Feature-Preserving Simpli cation of Point Cloud by Using ... · Feature-Preserving Simpli cation of Point Cloud by Using Clustering Approach Based on Mean Curvature Xi Yang1) Katsutsugu

The Journal of the Society for Art and Science Vol. 14, No. 4, pp. 117-128

Feature-Preserving Simplification of Point Cloud by UsingClustering Approach Based on Mean Curvature

Xi Yang1) Katsutsugu Matsuyama1) Kouichi Konno1)

Yoshimasa Tokuyama2)

1) Graduate School of Engineering, Iwate University2) Faculty of Engineering, Tokyo Polytechnic University

yangxi＠ lk.cis.iwate-u.ac.jp

AbstractFor point cloud data obtained from 3D scanning devices, excessively large storage and long post-processing time are required. Due to this, it is very important to simplify the point cloud to reducecalculation cost. In this paper, we propose a new point cloud simplification method that can maintainthe characteristics of surface shape for unstructured point clouds. In our method, a segmentation rangebased on mean curvature of point cloud can be controlled. The simplification process is completedby maintaining the position of the representative point and removing the represented points usingthe range. Our method can simplify results with highly simplified rate with preserving the formfeature. Applying the proposed method to 3D stone tool models, the method is evaluated preciselyand effectively.

Keywords: Point Cloud, Clustering, Mean Curvature, Simplification.

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1 Introduction

The better the performance of 3D scanningdevices[1] becomes year by year, the greater thenumber of generated point clouds will be. Forsome post-processing in the reverse engineering[2,3], however, a large number of scan points willgreatly consume the storage capacity for the pro-cess and the longer processing time is required.Simplification procedure for such point clouds isan efficient solution to these issues[4].

Point cloud simplification is a process that re-moves a large number of redundant points and itkeeps the feature points representing 3D modelsharp feature and boundaries. Currently, a lot ofpoint cloud simplification algorithms have beendeveloped. The algorithms are roughly dividedinto two categories: mesh-based methods andpoint-based ones. Simplification based on poly-gon meshes requires reconstruction of triangularmeshes from a point cloud and then redundantpoints are removed[5, 6, 7]. In contrast, simpli-fication based on points does not require recon-struction of a mesh model and it only relies on theinformation of points to simplify a point cloud.

On account of large amount of data to be pro-cessed and high computational complexity, theexecution time of mesh-based simplification al-gorithm is extremely long[5]. In addition, tri-angular mesh reconstruction from a point cloudis very complicated. Therefore, the point-basedsimplification algorithm is applied more widely atpresent.

In this paper, we propose a new pointcloud simplification algorithm based on curva-ture of points. Our simplification can be per-formed by using the pre-processing of Chida’smethod(see section 2.2). Simplification evalua-tion is optimized to find adjacent flakes with theirmethod[21].

2 Related Work

2.1 Previous Simplification

The simplification algorithms based on pointclouds have several typical methods. These meth-ods are not efficient or available for stone tool

models. Lee et al.[8] present a 3D point cloudsimplification method by using 3D grids. Paulyet al.[9] introduced and analyzed different strate-gies for surface simplification of geometric modelsfrom unstructured point clouds. Moenning andDodgson[10, 11] devised a coarse-to-fine uniformsimplification algorithm with user-controlled den-sity guarantee, based on the idea of progressiveintrinsic farthest point sampling of a surface inpoint clouds. Lee et al.[13] presented a novelsimplification method by adopting the DiscreteShape Operator to find the weight of the featuresof 3D models. Peng et al.[15] proposed a new sim-plification algorithm based on feature extractionfor unstructured point clouds with unit normalvectors.

In addition, the previous simplification meth-ods also have their own defects. For example,Song et al.[12] studied a global clustering pointcloud simplification approach by searching fora subset of the original input data set accord-ing to a specified data reduction ratio. In theirmethod, a global optimal result is obtained byminimizing the geometric deviation between theinput point sets and the simplified ones. Butwhen the number of points is reduced to becometoo small, the approximated point-to-surface dis-tances may no longer get accurate values, and itis hugely time-consuming. Thus, it is inefficientfor a large number of our stone tool models. Miaoet al.[14] proposed a curvature aware re-samplingapproach based on an adaptive mean-shift clus-tering scheme to simplify point clouds. An adap-tive mean-shift clustering scheme is designed togenerate a non-uniformly distributed simplifica-tion result. While it is difficult to incorporatethe simplified geometric error in their algorithm,its simplified rate is low. Thus, it is hard to ob-tain a consistent error result for each stone toolmodels. Shi et al.[16] presented a new adaptivecluster subdivision simplification method usingk-means clustering according to the two factors:user-defined space interval and normal vector tol-erance. For stone tools, however, flake surfacesare smooth and the others may be very rough.Thus, the ridge lines represented by flake surfacesare deleted when user-defined parameters becomelarge in order to raise the simplified rate. Further-

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more, these methods are not show the evaluationof normalized distance, therefore the simplifiedresults may not satisfy the requirements of ourfollow-up studies.

2.2 Search of Joining Material

Chida et al.[21] are studying on a method tosearch adjacent flake surfaces on stone imple-ments for generating a joining material. In theirmethod, a point cloud has to be simplified withmaintaining form features to extract edges. Sincethe method[21] is based on polygon-based ap-proach, it is necessary to generate a polygon meshfrom a point cloud. In addition, the method[21]is to search an adjacent stone tool through thegeometric matching score of flake surfaces. Ingeneral, however, since stone tools with match-ing flake surfaces are simplified respectively, thematching score is low.

For a simplification algorithm of a point cloud,it is important to evaluate the simplification er-rors. In general, the geometric errors between theoriginal point cloud and the simplified one shouldbe measured, such as the maximum error and theaverage error[9, 14, 16].

Our method, however, will be used in search-ing matching adjacent flake surfaces. Stone flakesare assembled on the same stone core and the po-sition and posture of the adjacent flake surfacesare restored to form a joining material of stonetools[21]. When flake surfaces are matched andthe candidates for adjacent surfaces are detected,a threshold value of the normalized distance[22]is a very important parameter for selecting theoptimal flake surface.

Then, the normalized distance is employed toevaluate the simplified result in this paper. Thenormalized distance D between the original pointcloud PC and the simplified one PC ′ is measuredby the following equations:

di = (Vi − gi) · ni (1)

D =1

S

n∑i=1

(di)2 (2)

For each point Vi ϵ PC, the geometric erroris distance di between original point Vi and its

Figure 1: Normalized distance

corresponding triangle Ti in simplified point cloudPC ′. Assuming that ni is the normal vector ofTi and gi is the geometric center of Ti, di can becalculated by equation(1). In order to get a valvethat does not depend on a polygonal mesh areaobtained from a point cloud, the sum of (di)

2 isdivided by the sum of triangular areas that belongto PC ′ like equation(2). Since a polygonal meshis required in equations (1) and (2), the simplifiedpoint cloud is temporally reconstructed by themethod described in [17].

3 Our Simplification Method

In this paper, a point cloud is simplified by themean curvature of points. The flowchart of themethod is shown in Figure 2. Firstly, the meancurvatures of points in a point cloud are com-puted. After that, all points in the entire pointcloud are sorted in descending order of the meancurvatures. Then the segmentation range of thepoint with the maximum mean curvature is cal-culated. Next, the points in the point cloud inthis range are removed. Finally, the point withthe second-largest mean curvature in the rest ofthe point cloud is taken and the steps describedearlier are repeated until all points in the pointcloud are processed. The simplification process iscompleted.

3.1 Curvature Calculation of Point Cloud

For curvature estimation of point cloud, sev-eral algorithms are proposed[18, 19]. In [20], amethod to calculate the curvature of the point

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Figure 2: Flowchart of our algorithm

cloud is proposed. Specifically, the method rep-resents shape features of the valley and the ridgelines by a frequency-domain. In addition, themethod can be applied to noisy non-aligned pointclouds. In our method, we calculate the princi-pal curvatures of a point cloud according to themethod[20].

Each point pi of an input point cloud is pro-cessed as shown in Figure 3. First, the twoprincipal directions of pi are selected from threeeigenvectors and calculated by the principal com-ponent analysis(PCA), according to K nearestneighbors of pi. In the experiment, the size ofneighbors K is set as 30. Let e1, e2, e3 be theeigenvectors of PCA, corresponding to the eigen-values λ1, λ2, λ3 (λ1 ≦ λ2 ≦ λ3). Eigenvectore1 estimates the normal vector of pi, eigenvectorse2 and e3 estimate the principal directions of pi.Thus, a local coordinate system (O′x′y′z′) can beconstructed at point pi, the axes lie on O′z′, O′y′

and O′x′ along the eigenvectors e1, e2, e3 respec-tively.

The point pi and its K nearest neighbors are

Figure 3: Mean curvature calculation in ourmethod

Figure 4: Equidistant sampling values generatedfrom nearest neighbors of pi

transformed to the local coordinate system andapproximated by a truncated Fourier series ineach principal direction. For example, a equidis-tant sampling points set ul (l = 0, ..., N − 1) iscreated along the O′x′ axis to calculate principalcurvature kix (N is set as 8 or 16 generally), asshown if Figure 4. The set center at pi and itsstep size is chosen to be smaller than the aver-age value d of the distances between each pointof the input point cloud and its closest point. Foreach point ul, its closest points qj (j = 0, 1, 2, 3)are selected from K nearest neighbors of pi in thefour spaces separated by plane x′O′z′ and planex′O′y′. The sampling value g(ul) is computedby equation (3), where, d(ul, qi) is a distance be-tween ul and pi and zqi is the z value of qi.

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g(ul) =

4∑i=1

1d(ul,qi)∑4j=1

1d(ul,qi)

zqi (3)

Next, the Fourier coefficients are derivedby Fast Fourier Transform(FFT) according tothe sampling values, as equation (4), whereRe(FFT (g(ui))) and Im(FFT (g(ui))) are realand imaginary parts of the coefficients ofFFT (g(ui)).

Z

a0 =1

NRe(FFT (g(u0))),

ai =2

NRe(FFT (g(ui))), (i = 1, ...,

N

2− 1),

ai =2

NIm(FFT (g(ui))), (i = 1, ...,

N

2− 1).

(4)

After that, with the expressions of curvatureand truncated Fourier series as equation (5), theprincipal curvature kix of pi is computed from theapproximated curves as equation (6), and kiy canbe computed in the same way along O′y′ axis.

f(x) ≃ 1

2a0+

N∑n=1

ancosnπx

L+

N∑n=1

bnsinnπx

L(5)

kix = −π2

L2

∑Nn=1 n

2an

(1 + ( πL∑N

n=1 nbn)2)

32

(6)

Finally, mean curvature Hi of pi is calculatedby the average of the two principal curvatures kixand kiy as the equation (7):

Hi =1

2(kix + kiy) (7)

3.2 Simplification Based on Curvature

For a 3D model, the feature points charac-terize sharp edges and surface boundaries thathave large curvature. On the other hand, thepoints with small curvature, which are not featurepoints, are redundant to represent the surfaces ofa model. A segmentation range of a point basedon the mean curvature is defined by the circles asshown in Figure 5, which can represent the corre-spondence with the range. Thus, if representativepoints are selected according to the mean curva-ture values, the point cloud can be simplified by

removing the points with small curvatures andmaintaining ones with large curvatures.

Firstly, all points in a point cloud are sorted bythe absolute value of their mean curvatures, andthe average mean curvature H of the point cloudis calculated as equation (8), H will be used tocalculate the segmentation range later.

H =1

n

n∑i=1

|Hi| (8)

Next, starting from the point pi of current max-imum curvature, the radius of its segmentationrange ri is calculated by equation (9).

ri = α · H

|Hi|(9)

According to equation (9), segmentation range

ri can be determined by H|Hi| . Therefore, in the

regions with larger curvatures the range is small,while in the regions with smaller curvatures, therange is large as shown in Figure 5. Additionally,users can control the scale of segmentation rangeby a positive value α. Points inside the circleof ri are the target ones for local simplification.The number of points in a simplified point cloudcan be estimated by α. Furthermore, in order tokeep the balance of the simplified point clouds,the minimum value of the range can be arbitrarilyset by users. In our implementation, this value isset to α/3.

Figure 5: Segmentation range

After the representative point and segmenta-tion range are determined, other points in the seg-mentation range are removed. Firstly the pointsin the range are searched by using the k-d tree[25].The computational complexity of k-d tree con-struction, insertion and deletion is high while

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search is low. Due to this complexity, deletionof elements in the k-d tree is replaced by markingelements in the list. Therefore, the k-d tree of apoint cloud needs to be built once in the begin-ning and deletion operation need not be repeatedfor other point clouds. Thus, the list and the k-dtree are retained in the memory during the oper-ation and points can be searched in the k-d treeand marked in list, as shown in Figure 6.

Figure 6: Removal of redundant points

After the points in the range are marked, thesegmentation range of the next unmarked pointis calculated, which has the currently maximummean curvature current. Repeat these steps untilall points in the point cloud are treated.

3.3 Evaluation

In the study of joining material searching[21], thenormalized distance of each pair of flake surfacesis employed to evaluate the matching score forsimilarity. Since our new simplification methodis used in the pre-processing stage of flake sur-face searching, the normalized distance for eachflake surface of a stone tool between the origi-nal point cloud and the simplified one should bemeasured in the same manner as the evaluation ofsimplification result. Our method prevents fromincreasing normalized distance because removedpoints have small curvature. Therefore, the shapeof simplified model is suitable for [21] by the pro-posed method.

4 Experimental Results andLimitation

We have implemented our algorithm using C++and OpenGL, and tested on a PC with an In-tel Core i5-3470 CPU and 8.00GB memory. Thesimplified point clouds are reconstructed by themethod described in [17]. The original pointclouds and the reconstructed simplified ones aredisplayed.

4.1 Clustering Result

(a) Original point cloud (b) Result of clustering

Figure 7: Result of clustering

Figure 7 shows the stone tool ofNo.L0197A0180 model and the result of clus-tering for this model, obtained by the proposedmethod. In the figure, the circles of differentcolors represent different clusters. Because ofcolor matching, the different adjacent clustersmay have the same color. The information ofthe model surface is shown clearly in Figure 7(a). As the two figures are compared, we canclearly see that small clusters are placed on theboundaries and ridges of the model, while largeclusters are placed for faces. So the method canmaintain the contour information of the model

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and the redundant information can be removedefficiently.

4.2 Simplified Results of Our Method

Figure 8 shows a group of No.L0197A0151model’s simplified results (triangulated) by thenew method with different segmentation ranges.The number of original points was 319K (a) andthe numbers of simplified ones were 160K (b),27K (c) and 9K (d). These examples demon-strated that the features of model were well main-tained by the proposed method, even though thenumber of points was reduced to 2.84% of thosein the original point cloud.

Figure 9 shows three stone tool models simpli-fied by using the proposed method. The numberof points of No.L0197A0178 model was reducedto 3.73% of those in its original model, that ofNo.S008A00010 model was reduced to 5.63%, andthat of the No.S008A0003 model was reduced to8.12%. In the right column of Figure 9, the pre-served points of models were clearly shown by sur-fels(surface elements proposed in [23]). The ex-perimental results indicate that the new methodcan maintain the boundaries and ridges of 3Dmodels.

Table 1 shows the number of original points,the execution time of each step, the numberof preserved points, and the maximum normal-ized distances of each flake surface. The execu-tion times show the method is efficient. For thematching study[21], the normalized distance ofeach flake surface segmented from the stone toolmodels is calculated. The evaluation results ofthe maximum normalized distance indicate thatthe error between simplified result and the orig-inal point cloud is very small, and the simpli-fied results are more than adequate requirementsof matching study[21]; that is, the new simpli-fication method has a good effect on the pre-processing of the matching study.

Stanford Dragon model was also tested andshown in Figures 10. In order to provide thestandard of evaluation, the normalized averagegeometric error[24, 14] was computed, shown inTable 2. In paper [14], Dragon model was testedand the number of simplified point cloud was

Table 2: Comparison of Dragon model.

Method Ours Miao et al.

Num. of original points 437,645 437,645

Eexecution time (sec.) 44.20 53.60

Num. of preserved points 34,861 34,049

Simplification ratio 92.03% 92.22%

Normalized average error 1.04 × 10−4 5.29 × 10−4

34,049, the normalized average error was 5.29 ×10−4. While the normalized average error of ourmethod was 1.04 × 10−4. According to paper[14],the total execution time of Miao et al. was 53.60seconds, and that of our method was 44.20 sec-onds. This contrast could be used as a referencefor the efficiency of our method, although theywere measured in different experiment environ-ments. As the result, normalized average errorwas smaller than one of the method[14], whenthe shape was simplified as same as the numberof the model. Our method prevents from increas-ing normalized distance because removed pointshave small curvature. Therefore, the superb eval-uation result of normalized average error is alsoobtained. This example demonstrated the effi-ciency of our method, and our method could ob-tain good simplified results for 3D models.

(a) Original point cloud (b) Simplified pointcloud

Figure 10: Dragon

Concerning the execution of our method forvery large data, an excavated relic model shownin Figure 11 by surfels[23] was tested. The num-ber of points of original model was 2,277,128 andthat of the preserved points was 10,144. The totalexecution time was approximately 253 seconds.This example indicated that our method has agood performance for big data.

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(a) Original point cloud (b) α = 0.1 (c) α = 0.5 (d) α = 1.1

Figure 8: Results of simplification with different segmentation ranges (triangulated)

Table 1: Status of three stone tool models.

Model Number of Execution time of each step (sec.) Number of Simplification Max normali-

original points preserved points ratio zation distance

File Curvature Simplification

reading calculation

L0197A0178 115030 0.522 12.676 0.353 4287 96.27% 0.0137

S008A00010 129145 0.566 12.001 0.417 7267 94.37% 0.0123

S008A0003 146214 0.688 13.834 0.508 11876 91.88% 0.0140

(a) Original point cloud (b) Simplified pointcloud

Figure 11: An excavated relic

4.3 Limitation

For a 3D model has thin flake shape as shown inFigure 12, the feature point p1 may be removedin the segmentation range of point p0 on the op-posite side, when the curvature of p1 smaller thanp0.

Figure 12: Thin flake shape section

5 Conclusions and Further Work

In this paper, a new curvature-based point cloudsimplification algorithm is proposed. The seg-mentation range is proposed for using one pointthat represents the others in the range. Thenthe method of removing redundant points in therange is applied according to the order of curva-tures to simplify the point cloud.

The normalized distance is employed to esti-mate the error of simplified results, and the usercan control the degree of simplification by a spaceinterval parameter. The larger parameters canlead to higher degree of simplification and largersimplification errors. Experiment results showthat the proposed new algorithm can obtain ef-

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L0197A0178 (α = 0.9)

S008A00010 (α = 0.8)

S008A0003 (α = 0.9)

Figure 9: Simplified results of the new method. Left: original point clouds(triangulated); middle:simplified point clouds(triangulated); right: simplified point clouds(surfel[23])

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ficient results that have high simplification ratesand low simplification errors.

This segmentation spherical range, however, isnot a good way for some extreme cases. Thereare three types of the local shape of point cloud: flat, rugged and ridge areas. These three typescan be expressed by principal curvatures in twodirections. Therefore, in the future, three dif-ferent ellipsoids will be designed to calculate thesegmentation spaces according to the two princi-pal curvatures against the different local shapes.This will obtain higher simplification accuracy.

The basic concept of our method has alreadybeen presented in NICOGRAPH 2014[26] and weextended the concept in this paper. We are ex-tremely grateful for lots of efficient advice fromthe paper reviewers.

Acknowledgements

We would like to thank the Tokyo National Mu-seum for providing data of their stone tool mod-els, and the Morioka City Study Museum ofArcheological Site for providing an excavated relicmodel in our experiment. This work was partiallysupported by KAKENHI(26420090).

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Xi Yang

Xi Yang is currently a postgraduate student inFaculty of Engineering at Iwate University. Hereceived the BE degree in College of InformationEngineering from Northwest A&F University in2012. His research interests include geometricmodeling and computer graphics.

Katsutsugu Matsuyama

Katsutsugu Matsuyama is currently an assistantprofessor at Iwate University. His research inter-ests include computer graphics, information vi-sualization and interactive systems. He receivedBE, ME, DE degrees in computer science from

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Iwate University in 1999, 2001 and 2005, respec-tively. He was a research associate at FutureUniversity-Hakodate from 2005 to 2011.

Kouichi Konno

Kouichi Konno is a professor of Faculty of Engi-neering at Iwate University. He received a BS inInformation Science in 1985 from the Universityof Tsukuba. He earned his Dr.Eng. in preci-sion machinery engineering from the Universityof Tokyo in 1996. He joined the solid modelingproject at RICOH from 1985 to 1999, and theXVL project at Lattice Technology in 2000. Heworked on an associate professor of Faculty of En-gineering at Iwate University from 2001 to 2009.His research interests include virtual reality, geo-metric modeling, 3D measurement systems, andcomputer graphics. He is a member of IEEE.

Yoshimasa Tokuyama

Yoshimasa Tokuyama is a professor of Depart-ment of Media and Image Technology, Faculty ofEngineering, Tokyo Polytechnic University. Hereceived his MS in Mechanical Engineering in1986 and doctor degree in Computer Graphicsin 2000 from The University of Tokyo. He was amember of the 3D CAD project at RICOH’s Soft-ware Division from 1986 to 2002. His areas of re-search interest include computer graphics, game,haptic interface, virtual reality, shape modeling,and their applications. He is a member of In-formation Processing Society of Japan, Instituteof Image Information and Television Engineers ofJapan, the Institute of Image Electronics Engi-neers of Japan, the Society for Art and Science.

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Feature-Preserving Simpli cation of Point Cloud by Using ... · Feature-Preserving Simpli cation of Point Cloud by Using Clustering Approach Based on Mean Curvature Xi Yang1) Katsutsugu

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