Affine iterative closest point algorithm for point set registration

Abstract

The traditional iterative closest point (ICP) algorithm is accurate and fast for rigid point set registration but it is unable to handle affine case. This paper instead introduces a novel generalized ICP algorithm based on lie group for affine registration of m-D point sets. First, with singular value decomposition technique applied, this paper decomposes affine transformation into three special matrices which are then constrained. Then, these matrices are expressed by exponential mappings of lie group and their Taylor approximations at each iterative step of affine ICP algorithm. In this way, affine registration problem is ultimately simplified to a quadratic programming problem. By solving this quadratic problem, the new algorithm converges monotonically to a local minimum from any given initial parameters. Hence, to reach desired minimum, good initial parameters and constraints are required which are successfully estimated by independent component analysis. This new algorithm is independent of shape representation and feature extraction, and thereby it is a general framework for affine registration of m-D point sets. Experimental results demonstrate its robustness and efficiency compared with the traditional ICP algorithm and the state-of-the-art methods.

Introduction

Point set registration is one of the important research issues in pattern recognition and computer vision due to its wide applications such as face recognition (Abate et al., 2008), computer aided design (Shammaa et al., 2007) and 3D reconstruction (Megyesi et al., 2006), etc. However, point set registration is of great difficulty in three aspects: (1) point correspondence is unknown, (2) transformation is unknown, and (3) there is not enough information about the physical properties of objects. One of the accurate and efficient approaches to solve the problem above is the iterative closest point (ICP) algorithm (Besl and McKay, 1992, Chen and Gerard, 1992, Zhang, 1994) which has been widely used in a variety of fields.

In the past decade, a number of scholars have studied the traditional ICP algorithm on how to speed it up. Fitzgibbon (2003) employed the Levenberg–Marquardt algorithm to accelerate ICP, and Jost and Hugli (2003) combined a coarse-to-fine multi-resolution technique with the neighbor search algorithm into ICP to improve the registration. Moreover, many scholars have introduced other methods into ICP for its more robustness. Lee et al. (2000) proposed a measure for estimating the reliability of ICP. Invariant features were described by Sharp et al. (2002) to decrease the probability of being trapped in a local minimum. Granger and Pennec (2002) added Expectation–Maximization principles to ICP and used a coarse-to-fine approach based on an annealing scheme to improve the robustness. Liu (2006) combined the SoftAssign and EMICP algorithms for the automatic registration of overlapping 3D point clouds. Phillips et al. (2007) proposed a fractional ICP algorithm which can identify and discard outliers. As for the overview of point set registration using ICP method, one can refer to (Planitz et al., 2005, Makadia et al., 2006, Smith et al., 2008) a step further.

The original ICP algorithm has been widely applied to rigid registration, but it does not work well in non-rigid registration problems. Du et al. (2007) proposed an extension of the ICP Algorithm for scaling registration. Amberg et al. (2007) used an additional stiffness term and a landmark term for locally affine regularization, which is based on the extended ICP algorithm. Some scholars managed to solve non-rigid registration problems but they avoided using the ICP algorithm. Jian and Vemuri (2005) proposed a robust approach for non-rigid registration by using mixture of Gaussian. Chui et al., 2004, Wang et al., 2008 presented algorithms to register multiple unlabeled point sets to an emerging mean shape.

In non-rigid registration, affine registration is an important and key problem. When affine transformation is considered to register two point sets without noise, the error of least square (LS) problem is 0, but affine transformation is not unique. For example, if two point sets are best matched with the true affine transformation, the error is 0. However, if the affine transformation is close to 0, that is, all points of one point set are transformed to register one point of the other point set, and then the error is also close to 0. To solve this ill-posed problem, the usual way is to add regularization terms. Feldmar and Ayache (1996) introduced the normal and the principal curvatures in points coordinates. It is apparently time-consuming to compute the normal and the principal curvatures and search the closet points. Meanwhile, much information of point sets is needed. For instance, the interior and the exterior of objects need to be known to obtain the orientation of the normal. Moreover, some scholars managed to avoid using optimization to guarantee uniqueness of affine transformation. Ho et al. (2007) reduced the general affine registration problem to that of the orthogonal case with covariance matrices. However, its result is not quite accurate. In addition, some scholars have tried to use probabilistic point set registration methods which soft assign correspondence for registration. Chui and Rangarajan (2003) proposed the thin plate spline – robust point match (TPS–RPM) algorithm which used the soft assignment of weights for non-rigid registration. Furthermore, probabilistic point matching (PPM) and coherent point drift (CPD) were proposed by McNeill and Vijayakumar, 2006, Myronenko et al., 2006, Myronenko and Song, 2009, respectively. These methods need to establish correspondence of all point sets, so their speed is much slower.

Different from above-mentioned works, we propose a new approach for this ill-posed problem without requiring any extra information of objects. This approach decomposes affine transformation into three special matrices with singular value decomposition (SVD) technique. To avoid the phenomenon that affine transformation is close to 0, these matrices are all given physical meaning and constraints are then added to these matrices. Hence, the affine registration problem turns out to be a constrained optimization problem, which is solved by affine iterative closest point algorithm, an extension of the ICP algorithm. At each iterative step, these special matrices are expressed, respectively by exponential mappings of lie group and their Taylor approximations. Ultimately, the constrained optimization problem is simplified to a quadratic programming problem which is easy to be solved. Similar to the ICP algorithm, the affine ICP algorithm based on lie group converges monotonically to a local minimum. To get the best matching between two m-D point sets, independent component analysis (ICA) is applied to estimate initial parameters and constraints. This novel proposed algorithm has been tested in experiments and the experimental results demonstrate that our presented algorithm is a robust technique to solve affine registration problems, and it can be used in a wide range of applications.

This paper is organized as follows. In Section 2, the ICP algorithm is reviewed briefly and lie group is stated. In Section 3, a general LS problem is presented and a proposed method – the affine ICP algorithm is given. Following that is Section 4 in which the proposed technique is evaluated on the experiments and a conclusion is finally drawn in the last section.