Simulate trajectories from the starting node to the end node:
Contextual Trajectory Forecasting (CTF) is used to simulate trajectories from the start node to the end node. Given the 3D geometry of the environment and the starting point and destination of a human, CTF is assembled on two assumptions. First, the human would follow a path that requires the shortest time to reach the destination, and second, the human would adhere to certain behavioral norms that are observed when walking. CTF assigns probabilities to points on the floor such that consecutive points can be sampled from start to destination and form a trajectory that represents the shortest path while conforming to observed behavioral norms.
Distance Map:
CTF algorithm takes as input a distance map to find points that are closer to the destination. This map calculates the distance to the destination from every other point on the floor. Euclidean distance between two points is not altered by the presence of inaccessible areas in the path. Hence using Euclidean distance can potentially be erroneous. Martinez et al. defined geodesic distances in [1], which is used instead. Geodesic distance is measured around the inaccessible areas along the ground plane and gives a more accurate sense of distance for human navigation. A rendering of the distance map for geometry A with a given destination is shown in Figure 1.
Figure 1.
Accessibility Map:
CTF also takes a human accessibility map as a second input. Hypothetically, if a large number of trajectories followed by human subjects from a particular start to destination are observed, it is possible that certain points on the ground plane are accessed more often then other points. This might imply the existence of a certain distribution or an accessibility map to the points on the ground plane. Estimating this map can assist the CTF in choosing points that are accessed often by complying with behavioral norms. Human motion is influenced by a multitude of factors, many of which are driven by perception. CTF specifically focuses on modeling this human motion by taking into account the constraints imposed by 3D geometry and the physical world. When traversing on the ground plane, the immediate decision of movement is influenced by the objects in the path and the surrounding geometry like walls. For example, the way humans navigate around tables and chairs when moving from one corner of a classroom to the opposite corner. Since the human behavior is assumed to be influenced by the 3D geometry, the aim is to model the relationship between them. This model would provide a means to estimate the accessibility map for any novel location based on its 3D geometry.
This relationship between them was modeled based on empirical data. The model first represents a point on the ground plane using a set of geometric features that capture the 3D geometry of the environment surrounding that point. Then establishes a linear relationship between geometric features of the point and its observed occupancy. Initially, the occupancy map of a known geometry was observed. Occupancy map is different from accessibility map. accessibility map depends only on the geometry of the environment and defines what areas are accessible by humans, on the other hand, occupancy map is generated by observing the trajectories taken by the human subjects in the environment and represents the frequency with which they are occupied. Occupancy map depends on the geometry and also the frequency with which the nodes are accessed in the geometry. Given a geometry, the accessibility map is fixed, but the occupancy map can change, if the location of the node or their purpose in the environment change. For example in a movie theater, if the location of the ticket counter changes, the occupancy map changes but the accessibility map remains the same. In this case, we are observing the actual trajectories of the humans and hence we are observing the occupancy map and not the accessibility map. We use this occupancy map to estimate the accessibility map of the environment. Consider a dataset of humans traversing the ground plane whose surrounding 3D geometry is known. The occupancy of the point on the floor is proportional to the number of times the humans in the dataset has occupied that point. The observed occupancy map in a hallway observed over a period of 5 days is shown in Figure 2.
Figure 2.
Since CTF assumes that the occupancy of a point on the ground plane is influenced by the 3D geometry surrounding it, the geometric features of a point on the ground plane in the 3D model are represented as a set of numbers, which are its distances from the walls and objects surrounding. So, to obtain the geometric features the distances are measured to walls or objects in the hallway along vectors pointing at a certain inclination from the ground plane at regular interval spanning an entire circle with its tail fixed at the point as shown in Figure 3.
Figure 3.
The distances are measured consistently in either clockwise or anti-clockwise direction always starting from the closet object or wall. In order to confine the effect to only objects with in the close vicinity of the point, the distances are thresholded by a hemisphere as shown in Figure 3. The radius of this hemisphere is inferred from the Theory of Proxemics [2]. This is a theory based on observation that defines how human beings unintentionally make use of physical space around them. Proxemics classifies the space close to a human subject into four broad regions, intimate, personal, social and public distance. It is assumed that the interaction between human subjects in closed hallways take place within the social distance.
We assume a linear relationship between the geometric features of a point and its observed occupancy. Linear regression is performed to estimate the values of the parameters by minimizing the sum of squares of the error term as described in LR. To determine the accessibility of any point on the ground plane in a new geometry, first the geometric features of that point are computed and then the estimated parameter values from linear regression are used to estimate accessibility. Figure 4 depicts the estimated accessibility map.
Figure 4.
It can be observed how the occupancy of the points in the center of the hallway is higher than those along the edges. The rotational in-variance of the features allow for the expected estimation of the occupancy even along curved hallways.
Trajectory Forecasting:
CTF combines these two maps and assigns an energy value to every point on the ground plane. Let O be the occupancy map function and let D be the distance map function. Then the energy of the point p is defined by the function E as:
E(p)= -D(p)/O(p)
Figure 5
The energy function for geometry A is shown in Figure 5. The energy is higher in the center of the hallway than along the edges, and the energy increases as the points get closer to the destination. To forecast the trajectory from the starting node to destination node, points are sampled consecutively with a probability defined by the energy map. The points closer to the destination are sampled with a probability which is proportional to the difference in there energies
~= E(i) - E(i-1)
Figure 6 (a) shows the distribution of simulating the trajectory prediction algorithm 5,000 times without the use of the accessibility map, but only using distance minimization. Figure 6 (b) simulates with the help of the accessibility map. We can see how the estimated occupancy map complements the geodesic distance minimization and forms a more desirable trajectory, that conforms to expected human behavior.
(a)
(b)
Figure 6.
References
[1] D. Martinez, L. Velho, and P. C. Carvalho, “Computing geodesics on triangular meshes,” Comp. Graph., 2005.
[2] E. T. Hall, The Hidden Dimension. Anchor Books. ISBN 0-385-08476-5, 1966.
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