The distance filter has specifically been designed for proximity sensors such as laser range-finders. Distance filters are based on a simple observation: In proximity sensing, unmodeled obstacles typically produce readings that are shorter than the distance expected from the map. In essence, the distance filter selects sensor readings based on their distance relative to the distance to the closest obstacle in the map.
To be more specific, this filter removes those sensor measurements s
which with probability higher than 
(this threshold is set to
0.99 in all experiments) are shorter than expected, and which
therefore are caused by an unmodeled object (e.g. a person).
To see, let 
be a discrete set of possible
distances measured by a proximity sensor. As in
Section 3.2, we denote by 
the probability of measuring distance 
if the robot is at
position l and the sensor detects the closest obstacle in the map
along the sensing direction. The distribution 
describes the
sensor measurement expected from the map. As described above,
this distribution is assumed to be Gaussian with mean at the distance
to the closest obstacle along the sensing direction. The dashed
line in Figure 9 represents , for a laser
range-finder and a distance of 230cm. We now can define the
probability 
that a measured
distance is shorter than the expected one given the robot
is at position l. This probability is obviously equivalent to the
probability that the expected measurement is longer than
given the robot is at location l and thus can be computed as
follows:
In practice, however, we are interested in the probability
that is shorter than expected,
given the complete current belief of the robot. Thus, we have to
average over all possible positions of the robot:
Given the distribution , we now can
implement the distance filter by excluding all sensor measurements
with 
. Whereas the
entropy filter filters measurements according to their effect on the
belief state of the robot the distance filter selects measurements
solely based on their value and regardless of their effect on the
robot's certainty.
It should be noted that [Fox1998] additionally developed a blockage filter for proximity sensors, which is based on a probabilistic description of situations in which a sensor is blocked by an unknown obstacle. We omit this filter here since its derivation is quite complex and the resulting filter is not significantly different from the distance filter described here.
