2011 18th IEEE International Conference on Image Processing

FAST AND ROBUST ISOTROPIC SCALING ITERATIVE CLOSEST POINT ALGORITHM Ce Li, Jianru Xue, Nanning Zheng, Shaoyi Du, Jihua Zhu, Zhiqiang Tian Institute of Artificial Intelligence and Robotics Xi’an Jiaotong University, Xi’an, Shaanxi Province 710049, P.R. China ABSTRACT The iterative closest point (ICP) algorithm is an accurate approach for the registration between two point sets on the same scale. However, it can not handle the case with different scales. This paper proposes a fast and robust ICP algorithm for isotropic scaling point sets registration (FRISICP). In order to accurately and directly estimate the scale factor without any constraints, we introduce a bidirection distance measurement method into the least square (LS) problem. Then to keep computational efficiency when the number of points in the set increasing, we further introduce a sparse-to-dense hierarchical model in ICP algorithm to speed up the isotropic scaling point set matching process. Experimental results demonstrate that the proposed FRISICP method outperforms other algorithms on both 2D and 3D point sets. Index Terms— Isotropic scaling registration, ICP, Sparseto-dense, Bidirection distance measurement. 1. INTRODUCTION Point sets registration is a hot topic in computer vision and image processing. The ICP algorithm is known for sloving rigid registration with its good accuracy and efficiency. It has been widely applied in many applications such as 3D reconstruction[1], face recognition[2] and medical image analysis[3] etc. Two major challenges of ICP are robustness to various types of point sets, especially to non-rigid point sets, and the algorithm’s efficiency while the number of point sets increasing. E.g. the original ICP algorithm does not take scale factor into account in the Least Squares (LS) problem while scale factor always exists in non-rigid point sets registration[4]. Prior Arts On one hand, some scholars have tried to improve ICP for scaling registration. Zha et al.[5] used extended signature images to estimate the scale of traditional ICP for registration, while Zinßer et al.[6] directly estimated the scale in the ICP algorithm. Du et al.[4] have proposed to introduce a scaling factor directly into the least square (LS) This work was supported by the National Natural Science Foundation of China under Grant Nos.60875008 and 61005014, the National Basic Research Program of China (973 Program) under Grant No.2010CB327902, and the XJTU Young Teachers Program.

978-1-4577-1302-6/11/$26.00 ©2011 IEEE

Fig. 1. An isotropic scaling point sets registration results by the proposed FRISICP algorithm. (a) Dense of data point set (blue) and model point set (red); (b) Sparse the two point sets; (c) FRISICP result by sparse the two point sets; (d) The final result of FRISICP using (c) provide the initial value of transformation matrix;(e) ICP’s result;(f) Zinßer’s result;(g) Du’s result. problem with a constraint of bounding scaling. However, when shapes deform greatly, it is difficult to accurately and robustly estimate the scaling factor. Furthermore, it also needs to estimate the boundaries of the scaling factor. On the other hand, while the number of points in the set increasing, the computational efficiency usually suffers a reduction, which limits the practical applications of ICP algorithm. Hence, some scholars also have tried to improve the speed of ICP. Rusinkiewicz et al.[7] combined multiple optimization strategies to improve the convergence speed of ICP . Fitzgibbon[8] used the Levenberg-Marquardt least square fitting method to speed up ICP. Meanwhile, Jost et al.[9] combined a coarse to fine multi-resolution technique with the neighbor search algorithm in ICP to improve the registration. All these methods tried to improve the searching strategy between the two point sets, and therefore it is still difficult for them to overcome the efficiency decline when the number of points grows. Overview Inspired by the previous algorithms, we propose a FRISICP algorithm, which utilizes a sparse-to-dense hierarchical model and a bidirection distance measurement method into the LS problem on isotropic scaling ICP algorithm. On the issue of rapid ICP, FRISICP is a sparse-to-dense piont sets machting process, which has simple formulas and

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2011 18th IEEE International Conference on Image Processing

effective performance. On the robustness of isotropic scaling ICP, FRISICP has a satisfied performance to accurately and directly estimate the scale factor without any constraint for introducing the bidirection distance feedback measurement. As shown in Fig.1, the iteration algorithm of FRISICP is a sparse(Fig.1b) to dense(Fig.1a) process with the bidirection distance measurement. It is an initial transformation matrix between two point sets acquired firstly through the fast iterative computation at lower resolution(Fig.1c). And then it is used as the initial value for the precise registration at high resolution to complete the overall isotropic scaling point sets registration process(Fig.1d). The rest of the paper is arranged as follows. In Section 2, a general objective function for isotropic scaling point sets registration with sparse-to-dense hierarchical model of bidirection distance measurement is described. Following that, the proposed FRISICP algorithm is given in detail. Then in Section 3, the proposed approach is evaluated by comparing our algorithm with the state-of-the-arts. Finally, the conclusion is drawn in the last section. 2. THE FRISICP ALGORITHM The registration of m-D point sets is a difficult problem. The standard ICP algorithm presented by Besl and McKay [10] is an efficient method to deal with rigid registration between two rigid point sets, which is presented below. Given ∆ two rigid point sets in Rm , one denotes a data shape A = Nn {ai }i=1 (Nn ∈ N) and the other denotes a model shape ∆

Nm B = {bj }j=1 (Nm ∈ N). The aim is to build up the correspondence and calculate the rigid transformation between two point sets A and B , so the formulation of the LS problem is given as:

min

N Pn

kRai + t − bj k22

R,t,j∈{1,...,Nm } i=1 s.t. RT R = Im ,

(1)

det(R) = 1

where R ∈ Rm×m is a rotation matrix, and t is a translation vector. The ICP algorithm is known to be an accurate approach for rigid registration between two point sets. However, scale factor may exist in two point sets. In practice, we often need to consider the following LS problem: min

s,R,t,j∈{1,...,Nm }

s.t.

N Pn i=1

ksRai + t − bj k22

bi-directional distance measurement for the isotropic scaling registration, which is expressed as follows: min

where s is a scale factor. Du’s ICP [4] ever defined that s is a scale factor and its boundaries must be estimated by the characteristics of the data sets, such as their covariance matrices. However, when shapes deform greatly, it is difficult to accurately and robustly estimate the scaling factor. From this point of Eq.(2), when s converges to 0, the point set A will converge to a small subset of B. Then, the isotropic scaling registration is an ill-posed problem. To avoid it, we propose a simple method to formulate the LS problem based on

N Pn

s,R,t,j∈{1,...,Nm } i=1 i∈{1,...,Nn } N m P

+

j=1

ksRai + t − bj k22 (3)

ksRai + t − bj k22 )

RT R = Im ,

s.t.

det(R) = 1, s 6= 0

In Eq.(3), the former term is the sum of the forward square distance,which is denoted the transformed data shape A can register with the model shape B. The latter term is the sum of backward square distance, which is denoted the transformed model shape B can register with the data shape A. From Eq.(3), we find the LS problem is not ill-posed. When s converges to 0, the sum of the forward square distance will converge to 0, but the sum of the backward square distance will be larger, in turn the situation is also same. On the other hand, Ezra et. al.[11] shows that the ICP algorithm is terminated after O(md nd ) iterations, where m, n is the number of point set A and B respectively in d-space. That is, the computational efficiency of ICP algorithm is affected by the number of point sets and the dimension of space. Therefore, Eq.(3) improves the robustness of ICP and and increase ICP run time at the same time. To speed up Eq.(3), we propose a sparse-to-dense hierarchical model for point set matching in this work, which gets an initial transformation matrix between two point sets acquired firstly. The less the number of point set is, the quicker the computational speed of the algorithm is. First of all, we use the sparse operator (fsample↓ ) to sparse points (e.g. Gaussian Pyramid), which could be replaced by other more sophisticated methods easily. After that, through the fast registration of sparse points, dense point get more precise matching of initial value matrix. This will greatly accelerate the process of points registration . The proposed fast and roubst isotropic scaling point sets registration algorithm (FRISICP) is summarized as follows. Step 1. Down-sampling the two point sets A and B in Rm , shown as follows: ∆

Nl

l−1 Al = fsample↓ (A ),



Nl

l−1 Bl = fsample↓ (B ),

n Al = {ali }i=1 ; m Bl = {blj }j=1 ;

(Nnl ∈ N;

(2)

RT R = Im , det(R) = 1, s 6= 0

(

l Nm ∈ N;

(4)

l = 1...L)

where l is the level of point sets resolution. Step 2. Find bidirection correspondence between two point sets Al and Bl based on the l level, (k − 1)th transformation (slk−1 , Rlk−1 ,tlk−1 ), which is shown as follows:

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clk (i) =

2

arg min ( slk−1 Rlk−1 ali + tlk−1 − blj ), 2

l } j∈{1,...,Nm

gkl (i) =

l f or i =

1, ..., Nn

l l arg min ( sk−1 Rk−1 ali l} i∈{1,...,Nn

f or

j=

l 1, ..., Nm

2

+ tlk−1 − blj ), 2

(5)

(6)

2011 18th IEEE International Conference on Image Processing

This step can be solved by many efficient methods such as the nearest point search based on Delaunay triangulation or k-d tree algorithm. Step 3. Compute a new transformation (sl∗ , Rl∗ , tl∗ ) based on the current bidirectional correspondence as follows:

Table 1. Compared RMS and run time results on two 2D shapes with respect to scale factors(s ∈ [0.1, 3.0], step = 0.2; RM S = 10−7 mm). Du’s FRISICP RMS 0.1 25.576 2.6888 0.6981 0.3657 Deer 0.8 3.6503 2.1511 0.5589 0.3657 3.0 9.9931 8.8665 2.0946 0.3656 Average* [0.1,3.0] 7.4562 2.1674 0.6267 0.2174 2D Shapes Scale Time(s) 0.1 2.5234 3.4314 3.4182 2.337 Deer 0.8 4.2189 3.4347 3.7613 1.8929 3.0 2.71 3.3387 4.1475 2.2083 Average* [0.1,3.0] 1.7904 1.7323 1.8690 1.0445 * The average value is each mean of four 2D shapes (Apple,Cock,Deer and Horse) point sets in fifteen isotropic scaling registration results.

2D Shapes

(sl∗ , Rl∗ , tl∗ ) = Nn

2 X

l l l

l l arg min(

s R a i + t − b cl (i) k

sl ,Rl ,tl

+

N m P j=1

s.t.

2

i=1

lT

(7)

2

l l l

s R a gl (j) + tl − blj ) k

l

R R =

Ilm ,

2

l

det(R ) = 1, sl 6= 0

Scale

ICP

Zinßer’s

Update slk ,Rlk and tlk , slk = sl∗ slk−1 ,

Rlk = Rl∗ Rlk−1 ,

tlk = sl∗ Rl∗ tlk−1 + tl∗

(8)

Step 4. Repeat Step2∼3, until the required error precision can be arrived in lth resolution. Let l=l-1, the new lth translation is received by up-sample as follows: Tl−1 = f sample↑ (Tlk )

(9)

Step 5. Repeat Step 4 until l=1. 3. EXPERIMENTAL RESULTS To demonstrate the robustness and efficiency of our proposed method, we test on the part B of CE-Shape-11 and the Stanford 3D Scanning Repository2 .We compare root mean square (RMS) error and run times with ICP[10], Zinßer’s[6] and Du’s[4]. All the tests are implemented in MATLAB 7.0 and are performed on the PC with P4 2.2G CPU and 2G RAM. 3.1. 2D Shapes matching In this part, to show the robustness and speed of registration with isotropic scaling 2D shapes, we conduct experiments and select Apple,Cock,Deer and Horse 2D shapes on the part B of CE-Shape-1. In our method, the sparse-to-dense operation is implemented by Gauss pyramid(l=2∼4 in this paper). For each kind of shapes, we select two 2D point sets, one is model point set; the other is data point set. Then we take a data point set of 15 scale factors (s ∈ [0.1, 3.0], step = 0.2) and its respectively registration with the model set of points. Therefore, we received 60 groups RMS and run times compared results of isotropic scale registration data on each four kinds of 2D shapes with state-of-art approaches and our FRISICP .The compared results are displayed in Table 1, Fig.1 and Fig.2. Observing Table 1, FRISICP always yields better registration with small RMS and run times. Especially, the average value shows in Table 1, which is each mean of four 1 http://www.cis.temple.edu/∼latecki/TestData/mpeg7shapeB.tar.gz 2 http://graphics.stanford.edu/data/3Dscanrep/

Fig. 2. Comparison of convergence speed for 2D ’Deer’ shapes (s = 0.8). 2D shapes point sets in fifteen isotropic scaling registration results. While ICP[10] can only handle the case with same scale. Zinßer’s[6] and Du’s[4] always work comparatively worse, for all cases points of data shape converge to a small subset of the model shape. The FRISICP works better under almost sacle factors than above three algrothims. In addition, Fig.2 shows that FRISICP rapidly converges and has almost the same accuracy as other approaches. Especially, under the same accuracy, by using our method, the initial transformation between two point sets can be completed within only 0.4s while the other methods need at least 2.5s. To compare the results in a more intuitive way, Fig.1 shows result of the Deer 2D shapes registration of isotropic scaling. It is easy to see form Fig.1, FRISICP works well both in sparse or dense cases. 3.2. 3D shapes matching Experiments on some 3D shapes (Buddha, Bunny and Dragon) from Stanford 3D Scanning Repository are also performed to demonstrate the robustness and speed of our FRISICP algorithm. The sparse operation is implemented by naive interlaced sampling (l=2∼4 in this paper); it is also easily replaced

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2011 18th IEEE International Conference on Image Processing

Table 2. Compared RMS and run time results on two 3D shapes with respect to scale factors(s ∈ [0.1, 3.0], step = 0.2; RM S = 10−7 mm). Du’s FRISICP RMS 0.1 5.3447 1.7782 1.7782 0.1832 Bunny 0.8 7.7392 1.4238 1.4226 0.1832 3.0 1.4278 0.5335 0.5355 0.1832 Average* [0.1,3.0] 3.8855 1.7027 1.7028 0.2348 3D Shapes Scale Time(s) 0.1 31.1947 30.3933 30.3943 16.7646 Bunny 0.8 39.9058 41.7233 41.3901 17.1353 3.0 48.0943 44.5314 44.5291 15.8928 Average* [0.1,3.0] 53.144 52.881 52.982 26.181 * The average value is each mean of three 3D shapes (Buddha,Bunny and Dragon) point sets in fifteen isotropic scaling registration results. 3D Shapes

Scale

ICP

Zinßer’s

Fig. 4. Registration results for 3D ’Bunny’ shapes (s = 0.1). (a) The original shapes; (b)The other view of original shapes; (c) Sparse of original shapes; (d) Sparse result of FRISICP; (e) Final result of FRISICP; (f) ICP’s result; (g) Zinßer’s result; (h) Du’s result. vious works, our algorithm can rapidly and accurately estimate the scale factor without any constraints, even shapes deform severely. A series of experiments demonstrate that our algorithm is fast and robust. In the future, we will try to apply our work to affine point sets complex matching process and in practical use, such as 3D reconstruction and the like. 5. REFERENCES

Fig. 3. Comparison of convergence speed for 3D ’Bunny’ shapes (s = 0.8). by more sophisticated methods,such as Quadric Error Metrics (QEM) [12]). After the sparse operation, the Bunny045 points number reduced from 400097 points to 5248 points. Fig.4(c) shows the down-sample result of example Bunny024 3D point sets (blue). We take a data point set with 15 scale factors (s ∈ [0.1, 3.0], step = 0.2) and its respectively registration with the model set of points. Therefore, we received 45 groups RMS and run times compared results of isotropic scale registration data on each four kinds of 3D shapes with our FRISICP and other approaches. The compared results are displayed in Table 2, Fig.3 and Fig.4. It is easy to see from above compared results, FRISICP works better than other three algorithms. In particular for large number points on 3D shapes, FRICISP on the running efficiency has better performance compared with other methods (see Fig.3). 4. CONCLUSION This paper proposes a novel approach for matching between two m-D isotropic scaling point sets. We introduce a sparseto-dense hierarchical model and a bidirection distance measurement method into the ICP algorithm. Compare with pre-

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[1] Z. Megyesi, G. K´os, and D. Chetverikov, “Dense 3D reconstruction from images by normal aided matching,” Machine Graphics & Vision International Journal, vol. 15, no. 1, pp. 3–28, 2006. [2] A.F. Abate, M. Nappi, D. Riccio, and G. Sabatino, “2D and 3D face recognition: A survey,” Pattern Recognition Letters, vol. 28, no. 14, pp. 1885–1906, 2007. [3] R.A. McLaughlin, J. Hipwell, D.J. Hawkes, J.A. Noble, J.V. Byrne, and T.C. Cox, “A comparison of a similarity-based and a feature-based 2-D-3-D registration method for neurointerventional use,” Medical Imaging, IEEE Transactions on, vol. 24, no. 8, pp. 1058–1066, 2005. [4] S. Du, N. Zheng, S. Ying, and J. Wei, “ICP with Bounded Scale for Registration of MD Point Sets,” in Multimedia and Expo, IEEE International Conference on, pp. 1291–1294, 2007. [5] H. Zha, M. Ikuta, and T. Hasegawa, “Registration of range images with different scanning resolutions,” in Systems, Man, and Cybernetics, IEEE International Conference on, vol. 2, pp. 1495–1500, 2002. [6] T. Zinßer, J. Schmidt, and H. Niemann, “Point set registration with integrated scale estimation,” in International Conference on Pattern Recognition and Image Processing, pp. 116–119, 2005. [7] S. Rusinkiewicz and M. Levoy, “Efficient variants of the ICP algorithm,” in 3DIM. Published by the IEEE Computer Society, pp. 145– 152, 2001. [8] A.W. Fitzgibbon, “Robust registration of 2D and 3D point sets,” Image and Vision Computing, vol. 21, no. 13-14, pp. 1145–1153, 2003. [9] T. Jost and H. Hugli, “A multi-resolution ICP with heuristic closest point search for fast and robust 3D registration of range images,” in 3DIM. Published by the IEEE Computer Society, pp. 427–433, 2003. [10] P.J. Besl and N.D. McKay, “A method for registration of 3-D shapes,” IEEE Transactions on pattern analysis and machine intelligence, pp. 239–256, 1992. [11] E. Ezra, M. Sharir, and A. Efrat, “On the performance of the ICP algorithm,” Computational Geometry, vol. 41, no. 1-2, pp. 77–93, 2008. [12] M. Garland and P.S. Heckbert, “Surface simplification using quadric error metrics,” in Proceedings of the 24th annual conference on Computer graphics and interactive techniques, pp. 209–216, 1997.

Fast and Robust Isotropic Scaling Iterative Closest ...

Xi'an Jiaotong University, Xi'an, Shaanxi Province 710049, P.R. China. ABSTRACT. The iterative closest point (ICP) ... of China under Grant Nos.60875008 and 61005014, the National Basic Re- search Program of China (973 ..... select Apple,Cock,Deer and Horse 2D shapes on the part B of CE-Shape-1. In our method, the ...

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