IEEE Robotics & Automation Magazine - June 2014 - 40

Y (x i) = )

with probability 1 - p D (x i)
" z i , with probability p D (x i) g (z i ; x i),
z

(1)

where g ^z i ; x ih is the measurement likelihood defined by the
sensor's noise characteristics. In addition, we define an RFS
C, which constitutes the clutter measurements. Typically, we
define the clutter to be a Poisson-distributed process. The
entire measurement set is, thus, an RFS defined as
Z = Y ^x 1h , g , Y ^x ih , C.

(2)

With this measurement model in hand, we can apply the tools
of multitarget calculus to derive a multiobject measurement
likelihood, which weighs each detection against the likelihood
of a missed detection or clutter in a principled fashion. This
likelihood can be used to formulate a multiobject Bayes filter,
but such a filter would be very difficult to implement for all
but the most trivial scenarios. This is because it would have to
propagate a multiobject probability density function, which is
defined over all possible configurations for every cardinality
of objects. The PHD filter is an approximation of this full
multiobject filter, which propagates the first moment of the
multiobject density function. This moment is known variously as the PHD or intensity function. The PHD can be
roughly conceived as the expected value of an RFS. It is a nonnegative function on X, where locations with high values correspond to likely locations of the objects. It also encodes the
likely number of objects; the integral of the PHD over a certain region is the expected number of objects in that region.
SC-PHD Filter
The derivation of the first moment filter involves an assumption on the nature of the prior distribution. For example, the
PHD filter assumes that the prior is distributed according to a
Poisson process, and the cardinalized PHD filter assumes an
independently and identically distributed process. The singlecluster PHD (SC-PHD) filter used for SLAM assumes a singlecluster process. A cluster process is a hierarchical relationship
between two point processes, where the outcome of a daughter
process is conditioned on the outcome of the parent process.
For example, it is the conditional relationship between the
daughter and the parent that defines a cluster process. In
SLAM, we consider the configuration of the map features to be
the daughter process, which is conditioned on the state of the
vehicle, which constitutes the parent process. As there is only
one vehicle, there is only one cluster, so we are dealing with a
single-cluster process. The full equations for the SC-PHD filter
can be found in [13], but, for the sake of completeness, we will
briefly summarize it here. Given a prior PHD for the vehicle
position X and map M
u k - 1 ( M X) .
D k - 1 ^ X, M h = s k - 1 ^ X h D

(3)

The first step in an iteration of the SC-PHD filter is the Chapman-Kolmogorov prediction
40

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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June 2014

D k k - 1 ^ X, M h =
# s k - 1 ^Xlhr k k - 1 ^X XlhDu k k - 1 (M Xl) dXl ,

(4)

u k k - 1 ^M X h is the predicted PHD of the map
where D
u k k - 1 ^M Xh = c k k - 1 ^M Xh
D
u k - 1 ^Ml Xlh p S ^Ml X lh ru k k - 1 ^M Ml ; X lh dMl . (5)
+#D
1 444444444444
2 444444444444 3
persistent
Similar to most SLAM filters, the parent and daughter PHDs
are convolved with Markov transition densities r () and ru (),
respectively, to propagate them forward in time. However,
there are also a couple of elements here that are not usually
found in SLAM filters. The map prediction is composed of
two parts. The additional term c k k -1 ^M X h is called the
birth intensity, which models the PHD of newly appearing
features. For practical purposes, the birth intensity is typically
modeled from the current measurements [14]. Furthermore,
the portion of the prediction corresponding to persistent targets incorporates a factor of p S ^Ml Xlh, which is a conditional probability distribution that models feature survival.
The measurement update equations are
D k ^X, Mh =

s k k - 1 ^Xh L Zk ^Xh

# s k k - 1 ^ X h L Z ^ X h dX
k

u k ^M X h,
D

(6)

u k ^M X h is the updated PHD of the map
where D
u k ^M Xh = D
u k k - 1 ^M Xh
D

g ^z M, X h p D ^M X h
(7)
# ^1 - p D ^M X hh + /
H
>1 444
h z ^M Xh
2 444 3 z d Z k
nondetection
1 444444 2 444444 3
detection

u k k -1 ^ M
h z ^Xh = k k ^ z h + D

Xh
# p D ^M X h g ^z M, X h dM

u k k - 1 ^ M X h dM .
L Z k ^X h = exp $ - # p D ^M X h D
#

%

z d Zk

h z ^Xh .

(8)

(9)

At first glance, these equations are understandably quite
daunting, but we can tease out some meaning with closer
inspection. In (6), we can see a Bayes' rule update for the parent PHD s ^X h, where L Z k ^X h is a multiobject observation
likelihood. The same Bayes' rule structure can be seen inside
the summation of (7). The remaining elements can be considered as a sort of bet hedging against the measurement difficulties previously discussed. The normalization term h z ^X h
incorporates a term l k ^ z h that models the intensity of clutter
measurements so that each received measurement is weighed
against the possibility of being spurious. The updated map
PHD is a sum of a detection and nondetection term. If a measurement is not received for a particular feature, it will remain
in the updated map, weighted by a factor of ^1 - p D ^M X hh .



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