Pairwise registration

Partial views acquisition

The first step in 3D modeling is to acquire the set of 3D partial views of the object to be registered. These views can be acquired by any Shape-from-X technique such as stereovision, laser triangulation, laser scanning and pattern projection (see Chapter 2).

Acquisition differs from the technique used but all of the techniques tend to deliver a cloud of 3D points referenced with respect to the pose of the measuring sensor.

Since the goal of this thesis is in the registration of the views instead of in their acquisition, we have preferred to use already acquired views available in the web¹. This decision has been also made to facilitate the comparison of the technique since these sets of views are nowadays used by several researchers. Besides, we have also simulate a virtual scanner to acquire synthetic partial views of a virtual object. Synthetic data is extremely

1http://www.csse.uwa.edu.au/ ajmal/3Dmodeling.html

4.2 Pairwise registration 77

Figure 4.1: Scheme of the multi-view registration process.

important during algorithm development since is the only way to analyze the accuracy of the method compared to ground truth.

Once all the partial views have been acquired, the next step is to align them in order to obtain the final model. In general, a former coarse registration should be used to get an initial estimation of the alignment from some surface features. Then, the coarse registration is considered as an initial solution for a fine registration minimization. Note that, the initial solution is crucial to avoid local minima.

However, in the present work, views have been acquired sequentially both in synthetic and real data (see Chapter 5), ensuring a sufficient overlapping region between them. In this case, the motion between each two consecutive views is very small (a huge amount of surface is shared) and the fine pairwise registration process can be initialized considering that this motion is null. This assumption avoids the expensive computational time required to compute the initial estimations (coarse registration) of the motion, without penalizing the accuracy of the process. However, not in all the real cases this small amount of motion can be assumed. In order to apply this assumption, the motion between consecutive views should be small enough to provide a significant overlapping when possible. In cases where such condition could not be assumed, additional techniques should be previously applied to pre-align the views, such as the mentioned coarse registration. The pairwise registration method used is explained in the following section.

Partial views registration

Pairwise registration consists in aligning the views sequentially while their are acquired.

The fast Point-to-plane strategy proposed by Park [68] has been used, though some mod- ifications have been made to increase accuracy and reduce computational time [53].

A common strategy to accelerate pairwise registration is to reduce the number of treated points. The idea is to consider only a sample of significative points from each view without losing surface information. Note also that the removal of despicable points should also increase registration accuracy so that the method use to perform the sampling is extremely important.

Several sampling techniques have been presented in the literature, including uniform sampling [50] [95] and random sampling [52], among others. In 2001, Rusinkiewicz et al.

presented a new approach called Normal space sampling [77]. The author proposed to discard the points that do not provide useful information by selecting more points from the regions where the normal is different from the other parts of the surface.

In a similar way, a variation of the method proposed by Rusinkiewicz has been used in our registration process. The strategy tries to eliminate redundant information by discarding points from the planar areas and keeping a high percentage of points on the uneven areas of the surface. Detailed information of this approach can be found in [53].

Once the new sets of points have been obtained, the next step is to perform a refined

4.2 Pairwise registration 79

alignment of the views. The method used is a variation of the fast point-to-plane approach proposed by Park [68], from the original Chen’s point-to-plane technique [16]. The idea is to compute the intersection between lines and surfaces using a recursive strategy. Initially, the points from one view (previous view) are projected orthographically onto the XY plane of the camera. Then, a grid composed of 50x50 square cells is generated and scaled so that it contains the projection of all the points. Next step is to project a pointp₀ from the current view to the grid and analyze all the projected points from theprevious view that reside in the same cell, looking for the closest one, obtainingq_p0. The projection of point q_p0 to the normal vector ofp₀ defines a new point p₁, which is actually an approximation of the intersection. This approximation is refined recursively by projecting new pointsp_i until norm(p_i−q_pi) is smaller than a threshold (see Figure 4.2). Finally, the process is repeated for all the points conforming to the current view and a set of correspondences is obtained.

Then, the motion between these views is computed by minimizing the function:

f = 1 N_p

Np

X

i=1

km_i−Rp_i−tk² (4.1)

where N_p is the number of correspondences; m_i is the set of points selected in the former view that have a correspondence in the present view; p_i are the correspondences ofm_i in the present view; andR andt are the rotation matrix and the translation vector that align both views.

The function of Equation 4.1 is minimized by iteratively refiningRandt, using quater- nions [10]. The algorithm stops when the mean of the square errors (distances between correspondences) is smaller than a given threshold. See Park [68] for a extended review of this method.

The pairwise registration method here explained aligns two consecutive views delivering the motion between them; a residue or aligning error is always present due to the intrinsic noise of the technique used in point acquisition. In the registration of small sequences of views, this error can be neglected. However, if the number of views to be registered increases, the propagated error may become larger leading to inaccuracies that produce important misalignments. Error propagation can be tackled by using a multiview regis- tration technique. As we have seen in Section 2.4, several techniques have been proposed

Figure 4.2: Strategy used to compute the intersection between the tangent plane and the surfaceS_q along the orthogonal vector ˆp.

that includemetaview,simultaneous minimization, graph and statistic approaches. Next section proposes a new multiview approach with the aim of dealing with error propagation.