Signal Processing - March 2017 - 89

Prediction Error Decomposition Calculation

State Prediction
Xi,tk |tk -1 = F (h, Di)Xi,tk -1|tk -1 + M (h, Di)
XX

YY
Ci,k |k -1

XX

Yk

Initialize

h

Xi,t 1, Ci,1 , h1

Xi,tk |tk

Ci,k |k -1 = F (h, Di)Ci,k -1|k -1F T (h, Di) + Q (h, Di)

XX

Ci,k |k

XX

=

Yi,k |k -1 = G Xi,tk |tk -1
XY
XX
G C i,k |k -1G T + Vk ; Ci,k |k -1

p (Yk |Y1:k -1, Di = DI) =

XX

= C i,k |k -1G T

YY
N(Yk |Yi,k |k -1, Ci,k |k -1)

State Estimation
XY

YY

Ki,k = C i,k |k -1(C i,k |k -1)-1
Xi,tk |tk = Xi,tk |tk -1 + Ki,k [Yk -Y i,k |k -1]
XX

XX

YY

p (Yk |Y1:k -1, DI = Di)

T

Ci,k |k = Ci,k |k -1 - Ki,k Ci,k |k -1Ki,k

Figure 5. The Kalman filter for sequentially evaluating the PED for end point D i at the arrival of observation Yk ; state prediction at t k relies on the state
estimation results, including covariance C XX
i, k - 1 | k - 1 , from the previous time step and h = t k - t k - 1.

distribution of the system state X t k, as in traditional tracking
applications [22]. Nonetheless, the latent state estimation,
which might be relevant in certain scenarios, is addressed below. Based on (6) and (7), a classical Kalman filter can be
employed to sequentially calculate the prediction error decomposition in (5) as depicted in Figure 5 and, thereby the
sought observation likelihood for the current set of measurements Y1:k conditioned on D i . The computationally efficient Kalman filter is particularly desirable since running,
concurrently, multiple Kalman filters for all D i ! D is
plausible in real-time, even in settings where limited computing power is available. This solution is also amenable
to parallelization.
For the bridging approach, it is shown in [27] and [28] how
the PED and observation likelihood in (5) from each constructed bridge, i.e., conditioned on T and D i, can be estimated
using a modified Kalman filter. As the true arrival time T is
unknown a priori in practice, approximating
p (Y1: K D I = D i) = #

T!T

p (Y1: k T, D I = D i) p (T D I = D i) dT,
(8)

is necessary, where p (T D) is the prior distribution of arrival
times at destination D i and T is the time interval of possible
arrival times T. In the simplest case, arrivals might be
assumed at some specific future time. This is a crude approximation; nevertheless, is often quite effective [28]. To improve
inference accuracy (and possibly also to learn about expected
arrival time), arrivals can be modeled as having a prior distribution, such as being expected uniformly within some time
period [t a, t b], giving p (T D) = U (t a, t b) . In this case,
numerical quadrature, for example, via Simpson's rule, can be
applied. Although BD-based intent inference involves running
multiple Kalman filters, and, hence, is more computationally
demanding, it can significantly improve the end-point inference capability of a predictive display and leads to a more
robust performance.

In summary, the introduced modeling approach for inferring
as early as possible the item that the user intends to select on
the display using the freehand pointing gesture is generic. Most
importantly, it offers considerable flexibility in terms of catering
to various sensing technology specifications (e.g., observation
error) as well as adaptability in terms of adjusting the motion
model parameters. The approach is simple and relatively computationally efficient, which makes it suitable for the requirements
of an automotive environment. In the developed predictive display prototype (an optimized C# implementation of the system
in Figure 1 on a typical automotive computing platform), prediction with Kalman filtering was tested with up to N = 64
destinations and an observations data rate H 30 Hz without any
noticeable delays in the system response in terms of the pointing
facilitation routine.

Handling perturbed pointing trajectories
When the user input is perturbed in a moving vehicle due to the
road and driving conditions, the predictive display system can
handle noisy freehand pointing gestures by setting the noise
covariance in the motion model in (6) relative to the measured
(experienced) in-vehicle vibrations/accelerations. This conforms
with the modeling assumptions, and a higher covariance corresponds to having less certainty in the inferred end-point-driven
latent state X i, t , i.e., pointing finger position, velocity, etc. This
technique is suitable for low to medium perturbation levels that
can be represented by Gaussian noise, for instance, driving on
smooth to moderately bumpy paved roads. The output of the filters, calculating the posterior of each nominal destination
p (D I = D i | Y1: k) at t k , can be used to estimate the posterior
probability of the system latent state X t k , including the perturbation-free pointing finger position. This is given by the Gaussian mixture
p (X t k Y1: k) =

N

/ p (X i,t

k

Y1: k) p (D I = D i Y1: k),

(9)

i= 1

where p (X i, t k | Y1: k) pertains to the ith destination and is also
calculated by the Kalman filter.

IEEE SIgnal ProcESSIng MagazInE

|

March 2017

|

89



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
Signal Processing - March 2017 - 3
Signal Processing - March 2017 - 4
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Signal Processing - March 2017 - Cover3
Signal Processing - March 2017 - Cover4
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