves with the scaled speed
Approach – Optimal Path Computation

It need to calculate the potential function for the group only once


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

Visibility Proximity of other pedestrians Other local factors


Different simulation parameters may be defined for each member. But…
Related work (continue)

Speed is a density-dependent variable.


Convert each person into an individual density field. The average velocity field
Approach – Speed

Low density
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

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.



The terrain is bounded to lie within the minimum and maximum slopes & is the slope of the height field h in direction Topographical speed