Estimating the pose from 3D correspondences, point-to-point, point-to-line and point-to-plane correspondences, is known as the 3D registration problem in the literature [25, 24, 26, 40, 4, 41]. It is one of the fundamental problems in computer vision with a wide range of applications, such as simultaneous localization and mapping (SLAM) [45, 44, 23, 30, 37], extrinsic calibration [47, 38, 22, 11, 49] and iterative closes point (ICP) framework . Besides, some camera pose estimation problems, such as the perspective-n-point (PnP) problem [12, 16] and the perspective-n-line (PnL) problem [42, 29], can be transformed to a 3D registration problem . However, the research on this problem is not as thorough as other pose estimation problems.
Previous works mainly focus on solving the least-squares problem. Although large progress has been made, achieving the optimal solution and real-time performance is still a challenge. Some algorithms [26, 4] are capable of finding the global minimum, however, their running time makes them not suitable for real-time applications. Recently, Wientapper  provided an efficient algorithm, but their algorithm can not achieve the global optimality in theory, and has the risk of no solution. Furthermore, previous works generally assume that the correct solution is the global minimizer. However, we find that the correct solution may come from a local minimizer in certain configurations.
Minimal solution is important to eliminate outliers in the RANSAC framework
. As the pose has 6 degrees of freedom (DoF), any combinations of 3D correspondences providing 6 independent constraints form a minimal configuration. Previous algorithms are proposed to solve these minimal configurations case by case[6, 32, 31]. The contributions of this paper are as follows:
First, we prove that there exist ambiguous configurations for any number of plane and line correspondences, which leads to multiple solutions. When a configuration approximates ambiguity, the correct solution of a problem may come from a local minimum. Therefore, previous works [25, 26, 4] that only compute the global minimizer will fail in this case. Revealing local minimizers is essential for an algorithm to handle all the configurations.
Second, we present an efficient and accurate solution for the least-squares 3D registration problem. We use the CGR parametrization to represent the rotation, which generates a rational cost function. We derive its first-order optimality conditions. They form a high order polynomial system, and are hard to solve. Four intermediate unknowns are introduced to relax the original problem, which results in much simpler first-order optimality conditions. Gröbner basis method  is applied to solve this equation system. Then we refine the solution by the Newton-Raphson method.
Third, we present an unified solution for the potential minimal configurations. Previous algorithms are proposed to solve the minimal problems case by case [6, 32, 31]. We also use the CGR parametrization to represent the rotation. which generates a third order equation system for the minimal configurations. Three intermediate unknowns are introduced to simplify the equation system. We introduce a novel hidden variable method  to solve the resulting second order equation system for the rotation matrix. Then the translation can be computed from a linear system.
We evaluate our algorithm with synthetic and real data. Extensive experimental results show that recovering the local minimal is essential, especially when the number of correspondences is small. For a small , our algorithm significantly outperforms previous works [25, 26, 4, 41]. Besides, experimental results verify that our algorithm can converge to the global minimizer with the same accuracy as the previous work with guaranteed globally optimality [26, 4], however, our algorithm is much faster.
2 Related Works
This paper focuses on the pose estimation from point-to-point, point-to-line and point-to-plane correspondences correspondences. Pose estimation from point-to-point correspondences has been solved in early works [1, 13, 14]. There exist closed-form solutions for this problem. However, estimating pose from point-to-line and point-to-plane correspondences is more complicated. These 3D correspondences actually yield similar distance function, which is employed by previous works to construct a general cost function [25, 24, 26, 40, 4, 41]. The difference of various cost functions lies in the parameterization of the rotation matrix. The raw rotation matrix is adopted in [24, 4], and the quaternion  is used in [25, 26, 40, 41]. Previous works mainly focus on finding the global minimizer of the resulting cost function. In , they proposed a provably optimal algorithm. They employed convex underestimators with branch-and-bound methods to iteratively compute the global minimum of the cost function. Although this algorithm can guarantee global optimality, it is very time-consuming. Olsson  reduced the computational complexity by applying the Lagrangian dual relaxation to approximate the cost function. Their experimental results showed that a single convex semidefinite program can well approximate the original problem. Recently, an improved result is obtained by a strengthened Lagrangian dual relaxation . Although without theoretical guarantees, their experimental results show that this algorithm can achieve the global minimum. Although these algorithms [24, 4] have made progress in efforts to reduce the computational time, they are still not suitable to demanding real-time applications. Recently, Wientapper  provided an efficient algorithm based on Gröbner basis polynomial solver. They introduced the first order derivatives of the norm-one constraint of the quaternion into the first order optimality conditions of the cost function as , rather than apply the Lagrangian formulation for the norm-one constraint of the quaternion. This results in an efficient solution, however, their result is not optimal in theory. Furthermore, as the number of equations is greater than the number of unknowns, the equation system may have no solution. They introduced 4 pre-rotations to solve the problem 4 times to handle this problem. The pre-rotation is to make the no-solution problem happen less likely, however, it can not fully solve this problem in theory.
Minimal solution is essential in the RANSAC framework  for robust pose estimation. Any combination of the 3D correspondences resulting in 6 independent constraints forms a minimal configuration. In the literature, specific algorithms have been proposed to solve certain configurations. In , a minimal solution for three line-to-plane correspondences were proposed. In [39, 48], they studied the minimal solution of a more specific line-to-plane configuration, , lines are all on a plane. As one line-to-plane correspondence can be treated as two point-to-plane correspondences, these problems can be formulated as a point-to-plane registration problem. Naroditsky  presented a solution to 6 point-to-plane correspondences. Ramalingam  solved all the point-to-plane minimal configurations. They designed specific intermediate coordinate systems to simplify each minimal configuration. This method is not convenient in the application, as it needs to implement a bunch of algorithms.
3D registration without correspondence is an related but more complicated problem. Several works [20, 27, 43, 50] presented globally optimal solution for 3D point registration without correspondence. The ICP framework  gives a general way to find correspondences and pose at the same time. Although this framework was originally designed for points, other 3D models can also be introduced into this framework [35, 5, 36]. The ICP framework iteratively finds the nearest elements as correspondences then calculates the pose until it converges. Efficiently and accurately estimating the pose from current nearest 3D correspondences is critical for the ICP framework.
3 Problem Formulation
In this paper, we use italic, boldfaced lowercase and boldfaced uppercase letters to represent scalars, vectors and matrices, respectively. The paper focuses on the problem of pose estimation from point-to-plane, point-to-line and point-to-point correspondences. We first consider the point-to-plane correspondence. Suppose we have a pointin one coordinate system and the corresponding plane in another coordinate system represented by the plane’s norm-one normal and a point lying on it. Then the scalar residual between a point and a plane can be written as
For the point-to-line correspondence, we can calculate the 3-dimensional residual vector between a point and the corresponding line represented by the norm-one direction and a point on it as
where is the identify matrix. Lastly, a point-to-point correspondence yields a -dimensional residual vector
and have 6 DoF. Our problem is to calculate and from any configuration of point-to-plane, point-to-line, and point-to-point correspondences that has at least 6 effective constraints. Let use define as the number of effective correspondences. When , this is a least squares problem described in section 4. When , this forms a minimal problem solved in section 5.
4 Least Squares Solution
where represents the determinant of .
4.1 Ambiguous Configurations
In previous works, the global minimizer is treated as the optimal solution. It is well known that there is an unique solution for at least three point-to-point correspondences, except for the degenerate configuration, all the points are collinear. However, for plane and line, there exist configurations which have several solutions. We call them as ambiguous configurations. The ambiguous configuration differs from the degenerate configuration. The ambiguous configuration has several solutions, but the degenerate configuration has infinite solutions. Formally, we have the following lemma for point-to-line correspondences. Proof is in the supplementary material.
For any points and any and , there exist lines to make and are exact solutions for the point-to-line correspondences.
Similarly, we have lemma 2 for point-to-plane correspondences.
For any points and any , and , there exist planes to make , and are exact solutions for the point-to-plane correspondences.
Finally, we have the following theorem for point-to-line and point-to-plane correspondences.
For any points on lines and points on planes and any and , there exist lines and planes to make and are exact solutions for the point-to-line and point-to-plane correspondences.
When measurements approximate an ambiguous configuration, a prior is required to identify the correct solution. The prior is generally available in real applications. For instance, we generally have a rough estimation of the pose between two sensors in the extrinsic calibration problem [47, 38, 22, 11, 49]. For the pose estimation in SLAM [45, 44, 23, 30, 37], current pose should be consistent with the previous motion trajectory.
4.2 Rotation Parameterization
Solving for the rotation matrix is the crux for pose estimation. Previous works [25, 26, 4, 41] adopt non-minimal representations for , which results in additional quadratic constraints in the minimization problem. This paper adopts the CGR parametrization [12, 21] which gives a minimal representation for the rotation matrix, removing the quadratic constraints in (4) . The CGR parametrization expresses a rotation matrix as
where is a 3-dimensional vector, and
is the skew-symmetric matrix of.
4.3 Rational Formulation of Residual
We will show that the residuals from point-to-point, point-to-line and point-to-plane correspondences have a general form
where and are two -dimensional vectors and is a scalar. We first consider the point-to-plane residual in (1). It is obvious that and . Let us then consider the point-to-line residual. We define and , where and are the three rows of . Using this notation, equation (2) can be written as
It is easy to find that and . Lastly, we consider the point-to-point correspondence. Similar to the point-to-line correspondence, we define and . Substituting this notation into (3), it is easy to find
4.4 First-Order Optimality Conditions
where is a 6th order polynomial in and . For the point-to-line distance in (4), we have
As mentioned above, and all have the same form as . Therefore, the point-to-line distance will have the same form as (10). Similarly, will also yield the same form as (10). After the summation of squared residuals in (4), we know that would have the same rational form as (10). We write it as
where is a 6th order polynomial function in and . To find the critical points of (12), we first calculate its first order optimality conditions as follows:
Canceling the denominator of (13) yields
and are of degree 5. Therefore, and are of degree 7 and 5, respectively. Although the Gröbner basis method gives a general way to solve the polynomial system, it is computationally demanding and numerically unstable to apply it to a high order polynomial system, as the experimental results in Fig. 1. The Newton-Raphson method provides an alternative way to find the roots of (14). Denote an equation system as . For the th iteration, the Newton-Raphson method updates the solution as
This iterative method requires an initial solution and . In the next section, we will describe how to calculate an accurate and .
4.5 Initial Estimation from Relaxation
Now we consider the least-squares problem (4) for the new unknowns. We stack the residuals from point-to-plane, point-to-line, and point-to-point correspondences to get
where is a matrix and is a matrix. Then (4) can be written as
As (19) is linear for , then has a closed-form solution as
where . The elements in the vector are second order monomials except for the constant term. Thus (22) is a fourth order polynomial function. We compute the first order optimality conditions to get all the stationary points. This first-order condition contains four third order polynomials for and as
According to Bézout’s theorem , there exist at most solutions. We find that the polynomial system in (23) only contains third and first degree monomials. This equation system stratifies the 2-fold symmetry [18, 2]. That is to say if any nontrivial is a solution, is also a solution. Thus there are at most 40 independent solutions. We adopt the algorithm introduced in  to generate the polynomial solver which can utilize this property to yield an efficient solution.
After solving (23), we are then able to compute from (21). The CGR parameters and can be recovered from the definitions in (16) and (17). The above formulation treats as independent unknowns. However, they are related as they are functions of and . To recover the minimizer of (12), we refine the solution using the Newton-Raphson iteration method in (15). We summarize our least-squares solution in Algorithm 1.
5 Minimal Solution
Unlike most pose estimation problems having an unique minimal configuration, our problem has multiple minimal configurations. Specifically, any combination of 3D correspondences with 6 effective constraints forms a minimal configuration. Table 1 lists the minimal configurations. We use the pattern PtLPl to represent point-to-point, point-to-line and point-to-plane correspondences. As one point-to-point, point-to-line and point-to-plane correspondence provide 3, 2 and 1 constraints, respectively, it is easy to verify that the configurations in Table 1 are minimal configurations, except for the case PtLPl (2 point-to-point and 1 point-to-plane correspondences). It seems that PtLPl has 7 constraints, as 2 point-to-point correspondences seemingly provide 6 constraints and 1 point-to-plane correspondence provides 1 constraint. However, the collinear configuration is degenerate for point-to-point correspondences . Two point-to-point correspondences form a collinear configuration. One additional constraint is required to recover and . We propose an unified solution for these minimal cases.
Let us first consider the rational equation from (9). Its numerator, , is a third order polynomial. Assume that we have 6 equations in the form of (9). This will lead to an equation system with 6 third order equations, which is hard to solve. We can not use the intermediate unknowns defined in (16) and (17) to the minimal problem, since this will result in 7 unknowns, but there are only 6 constraints. Here we introduce 3 new intermediate unknowns as
Define . Then (9) can be rewritten as
This is a quadratic equation in and a linear equation in the new unknowns . Compared to the original equation from the numerator of (9), the new one is much simpler. For the minimal problem, we have 6 effective equations as (26). Let us divide the 6 equations into two groups. Each group has 3 equations as
We can easily calculate using the first three equations as
where () is an element of .
5.1 Hidden Variable Polynomial Solver
Here we present a hidden variable method  to solve (29). We treat one unknown as a constant. This unknown is called the hidden variable. Without loss of generality, we treat and as unknowns, and as a constant. Thus (29) can be rewritten as an equation system in and as
where , , . Then we introduce an auxiliary variable to convert to a homogeneous equation , , every monomial in is of degree 2. This leads to the following homogeneous equation system:
If we treat , and as variables, we can rewrite (31) as a linear homogeneous system in , and as follows
where , , , . Let us denote the coefficient matrix of (32) as . If the polynomial system (31) has a nontrivial solution, there will exist a nontrivial solution for the homogeneous linear system (32). According to the algebraic theory, (32) has a nontrivial solution if and only if the determinant of is zero, , . Define . has the same form as in (31). Similarly, we can construct two homogeneous linear systems for and . This will yield two homogeneous equations and , respectively. Stacking () together, we have a homogeneous linear system as
where is a matrix with polynomials in as elements, and . If (33) has a nontrivial solution, if and only if . This is an eighth order polynomial equation in . We can get by solving this equation. Comparing (30) and (31), we can find that if is a solution of (30), is a solution of (31), and vice versa. After we get the solution of , we can back-substitute it into (33) and set , then solve the resulting linear system for and . After we get , we can recover by (5) and calculate using (28). Then can be recovered from (24). We summarize our minimal solution in Algorithm 2.
In this section, we compare our algorithm with the state-of-the-art algorithms, including Briales’s algorithm , BnB , Olsson’s algorithm  and Wientapper’s algorithm . We also consider the point-to-point case alone, as it has closed-form solution. We adopt Arun’s algorithm  implemented in OpenGV . The algorithms are evaluated in terms of accuracy and computational time using synthetic and real data.
|Method||KITTI 03||KITTI 04||KITTI 07|
|Err ()||Err ()||Time (s)||Err ()||Err ()||Time (s)||Err ()||Err ()||Time (s)|
6.1 Experiments with Synthetic Data
Our synthetic data is generated as . Specifically, each geometric element is determined by randomly sampling a point within a sphere of radius . For lines and planes, a random unit direction and normal are generated. We uniformly sample the Euler angles of the rotation matrix ( and
). The translation elements are uniformly distributed within. We use and to represent the estimated rotation and translation and use and to represent the ground truth. The rotation error is evaluated by the angle of the axis-angle representation of , and the translation error by . We consider the effective number of correspondences as . Specifically, the effective number of correspondences for point-to-plane, point-to-line, and point-to-point correspondences is calculated as . Given an , we randomly generate a combination of , and whose effective number of correspondences is .
|Our least-squares algorithm||2.95|
|Our minimal algorithm||0.296|
Effect of intermediate unknowns We introduce intermediate unknowns to simplify the least-squares and the minimal problems. To verify their benefit, we evaluate the performance of the direct least-squares solution (DLSSol) from solving (13), and the direct minimal solution (DMinSol) from solving the equation system from the numerator of (9) for the minimal problem. We employed the algorithm in  to generate the solvers. For the least-squares problem, we run 2000 trails for each . Fig. 1 shows the results. Directly solving (13) can recover the global minimizer of (4) in theory. However, the large mean errors of DLSSol verify that this polynomial solver is very unstable. For the minimal problem, we run 20000 trails for the 6 point-to-plane configuration. Fig. 2 shows the results. The DMinSol has a much longer tail than our algorithm.
Table 3 lists the average computational time of different algorithms. For the least-squares problem, we compare the computational time of solving the first order optimality conditions (13). This time is independent of the number of correspondences. As shown in Table 3, our least-squares solution is about 7 times faster than DLSSol, and our minimal solution is about 2 times faster than DMinSol.
The above results verify that our intermediate unknowns can increase the numeric stability as well as reduce the computational time.
Least-squares solution We conduct experiments to evaluate the performances of different algorithms under varying number of correspondences, increasing level of noise and computational time. The results of all experiments are from 100 independent trials as .
The first experiment considers a fixed noise level and an increasing number of correspondences. Let us denote the standard deviation of a zero mean Gaussian noise distribution as. We set . varies from to . Fig. 3 shows the result. For the pose-to-point case, most of the algorithms can recover the optimal solution except for  when the number of points is small. For the mixed correspondences, the results of BnB  and Briales’s algorithm  overlap. This is consistent to the results in . Our algorithm significantly outperforms previous works when is small as the ambiguous case more likely happens for a small . When is large, our algorithm achieves the same accuracy as  that has guaranteed global optimality. Wientapper’s algorithm  can not find the optimal solution even is large.
In the second experiment, is from to , stepping by 0.02m. The results are illustrated in Fig. 4. For the point-to-point case, most of the algorithms provide the optimal solution as , except for  when . For the mixed case, BnB, Briales’s algorithm and our algorithm present better results than other algorithms. Our algorithm gives a slightly better result than BnB and Briales’s algorithm when .
In the last experiment, we compare the computational time of different algorithms. We vary the effective number of correspondences from 10 to 2000. For every , we run each algorithm 100 times to calculate the average running time. Fig. 5 provides the result. We did not run the BnB algorithm , because it is too slow for a large . It is clear that our algorithm is the fastest one among the compared algorithms. Wientapper’s algorithm  is efficient. But their algorithm needs to compute a problem 4 times, thus the gap between the computational time of two algorithms increases as enlarges. Actually, could be 10 times larger in real applications than our simulation, as shown in Table 4.
Minimal solution The equation system (29) is important for the estimation of . Actually, Ramalingam  formulated some point-to-plane configurations as a quadratic equation system as (29). We first consider the numeric stability of the polynomial solver for (29). We compare our hidden variable method, E3Q3 , and the Gröbner basis polynomial solver generated by . We randomly generate a real solution and the coefficients of (29). Then we substitute this real solution into (29) to calculate the constant terms of (29). We run each algorithm 20000 trails and compute the estimation error of this real solution. Fig. 6 (a) shows the estimation error histograms of the compared algorithms. It is clear that our hidden variable method is more stable than other algorithms. The histograms of E3Q3 and the Gröbner basis method have long tails. This is because E3Q3 and the Gröbner basis method require to compute the inverse of a matrix. When a matrix approximates a singular matrix, the performance of these algorithms will degrade. Fig. 6
(b) demonstrates the performance of E3Q3 under a degenerate case. We generate nearly singular matrix for E3Q3, whose least singular value is in. In this situation, E3Q3 performs very bad.
We also evaluate the numeric stability of our unified minimal solution. We run 2000 independent trails for each minimal configuration. As demonstrated in Fig. 7, our algorithm provides accurate solutions for all minimal configurations. This avoids solving each configuration case by case.
|Sequences||Average Correspondences Per Frame|
6.2 Experiments with Real Data
We generated our real-world dataset from the KITTI dataset . Our dataset contains point-to-point, point-to-line and point-to-plane correspondences described in Table 4. There are more than 20000 correspondences in each frame, with the majority of them being point-to-plane correspondences. We evaluate the accuracy and running time of different algorithms on this dataset.
For each frame, 2D feature points are detected by the ORB feature detector . We project LiDAR points into the image plane, and select the LiDAR points around an ORB feature. Then, we fit a local plane to these LiDAR points. Finally, we calculate the 3D coordinates of an ORB feature by calculating the intersection of the back-projection ray of the ORB feature and the local plane. We use the ORB descriptors to match 2D feature points, and obtain the 3D point-to-point correspondences.
Next, we generate point-to-line correspondences. For each frame, 2D lines are detected by the Line Segment Detector (LSD)  and described by the Line Band Descriptor (LBD) . A 2D line is represented by two 2D endpoints. The 3D endpoints of a line is generated as the 2D feature points described above. We then generate 3D line correspondences by matching their LBD features. Given a line-to-line correspondence, two point-to-line correspondences are generated for each of its endpoints.
Finally, we calculated point-to-plane correspondences. For each frame, planes are extracted from LiDAR points by the region growing algorithm . To match a plane point to a previous plane, we find the nearest LiDAR point in previous planes.
We evaluate the performance of different algorithms on sequences 03, 04, and 07 of the KITTI dataset. We did not test BnB , as it is extremely slow on this large dataset. Our algorithm achieves the same or slightly better result as the state-of-the-art algorithm  while being around 7 times faster, as shown in Table 2.
In this paper, we present an efficient and accurate least-squares solution and an unified minimal solution for the 3D registration problem. We proof that there exist ambiguous configurations for any number of point-to-plane and point-to-line correspondences. This requires an algorithm has the ability to reveal local minimizers. We use the CGR parameterization to represent the rotation, removing the quadratic constraints on the rotation. However, this results in a rational form residual which is hard to solve. We introduce several intermediate variables to simplify the first-order optimality conditions of the least-squares problem, and the equation system of the minimal configuration. We evaluate our algorithm through synthetic and real data. The experimental results show that computing local minimizers is essential, especially when is small. Besides, our algorithm is as accurate as previous globally optimal solutions when is large, but is much faster.
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Lemma 1. For any points and any and , there exist lines to make and are exact solutions for the point-to-line correspondences.
Define the th point of an arbitrary point set with points. For any and , we can have and . Then we can use and to define a line with direct and a point on it, as demonstrated in Fig. 8 (a). According to how we construct , we know that passes through and . Therefore and are two solutions for the point-to-line . ∎
Lemma 2. For any points and any , and , there exist planes to make , and are exact solutions for the point-to-plane correspondences.
Define the th point of an arbitrary point set with points. For any , and , we can have , and . Then we can find a plane passing through , and , as demonstrated in Fig. 8 (b). According to how we construct , we know that , and are three solutions for the point-to-plane correspondences .
Theorem 1. For any points on lines, points on planes and any and , there exist lines and planes to make and are exact solutions for the point-to-line and point-to-plane correspondences.
We first consider the points on lines. Define is the th point. According to Lemma 1, we can find lines to make and are exact solutions for the point-to-line correspondences .
Then we consider the points on planes. For the th point within them, we can define and . Let us denote as a plane passing through the line defined by and . According to how we construct , we know that and are two solutions for the point-to-plane correspondences .
Therefore, and are the two solutions for the point-to-line correspondences and point-to-plane correspondences