2.4 Multiview Minimization
2.4.2 Simultaneous Minimization
a=
qx
√1−q02
qy
√1−q02
qz
√1−q02
; θ= 2arcos(q0) (2.4)
Note that the rotation by an axisa and the angle θ, are described by the quaternionq.
q=
q0
qx
qy
qz
=
1 2
ptrace(R)
1 2
r3,3−r3,2
√trace(R)
1 2
r2,1−r2,3
√trace(R)
1 2
r1,2−r1,1
√trace(R)
, whereRis the total error rotation matrix (2.5)
When compared to Pulli’s approach, this method does not require to acquire and reg- ister all the views previously, allowing to simultaneously localize de vehicle and mapping the environment in real time. In addition, Nuchter approach turns out to be more reli- able in large environments since loop constrains allow to reduce the accumulated error.
However the success of this method depends on a correct initial pose estimation of the vehicle. Another challenge appears when the robot has to deal with featureless environ- ments, requiring the use of specific methods such as Kalman Filter extensively explained in Section 2.4.4.
2.4 Multiview Minimization 27
Some authors have proposed several techniques in order to register multiples views and minimize the accumulated error simultaneously. These techniques make use of all the acquired views at the same time and have been classified as simultaneous minimization techniques.
Signed distance Field of Masuda
In 2002 Masuda proposed a new method that registers and integrates all views simultane- ously, rejecting outliers in the iterative process. The method is based on Matching Signed Distance Fields and works as follows: Initially a coarse registration is carried out in order to transform all views to a common reference frame, where the data shapes are integrated.
In the next step, data shapes are alternatively registered and integrated in an iterative process until convergence is achieved. The main idea is to generate a grid of arbitrary 3D points that are used as key points for the Signed Distance Field approach. The goal is to establish correspondences (the closest point) between the key points and the 3D points of the object surface, computing the distance between them. Once correspondences have been obtained the new motion parameters are calculated and the process is repeated again until residual error converges. Note that points are weighted depending on the computed distances in order to detect and reject outliers.
The main advantage of this method is that all views are registered simultaneously and consequently the error does not accumulate among them. In addition outliers are automatically removed leading to a robust method. On the other hand, the need of having in advance the complete set of 3D views of the object/scene leads to high memory requirements when dealing with large objects, together with the impossibility of working in real time.
Genetic Algorithms
Other authors have proposed the use of methods based on Genetic Algorithms (GA) for multiple view registration, as an alternative to the ICP-based approaches. In 2003 Silva et al. [83] proposed an algorithm to register multiple range images based on GA that allow to deal with low-overlapping views. The main contribution of their method is the introduction of a novel robust measure called the Surface Interpenetration Measure (SIM) that quantifies the visual registration error in order to determine the overlapping area between two partial views. Following this idea, they use the SIM along with a robust
estimator in order to implement a Genetic Algorithm in the transformation space, where the results of the SIM of each chromosome are used in each generation to choose the ones that yield the most accurate transformations. Note that each chromosome is composed by 6 parameters (three components for the translation vector and three components for the rotation angles).
Registration methods based on GAs provide accurate solutions in multiple view regis- tration, presenting interesting solutions to problems such as local minima or error prop- agation. In addition, this kind of algorithms presents good results in the presence of outliers and low-overlaping views. However, all this advantages are subject to the size of the population used. That is, the most important drawback of methods based on GA is the high computational time required to converge to a good solution.
Bundle Adjustment
Other simultaneous minimization methods has been recently proposed. In the last few years, a photogrammetric technique called Bundle Adjustment has increased popularity in the computer vision community and it is also growing in robotics. Bundle adjustment deals with the problem of refining a visual reconstruction to produce jointly optimal 3D structure and viewing parameters (camera pose and/or calibration) estimates [93]. There- fore, bundle adjustment techniques have been used in both robot/camera localization and 3D mapping in many fields such as camera calibration, robot navigation and scene recon- struction providing reliable solutions to the error accumulation in the registration process.
Since bundle adjustment is a non-linear minimization problem, it is solved by means of it- erative non-linear least squares or total squares methods such as Levenberg-Marquardt or M-estimator techniques [24] [53]. Recently, a new improvement of the bundle adjustment was proposed by Mouragnon et al. [58]. The authors proposed a generic real-time method based on bundle adjustment approach that permits an incremental 3D registration and reconstruction of the scene minimizing the angular error simultaneously.
Although bundle adjustment is commonly classified as a multiview technique, some authors have used it in consecutive pairwise alignment as a technique to reduce error propagation [72].
2.4 Multiview Minimization 29