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data point registration aims at finding the best transformation that places both the
given data set and corresponding model set into the same reference system. The
different approaches proposed in the literature can be broadly classified into two
categories, depending on whether an initial information is required (
fine registra-
tion
) or not (
coarse registration
); a comprehensive survey of registration methods
can be found in [23]. The approach followed in the current work for moving object
detection lies within the fine rigid registration category.
Typically, the fine registration process consists in iterating the following two
stages. Firstly, the correspondence between every point from the current data set
and the model set shall be found. These correspondences are used to define the
residual of the registration. Secondly, the best set of parameters that minimizes the
accumulated residual shall be computed. These two stages are iteratively applied
until convergence is reached. The Iterative Closest Point (ICP)—originally intro-
duced by [3] and [4]—is one of the most widely used registration techniques using
this two-stage scheme. Since then, several variations and improvements have been
proposed in order to increase the efficiency and robustness (e.g., [25], [8], [5]).
In order to avoid the point-wise nature of ICP, which makes the problem discrete
and non-smooth, different techniques have been proposed:
i
probabilistic represen-
tations are used to describe both data and model set (e.g. [31], [13]);
ii
)
in [8] the
point-wise problem is avoided by using a distance field of the model set;
iii
)
an im-
plicit polynomial (IP) is used in [36] to fit the distance field, which later defines a
gradient field leading the data points towards that model set;
iv
)
implicit polynomi-
als have been also used in [28] to represent both the data set and model set. In this
case, an accurate pose estimation is computed based on the information from the
polynomial coefficients.
Probabilistic-based approaches avoid the point-wise correspondence problem by
representing each set by a mixture of Gaussians (e.g., [13], [6]); hence, registration
becomes a problem of aligning two mixtures. In [13] a closed-form expression for
the
L
2
distance between two Gaussian mixtures is proposed. Instead of Gaussian
mixture models, [31] proposes an approach based on multivariate
t
-distributions,
which is robust to large number of missing values. Both approaches, as all mixture
models, are highly dependent on the number of mixtures used for modelling the sets.
This problem is generally solved by assuming a user defined number of mixtures or
as many as the number of points. The former one needs the points to be clustered,
while the latter one results in a very expensive optimization problem that cannot
handle large data sets or could get trapped in local minimum when complex sets are
considered.
The non-differentiable nature of ICP is overcome by using a derivable distance
transform—Chamfer distance—in [8]. A non-linear minimization (Levenberg -
Marquardt algorithm) of the error function, based on that distance transform, is
used for finding the optimal registration parameters. The main disadvantage of [8] is
the precision dependency on the grid resolution, where the Chamfer distance trans-
form and discrete derivatives are evaluated. Hence, this technique cannot be directly
applied when the point set is sparse or unorganized.
)
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