PERS_April2018_Public - page 205

it is equivalent to infer topics in word-document analysis.
Moreover, a traffic state is a combination of frequently co-
occurring activities (i.e. interactions). This makes it possible
to infer traffic states using topic model, too.
The HDP [20] is an unsupervised non-parametric hierarchical
Bayesian topic model and was originally proposed for word-
document analysis. It clusters the frequently co-occurring
words within the same documents into the same topics.
Furthermore, different from the other clustering topic models,
such as LDA [21], HDP is able to automatically determine the
number of clusters. The rest of this section will show how to
use HDP model to infer typical activities and traffic states from
the input video. Based on the output of HDP models, we pro-
pose a method to construct feature vectors to represent activi-
ties with visual words and traffic states with typical activities.
Afterward these will be used to train classifier to recognize
complicated traffic activities in surveillance video.
Learning Activities Using HDP
Figure 2 is a graphical representation of HDP model; it consists
of two Dirichlet Processes. The first one is used to generate a
global set of activities and the second one is used to sample
a subset of activities from the global set for a clip. Finally,
visual words are drawn from activities.
The possible activities are inferred by HDP whose standard
graphical representation is shown in Figure 2 [20]. The global
random measure G0={θ 1…θ∞} is a global list of activities that is
shared by all clips. Its distribution is a Dirichlet Process (DP)
with concentration parameter λ and Dirichlet prior H:
G 0|γ,HDP (γ, H) (1)
G0 can be expressed using the stick-breaking formulation [20]:
G k
k
k 0 0
1
=
=
∞
∑ π δϕ (2)
ϕk|γ~H, (3)
π π π
k k l
l
k
= ′ − ′ ( )
=
−
∏ 1
1
1
(4)
π ′
k ~Beta(1, λ) (5)
where {ϕk}k
∞=1 are parameters of multinomial distributions
over words in the codebook corresponding to activity θk, i.e.,
the word probability vector and the sum of its entries equals
1. θϕk is the Delta function at point ϕk. {π k} are random prob-
ability measures (mixtures over topics) and ∑k
∞
=1 π k = 1. For
convenience, the random probability measure of π defined
from Equations 1 to 5 is abbreviated with πk~GEM(γ), where
GEM stands for Griffiths-Engen-McCloskey distribution [22] .
The multinomial distribution ϕk over words in the codebook
is generated from H. Therefore, H is interpreted as a distribu-
tion over multinomial distributions and thus can be defined
as a Dirichlet distribution:
H = Dir(D 0) (6)
ϕk|γ~Dir(D0) (7)
G0 is the prior distribution for the second DP. For each clip t,
Gt is a random measure which is drawn from the second DP
with concentration parameter α and Dirichlet prior G 0:
Gt|α, G0~DP(α, G0) (8)
In our case describes the multinomial distribution of ac-
tive topics in clip t, i.e., it is a subset of the global activities .
We express it using the stick-breaking representation again:
G t tk
k
k π δφ
=
∞
∑
1
(9)
ϕk|α, G0~G 0, (10)
π π π
tk tk tl
l
k
= ′ − ′ ( )
=
−
∏ 1
1
1
(11)
π′
tk ~Beta(1,α) (12)
For the ith word in document t, a topic θti is first drawn from G t
and then the word xti is drawn from multi-nominal distribution
Multi(xti ;ϕθu ) (i.e. the multi-nominal distribution over words in
codebook corresponding to topic θu). We notice that, different
Gt has the same ϕk as G 0, i.e., different clips share the same set
of topics and statistical strength. We apply Gibbs sampling
schemes to do inference under an HDP model, which is a gener-
ally applied method in topic model. Figure 6 shows the learned
typical activities by HDP models for QMUL Junction Dataset [8].
The hyper-parameters γ and α are empirically predefined.
They are priors on the concentration of the word distribution
within topics. They influence the number of activities in G0
and Gt. The parameter D0 for the Dirichlet distribution is also
set empirically.
Although HDP models decide the number of topics automat-
ically, some of the explored activities are unrepresentative.
Because some very rare motion need to be explained by an
individual activity. They could be noise or rare events. Such
learned activities could lead to ambiguous or even mislead-
ing analysis of interactions. Therefore, the unrepresentative
activities need to be removed. The total number of words that
are assigned to activity k is noticed as nk throughout the train-
ing video. The occurrence ratio of activity k is computed as
r n
n n k k
k
= + +
1 
(13)
We rank {r1,…, rk} in decreasing order as {r ′
1≥,…, r′
k} and calcu-
late the accumulated sum as
′ = ′
=
∑
R rj i
i
j
1
(14)
The representative activity (topic) set is selected as
{ }
j jR j k ′ ≤ ≤ ≤ 0 99 1 . ,
= θ typical θ (15)
Learning States Using HDP-HMM
A busy traffic junction is normally regulated by traffic lights:
different traffic states occur sequentially and circulatory in
a certain order. Hidden Markov model (HMM) [23] is an effi-
cient method to explore the latent states and their transition
information. HMM can be explained as a doubly stochastic
Markov chain and is essentially a dynamic variant of a finite
mixture model. Teh et al. [20] replaced the finite mixture with a
Dirichlet process and proposed the HDP-HMM model which is
illustrated in Figure 3. Its stick-breaking formalism is:
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING April 2018 205