IEEE Robotics & Automation Magazine - March 2023 - 40

Py T 1ii = if tTi1 and (| )Py T 0ii = if tT .i2 However, in
(| )
a real-world environment, false detection and missing detection
often occur. To deal with this, we introduce two parameters
into our algorithm: the probability of missed detection
PM and the probability of false detection PF. Then, the formulation
of the map point detection likelihood is
Py T = ();
ii
) M
i
t
F
y
PP Tt
PP Tt
1-y
() ,
() ,
i
t
1 - M
1 - F
Furthermore, if [, ),ttN 3!
(| )( |) ().
k=1
PY Tt Py tP P1:: % M
N
M
11 y
i
i
$ == -
-
N
i
N
1
y
i
t
i
kkt
y
1-y
i
t
i
t
N
(4)
Hence, the likelihood at time tN 1+ can be defined recursively.
Our work capitalizes on the conditional independence
of map point detection given map point persistence. The likelihood
is constant over the intervals in the set, and the integral
is simplified by defining:
tt0 N01 3__+
N
,. Then, following
[11], the evidence is calculated as:
() (| )[ () ()]
PY PY TF tF tii1
1 N
i
::
i
==
/
t
where
FtTi () is the cumulative distribution function of PTi
which is described in (1).
Then, the evidence can be recursively defined as:
PYi () () (| )[
11 11N
i
i
where ()
LY :11N+
i
i
N+ =+ -
N
:: 1
:
LY PY TF t()]iT N
i
(6)
LY :1 N is the lower partition sum of evidence.
() can also be calculated recursively, resulting in a
complete algorithm, which we refer to as the BPF.
SURVIVAL ANALYSIS
For dynamic map points in the environments, we use the persistence
filter to evaluate the persistence of map points and
remove dynamic map points. Upon having no information of
the map point's activity pattern, we use the survival analysis
algorithm [11] to obtain the general-purpose survival priors
of the map points. The original probability density function
p ()T $
is defined as
pt = m
mT()$
TT te ()tT-K
() ()
is the hazard function and KT ()$
.
(7)
is the cumulative hazard
function. The principle of maximum entropy is introduced
into our method with exponential distribution. The
probability density function is simply defined as
()
pt e= m tmT
.
The parameters of the BPF are the hyperparameters
in our system. We tune the parameters according to
the category of dynamic objects in the environment. Moreover,
we fine-tune the parameters of the BPF in the training
dataset to achieve better performance and adapt to the realworld
environment.
40 IEEE ROBOTICS & AUTOMATION MAGAZINE MARCH 2023
+
1 N
iT NT N
(5)
i $
i 1
then we can obtain
(3)
DYNAMIC MAP POINTS REMOVAL
In the following sections, we introduce our dynamics-removal
algorithm in detail. We incorporate the BPF into front- and
back-end optimization modules to evaluate the persistence
probability of map points and remove the dynamic map
points. To accurately use the persistence filter, we have to
track observations of each map point.
TRACK OBSERVATIONS
In the track observations module, we can track all the
observations of the corresponding map points to update the
persistence filter. Whenever a new frame arrives, we match
the keypoints in the current frame with the candidate map
point. The maximum and minimum observation distance
dmax dmin are used to select the candidate map points that
are in the field of view (FoV). Then, we project the candidate
map points on the current frame and employ ORB
descriptors to match them with the keypoints. If the map
point is matched, it is defined as survive in this timestamp,
and we assign a hit observation to our persistence filter.
Otherwise, the map point is defined as death in this timestamp,
and we assign a missed observation to our persistence
filter. However, according to characteristics of the stereo/
RGB-D camera, the observation occlusion is really common
in the real-world environment, which can really make our
persistence filter ineffective. It will assign many wrong
observations to our persistence filter, leading to many erroneous
deletions of static map points, thus reducing the localization
ability of our LTVS system. To deal with it, we design an
algorithm for OD.
OD
For each candidate map point Mpi, it is projected into the current
frame using its depth and camera pose and finds the corresponding
keypoint in the KeyFrame. Then we calculate the
angle i between KeyFrame and current frame. We also calculate
the projected depth zp and compare it with the depth zc
of the keypoint in the current frame (from the depth measurement).
If the difference between zp and zc is greater than the
threshold zth and the angle i is greater than the threshold ,thi
the map point is defined as occluded. The detailed information
of our method is shown in Figure 3. The angle threshold
is set to
30 ,c and the depth threshold is set to 0.4 m.
LOCAL MAP UPDATE
In the persistence filters creation module, we create persistence
filters for each map point. Whenever a map point is created,
a persistence filter is initiated and assigned to it. With
the track observations module, the persistence probability of
map points is updated in each time stamp. If the persistence
probability of a map point is lower than the threshold pth, we
remove it from the global map in the map point removal
(MPR) module. To get the persistence of the map point, we
have to evaluate the persistence of the map point through several
frames, so we cannot immediately define it as a dynamic
map point. Each map point corresponds with some keypoints
IEEE Robotics & Automation Magazine - March 2023

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - March 2023
