Chapter 3 Human Body Part Detection and Labelling
4.3 Foundation of proposed methods
(a)
(b)
Figure 4.1: Concept of the Particle Filter for state prediction (a) No occlusion; and (b) Occlusion.
4.3
Foundation of proposed methods
4.3.1 Concept of proposed Particle Filter tracking
Let the state vector xt describe the tracked object parameters and the vector zt
denote all the observations z1, ..., zt up to timet. Baye’s estimator or rule, can be
used to estimate the current statextgiven all the data available up to and including
zt as
p(xt|zt) =
p(zt|xt)p(xt|zt−1)
p(zt|zt−1)
. (4.1)
Fig. 4.1 shows the conceptualization of standard Particle Filter behaviour with and without occlusion. When there is no occlusion, the particle weights are updated with respect to the observationzt−1 known from the last frame to estimate
the state vectorxtin the next frame. In occlusion, the last known observationzt is
used by the general Particle Filter to estimate the state vector, i.e.,xt+1, xt+2, xt+n−1
4.3 Foundation of proposed methods
Figure 4.2: Concept of proposed Particle Filter for state prediction in occlusion.
If the occlusion is for a small number of frames, then the state predicted using the last observation is quite close to the ground truth. However, if the occlusion continues for significant number of frames, then the predicted state diverges from the ground truth. A Particle Filter adjusts the weights of the particles based on the most current observation to predict the next state. Hence, the lack of current observation is a clear reason for error in estimation of the state for frames at time t+ 1, t+ 2, t+n−1. This can be seen using the qualitative results in Section 4.5 on SBP tracking.
In stochastic dynamics a somewhat general assumption is made for the prob- abilistic framework that the object dynamics form a temporal Markov chain so that p(xt|Xt−1) =p(xt|xt−1). (4.2)
This means that the new state is conditioned directly only on the immediately preceding state independent of the earlier history.
In Fig. 4.2 a new Particle Filter strategy or concept is illustrated to estimate the state in occlusion. During occlusion the last known observationzt is only used
to estimate the statext+1 at time t+ 1. This state xt+1 is used as an observation
to generate the next subsequent state xt+2. Similarly, the state xt+2 is used as an
observation to next state xt+n−1 and so on until an observation zt+n is obtained.
This strategy works because the most recent state is used as an observation to generate the next state in all time frames during occlusion. The proposed Particle Filter algorithm is described in Algorithm. 4.3.1 (see Section 4.4.1 for details).
4.3 Foundation of proposed methods
Algorithm 4.3.1: Proposed Particle Filter Algorithm(x, z, s, π)
Construct a new weighted particle setS ={(s(tn), πt(n))}Nn=1 for timet from the old weighted particle setS={(s(t−n)1, πt(−n)1)}Nn=1 at timet−1. SelectN particles from the setS ={(s(tn−)1, πt(−n)1)}Nn=1 to give
S={(st′(−n1),1/N)}N n=1.
Predicteach particle using the dynamic modelp(xt|xt−1) =s
′(n)
t−1 to give {(s′t−(n1),1/N)}Nn=1.
No Occlusion:
Measureand weight the particles asπ(tn)∝p(zt|xt=s
′(n)
t ) to give
S={(s(tn), π(tn))}nN=1. Normalizeπ(tn) so that ∑Nn πt(n)= 1. Estimatethe tracking result for time tasE[xt] =
∑N n=1π (n) t s (n) t . Occlusion:
Fornon-consecutive occlusion, use the last known measurement S={(s(tn), πt(n))}Nn=1 to estimate the tracking for next time step. Forconsecutiveocclusion, use the last estimation E[xt] =
∑N n=1π (n) t s (n) t
as measurementS ={(s(tn), πt(n))}Nn=1 for estimation in next time step.
4.3.2 Concept of Motion Flow (MFL) tracking
The direction of the instantaneous angular velocity (which is measured over an extremely small time interval [90]) is the basis for motion flow prediction. Consider the human arm as a pendulum attached at the shoulder joint producing curvilinear motion (incurring an angular displacement) as shown in Fig. 4.3. As the pendulum (arm) swings from its equilibrium position (vertical) to its maximum displacement, the magnitude and direction of angular velocity vector change. Two geometric constraints are proposed for predicting arm location based on pendulum motion. For an extremely small time interval in consecutive time frames:
Conjecture 1:
The direction of the instantaneous angular velocity must be the same until the arm reaches its maximum displacement.
Conjecture 2:
A large instantaneous angular displacement shows that the arm has passed its maximum displacement.
Based on first conjecture the point to be predictedA(t+ 1) should be close to the last arm point and continue in the direction of the previous two arm points, i.e.,
4.3 Foundation of proposed methods
(a) (b)
Figure 4.3: Motion flow based arm predictionAusing previous armApand current armAc during occlusion (see Section 4.3.2).
follows the swing of arm for cyclic activities, as shown in Fig. 4.3 (a). The second conjecture leads to identify the change in direction of arm swing.
Consider the arm motion as a pendulum swing which draws a small dotted curve f in each frame as shown in Fig. 4.3 (b). Denote (xAt−1, ytA−1) and (xAt , yAt ), respectively, as coordinates of labelled arm points in the previous and current frames. For every frame, the linear displacement between the current and previous arm points is
dx=xAt −xAt−1 , dy=yAt −yAt−1. (4.3) The lengthL of the entire curvef (i.e., angular displacement) traced by the arm movement on the interval [P1-P2] can be approximated as a summation of all the line segments of the entire piecewise linear curve. Theath line segment is the
hypotenuse of a triangle with basedx and heightdy, and has length La=
√
(xAt −xAt−1)2+ (yA
t −ytA−1)2. (4.4)
By the Mean Value Theorem, there existsx∗ ∈[xAt−1, xAt] such that yA t −ytA−1 xA t −xAt−1 =f′(x∗). (4.5) yAt −yAt−1 =f′(x∗)dx (4.6)