On the representation of rigid body transformations for accurate registration of free-form shapes

Abstract

In this paper we consider representation issues of rigid body transformations based on geometric properties of reflected correspondence vectors. A sufficient and necessary representation of rigid body transformations is put forward followed by an accurate and robust algorithm for the registration of free-form surfaces. The algorithm makes full use of global rigid motion constraints derived from the representation of rigid body transformations and can effectively deal with occlusion, noise, and appearance and disappearance of points. A comparative study based on both synthetic data and real images show that the registration method is accurate and robust.

Introduction

Recent developments in electronics and optic devices have enabled depth information to be directly incorporated into image acquisition systems. The availability of range images is leading to a wide range of useful applications, from object recognition to autonomous navigation, to a full integration of computer vision and graphics. However, unavoidable limitations exist in range image acquisition and processing. First, due to physical constraints such as limited field of view of range cameras, the acquisition of object surfaces often requires a number of images to be taken from different viewpoints. Second, these different views need to be put together into a full 3D model. Thus, techniques for image registration and fusion are required so that quality models can be constructed. In this paper, we limit our attention to the registration of two overlapping range images of free-form surfaces which are represented as two sets of unorganised points described in two different co-ordinate frames. Such sets of points typically represent range image acquisition from a stationary camera with a moving object or, say, a sequence of images acquired from a moving robot platform at different time instants.

Many methods have been proposed to solve the registration problem based on techniques such as scatter matrix [11], iterative closest point (ICP) algorithm [3], [5], [27], extreme points [21], crest line [20], reverse calibration [4], interactive method [23], and geometric histogram [1], among many others. Among these methods, the ICP algorithm implements a natural and practical idea and has attracted much attention from the machine vision community since it was independently proposed in 1992 by several researchers. A version of the ICP algorithm was proposed in [3] to register model and scene data sets. This algorithm minimises the squared distance between the transformed points in one co-ordinate frame and the closest points in another. The advantage of this method is that it can be used for any type of object surfaces and it is generally accurate. The disadvantage is that the algorithm assumes that the description of the scene must be a subset of the model and requires a good initialisation of motion parameters.

In [5], another iterative algorithm was proposed with the same purpose. However, this algorithm instead minimises the squared distance from the transformed points in one co-ordinate frame to the tangent planes at the intersection points between the normal vectors at the transformed points and the surfaces in the second frame. It computes the variation of rigid body transformation at each iteration. The advantage of this algorithm is that it does not require point to point correspondences and is generally efficient [19]. The disadvantage is that it requires organised points and accurate estimation of the intersection points where the tangent planes lie. In [27], another version of ICP algorithm was proposed. In order to speed up the search for the closest points, the algorithm uses a K-D tree representation of image data. The advantage of this algorithm is that it can deal with occlusion, appearance and disappearance of points. The disadvantage is that it requires the experimenter to set a threshold for maximum distance for matched points.

A flowchart depicting the steps in the ICP algorithm is shown in Fig. 1. Since 1992, a number of methods have been proposed to improve every step of this algorithm. For instance, in order to improve step 1, the distances between bitangent points [7], bitangent curves [24], surface signatures [26], and spin images [9] were matched. Also, the correlation of spherical harmonic projections [6] were performed for correspondences from which a crude motion can be estimated. In order to improve step 2, colour information [10], normal vector [7], and a number n of invariants [18] associated with each point were incorporated so that distances were defined in 6 or (3+n)-dimensional space rather than just in 3D space.

Step 3 represents the crucial aspect of the algorithm and, in order to improve it, the K-D tree representation of image data [10] and the grid closest point transform [25] were used to speed up the search for closest points. In this step, once possible matching points are found, their correct evaluation (as true or false matches) will determine the overall robustness and accuracy of the registration. For this reason, most work so far on variants of the ICP method have focused on the evaluation of whether matching points represent plausible point correspondences or not. Unfortunately, such evaluation has proven difficult and various schemes have been proposed, highlighted as follows. In [10], a threshold is required for the maximum distance for matched points. In [17], [27], the orientation consistency and the normal vectors associated with the matched points were validated. In [15] and [12], [13], respectively, geometric properties of correspondence vectors and reflected correspondence vectors were used to cope with false matches. In [22], the boundary points are discarded since they are more likely to yield false matches.

Once more accurate sets of correspondences have been established (here more accurate means that points are determined by a method that yields better results than simply taking the point’s nearest neighbour as its correspondence), motion parameters can be estimated using algorithms such as based on the constraint least squares method, quaternion method [3], or dual quaternion method [27]. In practice, due to false matches and noise corrupting the data, the objective function usually has a large number of local minima. Thus, iterative algorithms such as the ICP are very likely to converge to a local minimum. In order to overcome this, a number of techniques can be used with good results, such as simulated annealing as proposed in [16].

A summary of the techniques highlighted above used to improve the standard ICP algorithm is depicted in Table 1. An overall analysis reveals that these techniques are mainly based on local invariants described in a single co-ordinate frame. In theory, local invariants can be extracted from image data but, unfortunately, these invariants are sensitive to noise, occlusion, appearance and disappearance of points. In contrast, invariants defined by points described in different co-ordinate systems are termed global invariants such as the ones used in this paper. Since the estimation of global invariants makes use of redundant data their estimation is more robust to noise, occlusion, appearance and disappearance of points. The method presented in this paper is based on geometric relationships described by two co-ordinate frames bridging the gap between correspondences before and after a rigid motion. This proposed method is consistent with the nature of the data which are normally available leading to accurate and robust registration results.

The standard ICP algorithm uses only a distance constraint to define the position of correspondences and this leads to a number of false matches at every iteration. Thus, the key factor for a successful application of the ICP method is the rejection of false matches. Since false matches are created by the motion, we argue that their elimination must consider the properties of the motion. Thus, in this paper, we first reconsider the representation of rigid body transformations aiming at developing new constraints to evaluate correspondences and eliminate false matches. While the main idea of this paper follows similar reasoning as described in [14] where the projected distance, angle, and projection information are used, in this paper the distance and the projected distance between a point and its reflected correspondence are used. Doing so is justified as such distances are geometrically intuitive and thus can further improve our understanding of rigid body motions which, in turn, lead to the specification of effective constraints for the elimination of false matches.

An earlier version of the extended algorithm presented here has been described in [13]. Also, while the algorithms presented in [12], [13], [15] employ global rigid motion constraints, this paper makes the further distinction between correspondences established by different mappings implied by the traditional ICP criterion. These mappings are one-to-one, many-to-one and one-to-many. To our knowledge, existing ICP-based registration algorithms treat equally all correspondences established by these different mappings. Even though it cannot be said for certain that the one-to-one mapping of established correspondences are plausible, it is certain, however, that the vast majority of the established correspondences by other mappings are not plausible. Thus, such distinction greatly facilitates accurate and robust estimation of the parameters of interest especially for data that are highly corrupted by outliers. This, in turn, reduces the complexity of the process of eliminating false matches.

Similar observation was made in [22] where either point of a correspondence lying on a triangular mesh boundary is considered to yield false matches and is, thus, discarded. However, this assumption does not imply for certain that it is a false match, so at least one correspondence established by many-to-one and one-to-many mappings implies a false match. This paper considers this situation in detail through a comparative study based on both synthetic data and real images. It is shown that the extended algorithm presented here is accurate and robust for the registration of free-form shapes with a small motion and that it is superior to the algorithms presented in [12], [13], [15].

The rest of this paper is structured as follows. Section 2 provides a representation of rigid body transformations derived from the geometric properties of reflected correspondence vectors. Such representation provides explicit constraints to rigid motion forming the basis for the proposed registration algorithm. Section 3 presents the extended registration algorithm, and Section 4 presents experimental results. Finally, Section 5 presents a detailed discussion on some relevant issues in range image registration and some conclusions are drawn.