Social Force Model SFM Detailed application & anomaly detection on population

Social force model (Social Force Model, SFM)

The theoretical model of Dirk Helbing, published in 1998 "Social force model for pedestrian dynamics" proposed. In this paper the removal of the theoretical model of Helbing this paper detailed interpretation.

Three necessary force:

1. desired acceleration to a target speed (force acting in a direction of the target)

Suppose pedestrian $\alpha$ needs to reach the target position as much as possible $\ Begin {R} ^ {0} _ {a}$ , pedestrian path to the target may be generally abstracted as an edge along the polygon advances $\ Begin {R} ^ {} _ {1}, and$ , ......, $\ Begin {R} ^ {N} _ {a}$ : = $\ Begin {R} ^ {0} _ {a}$ . It is assumed that in a pedestrian $\ Begin {r} ^ {k} _ {a}$ , the target direction can be calculated according to the formula $\ Begin {e} and {_} (t)$ :

$\vec{e}_{a}(t) = \frac{\vec{r}^{k}_{a} - \vec{r}_{a}(t)}{\left \| \vec{r}^{k}_{a} - \vec{r}_{a}(t) \right \|}$

Here a pedestrian $\alpha$ desired speed: $\ Begin {of} ^ {0} _ {a} (t): = v_ {a} ^ {0} \ begin {e_ {a}} (t)$

We can come to this force derived formula:

$\ Begin {m} ^ {0} _ {a} (\ vec {v} _ {a}, the ^ {0} _ {a} \ case {e} _ {a}) = \ frac {1} T_ {{a}} (a ^ {0} _ {a} \ case {e} and {_} - \ begin {of} _ {a})$

2. maintain a repulsive force between the entities (always keep a certain distance between the entity and entity)

Here, the range of the privacy of each pedestrian, can be interpreted as local effects. If an entity is too close, it will cause other pedestrians β repelling effect, it can be represented by the vector:

$\ Begin {m} ^ {\ alpha \ beta} (\ vec {r} _ {\ alpha \ beta}) = - \ Delta _ {\ vec {r} _ {\ alpha \ beta}} W _ {\ alpha \ beta} [b (\ vec {r} _ {\ alpha \ beta})]$

We assume repulsive potential $V _ {\ alpha \ beta} (b)$ is a decreasing monotonic function of b, etc. is elliptical form equipotential lines in the direction of movement points. The reason is that other pedestrians will take into account the need for space next move. b represents the semi-minor axis of the ellipse:

$2b: = \ sqrt {(\ left \ | \ more {r} _ {\ alpha \ beta} \ right \ | + \ left \ | \ more {r} _ {\ alpha \ beta} - in _ {\ beta} \ Delta t \ more {e} _ {\ beta} \ right \ |) ^ {2} - (v _ {\ beta} \ Delta t) ^ {2}}$

Wherein $\ Begin {r} _ {\ alpha \ beta} = \ vec {r} _ {\ alpha} - \ begin {r} _ {\ beta}$ , $s _ {\ alpha \ beta} = v _ {\ beta} \ Delta t$ pedestrians β-step step size.

In addition, pedestrians still maintain a certain distance from the boundary of the building, walls, streets, and other obstacles. The more pedestrians will feel more discomfort he walked close to the border, because they will pay attention to pedestrians reduce the risk of injury, for example, be careful not to run into a wall. Thus, the boundary B causing one kind repelling effect, this force may be expressed by the following:

$\vec{F}_{\alpha B}(\vec{r}_{\alpha B}) := -\Delta_{\vec{r}_{\alpha B}}U_{\alpha B}(\left \| \vec{r}_{\alpha B} \right \|)$

3. attractiveness (probably between entities, it may be between entities with the surrounding environment)

Modeling in position $\ Vec {r} _ {i}$ attraction at $\vec{f}_{\alpha i}$ :

$\vec{f}_{\alpha i}(\left \| \vec{r}_{\alpha i} \right \|, t) := -\Delta_{\vec{r}_{\alpha i}}W_{\alpha i}(\left \| \vec{r}_{\alpha i} \right \|, t)$

Due to the decline in interest, attraction $\left \| \vec{f}_{\alpha i} \right \|$ usually decreases over time t. However, the above formula for attraction and repulsion effect applies only to the desired direction of motion in $\ Begin {e} and {_} (t)$ the case of perceived on. Where a pedestrian is located behind little effect on c (0 <c <1) to. To account for this perception effect (ie, the effective angle of view 2 $\varphi$ ), we have to introduce the right direction-dependent weight:

$w(\vec{e}, \vec{f}) := \left\{\begin{matrix} 1 & if \vec{e}\cdot \vec{f} \geq \left \| \vec{f} \right \|cos\varphi \\ c & otherwise. \end{matrix}\right.$

In summary, the exclusion of pedestrian behavior and attracting forces are:

$\ Vec {F} _ {\ alpha \ beta} (\ vec {e} _ {\ alpha} \ case {r} _ {\ alpha} - \ vec {r} _ {\ beta}): = w ( \ begin {e} _ {\ alpha}, - \ begin {m} ^ {\ alpha \ beta}) \ begin {m} ^ {\ alpha \ beta} (\ vec {r} _ {\ alpha} - \ vec {r} _ {\ beta})$

$\ Vec {F} _ {\ alpha i} (\ vec {e} _ {\ alpha} \ case {r} _ {\ alpha} - \ vec {r} _ {i}, t): = w ( \ begin {e} _ {\ alpha}, - \ begin {m} ^ {\ alpha i}) \ begin {m} ^ {\ alpha i} (\ vec {r} _ {\ alpha} - \ begin { r}} _ {i, t)$

You can now get a total power of pedestrians $\vec{F}_{\alpha }(t)$ :

$\ Vec {F} _ {\ alpha} (t): = \ vec {F} _ {\ alpha} ^ {0} (\ vec {v} _ {\ alpha} in _ {\ alpha} ^ {0} \ begin {e} _ {\ alpha}) + \ sum _ {\ beta} \ case {F} _ {\ alpha \ beta} (\ vec {e} _ {\ alpha} \ case {r} _ {\ alpha} - \ vec {r} _ {\ beta}) + \ frac {B} \ case {F} _ {\ alpha B} (\ vec {e} _ {\ alpha} \ case {r} _ { \ alpha} - \ vec {r} _ {B} ^ {\ alpha}) + \ sum _ {i} \ case {F} _ {\ alpha i} (\ vec {e} _ {\ alpha} \ case {r} _ {\ alpha} - \ begin {r}} _ {i, t)$

This social force model has been derived out:

$\frac{d\vec{w}_{\alpha }}{dt} := \vec{F}_{\alpha }(t) + fluctuations$

Fluctuations random fluctuations which is variable. In one aspect, these fluctuations derived from two or more alternative behavior similar blurring (e.g. utility by the same right or left side of the obstacle). On the other hand, accidental or deliberate fluctuation pedestrian from aberrant motion law.

Then consider the existence of a maximum speed of pedestrians $v_{\alpha }^{max}$ , so the estimated real motion model is:

$\frac{d\vec{r}_{\alpha }}{dt} = \vec{v}_{\alpha }(t) := \vec{w}_{\alpha }(t)g(\frac{v_{\alpha }^{max}}{\left \| \vec{w}_{\alpha } \right \|})$

among them,

$g(\frac{v_{\alpha }^{max}}{\left \| \vec{w}_{\alpha } \right \|}) := \left\{\begin{matrix} 1 & if \left \| \vec{w}_{\alpha } \right \| \leq v_{\alpha }^{max} \\ v_{\alpha }^{max} / \left \| \vec{w}_{\alpha } \right \| & otherwise. \end{matrix}\right.$

Application anomaly detection on SFM in the crowd

The following is an excerpt from Ramin Mehran et al, published in 2009, the paper "Abnormal Crowd Behavior Detection using Social Force Model"

Here simply talk about the method mentioned article: first with a grid overlaid on the image point, and calculates an average of the optical flow based on temporal dimension. Then based on the pixel optical flow tracking movement, calculate the interaction force between them, here with a social force model to model (here also improve the social force model based on the introduction of panic factor Helbing made in 2000). Then randomly select a plurality of regions in the force flow to model the normal movement pattern. Finally, a bag of words method of classifying normal or abnormal.

The picture shows the use of the population do SFM abnormality detection method for a flowchart.

The above method of using paper test results, normal and abnormal have been correctly classified.

The figure on the anomaly detection UMN dataset ROC area, contrast pure optical flow, perform better social force model.

ROC calculation results on network datasets show that social force model compared to optical flow still more advantages.

The method proposed in 2009, at the time effect is very good. Social force model applications are still many researchers continue to do thorough research in the back of the crowd anomaly detection areas, including social force model 2012 Y Zhang et al based on the perceived attributes (attribute-aware) made the crowd anomaly detection, while the introduction of social disorder and crowded attribute properties using statistical characteristics describe the real context of social behavior. Perception enhanced by the semantic properties have been improved based on a model of social forces. This method has an advantage of some other method.

Reference material

1. Helbing D , Molnar P . Social Force Model for Pedestrian Dynamics[J]. Physical Review E Statistical Physics Plasmas Fluids & Related Interdisciplinary Topics, 1998, 51(5):4282.

2. Helbing D , Farkas I J , Vicsek T . Simulating Dynamical Features of Escape Panic[J]. Social ence Electronic Publishing, 2000, 407(6803):487-90.

3. Mehran R , Oyama A , Shah M . Abnormal crowd behavior detection using social force model[C]// 2009 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2009), 20-25 June 2009, Miami, Florida, USA. IEEE, 2009.

4. Zhang Y , Qin L , Yao H , et al. Abnormal crowd behavior detection based on social attribute-aware force model[C]// IEEE International Conference on Image Processing. IEEE, 2012.

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