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Our work use Kalman filter (KF) estimation as in [18,4,16]. Also it uses a
clustering procedure prior to line estimation, this procedure is used in several
works cited previously. Clustering provides a representation of the world at a
coarse scale, the problem is to select the proper scale level to represent it in
an adequate way. This question leads to the concept of natural (most signifi-
cant) scales [12]. The selected method has been Interval Tree by Witkin [17],
this is the pioneer paper about scale-space. Scale space theory [8] provides a
multi-scale representation of a signal formed by smoothed versions of the sig-
nal. Smoothing is performed convoluting the signal with gaussian kernels with
different variances, the changing scale parameter. Features are normally discov-
ered as extrema or inflexion points of the signal, thus they are represented as
zero-crossing level curves in the scale-space representation, see fig. 1 in section 4.
Witkin's Interval Tree procedure associates a rectangle to each curve and defines
a stability criterion for rectangles, most stable rectangles define natural scales.
3 Line Estimation
3.1 Measures Model
The problem about using rectangular coordinates with noisy measures taken
radially is that both variables,
x
and
y
, are random and correlated [7], and there
is not a non-stochastic independent variable. In this case many of the properties
and results from regression theory. The only case where stochastic independent
variables are allowed is when they have to take equally spaced values. This is the
case for angles in a laser scan, so polar coordinates are an adequate statistical
frame for line estimation from laser measures.
The equation of a line, not crossing the coordinates origin (pole), in polar
coordinates is:
ρ
π
2
θ<
π
2
r
=
f
(
ϕ
)=
−
<ϕ
−
(1)
cos(
ϕ
−
θ
)
being
r
the measured distance,
ϕ
the angle of the measure,
ρ
line distance to
pole, and
θ
angle between
x
axis and the line.
Our measure model is:
r
n
=
E
[
r
n
]+
ε
n
,being
E
[
r
n
]
the expected value of
the measure and
ε
n
the noise. Noise has a gaussian probability distribution
with zero mean and variance
σ
n
∝
h
2
(
,being
h
a known function. In
our experiments we use a noise standard deviation proportional to the expected
measure, so
σ
n
=
E
[
r
n
])
σ
2
E
2
[
,being
σ
an unknown constant representing the
deviation per unit measure, or roughly the noise proportion.
r
n
]
3.2 EKF for Line Estimation in Polar Coordinates
As we commented in section 2, in order to detect outliers it is necessary to
estimate the line sequentially and to have a test for outliers detection. Sequential
linear regression estimation studied by Plackett in the 50's [11]. Later Peña [10],
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