Crowd Equations

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CROWD MODEL VERIFICATION USING VIDEO DATA By
Thomas L. Clarke, D. J. Kaup, Linda Malone, Rex
Oleson, and Mario Rosa Institute for Simulation
and Training, Mathematics Department, Department
of Industrial Engineering and Management Universit
y of Central Florida, 4300 Technology Parkway,
Orlando, FL 32826
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Outline

• Purpose
• Crowd Model
• Recorded Videos
• Capturing Video Data
• Luca-Kanade Algorithm
• Optical Flow to Velocity
• Comparison to Hand Count Data

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Purpose

• A crowd could be any collection of living
  beings,
• traffic flow, etc.
• How good is any given crowd simulation?
• How well do the motions and actions observed,
• model actual systems?
• Given people crowds, videos are best for
• obtaining raw data.
• How to turn video data into quantitative data?
• How to compare video data and simulation data?

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Crowd Model Verification

• Desire for automated techniques to extract
  validation data from real world videos

• Need quantitative techniques to compare real
  world videos with simulation outputs.

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Equations of the model Social forces

Repulsion
Attraction to exit(s)
Preferred speed
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4/2/2015
UCF IST - CECS
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Equations of the model Physical forces

Pushing and Friction (contact
forces)

• Note
• Physical forces do not depend on
• relative orientation of pedestrians
• - By themselves, the pushing forces
• do NOT prevent pedestrians from
• walking through each other !

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4/2/2015
UCF IST - CECS
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Example of HMFV simulation of crowd exiting
Citrus Bowl football stadium in Orlando, Florida.
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Example video frame of crowd exiting UCF-Tulane
game at Citrus Bowl, Orlando Florida, Oct 22,
2005.
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Relating Optical Flow to Crowd Motion
Assumptions (naïve case) 1. All motion due to
crowd movement 2. 3. Reflectivity and
illumination of video elements is
constant.
Physical space
Image space
Related by simple scaling
Assumptions 2 and 3 may be relaxed at cost of
adding complexity to equations
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Optical Flow and Physical Parameters
The ideal relation of the ratio between optical
flow (apparent velocity in image) and physical
velocity
where fs is the frame rate of the video. M
was units of pixels/meter.
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Example of averaged optical flow of crowd exiting
UCF-Tulane game at Citrus Bowl, Orlando Florida
Oct 22, 2005.
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Lucas - Kanade Algorithm
where Ix, and Iy are partial derivatives of image
intensity, It is usually assumed zero. Higher
order terms are ignored.
Optical flow is assumed locally constant in time
so that near (xI, yI)
For a processing window of size mxm pixels, there
are m2 equations in unknowns Vx and Vy.
Lucas-Kanade solves this overdetermined system by
a pseudo-inverse technique.
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Lucas - Kanade in Crowds

• Optical flow assumption only good when the
  neighborhood is entirely inside the image of a
  person.
• When the neighborhood is on the edge of a
  person, the Lucas-Kanade relation for Vx and Vy
  holds with probability p and for zero velocity
  with probability (1 p). (The probability is due
  to using edge detection.)
• Thus the Lucas-Kanade estimate will be pVx and
  pVy .
• So the Lucas-Kanade estimate will be weighted by
  the apparent area of the moving individuals.

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Relating Flow to Velocity
Thus, the physical velocity of individuals in the
crowd relates to optical flow via
where Ap is the apparent area of individuals, Nk
is the number of individuals in the Lx by Ly
neighborhood. Vx and Vy , vx and vy are the
optical flow and physical velocity respectively.
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Dark box MC box
Optical flow count requires doubling.
which results in
Final views
Overlapping grids used to compare optical flow to
hand counted fluxes.
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Confirmation by Hand Count
The video frames were divided into cells or
neighborhoods as the previous figure suggests.
Individuals entering and leaving each cell were
counted by hand while viewing the video. With
the notation for individuals crossing boundaries
suggested in the figure, the change of number of
individuals in cell k from time t to t?t is
The fluxes across the cell edges are related to
physical velocity by
where Nk(k-) denotes the number individuals in
the cell above(below) k and Nk(-k) to the
right(left). Lx and Ly are needed for
dimensional reasons but can also be thought of as
describing a queue of length Lx or Ly within the
cell.
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Hand count flux data, MC-Comb (manual count
combined), and optical flow data, OF-Comb
(optical flow combined), frames 710 to 1260
Citrus bowl video as organized in a spread sheet.
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Hand Count to Flow Relation
Combining the velocity/optical-flow and
velocity/hand-count-flux relations, gives the
optical-flow/hand-count-flux relations
It is significant that the optical flow does not
depend on the number of individuals Nk except
through the fluxes. If the cells are square,
only a single constant C is needed
The constant C is calculated from density on the
ground. From the viewpoint of a fixed camera,
Lx and Ly will be proportional to range R so that
calibration at range R1 can be transferred to
range R2 though scaling by R1 / R2.
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Combined horizontal and vertical direction hand
count flux data plotted versus optical flow data
frames 710 to 1260 Citrus bowl video
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Table 1. Results from Citrus Bowl.
Central All Edges
C 19.0 20.4
? 0.79 0.53
N 5.21 5.21
C is coefficient relating hand-count-flux to
optical-flow. ? is correlation coefficient. N is
average number of individuals in cells.
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Combined horizontal and vertical direction hand
count flux data plotted versus optical flow data
frames 900 to 1000 of church narthex video.
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Table 2. Results from church.
Central All Edges
C 76.5 108.4
? 0.65 0.33
N 3.15 3.15
C is coefficient relating hand-count-flux to
optical-flow. ? is correlation coefficient. N is
average number of individuals in cells.
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CONCLUSION

• Correlation between hand count and optical flow
  is statistically significant.
• Optical flow can thus be used in place of time
  consuming manual counts.
• The correlation of hand flux count with optical
  flow is highest for fluxes across internal
  boundaries.
• The calibration constant C is essentially the
  same for interior and exterior edges.
• The values of C when corrected for range, are
  identical within error for the venues of a
  football stadium and a church.
• Technique should be extendable to other
  circumstances with only a simple allowance for
  camera distance.

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Acknowledgements
We thank P. Kincaid, B. Goldiez and R. Shumaker
of the Institute for Simulation and Training for
their interest and discussions throughout
different stages of this work. Valuable
programming assistance was provided by Sivakumar
Jaganathan and Jittendrakumar Koshti, also of the
Institute for Simulation and Training. This
research has been supported in part by the
National Science Foundation Grant No. BCS-0527545
and by the U.S. Army Research Development and
Engineering Command, Simulation Technology
Center, Contract N61339-02-C-0107.
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