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An automatic 3D point cloud registration method based on regional curvature maps
An Automatic 3D Point Cloud Registration Method Based on Regional Curvature Maps Junhua Sun, Jie Zhang, Guangjun Zhang PII: DOI: Reference:
S0262-8856(16)30159-7 doi:10.1016/j.imavis.2016.09.002 IMAVIS 3553
To appear in:
Image and Vision Computing
Received date: Revised date: Accepted date:
14 August 2012 7 June 2016 22 September 2016
Please cite this article as: Junhua Sun, Jie Zhang, Guangjun Zhang, An Automatic 3D Point Cloud Registration Method Based on Regional Curvature Maps, Image and Vision Computing (2016), doi:10.1016/j.imavis.2016.09.002
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ACCEPTED MANUSCRIPT An Automatic 3D Point Cloud Registration Method Based on Regional Curvature Maps Junhua Sun, Jie Zhang, *Guangjun Zhang
Address: XueYuan Road No. 37, HaiDian District, 100191, Beijing, China Tel: +86-10-82338768
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Fax: +86-10-82316930
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Affiliation: Key Laboratory of Precision Opto-mechatronics Technology, Beihang University, Beijing, China
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*Corresponding author: Guangjun Zhang; E-mail: [email protected] Abstract
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3D point cloud registration is a fundamental and critical issue in 3D reconstruction and object recognition. Most of the existing methods are based on local shape descriptor. In this paper, we propose a discriminative and robust local shape
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descriptor-Regional Curvature Map (RCM). The keypoint and its neighboring points are firstly projected onto a 2D plane according to a robust mapping against normal errors. Then, the projection points are quantized into corresponding bins of the 2D support region and their weighted curvatures are encoded into a curvature distribution
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image. Based on the RCM, an efficient and accurate 3D point cloud registration method is presented. We firstly find 3D
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point correspondences by a RCM searching and matching strategy based on the sub-regions of the RCM. Then, a coarse registration can be achieved with geometrically consistent point correspondences, followed by a fine registration based on a modified iterative closest point (ICP) algorithm. The experimental results demonstrate that the
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RCM is distinctive and robust against normal errors and varying point cloud density. The corresponding registration method can achieve a higher registration precision and efficiency compared with two existing registration methods. Keywords: 3D point cloud; local shape descriptor; regional curvature distribution; 3D surface registration
1. Introduction
3D surface registration is a crucial and active issue in computer vision with numerous applications including 3D modeling, object recognition, scene understanding, 3D shape detection, etc. [1, 2] Due to the self-occlusion of 3D object, 3D scanner can only obtain a partial 3D point cloud associated with a single coordinate system from one viewpoint. In order to reconstruct the whole 3D shape, we must transform the 3D point clouds captured from different viewpoints into a common coordinate system according to rigid transformations. The aim of 3D surface registration is to compute the rigid transformation between 3D point clouds and recover the complete 3D shape of the object automatically. Most of the existing shape-based 3D registration methods consist of initial registration and fine registration. In initial registration, local shape representation and matching [3-8] are crucial steps for recovering a coarse rigid transformation. An accurate initial transformation can improve the optimization efficiency and reduce the optimization error in the following fine registration [9, 10]. So far, fine registration methods are well-developed with the wide 1
ACCEPTED MANUSCRIPT application of iterative closest point (ICP) algorithm [11] and its improved algorithms [12-15]. However, shape-based initial registration is still a challenging and active issue. The existing shape descriptors can be generally classified as global descriptors or local descriptors according to the description region. Global shape descriptor [3, 4, 16] performs good discrimination for complete object
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representation, while local shape description [17-21] is more robust against clutter and occlusion. Among local shape descriptors, 3D point-based descriptor has been widely applied to represent partial object due to its good
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generalization. It encodes the information of neighboring points of an interest point in a compact and distinctive fashion. Then, the 3D points with similar local shapes can be identified from cluttered scenes by descriptor matching.
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Earlier local shape descriptors, such as point signatures [17] and fingerprints [18] encoded the information of 3D points along a 1D support region. For richer representation, 2D and 3D support regions have been employed. For
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example, Spin image [19] counted the points falling into the bins of a 2D histogram on a specific plane, forming a rotationally invariant description image. 3D shape context [20] divided its spherical support region into several bins along the three axes of the local reference frame, and counted the number of points in each bin. Harmonic shape
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context [20] transformed the 3D shape context to a 2D descriptor with a spherical harmonic transformation. Whereas, the measurement error of the normal vector of 3D point and the neighboring area with missing data easily influence
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the local shape representation, leading to incorrect point matching and low registration precision.
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In this paper, we propose a distinctive and robust local shape descriptor-Regional Curvature Map (RCM) for 3D point cloud registration. The neighboring points of the interest point are firstly projected onto a specific 2D plane according to a mapping which is robust against the normal error. Then, the position of each point in the 2D support
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region is quantified and its curvature is encoded into a curvature distribution map of the local shape. The RCM exhibits high discrimination and robustness to normal errors and varying point cloud density. Based on the RCM, an automatic 3D point cloud registration method is introduced. We present a RCM searching and matching strategy based on the sub-regions of the RCM, which reduce the influence of the area with missing data on RCM similarity measuring and improve the matching efficiency as well. Lastly, we recover an initial rigid transformation using geometrically consistent point correspondences and achieve a fine registration with a modified ICP algorithm. Experimental results demonstrate the superiority of the RCM-based registration method. The rest of this paper is organized as follows. Section 2 describes the establishment of the RCM descriptor. Section 3 presents the automatic 3D point cloud registration based on RCMs. Section 4 shows experimental results and discussion. Section 5 concludes the paper.
2. Regional Curvature Map In this section, the generation process of the RCM descriptor is detailed. We construct the RCM descriptor of an interest point by capturing a neighboring shape patch centered with the interest point, projecting all the points to a specific 2D plane, and encoding the curvatures of the projection points into a curvature distribution map.
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ACCEPTED MANUSCRIPT 2.1 A Distribution Map of Neighboring Points A 3D shape patch P is extracted from the 3D point cloud as illustrated in Fig. 1a. Each shape patch contains an interest point p and its neighboring points {pi}. We define a local reference frame (LRF) op-xpypzp of the 3D shape
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patch with p as the origin of the LRF, the surface normal of p as the z axis, and the maximum principal curvature orientation dκ1 of p as the x axis. As shown in Fig. 1b, we project all the neighboring point pi(xpi, ypi, zpi) to a specific
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2D plane with a 2D coordinate system op'-xp'yp' according to a mapping M, and form a distribution image of the projection points on the 2D support region. The mapping M is defined as
(1)
x 2pi y 2pi z 2pi is the distance between the neighboring point pi and the origin op of the sphere coordinate
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where
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M( x pi , y pi , z pi ) ( cos , sin ) ,
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system op-xpypzp, and arctan( y pi , x pi ) is the azimuth angle of pi. Therefore, a projection point pi' in the 2D coordinate system op'-xp'yp' is ( x pi , y pi ) ( cos , sin ) . Since the surface normal of the point pi mainly
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determines the pitching angle of pi instead of the azimuth angle θ of pi, the mapping M is less affected by the normal
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error.
Fig. 1 (a) The interest point p and its neighboring points {pi} in the local coordinate system op-xpypzp; (b) the projection points on a 2D plane with the local coordinate system op'-xp'yp'.
2.2 RCM Generation The RCM of the interest point is generated as follows. We resample the original 3D point cloud (shown in Fig. 2(a)) to reduce its density and then obtain a well-organized 3D point cloud (shown in Fig. 2(b)). All the neighboring points {pi} of the interest point p are projected onto a specific 2D plane according to the mapping M, forming a 2D distribution image as shown in Fig. 2c. The maximum principal curvatures of all the 3D points are computed and represented with gray values. Then, the RCM is established by information quantization and encoding as shown in Fig. 2d. 3
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Fig. 2 RCM generation process: (a) original 3D point cloud; (b) sparse 3D point cloud and an interest point p with its local coordinate system op-xpypzp ; (c) The distribution of projection points with maximum principal curvatures on the 2D support region op'-xp'yp'; (d) RCM representation.
Since the projection points {pi'} on the 2D distribution image (Fig.2c) are disordered, we divide the 2D support
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region into ns×ns average bins along the x' axis and y' axis and then quantize the projection points into corresponding bins for unified representation. The center of the bin can be expressed as
Ps {( x, y), x u ds , y v ds , u 0, 1, 2,
ns , v 0, 1, 2,
ns} ,
(2)
where ds is the size of each bin, and Rs = ns∙ds is the radius of the 2D support region.
d d d d The projection points pi'(xi', yi') locating in the area Q | xi x 2s , x 2s , yi y 2s , y 2s
are counted into
the bin (x, y). For each bin, we sum up the weighted curvatures of all the points falling in the bin. Specifically, the curvature of each point is computed as follows. We employ a second order Taylor expansion on the point p to approximate the neighboring surface of the point. The parameters of the approximate neighboring surface can be obtained with the neighboring points pi(xi, yi, zi) according to
x12 ... xi2
x1 y1 ... xi yi
y12 hxx
z1 ... hxy ... . yi2 hyy zi
Then, we can obtain the Hessian matrix of the approximate surface as
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(3)
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2h h
hxy
xx
2 hyy
xy
,
(4)
The maximum eigenvalue of the Hessian matrix is regarded as the maximum principal curvature κ of the point,
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and the corresponding eigenvector of the maximum eigenvalue is the principal direction x of the point. They can be expressed explicitly as
H x x with hxx hyy (hxx hyy )2 hxy 2
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(5)
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Then, the curvatures of the points falling in the bin (x, y) are encoded into a curvature distribution map as 1 nq wii ˆ k ( x, y) Sw i1
pi
if
pi
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if
,
(6)
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where wi exp ( xi x ) 2 ( yi y ) 2 / d s is a Gaussian weight assigned to the projection point pi'(xi, yi), emphasizing the points closer to the center. κi is the principal curvature of the projection point. S w
nq
w is a i
i 1
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normalizer. The area without projection points on the 2D support region is represented as . Finally, we can obtain a
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(2ns+1)×(2ns+1) matrix as the RCM descriptor of the interest point by aggregating the encoding values of all the bins.
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3 Automatic 3D Point Cloud Registration Based on RCMs Given two partial 3D point clouds (S1 and S2) with overlapping areas, our aim is to find a unique rigid transformation between the two views based on the RCMs and then to transform the two views into a common coordinate system automatically. The registration is achieved following three main stages. Firstly, 3D interest points with their neighboring points are extracted and represented as RCM descriptors respectively. Then, the 3D points with similar local shapes in different views are matched based on the RCM representation. In this stage, we employ a point correspondence searching and matching strategy based on the sub-regions of the RCM to reduce the effect of missing areas and improve the searching efficiency as well. Finally, we recover a coarse rigid transformation of the two views with no less than three 3D point correspondences satisfying the geometry consistency, and then determine a fine rigid transformation using a modified iterated closest point (ICP) method.
3.1 Similarity Measure The similarity of two 3D points is measured with the l2 distance between the overlapping filled areas of their RCMs. Thus, the effect of the area with missing data on the RCM descriptor can be eliminated. For a RCM matrix X=[xij], the set F {(i, j ) | xij c } of the RCM is defined as filled area. For two RCMs X and X', the overlapping set F∩F' is regarded as valid area in measuring the l2 distance between the RCMs. Thus, the 5
ACCEPTED MANUSCRIPT valid distance is expressed as 1 2 2 1 xij xij d ( X, X) n ( i , j )F F 0
if
n0 r0 (2ns 1) 2
,
(7)
others
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where n0 is the number of the elements in F∩F', ns is the radius of the 2D support region, and r0 is a threshold of the
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number of valid bins.
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Considering the opposite direction of the principal curvature, we determine the valid distance between the RCMs X and X' as:
(8)
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d X ( X, X ') min(d X, X ' , d ( X, X ')) ,
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where X is obtained by rotating X 180 degrees clockwise.
3.2 Corresponding Point Searching and Matching
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To improve the searching efficiency, we search for the corresponding point of 3D interest point of S1 in S2 by
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matching the sub-regions of their RCMs. The 2D support region of the RCM descriptor is divided into 8 sub-regions R1, R2,…, R8, as shown in Fig. 3. Each sub-region Ri occupies a quarter of the support region. We define the bin with
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enough projection points as a valid bin. When the number of valid bins in a sub-region is more than a certain proportion rv, the sub-region is defined as a valid sub-region. The sub-RCM of a valid sub-region Ri is represented as Xp(Ri). Fig. 4 shows a RCM descriptor with 3 valid sub-regions R1, R2 and R8 as examples.
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Fig. 3
Eight sub-regions of a RCM
A RCM with 3 valid sub-regions R1, R2, R8
For a 3D point correspondence (p1, p2), the RCMs Xp1 and Xp2 representing their local shapes are similar. Correspondingly, their valid sub-regions Ri together with their RCM representation Xp1(Ri) and Xp2(Ri) are similar. For examples, Fig. 5 shows the RCMs of a correct point correspondence with same valid sub-regions R1, R2 and R8 and similar RCM representations X(R1), X(R2) and X(R8). Fig. 6a shows the RCMs of an incorrect point correspondence with different valid sub-regions Ri, and Fig. 6b shows the RCMs of another incorrect correspondence with same valid 6
ACCEPTED MANUSCRIPT sub-regions Ri but different valid sub-RCMs. The candidate corresponding points of p1 are searched in the view S2 as follows. Firstly, the valid sub-regions of the interest point p1 are determined as Ri1, Ri2,…Rin. For each sub-region Rin, We group the sub-RCMs X(Rin) of all the points in S2 as a single set {XRin(1), XRin(2),…,XRin(m)}. Then, we search for the valid sub-regions similar with Xp1(Rin) in the candidate set {XRin(m)} using the Locality Sensitive Hashing
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(LSH) method [22, 23] respectively. The LSH method projects Xp1(Rin) and the elements of the candidate set {XRin(m)} to a series of buckets, and searches for the similar sub-RCMs of the query sub-RCM in the corresponding bucket. It
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greatly improves searching efficiency and reduces false matching as well. Then, the 3D points of S2 corresponding to the selected sub-region XRin(m) are grouped as a single point set {CRin(1), CRin(2),…,CRin(m)}. Finally, the points in
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Sub-regions and sub-RCMs of a correct point correspondence
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the set Cp1 = CRi1∩CRi2∩…CRin are determined as the candidate corresponding points of p1.
Fig. 6 Sub-regions and sub-RCMs of two incorrect point correspondences: (a) a correspondence with different valid sub-regions Ri; (b) a correspondence with same valid sub-regions Ri but different X(Ri).
For each candidate corresponding point qi in Cp1, we calculate the valid l2 distance between qi and p1 as a similarity score of the two points and determine the final corresponding point of p1 using a threshold method or a nearest point method. For the threshold method, if the similarity score of the points is higher than a threshold, qi is accepted as the corresponding point of p1. For the nearest point method, if the similarity score of the points is the highest and higher than a threshold as well, qi is considered to be the corresponding point of p1. Fig. 7a shows the candidate point correspondences and Fig. 7b shows the point correspondences filtered by the threshold method.
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Fig. 7(a) Candidate 3D point correspondences; (b) filtered 3D point correspondences.
3.3 Recovering Rigid Transformation
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An initial rigid transformation between the point clouds S1 and S2 is recovered with a hypothesis-verification mode. To generate a hypothesized rigid transformation, we extract at least three point correspondences satisfying the geometric consistency from the candidate point correspondence set. Specifically, for three point correspondences Mi{pi,
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qi}(i=1, 2, 3), if there is a closed triangle formed by pi in S1 or by qi in S2, we verify whether the area of the closed triangle is consistent with that of its corresponding triangle in another view. If the consistent area is larger than a threshold, the three correspondences can be accepted as positive correspondences and used for generating a
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hypothesized rigid transformation. Then we count the number of the point correspondences consistent with the
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hypothesized transformation in the candidate set and define the number as the confidence score of the hypothesized transformation. We search iteratively for a rigid transformation with the highest confidence score and the
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corresponding geometrically consistent point correspondences. As a result, most of the outliers in the candidate point correspondence set are eliminated and the rigid transformation between the point clouds is recovered simultaneously. After the initial registration, the fine registration of the 3D point clouds can be achieved by a modified ICP algorithm [15].
4 Results and Discussion
In this section, we verified the effectiveness and robustness of the RCM descriptor with varying sampling density and normal error, respectively. The results were compared with respect to two existing 3D shape descriptors (Spin Image and 3D Shape Context). Then, the performance of the RCM-based registration method was compared with the method based on 3D Shape Contexts and the RANSAC method [24] respectively.
4.1 Descriptor Evaluation The descriptor evaluation experiments were conducted on the Happy Buddha model from Standford 3D Scanning Repository. Two views of the Happy Buddha model are shown in Fig. 8. After pretreatment, they are made up of about 40000 points and their sizes are 150mm×150mm. To avoid the influence of interest point detection [25], we chose 300 interest points by random sampling on the Scan1 and searched for their corresponding points on the Scan2. The parameters of the descriptors are listed in Table 1. 8
ACCEPTED MANUSCRIPT The performance of each descriptor was evaluated using recall and precision rate, similar to the criterion used in [26]. The precision rate is the ratio of the correct point correspondences to the candidate point correspondences. The recall rate is the ratio of the correct point correspondences to the total point correspondences. For an obtained point correspondence M(p, q), if the distance between the true corresponding point p' of p and the obtained point q is less
Happy Buddha scanning models (Scan1 and Scan2) Table 1
Parameters of the three descriptors
Support R/mm
Bin Size/mm
Number of Bins
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Descriptor
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than 0.5mm, the point correspondence is determined as a correct point correspondence.
3
0.2
900
3D Shape Context
2
0.25
4096
Spin Image
3
0.1
900
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RCM
4.1.1 Point cloud density test
Point cloud density is a significant factor on the generation of descriptor. In this test, the original point clouds were resampled according to a varying sample interval. The average point number of each point cloud varies from 1300 to 40k. For each point cloud density, we conducted descriptor matching test 100 times repeatedly and calculated the mean recall rate. The recall rates of the descriptors with respect to varying point cloud density were shown in Fig.9(a). From the results, we can find that the RCM descriptor achieves the highest recall rate under all levels of point cloud density, followed by the 3D Shape Context and the Spin Image. Specifically, at the point cloud density of 40k, the RCM descriptor can achieve a recall rate of 91% which is higher than the 3D Shape Context and the Spin Image with 12% and 25%, respectively. When the point cloud density falls to 1300, the RCM descriptor still outperforms the other descriptors with a recall rate of about 19%. Therefore, we can conclude that the RCM descriptor is more robust against varying point cloud density than the other descriptors.
4.1.2 Normal error test In this test, the influence of the normal errors of 3D points on the robustness of RCM descriptor was evaluated. We added a Gaussian noise with a standard deviation varying from 0.1cm to 1 cm to the point clouds, generating varying levels of normal measurement error. The point numbers of two point clouds are around 4000. For each normal 9
ACCEPTED MANUSCRIPT error level, we conducted descriptor matching test 100 times repeatedly and calculated the mean recall rate. The results are shown in Fig.9(b). It can be seen that the RCM performs better than the other two descriptors by a large margin. As the noise level increased, the recall rate of the 3D Shape Contexts and the Spin Image declined rapidly while the performance of the RCM is relatively stable. Specifically, at a high noise level of 1cm, the RCM can still achieve a
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recall rate of about 73%, which is higher than the Spin image and 3D shape context with 61% and 73% respectively. Therefore, the RCM demonstrates a strong robustness against the normal evaluation error due to the 2D mapping
Fig. 9
Recall rate versus (a) point cloud density and (b) noise level
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4.1.3 Precision evaluation
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which is insensitive to the normal error.
In this test, we investigated the comprehensive performances of the three descriptors using two types of curves. The sampling density of the point clouds is 4000 and the noise level is 0. The first curve is precision-similarity curve. We calculated the precisions of the three descriptors with respect to varying similarity threshold, using two similarity measuring methods respectively. Then, we also compared the recall-precision curves of the three descriptors. The results are shown in Fig.10 and Fig.11 respectively.
Fig. 10 Precision vs. similarity: (a) RCM; (b) 3D Shape Context; and (c) Spin Image.
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1-Precision vs. recall
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The precision-similarity curve can be divided into three regions. The region with the precision more than 0.95 is defined as high precision region. The region with the precision between 0.05 and 0.95 is defined as transition region.
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The rest region with the precision less than 0.05 is regarded as low precision region. It is considered that the larger the transition region is, the lower the recognition ability of the descriptor is. That is because two descriptors with the similarity score between the transition region cannot be determined to be a point correspondence with a high
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confidence. From Fig.10, we can find that the transition region of the RCM is the smallest with 30% of the total similarity region, while the transition regions of the 3D Shape Context and the Spin Image are 35.7% and 50%,
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respectively. Therefore, the recognition ability of the RCM is higher than the other two descriptors. In addition, Fig.11 shows that the RCM outperforms the other descriptors with a higher recall rate and precision rate. Specifically, when
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the precision is 0.65, the RCM can achieve a recall rate of 0.55, which is higher than the 3D Shape Context and the Spin Image with 63% and 87% respectively. In sum, the RCM descriptor is more discriminative than the other two descriptors.
4.2 Automatic 3D Point Cloud Registration Results The automatic 3D point cloud registration experiments were conducted on four 3D scanning models. The Happy Buddha model and the Bunny model from the Standford 3D Scanning Repository were obtained by a Cyberware 3030MS laser scanner. The Chicken model from the UWA 3D Scanning Dataset was acquired by a Minolta Vivid 910 scanner. The outdoor scene model from the Oakland 3D point cloud dataset [27] was collected around CMU campus in Oakland by a SICK LMS laser scanner. It is more challenging than single object models with cluttered scene and incomplete building.
4.2.1 Comparison of registration methods based on descriptors We investigate the performances of RCM-based registration method and 3D shape context-based registration method on single object models. We chose 30 pairs of incomplete point clouds to test the registration methods, using a MATLAB implementation on a 512M Memory, 2.4 GHz PC. For each pair of point clouds, the overlapping area is more than 50% of the total area. All the point clouds were normalized to be the same size of about 150mm×150mm, and 11
ACCEPTED MANUSCRIPT the average distance between two points is 3mm after the point clouds were resampled sparsely. We employed a neighboring area with the radius of 10mm to compute curvature. The proportion rv of judging a valid sub-region is 0.8. The threshold of judging vacancies of a RCM descriptor is 0.5. The parameters of the two descriptors were selected as shown in Table 2.
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In the test, the point correspondences were firstly obtained by RCM matching. The initial rigid transformation between two point clouds was recovered with m(where m=3) point correspondences. Then, the fine registration was
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achieved by ICP algorithm. Fig. 12 show the qualitative registration results of RCM-based method on the Happy Buddha scanning models and the Chicken scanning models. From left to right, the figure shows two original scanning
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models, initial registration model and fine registration model respectively. The results demonstrate that the incomplete models can be well aligned with each other based on the RCM descriptors.
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The RCM-based registration method was also tested on the outdoor scene dataset to verify its applicability on a more cluttered scene. The point numbers of the two original point clouds are 14k and 18k, respectively. The registration
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results are shown in Fig. 13. Fig. 13(c) and (d) show the initial registration result and final registration result, respectively. It can be seen that good performance is achieved on the outdoor scene data by the proposed registration
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Fig. 12
Automatic registration results: (a)Happy Buddha; (b) Chicken.
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Automatic registration results of outdoor data (a) (b) original point clouds(c) initial registration result (d) fine registration result.
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Fig. 13
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Fig. 14
Automatic registration results with noise: (a) Happy Buddha; (b) Chicken. Table 2 Descriptor parameters
Descriptor
Rs/mm
ds/mm
RCM
30
6
3D Shape Context
30
6
Table 3 The number of successfully registered point cloud pairs Without noise
With Gaussian noise
The proposed method
29
26
The method based on 3D Shape Context
27
22
13
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Med(t)/s
Maj(t)/s
The proposed method
13
18
4.7
6.6
The method based on 3D Shape Context
23
45
6.0
10.8
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Med(Nx)
Table 5 The minimum number of points Nx and registration time t with Gaussian noise
The proposed method
17
26
The method based on 3D Shape Context
31
56
Med(t)/s
Maj(t)/s
5.5
7.4
7.6
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Med(Nx)
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For detailed quantitative comparison, we compared two registration methods by analyzing three factors as follows. (1) The number of successfully registered point cloud pairs in 30 pairs; (2) the minimum number Nx of the points used for registering a point cloud pair. Since the majority of registration time is spent on the interest point matching, Nx can
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reflect the registration efficiency of the registration method. The lower Nx is, the higher the registration efficiency is. (3) The registration time t. The comparison results are shown in Table 3 and Table 4 respectively.
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From Table 3, we can find that the proposed method can achieve 3D point cloud registration with a higher registration rate compared with the method based on 3D Shape Contexts. In Table 4, Med(Nx) is the mean Nx for
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registering all the point cloud pairs successfully, and Maj(Nx) is the Nx used for registering 80% of all the point cloud pairs successfully. Correspondingly, Med(t) and Maj(t) are the mean time for registering all the pairs and the mean
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time for registering 80% of all the pairs, respectively. Without noise, both the Med(Nx) and Maj(Nx) of the RCM-based method are less than those of the method based on 3D Shape Contexts by a large margin, with 43% and 60% respectively. Similarly, the registration time Med(t) and Maj(t) of the RCM-based method are less than the other method with 22% and 39% respectively. The results demonstrate that the RCM-based registration method is more effective due to the good discrimination of the RCM representation. Since the RCM-based registration method can be achieved with less point correspondences, the registration efficiency is greatly improved. To test the robustness of the proposed method, we added a Gaussian noise with a standard deviation of 0.2cm to all the point clouds and completed the point cloud registration using the proposed method and the method based on 3D Shape Contexts respectively. Fig.14 illustrates the qualitative registration results of the Happy Buddha models and the Chicken models achieved by the proposed method. The partial object models can be registered well even under the noise. Table 3 compares the registration rates of the methods. In 30 pairs of point cloud with Gaussian noise, 26 pairs of point cloud can be registered successfully by the proposed method, while only 22 pairs of point cloud registration were achieved successfully by the method based on 3D Shape Contexts. We also tested the minimum number of points Nx and the registration time t of the methods under the noise. Comparing Table 4 and Table 5, we can find that although both the Nx and t of the two methods increased obviously under the noise, the Med(Nx), Maj(Nx), Med(t) and Maj(t) of the RCM-based method are still less than the other method with 45%, 53%, 28% and 44% respectively.
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the points remains invariant in the rigid transformation.
In this test, we compared the precision and robustness of the RCM-based method with those of the
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RANSAC-based method. The approximate level of congruent point extraction and alignment were set as 0.005cm and
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0.1cm, respectively. The iteration number is 17. The overlapping area of the two point clouds is about 80%. We define the mean registration error of 3D point correspondences as registration score for comparison. A Gaussian noise with a standard deviation varying from 0.05cm to 0.2cm was added to the point clouds. The registration scores were calculated,
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as shown in Table 6. From the results, we can find that the mean registration score of the RANSAC-based method is higher than that of the RCM-based method, especially under a high level of noise. It demonstrates that the RCM
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correspondences can provide a more precise and robust initial value for the fine registration, compared with the ratio constraint. The point-based local shape not only contains the geometric consistency of the keypoint, but also encodes the neighboring points in a distinctive and robust way. Therefore, it provides a stronger geometric constraint so that
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the RCM-based method can achieve a better registration performance than the RANSAC-based method.
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Noise level(cm)
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Table 6 Registration scores of the methods based on RCMs and RANSAC 0.1
0.15
0.2
Initial/Fine Registration Score of RCMs
1.48×10-5
1.29×10-5
0.1614
0.0358
0.1808
0.0377
0.2309
0.0955
0.3781
0.1233
Initial/Fine Registration Score of RANSAC
0.0434
1.65×10-4
0.1598
0.0725
0.2163
0.1589
0.3068
0.2082
0.5614
0.2095
5. Conclusions In this paper, we proposed a novel local shape descriptor—RCM for automatic 3D point cloud registration. The keypoint and its neighboring points are firstly projected onto a 2D plane according to a mapping robust against the normal errors. Then, all the projection points are quantized into the bins of the 2D support region, and their weighted curvatures are counted and encoded into a curvature distribution map. Thorough experiments were conducted to verify the effectiveness and robustness of the RCM. The RCM descriptor outperforms the two existing descriptors (3D Shape Context and Spin Image) with a higher discrimination and robustness against normal error and varying point cloud density. Based on the RCM descriptor, we achieved 3D point cloud registration by matching RCMs, recovering initial transformation with geometrically consistent point correspondences and refinement. We presented an effective point correspondence searching and matching strategy based on the sub-regions of the RCM to reduce the effect of 15
ACCEPTED MANUSCRIPT missing areas and improve the searching efficiency. Experimental results indicate that the RCM-based registration method can achieve a superior performance with a higher registration rate and efficiency, compared with the method based on 3D Shape Contexts and RANSAC.
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Acknowledgements
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This work is supported by National Natural Science Foundation of China under Grant No.61275162 and No.61127009 and the Innovation Foundation of BUAA for Ph.D Graduates. We would like to thank the Computer
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Science Department of Stanford University for providing the 3D Scanning Repository, the Vision Laboratory of University of Western Australia for providing the 3D Scanning Dataset and the Oakland 3D point cloud dataset.
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References
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[1] J. Salvi, “A Review of Recent Range Image Registration Methods with Accuracy Evaluation,” C. Matabosch and D. Fofi, et al., Image and Vision Computing, vol. 25, no.5, pp. 578-596, 2007. [2] G. K. Tam, Z. Cheng, Y. Lai, F. C. Langbein, Y. Liu, D. Marshall, R. R. Martin, X. Sun, and P. L. Rosin, "Registration
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ACCEPTED MANUSCRIPT [11] P.J. Besl and N.D. McKay, “A Method for Registration of 3-D Shapes,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, no. 2, pp. 239-256, 1992. [12] A.E.Johnson, S.B.Kang. “Registration and Integration of Textured 3D Data,” Image and Vision Computing,vol. 17, no.2, pp. 135-147, 1999.
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descriptors," in Proc. European Conf. Computer Vision (ECCV), vol. 3023, pp. 224-237, 2004 [21] Z. Yu, "Intrinsic shape signatures: A shape descriptor for 3D object recognition," in Proc. Int’l Conf. Computer Vision Workshops (ICCV Workshops), pp. 689-696, 2009. [22]J. Kybic and I. Vnucko, “Approximate All Nearest Neighbor Search for High Dimensional Entropy Estimation for Image Registration,” Signal Processing, vol. 92, no. 5, pp.1302-1316, 2011 [23]A. Andoni and P. Indyk, “Near-Optimal Hashing Algorithms for Approximate Nearest Neighbor in High Dimensions,” Proc. IEEE Symposium on Foundations of Computer Science, pp. 459-468, 2006 [24] D. Aiger, N.J. Mitra, D. Cohen-Or, “4-Points Congruent Sets for Robust Pairwise Surface Registration,” ACM Transactions on Graphics, vol. 27, no. 3, 2008 [25] T.H. Yu, O.J. Woodford and R. Cipolla, “A Performance Evaluation of Volumetric 3D Interest Point Detectors,” Int. J. Comput. Vis., vol. 102, no.1-3, pp. 180-197, 2013 [26] Y. Ke, R. Sukthankar, “PCA-SIFT: A More Distinctive Representation for Local Image Descriptors,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), pp. 506-513, 2004 [27] D.Munoz, J.A. Bagnell, N. Vandapel, M. Hebert, “Contextual Classification with Functional Max-Margin Markov Networks,” Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), vol.1-4, pp. 975-982, 2009
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ACCEPTED MANUSCRIPT Statement We have carefully read the comments from editor and reviewers and agreed with those suggestions. We have improved the overall presentation and made revisions according to the suggestions from editor and reviewers for
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possible publication in this journal. For more details on revisions, please see Responses to Editor and Reviewers.
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Finally, we greatly appreciate your time and efforts!
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ACCEPTED MANUSCRIPT A novel 3D shape descriptor-Regional Curvature Map (RCM) is proposed.
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An automatic 3D point cloud registration method based on RCMs is introduced.
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The RCM is discriminative and robust against normal errors and varying point cloud density.
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The RCM searching and matching strategy reduces the influence of area with missing data.
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The RCM-based registration method achieves a higher registration rate and efficiency.
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S0262-8856(16)30159-7 doi:10.1016/j.imavis.2016.09.002 IMAVIS 3553
To appear in:
Image and Vision Computing
Received date: Revised date: Accepted date:
14 August 2012 7 June 2016 22 September 2016
Please cite this article as: Junhua Sun, Jie Zhang, Guangjun Zhang, An Automatic 3D Point Cloud Registration Method Based on Regional Curvature Maps, Image and Vision Computing (2016), doi:10.1016/j.imavis.2016.09.002
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ACCEPTED MANUSCRIPT An Automatic 3D Point Cloud Registration Method Based on Regional Curvature Maps Junhua Sun, Jie Zhang, *Guangjun Zhang
Address: XueYuan Road No. 37, HaiDian District, 100191, Beijing, China Tel: +86-10-82338768
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Fax: +86-10-82316930
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Affiliation: Key Laboratory of Precision Opto-mechatronics Technology, Beihang University, Beijing, China
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*Corresponding author: Guangjun Zhang; E-mail: [email protected] Abstract
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3D point cloud registration is a fundamental and critical issue in 3D reconstruction and object recognition. Most of the existing methods are based on local shape descriptor. In this paper, we propose a discriminative and robust local shape
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descriptor-Regional Curvature Map (RCM). The keypoint and its neighboring points are firstly projected onto a 2D plane according to a robust mapping against normal errors. Then, the projection points are quantized into corresponding bins of the 2D support region and their weighted curvatures are encoded into a curvature distribution
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image. Based on the RCM, an efficient and accurate 3D point cloud registration method is presented. We firstly find 3D
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point correspondences by a RCM searching and matching strategy based on the sub-regions of the RCM. Then, a coarse registration can be achieved with geometrically consistent point correspondences, followed by a fine registration based on a modified iterative closest point (ICP) algorithm. The experimental results demonstrate that the
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RCM is distinctive and robust against normal errors and varying point cloud density. The corresponding registration method can achieve a higher registration precision and efficiency compared with two existing registration methods. Keywords: 3D point cloud; local shape descriptor; regional curvature distribution; 3D surface registration
1. Introduction
3D surface registration is a crucial and active issue in computer vision with numerous applications including 3D modeling, object recognition, scene understanding, 3D shape detection, etc. [1, 2] Due to the self-occlusion of 3D object, 3D scanner can only obtain a partial 3D point cloud associated with a single coordinate system from one viewpoint. In order to reconstruct the whole 3D shape, we must transform the 3D point clouds captured from different viewpoints into a common coordinate system according to rigid transformations. The aim of 3D surface registration is to compute the rigid transformation between 3D point clouds and recover the complete 3D shape of the object automatically. Most of the existing shape-based 3D registration methods consist of initial registration and fine registration. In initial registration, local shape representation and matching [3-8] are crucial steps for recovering a coarse rigid transformation. An accurate initial transformation can improve the optimization efficiency and reduce the optimization error in the following fine registration [9, 10]. So far, fine registration methods are well-developed with the wide 1
ACCEPTED MANUSCRIPT application of iterative closest point (ICP) algorithm [11] and its improved algorithms [12-15]. However, shape-based initial registration is still a challenging and active issue. The existing shape descriptors can be generally classified as global descriptors or local descriptors according to the description region. Global shape descriptor [3, 4, 16] performs good discrimination for complete object
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representation, while local shape description [17-21] is more robust against clutter and occlusion. Among local shape descriptors, 3D point-based descriptor has been widely applied to represent partial object due to its good
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generalization. It encodes the information of neighboring points of an interest point in a compact and distinctive fashion. Then, the 3D points with similar local shapes can be identified from cluttered scenes by descriptor matching.
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Earlier local shape descriptors, such as point signatures [17] and fingerprints [18] encoded the information of 3D points along a 1D support region. For richer representation, 2D and 3D support regions have been employed. For
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example, Spin image [19] counted the points falling into the bins of a 2D histogram on a specific plane, forming a rotationally invariant description image. 3D shape context [20] divided its spherical support region into several bins along the three axes of the local reference frame, and counted the number of points in each bin. Harmonic shape
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context [20] transformed the 3D shape context to a 2D descriptor with a spherical harmonic transformation. Whereas, the measurement error of the normal vector of 3D point and the neighboring area with missing data easily influence
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the local shape representation, leading to incorrect point matching and low registration precision.
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In this paper, we propose a distinctive and robust local shape descriptor-Regional Curvature Map (RCM) for 3D point cloud registration. The neighboring points of the interest point are firstly projected onto a specific 2D plane according to a mapping which is robust against the normal error. Then, the position of each point in the 2D support
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region is quantified and its curvature is encoded into a curvature distribution map of the local shape. The RCM exhibits high discrimination and robustness to normal errors and varying point cloud density. Based on the RCM, an automatic 3D point cloud registration method is introduced. We present a RCM searching and matching strategy based on the sub-regions of the RCM, which reduce the influence of the area with missing data on RCM similarity measuring and improve the matching efficiency as well. Lastly, we recover an initial rigid transformation using geometrically consistent point correspondences and achieve a fine registration with a modified ICP algorithm. Experimental results demonstrate the superiority of the RCM-based registration method. The rest of this paper is organized as follows. Section 2 describes the establishment of the RCM descriptor. Section 3 presents the automatic 3D point cloud registration based on RCMs. Section 4 shows experimental results and discussion. Section 5 concludes the paper.
2. Regional Curvature Map In this section, the generation process of the RCM descriptor is detailed. We construct the RCM descriptor of an interest point by capturing a neighboring shape patch centered with the interest point, projecting all the points to a specific 2D plane, and encoding the curvatures of the projection points into a curvature distribution map.
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ACCEPTED MANUSCRIPT 2.1 A Distribution Map of Neighboring Points A 3D shape patch P is extracted from the 3D point cloud as illustrated in Fig. 1a. Each shape patch contains an interest point p and its neighboring points {pi}. We define a local reference frame (LRF) op-xpypzp of the 3D shape
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patch with p as the origin of the LRF, the surface normal of p as the z axis, and the maximum principal curvature orientation dκ1 of p as the x axis. As shown in Fig. 1b, we project all the neighboring point pi(xpi, ypi, zpi) to a specific
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2D plane with a 2D coordinate system op'-xp'yp' according to a mapping M, and form a distribution image of the projection points on the 2D support region. The mapping M is defined as
(1)
x 2pi y 2pi z 2pi is the distance between the neighboring point pi and the origin op of the sphere coordinate
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where
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M( x pi , y pi , z pi ) ( cos , sin ) ,
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system op-xpypzp, and arctan( y pi , x pi ) is the azimuth angle of pi. Therefore, a projection point pi' in the 2D coordinate system op'-xp'yp' is ( x pi , y pi ) ( cos , sin ) . Since the surface normal of the point pi mainly
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determines the pitching angle of pi instead of the azimuth angle θ of pi, the mapping M is less affected by the normal
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error.
Fig. 1 (a) The interest point p and its neighboring points {pi} in the local coordinate system op-xpypzp; (b) the projection points on a 2D plane with the local coordinate system op'-xp'yp'.
2.2 RCM Generation The RCM of the interest point is generated as follows. We resample the original 3D point cloud (shown in Fig. 2(a)) to reduce its density and then obtain a well-organized 3D point cloud (shown in Fig. 2(b)). All the neighboring points {pi} of the interest point p are projected onto a specific 2D plane according to the mapping M, forming a 2D distribution image as shown in Fig. 2c. The maximum principal curvatures of all the 3D points are computed and represented with gray values. Then, the RCM is established by information quantization and encoding as shown in Fig. 2d. 3
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Fig. 2 RCM generation process: (a) original 3D point cloud; (b) sparse 3D point cloud and an interest point p with its local coordinate system op-xpypzp ; (c) The distribution of projection points with maximum principal curvatures on the 2D support region op'-xp'yp'; (d) RCM representation.
Since the projection points {pi'} on the 2D distribution image (Fig.2c) are disordered, we divide the 2D support
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region into ns×ns average bins along the x' axis and y' axis and then quantize the projection points into corresponding bins for unified representation. The center of the bin can be expressed as
Ps {( x, y), x u ds , y v ds , u 0, 1, 2,
ns , v 0, 1, 2,
ns} ,
(2)
where ds is the size of each bin, and Rs = ns∙ds is the radius of the 2D support region.
d d d d The projection points pi'(xi', yi') locating in the area Q | xi x 2s , x 2s , yi y 2s , y 2s
are counted into
the bin (x, y). For each bin, we sum up the weighted curvatures of all the points falling in the bin. Specifically, the curvature of each point is computed as follows. We employ a second order Taylor expansion on the point p to approximate the neighboring surface of the point. The parameters of the approximate neighboring surface can be obtained with the neighboring points pi(xi, yi, zi) according to
x12 ... xi2
x1 y1 ... xi yi
y12 hxx
z1 ... hxy ... . yi2 hyy zi
Then, we can obtain the Hessian matrix of the approximate surface as
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2h h
hxy
xx
2 hyy
xy
,
(4)
The maximum eigenvalue of the Hessian matrix is regarded as the maximum principal curvature κ of the point,
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and the corresponding eigenvector of the maximum eigenvalue is the principal direction x of the point. They can be expressed explicitly as
H x x with hxx hyy (hxx hyy )2 hxy 2
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(5)
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Then, the curvatures of the points falling in the bin (x, y) are encoded into a curvature distribution map as 1 nq wii ˆ k ( x, y) Sw i1
pi
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if
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(6)
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where wi exp ( xi x ) 2 ( yi y ) 2 / d s is a Gaussian weight assigned to the projection point pi'(xi, yi), emphasizing the points closer to the center. κi is the principal curvature of the projection point. S w
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w is a i
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normalizer. The area without projection points on the 2D support region is represented as . Finally, we can obtain a
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(2ns+1)×(2ns+1) matrix as the RCM descriptor of the interest point by aggregating the encoding values of all the bins.
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3 Automatic 3D Point Cloud Registration Based on RCMs Given two partial 3D point clouds (S1 and S2) with overlapping areas, our aim is to find a unique rigid transformation between the two views based on the RCMs and then to transform the two views into a common coordinate system automatically. The registration is achieved following three main stages. Firstly, 3D interest points with their neighboring points are extracted and represented as RCM descriptors respectively. Then, the 3D points with similar local shapes in different views are matched based on the RCM representation. In this stage, we employ a point correspondence searching and matching strategy based on the sub-regions of the RCM to reduce the effect of missing areas and improve the searching efficiency as well. Finally, we recover a coarse rigid transformation of the two views with no less than three 3D point correspondences satisfying the geometry consistency, and then determine a fine rigid transformation using a modified iterated closest point (ICP) method.
3.1 Similarity Measure The similarity of two 3D points is measured with the l2 distance between the overlapping filled areas of their RCMs. Thus, the effect of the area with missing data on the RCM descriptor can be eliminated. For a RCM matrix X=[xij], the set F {(i, j ) | xij c } of the RCM is defined as filled area. For two RCMs X and X', the overlapping set F∩F' is regarded as valid area in measuring the l2 distance between the RCMs. Thus, the 5
ACCEPTED MANUSCRIPT valid distance is expressed as 1 2 2 1 xij xij d ( X, X) n ( i , j )F F 0
if
n0 r0 (2ns 1) 2
,
(7)
others
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where n0 is the number of the elements in F∩F', ns is the radius of the 2D support region, and r0 is a threshold of the
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number of valid bins.
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Considering the opposite direction of the principal curvature, we determine the valid distance between the RCMs X and X' as:
(8)
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d X ( X, X ') min(d X, X ' , d ( X, X ')) ,
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where X is obtained by rotating X 180 degrees clockwise.
3.2 Corresponding Point Searching and Matching
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To improve the searching efficiency, we search for the corresponding point of 3D interest point of S1 in S2 by
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matching the sub-regions of their RCMs. The 2D support region of the RCM descriptor is divided into 8 sub-regions R1, R2,…, R8, as shown in Fig. 3. Each sub-region Ri occupies a quarter of the support region. We define the bin with
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enough projection points as a valid bin. When the number of valid bins in a sub-region is more than a certain proportion rv, the sub-region is defined as a valid sub-region. The sub-RCM of a valid sub-region Ri is represented as Xp(Ri). Fig. 4 shows a RCM descriptor with 3 valid sub-regions R1, R2 and R8 as examples.
Fig. 4
Fig. 3
Eight sub-regions of a RCM
A RCM with 3 valid sub-regions R1, R2, R8
For a 3D point correspondence (p1, p2), the RCMs Xp1 and Xp2 representing their local shapes are similar. Correspondingly, their valid sub-regions Ri together with their RCM representation Xp1(Ri) and Xp2(Ri) are similar. For examples, Fig. 5 shows the RCMs of a correct point correspondence with same valid sub-regions R1, R2 and R8 and similar RCM representations X(R1), X(R2) and X(R8). Fig. 6a shows the RCMs of an incorrect point correspondence with different valid sub-regions Ri, and Fig. 6b shows the RCMs of another incorrect correspondence with same valid 6
ACCEPTED MANUSCRIPT sub-regions Ri but different valid sub-RCMs. The candidate corresponding points of p1 are searched in the view S2 as follows. Firstly, the valid sub-regions of the interest point p1 are determined as Ri1, Ri2,…Rin. For each sub-region Rin, We group the sub-RCMs X(Rin) of all the points in S2 as a single set {XRin(1), XRin(2),…,XRin(m)}. Then, we search for the valid sub-regions similar with Xp1(Rin) in the candidate set {XRin(m)} using the Locality Sensitive Hashing
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(LSH) method [22, 23] respectively. The LSH method projects Xp1(Rin) and the elements of the candidate set {XRin(m)} to a series of buckets, and searches for the similar sub-RCMs of the query sub-RCM in the corresponding bucket. It
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greatly improves searching efficiency and reduces false matching as well. Then, the 3D points of S2 corresponding to the selected sub-region XRin(m) are grouped as a single point set {CRin(1), CRin(2),…,CRin(m)}. Finally, the points in
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Sub-regions and sub-RCMs of a correct point correspondence
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the set Cp1 = CRi1∩CRi2∩…CRin are determined as the candidate corresponding points of p1.
Fig. 6 Sub-regions and sub-RCMs of two incorrect point correspondences: (a) a correspondence with different valid sub-regions Ri; (b) a correspondence with same valid sub-regions Ri but different X(Ri).
For each candidate corresponding point qi in Cp1, we calculate the valid l2 distance between qi and p1 as a similarity score of the two points and determine the final corresponding point of p1 using a threshold method or a nearest point method. For the threshold method, if the similarity score of the points is higher than a threshold, qi is accepted as the corresponding point of p1. For the nearest point method, if the similarity score of the points is the highest and higher than a threshold as well, qi is considered to be the corresponding point of p1. Fig. 7a shows the candidate point correspondences and Fig. 7b shows the point correspondences filtered by the threshold method.
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Fig. 7(a) Candidate 3D point correspondences; (b) filtered 3D point correspondences.
3.3 Recovering Rigid Transformation
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An initial rigid transformation between the point clouds S1 and S2 is recovered with a hypothesis-verification mode. To generate a hypothesized rigid transformation, we extract at least three point correspondences satisfying the geometric consistency from the candidate point correspondence set. Specifically, for three point correspondences Mi{pi,
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qi}(i=1, 2, 3), if there is a closed triangle formed by pi in S1 or by qi in S2, we verify whether the area of the closed triangle is consistent with that of its corresponding triangle in another view. If the consistent area is larger than a threshold, the three correspondences can be accepted as positive correspondences and used for generating a
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hypothesized rigid transformation. Then we count the number of the point correspondences consistent with the
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hypothesized transformation in the candidate set and define the number as the confidence score of the hypothesized transformation. We search iteratively for a rigid transformation with the highest confidence score and the
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corresponding geometrically consistent point correspondences. As a result, most of the outliers in the candidate point correspondence set are eliminated and the rigid transformation between the point clouds is recovered simultaneously. After the initial registration, the fine registration of the 3D point clouds can be achieved by a modified ICP algorithm [15].
4 Results and Discussion
In this section, we verified the effectiveness and robustness of the RCM descriptor with varying sampling density and normal error, respectively. The results were compared with respect to two existing 3D shape descriptors (Spin Image and 3D Shape Context). Then, the performance of the RCM-based registration method was compared with the method based on 3D Shape Contexts and the RANSAC method [24] respectively.
4.1 Descriptor Evaluation The descriptor evaluation experiments were conducted on the Happy Buddha model from Standford 3D Scanning Repository. Two views of the Happy Buddha model are shown in Fig. 8. After pretreatment, they are made up of about 40000 points and their sizes are 150mm×150mm. To avoid the influence of interest point detection [25], we chose 300 interest points by random sampling on the Scan1 and searched for their corresponding points on the Scan2. The parameters of the descriptors are listed in Table 1. 8
ACCEPTED MANUSCRIPT The performance of each descriptor was evaluated using recall and precision rate, similar to the criterion used in [26]. The precision rate is the ratio of the correct point correspondences to the candidate point correspondences. The recall rate is the ratio of the correct point correspondences to the total point correspondences. For an obtained point correspondence M(p, q), if the distance between the true corresponding point p' of p and the obtained point q is less
Happy Buddha scanning models (Scan1 and Scan2) Table 1
Parameters of the three descriptors
Support R/mm
Bin Size/mm
Number of Bins
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Descriptor
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Fig. 8
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than 0.5mm, the point correspondence is determined as a correct point correspondence.
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4096
Spin Image
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4.1.1 Point cloud density test
Point cloud density is a significant factor on the generation of descriptor. In this test, the original point clouds were resampled according to a varying sample interval. The average point number of each point cloud varies from 1300 to 40k. For each point cloud density, we conducted descriptor matching test 100 times repeatedly and calculated the mean recall rate. The recall rates of the descriptors with respect to varying point cloud density were shown in Fig.9(a). From the results, we can find that the RCM descriptor achieves the highest recall rate under all levels of point cloud density, followed by the 3D Shape Context and the Spin Image. Specifically, at the point cloud density of 40k, the RCM descriptor can achieve a recall rate of 91% which is higher than the 3D Shape Context and the Spin Image with 12% and 25%, respectively. When the point cloud density falls to 1300, the RCM descriptor still outperforms the other descriptors with a recall rate of about 19%. Therefore, we can conclude that the RCM descriptor is more robust against varying point cloud density than the other descriptors.
4.1.2 Normal error test In this test, the influence of the normal errors of 3D points on the robustness of RCM descriptor was evaluated. We added a Gaussian noise with a standard deviation varying from 0.1cm to 1 cm to the point clouds, generating varying levels of normal measurement error. The point numbers of two point clouds are around 4000. For each normal 9
ACCEPTED MANUSCRIPT error level, we conducted descriptor matching test 100 times repeatedly and calculated the mean recall rate. The results are shown in Fig.9(b). It can be seen that the RCM performs better than the other two descriptors by a large margin. As the noise level increased, the recall rate of the 3D Shape Contexts and the Spin Image declined rapidly while the performance of the RCM is relatively stable. Specifically, at a high noise level of 1cm, the RCM can still achieve a
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recall rate of about 73%, which is higher than the Spin image and 3D shape context with 61% and 73% respectively. Therefore, the RCM demonstrates a strong robustness against the normal evaluation error due to the 2D mapping
Fig. 9
Recall rate versus (a) point cloud density and (b) noise level
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4.1.3 Precision evaluation
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which is insensitive to the normal error.
In this test, we investigated the comprehensive performances of the three descriptors using two types of curves. The sampling density of the point clouds is 4000 and the noise level is 0. The first curve is precision-similarity curve. We calculated the precisions of the three descriptors with respect to varying similarity threshold, using two similarity measuring methods respectively. Then, we also compared the recall-precision curves of the three descriptors. The results are shown in Fig.10 and Fig.11 respectively.
Fig. 10 Precision vs. similarity: (a) RCM; (b) 3D Shape Context; and (c) Spin Image.
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Fig.11
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1-Precision vs. recall
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The precision-similarity curve can be divided into three regions. The region with the precision more than 0.95 is defined as high precision region. The region with the precision between 0.05 and 0.95 is defined as transition region.
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The rest region with the precision less than 0.05 is regarded as low precision region. It is considered that the larger the transition region is, the lower the recognition ability of the descriptor is. That is because two descriptors with the similarity score between the transition region cannot be determined to be a point correspondence with a high
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confidence. From Fig.10, we can find that the transition region of the RCM is the smallest with 30% of the total similarity region, while the transition regions of the 3D Shape Context and the Spin Image are 35.7% and 50%,
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respectively. Therefore, the recognition ability of the RCM is higher than the other two descriptors. In addition, Fig.11 shows that the RCM outperforms the other descriptors with a higher recall rate and precision rate. Specifically, when
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the precision is 0.65, the RCM can achieve a recall rate of 0.55, which is higher than the 3D Shape Context and the Spin Image with 63% and 87% respectively. In sum, the RCM descriptor is more discriminative than the other two descriptors.
4.2 Automatic 3D Point Cloud Registration Results The automatic 3D point cloud registration experiments were conducted on four 3D scanning models. The Happy Buddha model and the Bunny model from the Standford 3D Scanning Repository were obtained by a Cyberware 3030MS laser scanner. The Chicken model from the UWA 3D Scanning Dataset was acquired by a Minolta Vivid 910 scanner. The outdoor scene model from the Oakland 3D point cloud dataset [27] was collected around CMU campus in Oakland by a SICK LMS laser scanner. It is more challenging than single object models with cluttered scene and incomplete building.
4.2.1 Comparison of registration methods based on descriptors We investigate the performances of RCM-based registration method and 3D shape context-based registration method on single object models. We chose 30 pairs of incomplete point clouds to test the registration methods, using a MATLAB implementation on a 512M Memory, 2.4 GHz PC. For each pair of point clouds, the overlapping area is more than 50% of the total area. All the point clouds were normalized to be the same size of about 150mm×150mm, and 11
ACCEPTED MANUSCRIPT the average distance between two points is 3mm after the point clouds were resampled sparsely. We employed a neighboring area with the radius of 10mm to compute curvature. The proportion rv of judging a valid sub-region is 0.8. The threshold of judging vacancies of a RCM descriptor is 0.5. The parameters of the two descriptors were selected as shown in Table 2.
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In the test, the point correspondences were firstly obtained by RCM matching. The initial rigid transformation between two point clouds was recovered with m(where m=3) point correspondences. Then, the fine registration was
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achieved by ICP algorithm. Fig. 12 show the qualitative registration results of RCM-based method on the Happy Buddha scanning models and the Chicken scanning models. From left to right, the figure shows two original scanning
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models, initial registration model and fine registration model respectively. The results demonstrate that the incomplete models can be well aligned with each other based on the RCM descriptors.
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The RCM-based registration method was also tested on the outdoor scene dataset to verify its applicability on a more cluttered scene. The point numbers of the two original point clouds are 14k and 18k, respectively. The registration
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results are shown in Fig. 13. Fig. 13(c) and (d) show the initial registration result and final registration result, respectively. It can be seen that good performance is achieved on the outdoor scene data by the proposed registration
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Fig. 12
Automatic registration results: (a)Happy Buddha; (b) Chicken.
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Automatic registration results of outdoor data (a) (b) original point clouds(c) initial registration result (d) fine registration result.
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Automatic registration results with noise: (a) Happy Buddha; (b) Chicken. Table 2 Descriptor parameters
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3D Shape Context
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Table 3 The number of successfully registered point cloud pairs Without noise
With Gaussian noise
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The method based on 3D Shape Context
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Maj(t)/s
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Table 5 The minimum number of points Nx and registration time t with Gaussian noise
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Med(t)/s
Maj(t)/s
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For detailed quantitative comparison, we compared two registration methods by analyzing three factors as follows. (1) The number of successfully registered point cloud pairs in 30 pairs; (2) the minimum number Nx of the points used for registering a point cloud pair. Since the majority of registration time is spent on the interest point matching, Nx can
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reflect the registration efficiency of the registration method. The lower Nx is, the higher the registration efficiency is. (3) The registration time t. The comparison results are shown in Table 3 and Table 4 respectively.
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From Table 3, we can find that the proposed method can achieve 3D point cloud registration with a higher registration rate compared with the method based on 3D Shape Contexts. In Table 4, Med(Nx) is the mean Nx for
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registering all the point cloud pairs successfully, and Maj(Nx) is the Nx used for registering 80% of all the point cloud pairs successfully. Correspondingly, Med(t) and Maj(t) are the mean time for registering all the pairs and the mean
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time for registering 80% of all the pairs, respectively. Without noise, both the Med(Nx) and Maj(Nx) of the RCM-based method are less than those of the method based on 3D Shape Contexts by a large margin, with 43% and 60% respectively. Similarly, the registration time Med(t) and Maj(t) of the RCM-based method are less than the other method with 22% and 39% respectively. The results demonstrate that the RCM-based registration method is more effective due to the good discrimination of the RCM representation. Since the RCM-based registration method can be achieved with less point correspondences, the registration efficiency is greatly improved. To test the robustness of the proposed method, we added a Gaussian noise with a standard deviation of 0.2cm to all the point clouds and completed the point cloud registration using the proposed method and the method based on 3D Shape Contexts respectively. Fig.14 illustrates the qualitative registration results of the Happy Buddha models and the Chicken models achieved by the proposed method. The partial object models can be registered well even under the noise. Table 3 compares the registration rates of the methods. In 30 pairs of point cloud with Gaussian noise, 26 pairs of point cloud can be registered successfully by the proposed method, while only 22 pairs of point cloud registration were achieved successfully by the method based on 3D Shape Contexts. We also tested the minimum number of points Nx and the registration time t of the methods under the noise. Comparing Table 4 and Table 5, we can find that although both the Nx and t of the two methods increased obviously under the noise, the Med(Nx), Maj(Nx), Med(t) and Maj(t) of the RCM-based method are still less than the other method with 45%, 53%, 28% and 44% respectively.
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ACCEPTED MANUSCRIPT 4.2.2 Comparison of descriptor-based method and RANSAC method The point cloud registration method based on random sample consensus (RANSAC) makes no assumption about the coarse rigid transformation of the two 3D point clouds. When 4 coplanar points are extracted from one point cloud, we can find their approximately congruent points from the other point cloud according to the fact that certain ratio of
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the points remains invariant in the rigid transformation.
In this test, we compared the precision and robustness of the RCM-based method with those of the
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RANSAC-based method. The approximate level of congruent point extraction and alignment were set as 0.005cm and
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0.1cm, respectively. The iteration number is 17. The overlapping area of the two point clouds is about 80%. We define the mean registration error of 3D point correspondences as registration score for comparison. A Gaussian noise with a standard deviation varying from 0.05cm to 0.2cm was added to the point clouds. The registration scores were calculated,
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as shown in Table 6. From the results, we can find that the mean registration score of the RANSAC-based method is higher than that of the RCM-based method, especially under a high level of noise. It demonstrates that the RCM
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correspondences can provide a more precise and robust initial value for the fine registration, compared with the ratio constraint. The point-based local shape not only contains the geometric consistency of the keypoint, but also encodes the neighboring points in a distinctive and robust way. Therefore, it provides a stronger geometric constraint so that
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Initial/Fine Registration Score of RCMs
1.48×10-5
1.29×10-5
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0.1808
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Initial/Fine Registration Score of RANSAC
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1.65×10-4
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0.5614
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5. Conclusions In this paper, we proposed a novel local shape descriptor—RCM for automatic 3D point cloud registration. The keypoint and its neighboring points are firstly projected onto a 2D plane according to a mapping robust against the normal errors. Then, all the projection points are quantized into the bins of the 2D support region, and their weighted curvatures are counted and encoded into a curvature distribution map. Thorough experiments were conducted to verify the effectiveness and robustness of the RCM. The RCM descriptor outperforms the two existing descriptors (3D Shape Context and Spin Image) with a higher discrimination and robustness against normal error and varying point cloud density. Based on the RCM descriptor, we achieved 3D point cloud registration by matching RCMs, recovering initial transformation with geometrically consistent point correspondences and refinement. We presented an effective point correspondence searching and matching strategy based on the sub-regions of the RCM to reduce the effect of 15
ACCEPTED MANUSCRIPT missing areas and improve the searching efficiency. Experimental results indicate that the RCM-based registration method can achieve a superior performance with a higher registration rate and efficiency, compared with the method based on 3D Shape Contexts and RANSAC.
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Acknowledgements
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This work is supported by National Natural Science Foundation of China under Grant No.61275162 and No.61127009 and the Innovation Foundation of BUAA for Ph.D Graduates. We would like to thank the Computer
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Science Department of Stanford University for providing the 3D Scanning Repository, the Vision Laboratory of University of Western Australia for providing the 3D Scanning Dataset and the Oakland 3D point cloud dataset.
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References
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ACCEPTED MANUSCRIPT Statement We have carefully read the comments from editor and reviewers and agreed with those suggestions. We have improved the overall presentation and made revisions according to the suggestions from editor and reviewers for
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possible publication in this journal. For more details on revisions, please see Responses to Editor and Reviewers.
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Finally, we greatly appreciate your time and efforts!
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ACCEPTED MANUSCRIPT A novel 3D shape descriptor-Regional Curvature Map (RCM) is proposed.
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An automatic 3D point cloud registration method based on RCMs is introduced.
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The RCM is discriminative and robust against normal errors and varying point cloud density.
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The RCM searching and matching strategy reduces the influence of area with missing data.
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The RCM-based registration method achieves a higher registration rate and efficiency.
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