Model Matching

After the 3D data (set $\cal D$) that contain the object is found, a given 3D model from the object database is matched into the point cloud. The model $\cal M$ is also saved as 3D point cloud in the database. The well known iterative closest points algorithm (ICP) is used to find a matching [4]. The ICP algorithm calculates iteratively the point correspondences. In each iteration step, the algorithm selects the closest points as correspondences and calculates the transformation, i.e., rotation and translation ( $\M R, \V t$) for minimizing the equation

$$E(\M R, \V t) = \sum_{i=1}^{N_m}\sum_{j=1}^{N_d}w_{i,j}\norm {\V d_{i}-(\M R \V m_j+\V t)}^2,$$

$$end{tex2html_deferred}$$ (1)

$$end{tex2html_deferred} \propto \frac{1}{N} \sum_{i=1}^N \norm {\V m_i - (\M R \V d_i + \V t)}^2$$ (2)

where $N_m$ and $N_d$, are the number of points in the model set $\cal M$ or data set $\cal D$, respectively, and $w_{ji}$ are the weights for a point match. The weights are assigned as follows: $w_{ji} = 1$, if $\V m_i$ is the closest point to $\V d_j$ within a close limit, $w_{ji} = 0$ otherwise.

It is shown that the iteration terminates in a minimum [4]. In each iteration, the transformation is calculated by the quaternion based method of Horn [9]. The assumption is that the point correspondences are correct in the last iteration step. Finally, the pose of the model corresponds to the one in the data set.