Signal Processing - March 2017 - 88
the relationship between the system state at times t and t + h
can be written as
X i, t + h = F (h, D i) X i, t + M (h, D i) + f t,
(6)
where f t ~N (0, Q (h, D i)) is the dynamic noise embodying the
randomness in the motion model. The matrices F and Q as
well as the vector M , which together define the state transition
from one time to another, are functions of the time step h and,
notably, the destination D i ! D. Thereby, N such models are
constructed to establish the end point of the pointing gesture.
The kth observation, for example, the pointing finger position as provided by the gesture tracking device, is also modeled
as a linear function of the time t k state perturbed by additive
Gaussian noise,
Yk = GX i, t k + o k,
(7)
where G is a matrix mapping from the hidden state to the
observed measurement and o n ~N (0, Vn) . For instance, if the
gesture tracker provides the pointing finger positions directly
and the system state includes only position, then G is a 3 × 3
identity matrix. The noise covariance can be utilized to set
the level of noise in each of the x, y, and z axes as per the
gesture tracker specifications; e.g., a time-of-flight-based
tracker such as the SoftKinetic DepthSense camera exhibits
higher inaccuracies in observations along the depth axis. It is
noted that no assumption is made about the observation arrival times t k and irregularly spaced, asynchronous measurements can naturally be addressed within this formulation. The
system structure, for each nominal end point D i, is depicted
graphically in Figure 4, where the destination D i influences
the end-point-driven state at all times.
Among linear Gaussian models, linear destination reverting models, such as the mean reverting diffusion (MRD) and
equilibrium reverting velocity (ERV) models, make particularly suitable candidates for the pointing finger motion in (6), as
discussed in [10]. Their state evolution explicitly incorporates
the destination information. For example, the governing SDE
for the MRD model is given by dX i, t = K ^d i - X i, t h dt + vdw t .
It indicates an attraction of the motion toward the location of
Xt 1
Prior
F (tk-tk-1, Di)
M (tk-tk-1, Di)
F (T-tk, Di)
M (T-tk, Di)
Xt 1
Xt k - 1
Xt k
Yt 1
Yt k - 1
Yt k
Di
XT
Intent inference: Sequential likelihood evaluation
Figure 4. The system graphical structure; end point D i acts as a prior
and affects the state transition.
88
destination d i (e.g., the mean of the Gaussian distribution representing D i), with K (a design parameter) capturing the strength
of this reversion for each axis in 3-D and w t being a Wiener
process. While the MRD is based on a multivariate Ornstein-
Uhlenbeck process [29] and the system state includes only the
position information in 3-D, the state of the ERV model proposed in [10] additionally includes the velocity of the pointing
finger, in 3-D, driven by the end point. This facilitates modeling
pointing velocity profiles like those shown in Figure 2(b). Integrating the SDE of the MRD and ERV results in (6), each with
specific F, M, and Q matrices.
During a pointing task, the path of the pointing finger, albeit
random, must end at the intended destination at time T (i.e., the
pointing finger reaches its end point on the display). This can be
modeled by an artificial prior probability distribution for X T corresponding to the geometry of the destination; alternatively, it can be
treated as a pseudo-observation at T. To maintain the linear Gaussian structure of the system in (6) and (7), this distribution is assumed
to be Gaussian, such that p (X T D I = D i) = N (X T ; a i, R i); see
[28] for a discussion on this construct. The mean vector a i specifies the constrained system state at the destination, whereas R i is a
covariance matrix of the appropriate dimension. For instance, for
the MRD model, in which only pointing finger position is considered, a i = n i = d i representing the location-center of the destination in 3-D. In the case of the ERV model, defining the final state
distribution also involves specifying a distribution of the pointing
finger velocity at the end point. A large-scale prior covariance can
be used to model the uncertainty in this; however, certain properties might be assumed, e.g., relatively high velocity in the direction
toward the screen.
Exploiting the artificial prior on the distribution of X T requires that the state of the motion models in (6) to be conditioned not only on D i ! D but also on the arrival time T.
Including this permits the posterior of the system state at
time t k to be expressed as p (X t k Y1: k, T, D I = D i), and the
sought observation likelihood in (5) is subsequently given
by p (Y1:k T, D I = D i) after k measurements. The inclusion
of the prior on X T in the motion model changes the system
dynamics (even for MRD and ERV models), where the predictive distribution of the pointing finger state changes from
a fully random walk to a bridging distribution (BD), terminating at the end point. This encapsulates the long term dependencies in the pointing finger trajectory due to premeditated
actions guided by intent. Since the intended destination is not
known, N such bridges are constructed, one per nominal end
point. Consequently, all Gaussian linear models, including the
nondestination reverting ones, whose dynamic models are not
dependent on D i like Brownian motion (BM) and CV, can be
utilized for destination prediction within the presented Bayesian framework. This technique of conditioning on the end
point is dubbed BD-based inference.
We recall that the primary objective of the intent inference
routine is to determine the observation likelihoods
p (Y1:k | D I = D i), D i ! D, at t k , rather than the posterior
IEEE SIgnal ProcESSIng MagazInE
|
March 2017
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Table of Contents for the Digital Edition of Signal Processing - March 2017
Signal Processing - March 2017 - Cover1
Signal Processing - March 2017 - Cover2
Signal Processing - March 2017 - 1
Signal Processing - March 2017 - 2
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Signal Processing - March 2017 - Cover3
Signal Processing - March 2017 - Cover4
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