Predicting Pedestrian Trajectories

Abstract

Pedestrians do not walk randomly. While they move toward their desired destination, they avoid static obstacles and other pedestrians. At the same time they try not to slow down too much as well as not to speed up excessively. Studies coming from the field of social psychology show that pedestrians exhibit common behavioral patterns. For example the distance at which one individual keeps himself from others is not uniformly random, but depends on the acquaintance level of the individuals, the culture and other factors. Our goal here is to use this knowledge to build a model that probabilistically represents the future state of a pedestrian trajectory. To this end, we focus on a stochastic motion model that caters for the possible behaviors in an entire scene in a multi-hypothesis approach, using a principled modeling of uncertainties.

Appendix A: Derivation of the marginal probabilities

In this appendix we are going to show a derivation for $p(\mathbf{x}_{i}^{t},\mathbf{v}_{i}^{t})$, and $p(\mathbf{x}_{i}^{t})$.

We start by factorizing the probability of the state S ^t

$$p\bigl(\mathbf{S}^{t}\bigr)=\prod_{i=1}^{N}p\bigl(\mathbf{x}_{i}^{t},\mathbf {v}_{i}^{t}\bigr). $$

(24.19)

Now, we can assume that $p(\mathbf{x}_{i}^{0},\mathbf{v}_{i}^{0})$ is initially given as a mixture of Gaussians (possibly with a single component). To show, by induction, that the marginal $p(\mathbf {x}_{i}^{t},\mathbf{v}_{i}^{t})$ will have the form of a mixture of Gaussians, we assume that at time t−1 the distribution $p(\mathbf{x}_{i}^{t-1},\mathbf{v}_{i}^{t-1})$ is already a mixture of Gaussians and prove that this is sufficient for $p(\mathbf{x}_{i}^{t},\mathbf{v}_{i}^{t})$ to have the same form.

For the moment, the fact that the each factor of (24.19) is a mixture of Gaussians, say with H ^t component, means that P(S ^t) will be itself a mixture of Gaussians with M ^t=(H ^t)^N components

(24.20)

(24.21)

where we use a mapping function φ:{1,…,H}^N→{1,…,M} from each N-tuple in the Cartesian product of the h indices to a single m and we define $w_{m}=w_{1h_{1}} w_{2h_{2}}\cdots w_{3h_{3}}$. $\boldsymbol{\mu}_{ih}^{t}$ and $\boldsymbol{\varSigma }_{ih}^{t}$ are the mean and the covariance matrix for the hth component of the mixture for subject i at time t, and $\mu_{\mathbf{S}_{m}}^{t}$ and $\boldsymbol{\varSigma }_{\mathbf{S}_{m}}^{t}$ are the mean and the covariance matrix for the joint state S ^t. Note that $\boldsymbol{\mu}_{\mathbf{S}_{m}}^{t}$ is obtained by simply concatenating the $\boldsymbol{\mu}_{ih_{i}}^{t}$ for each subject, and the covariance matrix $\boldsymbol{\varSigma }_{\mathbf{S}_{m}}^{t}$ is a block diagonal matrix with blocks $\boldsymbol{\varSigma }_{ih_{i}}^{t}$. Also, note that the φ mapping describes the possible world models for the set of subjects, as in each component of the mixture in (24.21) there is a single component h _i selected from the mixture $p(\mathbf{x}_{i}^{t},\mathbf{v}_{i}^{t})$.

Now let us see how to derive a single marginal $p(\mathbf {x}_{i}^{t},\mathbf{v}_{i}^{t})$:

(24.22)

to meet real-time requirements, here we simplify the mixture of Gaussians in (24.21) by substituting each normal distribution with a Dirac function:

$$p\bigl(\mathbf{S}^{t-1}\bigr)\approx\sum_{m=1}^{M^{t-1}}w_{m}\delta\bigl(\mathbf {S}_{m}^{t-1}-\mu_{\mathbf{S}_{m}}^{t-1}\bigr), $$

(24.23)

so that we can rewrite the integral in (24.22) as

$$p\bigl(\mathbf{x}_{i}^{t},\mathbf{v}_{i}^{t}\bigr)=\sum _{m=1}^{M^{t-1}}w_{m}p\bigl(\mathbf{x}_{i}^{t},\mathbf{v}_{i}^{t}|\mu _{\mathbf{S}_{m}}^{t-1}\bigr).$$

(24.24)

By this approximation, we retain the mean and weight of the mixture components, but we discard the covariances. We will compensate for this in the empirical covariance that we will introduce later on. Continuing with the derivation we have

(24.25)

(24.26)

(24.27)

(24.28)

where

(24.29)

(24.30)

and where we define w _mk =w _m w _k . So we show that $p(\mathbf {x}_{i}^{t},\mathbf{v}_{i}^{t})$ has a mixture of Gaussian form with H ^t=M ^t−1 K components. As we noted above, because of the approximation in (24.23), we are discarding the uncertainty information included in the covariance matrix $\boldsymbol{\varSigma }_{\mathbf{S}}^{t-1}$. We can partly compensate for this by appropriately modifying the covariance $\boldsymbol{\varSigma }_{mk}^{t}$. In particular, we modify (24.30) by adding the position covariance at the previous time step as

$$\boldsymbol{\varSigma }_{imk}^{t}=\left[\begin{array}{c@{\quad }c}\varGamma +\varDelta ^{2}\tilde{\varPsi }_{imk}^{t}+\boldsymbol{\varSigma }_{imk}^{t-1} &\varDelta \tilde{\varPsi }_{imk}^{t}\\[3pt](\varDelta \tilde{\varPsi }_{imk}^{t})^{T} & \tilde{\varPsi }_{imk}^{t}\end{array}\right].$$

(24.31)

Note that the velocity components of the covariance are in fact discarded, but we partially account for this in the empirical setting of Γ.

Finally, let us derive $p(\mathbf{x}_{i}^{t})$ as

(24.32)

(24.33)

where we made use of the marginalization property of the Gaussian distribution.