Research ArticleA Point Cloud Registration Algorithm Based on FeatureExtraction and Matching
Yongshan Liu1 Dehan Kong 2 Dandan Zhao1 Xiang Gong1 and Guichun Han1
1Department of Information Science and Engineering Yanshan University Hebei Street No 438 Qinhuangdao Hebei 066000China2Department of Information Engineering Hebei University of Environmental Engineering Jingang Street No 8 QinhuangdaoHebei 066000 China
Correspondence should be addressed to Dehan Kong kdh0312163com
Received 23 August 2018 Revised 18 November 2018 Accepted 9 December 2018 Published 24 December 2018
Academic Editor Kauko Leiviska
Copyright copy 2018 Yongshan Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The existing registration algorithms suffer from lowprecision and slow speedwhen registering a large amount of point cloud data Inthis paper we propose a point cloud registration algorithm based on feature extraction andmatching the algorithm helps alleviateproblems of precision and speed In the rough registration stage the algorithm extracts feature points based on the judgment ofretention points and bumps which improves the speed of feature point extraction In the registration process FPFH features andHausdorff distance are used to search for corresponding point pairs and the RANSAC algorithm is used to eliminate incorrectpoint pairs thereby improving the accuracy of the corresponding relationship In the precise registration phase the algorithm usesan improved normal distribution transformation (INDT) algorithm Experimental results show that given a large amount of pointcloud data this algorithm has advantages in both time and precision
1 Introduction
The continuous development of three-dimensional (3D)scanning equipment has promoted the application of 3Dpoint cloud reconstruction technology in the fields ofmechanical manufacturing medicine robot navigation andother industries Registration technology plays an importantrole in 3D point cloud reconstruction Due to the influenceof the measuring equipment object shape and environmenta complete point cloud data model must be measured frommultiple perspectives It is an important step to integrate thepoint cloud in different coordinate systems into a unifiedcoordinate system to obtain the complete data model of theobject being tested
With a large amount of point cloud data the existing reg-istration algorithms either improve the registration accuracyat the cost of time or increase the registration speed How toachieve a balance between time and accuracy when dealingwith a large volumeof cloud data is a problemworth studyingIn this paper a point cloud registration algorithm based onfeature extraction and matching is proposed to improve thespeed while maintaining the accuracy of the registration of
a large amount of cloud data The algorithm uses the fastpoint feature histogram (FPFH) descriptor [1] and Hausdorffdistance [2] to extract and locate the corresponding featurepoints in order tominimize the number of registration pointsthereby improving its speedWepropose an improved normaldistribution transform (INDT) algorithm for precise regis-tration and a mixed probability density function (PDF) toreplace the single PDF which further improves the accuracyof the algorithm
The remainder of this article is organized as followsThe research situation is introduced and analyzed in Sec-tion 2 Section 3 introduces our rough registration algorithmSection 4 provides the improved algorithm of the normaldistribution transformation Section 5 verifies the speed andaccuracy of the registration algorithm based on experimentalsimulation data
2 Related Works
The research on registration technology started in 1960 andit experienced a transition from the two-dimensional image
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7352691 9 pageshttpsdoiorg10115520187352691
2 Mathematical Problems in Engineering
registration to a 3D model registration Automatic registra-tion is one of the three types of point cloud registration meth-ods which includes instrument-based and manual registra-tion Automatic registration has greater flexibility and doesnot require human intervention and auxiliary equipmentTherefore most researchers study automatic registrationPoint cloud registration technology mainly has two partsrough registration and accurate registration The registrationalgorithm in this paper is also composed of two parts roughregistration and precise registration while registration ofpoint clouds is mainly divided into rigid registration andnonrigid registrationThedeformation of point clouds whichwill directly affect the extraction of feature points mustbe considered in nonrigid registration This study focusesmainly on rigid registration
21 Initial Registration of Point Cloud The four current typesof rough registration of point clouds are based on the follow-ing (1) the measurement equipment (2) human-computerinteraction (3) exhaustive thinking and (4) features Thefirst two methods rely on the limitations of the auxiliaryequipment and the need for manual intervention The lattertwo constitute the main research direction
The registration method based on exhaustive thinkingcan be divided into methods of (1) maximizing the pointof rigid body transformation to obtain registration param-eters and (2) finding the minimum transformation of theerror function by traversing the transformation space quasi-parameter These include the voting method [3] geomet-ric hashing search transformation space method [4] andRANSAC (RANdom SAmple Consensus) algorithm In 1981Fischler proposed theRANSACalgorithmwhich is amethodof fitting a mathematical model based on a sample Meng [5]proposed the application of a sampling sphere and improvedthe original registration algorithm based on it this method isbased on the RANSAC idea By setting up a sampling spheremodel the searching efficiency of the corresponding pointpair is improved and the time complexity of the algorithm isreduced In 2015 Zhang Yonging [6] proposed a point cloudrough registration method based on the feature descriptors ofa point feature histogram and FPFH along with the principleof sampling consistency
The feature-based registration method obtains the coor-dinate transformation parameters by performing a corre-sponding point pair search on the overlapping regions of thepoint clouds thereby completing the point cloud registrationWhen looking for a corresponding point themethod choosesthe point with the most similar geometry In 2000 Chua[7] and others proposed the point signature (PS) methodThis method constructs a point signature through extensivecalculation it is a complex process that is sensitive to noiseIn 2005 through the unremitting efforts of Gelfand [8] andothers feature descriptors based on integral invariants wereapplied to the field of 3D registration thereby ensuringthe reliability of the registration results Yan Jianfeng [9]proposed a method of rough registration using the curvatureof the point cloud to extract feature points and usingsimilarity to find pairs of points this shortened the time of
registration but the accuracy remained low Yang Xiaoqing[10] improved the point cloud coarse registration algorithmbased on the normal vector based on a point geometricfeature extraction algorithm in 3D space The key points areselected by comparing the characteristic degree of each pointin the point cloud data (ie the angle between the normalpoints and adjacent points) and the threshold valueThemaincurvature is then calculated only for the key points therebyincreasing the speed of calculating the curvature and theefficiency of the corresponding point pair search Howeverthis method facing the registration of a large volume ofpoint cloud data has a long running time or low precisionTo solve this problem we propose a rough point cloudregistration algorithm to ensure a shorter running time andmore accurate calculations
22 Point CloudAccurate Registration Themost classic pointcloud accurate registration technology is the iterative closestpoint (ICP) algorithm However because ICP is used tofind the corresponding point using the nearest neighborsearch the operation is too long and the initial positionof the point cloud is too high so the normal distributionstransform (NDT) algorithm is the main precision Quasi-technical research
The NDT algorithm was proposed in 2003 by Biber etal [11] In 2006 Martin Magnusson [12] summarized 2D-NDT and extended it to the registration of 3D data through3D-NDT Magnussonrsquos algorithm is faster than the currentstandard for 3D registration and is often more accurate In2011 Cihan [13] proposed a multilayered normal distributiontransformation algorithm called MLNDT The algorithmdivides a point cloud into 8n cells where n is the numberof layers replaces the original Gaussian probability functionwith the Mahalanobis distance function as a fractionalfunction and uses the Newton and Levenberg-Marquardt(LM) optimization method to optimize the fractional func-tion with better registration speed than the previous NDTalgorithm In 2013 Cihan [14] further explained the MLNDTalgorithm In 2015 Hyunki Hong and BH Lee [15] proposedthe key-layered normal distributions transform algorithm(KLNDT) based on the key layer normal distribution trans-formation it has a good success rate and accuracy In 2016Hyunki Hong and BH Lee [16] proposed to convert thereference point cloud into a disk-like distribution suitable forthe point cloud structure to improve the registration accuracyof the NDTalgorithm However there are few studies of NDTalgorithms in China Zhang Xiao [17] performed a featurepoint search based on the speeded up robust features (SURF)algorithm an improvement of the 3D-NDT algorithm HuXiuxiang [18] proposed the normal aligned radial feature(NARF) algorithmwhich improvedNDTChenQingyan [1920] study is based on curvature feature of NDT registrationmethod Zheng Fen [21] et al improved 3D scale invariantfeature transform (3DSIFT) algorithm and 3D-NDT algo-rithm The above methods all improve the original NDTalgorithm in different ways with various accuracy and timeadvantages However it has always been a challenge to obtaina balance between time and accuracy when registering large
Mathematical Problems in Engineering 3
amounts of point cloud data The registration algorithm inthis paper improves the normal distribution transformationINDT algorithm in the accurate registration stage improvesthe speed of the registration algorithm and ensures goodregistration accuracy
Existing initial and precise registration algorithms faceproblems when dealing with a large volume of point clouddata This paper proposes an algorithm that is based onfeatures andmatching which improves the registration speedwhile ensuring good registration accuracy
3 Rough Registration of Point Clouds
The main ideas of the initial algorithm are as followsFirst the feature points are extracted from the source pointcloud and the target point cloud respectively reducing thenumber of points involved in registration and removing theredundant points and thus greatly reducing the registrationtime Second the high dimensional descriptor FPFH is usednot only as a standard but also the Moorhouse multihusbanddistance to further determine the correspondence betweenpoints The RANSAC algorithm is then used to remove thecorresponding pairs of errors Finally we use singular valuedecomposition (SVD) to solve the coordinate transformationmatrix and apply it to the source point cloud to complete thecoarse registration Next we introduce the main links of therough registration algorithm
31 Feature Point Extraction The extraction of feature pointsreduces the number of point sets involved in registrationthereby improving the speed of registration With fewer pointcloud data points the introduction of feature points greatlyimproves the algorithms speed But a large volume of pointcloud data will increase the running time of feature pointextraction This can easily burden the entire registrationalgorithm which leads to long running time To solve thisproblem we judge the retention points before judging thefeature points The judgment of reserved points cannot onlyremove obvious outliers but can improve the efficiency of thewhole feature point extraction algorithmMoreover themorepoint cloud data the greater the time advantage of the featurepoint extraction algorithm
The method of extracting feature points in this paper isdivided into two steps (1) removing somepoints according tothe mean curvature of the point ie selecting the reservationpoint and (2) determining the feature points by the concaveand convex points as described below
The first step is the judgment of the reservation pointThemean curvature of each point in the point cloud is calculatedand the mean 120583 and variance 120590 of the point average meancurvature are calculated Second the points with averagecurvature greater than 120583+119901120590 are chosen as reservation pointsto remove the other points in the point cloud Among them119901 is the proportional coefficient whose value controls thenumber of reserved points When 119901 is large there are fewerreserved points and vice versa
Curvature generally includes the principal normalmean and Gaussian curvature We use the mean curvature
to filter feature points The normal curvature measures thedegree of curvature of a surface in a certain direction andthe principal curvature represents the extreme value of thenormal curvature Gaussian curvature is an intrinsic measureof curvature ie its value depends on how the distance on thesurface is measured rather than how the surface is embeddedin space Average curvature is an ldquoexternalrdquo bendingmeasure-ment standard which describes the curvature of a surfaceembedded in the surrounding space locally Therefore it canbetter reflect the degree of change and particularity of thecurrent point so in this paper average curvature is used asthe measure of screening feature points
The second step is the judgment of the concave andconvex points If a point on any surface is concave or convexit is taken as a feature point [22] Based on this principle wecalculate the value of 119878(119901)(119904(119901) isin (0 1)) using the followingequation
119878 (119901) = 12 minus 1120587 arctan1198961 (119901) + 1198962 (119901)1198961 (119901) minus 1198962 (119901) (1)
If 119878(119901119894) gt max(119878(1199011198941) 119878(1199011198942) 119878(119901119894119896)) then point 119901119894 isa local convex point If 119878(119901119894) lt min(119878(1199011198941) 119878(1199011198942) 119878(119901119894119896))then 119901119894 is a local concave point In this the 119878(1199011198941)119878(1199011198942) 119878(119901119894119896) are the 119878(119875) value of the point 119901119894 neigh-borhood points Points in point cloud data that are concavepoints or bumps are characteristic points
The steps to extract feature points from point cloud 119875 areas follows(1) Calculate the mean curvature of each point(2) Calculate the mean 120583 and variance 120590 of meancurvature and select points with mean curvature greater than120583+119901120590 as retention points (119901 is proportional to the coefficientand the 119901 is used to control the number of reserved points)(3)The concave points are judged by the reserved pointsand the feature points are extracted
32 Corresponding Point Pair Lookup Based on FPFH andHausdorff Distance To find the corresponding point is tofind the nearest point of the query point in another pointcloud If we calculate the Euclidean distance from eachpoint in the other point cloud to the query point wewill find the corresponding point pair but this requiresextensive calculation with little accuracy in the results Theaccuracy of the corresponding point pairs directly affects theregistration effect of point clouds In this paper the robustFPFH descriptor is added to the corresponding point pairsearch algorithm The fast point feature histogram (FPFH)is based on the relationship between the sampling pointand the neighborhood point normal which is representedby 33 dimensional descriptors In addition to find thecorresponding pointsmore accurately and reduce the effect oferror correspondences the Hausdorff distance is introducedto the algorithm
We use FPFH points and Hausdorff distances to findcorresponding points First the feature points set in thesource point cloud and target point cloud are describedby FPFH Next we find the point of minimum differencebetween the FPFH points of each point of the source point
4 Mathematical Problems in Engineering
Input source cloud 119875 target cloud119876Output Registered point cloud 1199001199061199051198621198971199001199061198891 Extract feature points from the source cloud and the target cloud separately2 for all points 119901119894 isin 119875 do3 if 119901119894 is a key point then4 119875119896119890119910 larr997888 1199011198945 end if6 end for7 for all 119901119894 isin 119875119896119890119910 do8 119875119889119890119904minus119896119890119910 larr997888 1199011198949 end for10 gets 119876119896119890119910 and 119876119889119890119904minus119896119890119910 in the same way11 for point 119902119895 isin 119876119889119890119904minus119896119890119910 do12 gets 11987911990213 end for14 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do15 119878 larr997888most similar point to 119901119894 in 11987911990216 119862119888119900119899 larr997888 11987811989417 end for18 getRemainingCorrespondences()19 estimateRigidTransformation(119875119896119890119910119876119896119890119910)20 transformPointCloud(119875119876transform matrix)21 Return 119877119878
Algorithm 1 Point cloud initial registration algorithm
cloud feature point through the nearest neighbor search inthe target point cloud feature point set Finally we calculatethe Hausdorff distance if it is less than the threshold valuethen the two are corresponding points
33 RANSAC Culling Error Corresponding Point Pair Thecorresponding error pairs will affect the estimation of thefinal rigid transformation matrix thus affecting the pointcloud registration Therefore wrong corresponding pointpairsmust be eliminated By finding the corresponding pointsbased on FPFH and the Hausdorff distance in the previoussection we obtained a corresponding point pair with a highermatching degree however with a certain gap in the actualapplication of the distance point cloud registration This ismainly due to noise in the gathering process of the pointcloud which leads to the topology of the point cloud data ofthe two times of obtaining the same area being not completelyconsistent so there are some wrong correspondences In thispaper the RANSAC (random sample consistency) algorithmis used to eliminate error correspondences
34 Description of the Algorithm The rough point cloudregistration algorithm for feature extraction and matchingmainly uses the FPFH description Hausdorff distance andRANSAC algorithm to perform pairwise registration of pointclouds aiming to provide their accurate registration of pointclouds and good initial position The specific algorithmdescription is shown as Algorithm 1
In lines 1-6 the feature points are extracted according tothe method mentioned in Section 31 The reserved points
are obtained in lines 2-4 and concave and convex points arejudged in lines 5 and 6 Finally the feature points of the sourcepoint cloud are obtained Line 7 performs similar operationson the target point cloud to get the feature points set Lines8-12 as mentioned in Section 32 perform a lookup basedon corresponding points Line 13 uses RANSAC to eliminatethe corresponding error points Line 14 solves the rigid bodytransformation matrix Line 15 transforms the source pointcloud to complete the initial registration
4 Accurate Registration Based on ImprovedNDT Algorithm
Weuse the improvedNDTalgorithm INDT for accurate reg-istration First only the feature point sets of the source cloudand the target cloud are involved in the algorithm during theregistration process Second the algorithm replaces a singleprobability density function with a mixed probability densityfunction Finally the algorithm updates the Newton iterativealgorithm through a linear searchWewill describe the INDTalgorithm in detail
41 Mixed Probability Density Function For noise pointsthe negative logarithm probability function of the normaldistribution will grow without limit which will greatlyaffect the result Therefore using a single probability densityfunction as shown in Formula (1) the registration effect isnot good for noisy point cloud data Like Lu Ju [23] we use
Mathematical Problems in Engineering 5
04
035
03
025
02
015
01
005
0
minus4 minus2 0 2 4
xminus4 minus2 0 2 4
x
p(x)
5
4
3
2
1
0
-log(p(x))(x)p -log( )(x)p
Figure 1 Comparison of single probability density function and mixed probability density function
the PDF of the PDF which consists of a normal distributionand a uniform distribution
119901 (119909) = 1198881 exp((119909 minus 120583)119879summinus1 (119909 minus 120583)2 ) + 11988821199010 (2)
1199010 is the expected ratio of noise and 1198881 1198882 are usedto determine the normalization of the probability densityfunction by the spatial span of voxelsThe effect of noise whenusing the mixed probability density function is bounded asshown in Figure 1
The form of 997888rarr119901(119909) = minus log(1198881 exp((119909minus120583)119879summinus1(119909minus120583)2)+1198882) is a logarithmic function to be optimized These termshave no simple first and second derivatives However thenegative logarithm probability function can be approximatedby a Gaussian function This form of function is similar tothe Gauss function 119901(119909) = 119889119894 exp(minus1198892119909221205902) The requiredfitting parameters should be consistent with x=0 x=120590 andx=infin as shown in Formula (3) Using the Gaussian functionto estimate the influence of a point in the current point cloudon the NDT fractional function is shown in Formula (4)The NDT fractional function is simpler than the derivativeof Formula (2) but it still has the same general properties inthe optimization Since 1198893 only adds a constant offset to thefractional function when registering with NDT and does notchange its shape or its optimized parameters 1198893 is roundedoff from Formula (4)
1198893 = minus log (1198882)1198891 = minus log (1198881 + 1198882) minus 1198893
119901 (119909119896) = minus1198891 exp(minus11988922 (119909119896 minus 120583119896)119879)minus1sum119896
(119909119896 minus 120583119896) (4)
In Formula (4)119909119896minus120583119896is the average number ofNDTcellssumminus1119896 (119909119896 minus 120583119896) is the covariance of the NDT cell
42 Newton Iteration Algorithm for Parameter SolutionGiven point set 119909 = 1199091 1199092 119909119899 pose 119901 and the transformfunction 119879(119901 119909) that transforms 119909 by 119901 for the currentparameter vector the NDT fractional function is shown in
119904 (119901) = minus 119899sum119896minus1
119901 (119879 (119901 minus 119909119896)) (5)
This corresponds to the possibility of point 119909119896 beinglocated on the surface of the target point cloud after 119901transformation
The probability function requires the inverse of thecovariance matrix summinus1 Where the points in a cell are com-pletely coplanar or collinear the covariance matrix is singularand cannot be inverted In the 3D case the covariance matrixcalculated from three or fewer points is singular ThereforePDFs only calculate cells that contain more than three pointsIn addition to prevent numerical problems as long as it isfound that the sum is almost singular sum is slightly enlarged Ifthe maximum eigenvalue of sum 1205823 is 100 times greater thanthe eigenvalue 1205821 or 1205822 the smaller eigenvalue 120582119895 will bereplaced by 120582119895 = 1205823100 Replace the covariance matrix sum
6 Mathematical Problems in Engineering
Input two point clouds 119875 and 119876 to be registeredOutput Point cloud 1198751015840 and 1198761015840 after accurate registration1 Initialization2 Divide the ell structure B3 for all points 119902119894 isin 119876 do4 find the cell 119887119894 containing the point 119902119894119887119894 isin 1198615 store 119902119894 in 1198871198946 end for7 for all cells 119887119894 isin 119861 do8 1199011015840 997888rarrall points in 1198871198949 119898119906119894 997888rarr 1119899 sum119898119896=1 119910101584011989610 sum119894 997888rarr 1119898 minus 1 sum119898119896=1(1199101015840119896 minus 120583)(1199101015840119896 minus 120583)11987911 end for12 Registration13 while not converged do14 119904119888119900119903119890 997888rarr 015 119892 997888rarr 016 119867 997888rarr 017 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do18 find the cell 119887119894 containing the conversion function 119879(119901 119909119896)19 119904119888119900119903119890 997888rarr 119904119888119900119903119890+119901(119879(119901 119909119896))20 update 11989221 update11986722 end for23 solve119867Δ119901 = minus11989224 119901 997888rarr 119901 + Δ11990125 end while
Algorithm 2 INDT algorithm
with a matrix sum1015840 = orand1015840or where or contains the eigenvalue ofsum and and1015840 is shown in
The Newton algorithm can be used to find the parameter119901 for optimization 119904(119901) The algorithm iteratively solves theequation 119867Δ119901 = minus119892 where119867 and 119892 are the Hessian and thegradient vector of the fractional function 119904(119901) respectivelyDuring each iteration119901 997888rarr 119901+Δ119901 (the increment isΔ119901) Let119909119896 equiv 119879(119901 119909119896)minus120583119896 in other words 119909119896) is the transformation ofpoint 119909119896) by the current pose parameter relative to the centerof the cell PDF to which it belongs The element 119892119894 of thegradient vector 119892 is as shown as
Regardless of the dimension of registration the gradientof the NDT fractional function and the expression of theHessian matrix are the same and are similarly independentof the transformation representation
43 Description of the Algorithm The algorithm uses thefeature point set of the source point cloud and the target pointcloud to register and replaces the single probability densityfunction with the mixed probability density function witha normal distribution and a uniform distribution and usesthe Newton algorithm to iteratively solve the transformationparameters The algorithm is shown as Algorithm 2
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
2 Mathematical Problems in Engineering
registration to a 3D model registration Automatic registra-tion is one of the three types of point cloud registration meth-ods which includes instrument-based and manual registra-tion Automatic registration has greater flexibility and doesnot require human intervention and auxiliary equipmentTherefore most researchers study automatic registrationPoint cloud registration technology mainly has two partsrough registration and accurate registration The registrationalgorithm in this paper is also composed of two parts roughregistration and precise registration while registration ofpoint clouds is mainly divided into rigid registration andnonrigid registrationThedeformation of point clouds whichwill directly affect the extraction of feature points mustbe considered in nonrigid registration This study focusesmainly on rigid registration
21 Initial Registration of Point Cloud The four current typesof rough registration of point clouds are based on the follow-ing (1) the measurement equipment (2) human-computerinteraction (3) exhaustive thinking and (4) features Thefirst two methods rely on the limitations of the auxiliaryequipment and the need for manual intervention The lattertwo constitute the main research direction
The registration method based on exhaustive thinkingcan be divided into methods of (1) maximizing the pointof rigid body transformation to obtain registration param-eters and (2) finding the minimum transformation of theerror function by traversing the transformation space quasi-parameter These include the voting method [3] geomet-ric hashing search transformation space method [4] andRANSAC (RANdom SAmple Consensus) algorithm In 1981Fischler proposed theRANSACalgorithmwhich is amethodof fitting a mathematical model based on a sample Meng [5]proposed the application of a sampling sphere and improvedthe original registration algorithm based on it this method isbased on the RANSAC idea By setting up a sampling spheremodel the searching efficiency of the corresponding pointpair is improved and the time complexity of the algorithm isreduced In 2015 Zhang Yonging [6] proposed a point cloudrough registration method based on the feature descriptors ofa point feature histogram and FPFH along with the principleof sampling consistency
The feature-based registration method obtains the coor-dinate transformation parameters by performing a corre-sponding point pair search on the overlapping regions of thepoint clouds thereby completing the point cloud registrationWhen looking for a corresponding point themethod choosesthe point with the most similar geometry In 2000 Chua[7] and others proposed the point signature (PS) methodThis method constructs a point signature through extensivecalculation it is a complex process that is sensitive to noiseIn 2005 through the unremitting efforts of Gelfand [8] andothers feature descriptors based on integral invariants wereapplied to the field of 3D registration thereby ensuringthe reliability of the registration results Yan Jianfeng [9]proposed a method of rough registration using the curvatureof the point cloud to extract feature points and usingsimilarity to find pairs of points this shortened the time of
registration but the accuracy remained low Yang Xiaoqing[10] improved the point cloud coarse registration algorithmbased on the normal vector based on a point geometricfeature extraction algorithm in 3D space The key points areselected by comparing the characteristic degree of each pointin the point cloud data (ie the angle between the normalpoints and adjacent points) and the threshold valueThemaincurvature is then calculated only for the key points therebyincreasing the speed of calculating the curvature and theefficiency of the corresponding point pair search Howeverthis method facing the registration of a large volume ofpoint cloud data has a long running time or low precisionTo solve this problem we propose a rough point cloudregistration algorithm to ensure a shorter running time andmore accurate calculations
22 Point CloudAccurate Registration Themost classic pointcloud accurate registration technology is the iterative closestpoint (ICP) algorithm However because ICP is used tofind the corresponding point using the nearest neighborsearch the operation is too long and the initial positionof the point cloud is too high so the normal distributionstransform (NDT) algorithm is the main precision Quasi-technical research
The NDT algorithm was proposed in 2003 by Biber etal [11] In 2006 Martin Magnusson [12] summarized 2D-NDT and extended it to the registration of 3D data through3D-NDT Magnussonrsquos algorithm is faster than the currentstandard for 3D registration and is often more accurate In2011 Cihan [13] proposed a multilayered normal distributiontransformation algorithm called MLNDT The algorithmdivides a point cloud into 8n cells where n is the numberof layers replaces the original Gaussian probability functionwith the Mahalanobis distance function as a fractionalfunction and uses the Newton and Levenberg-Marquardt(LM) optimization method to optimize the fractional func-tion with better registration speed than the previous NDTalgorithm In 2013 Cihan [14] further explained the MLNDTalgorithm In 2015 Hyunki Hong and BH Lee [15] proposedthe key-layered normal distributions transform algorithm(KLNDT) based on the key layer normal distribution trans-formation it has a good success rate and accuracy In 2016Hyunki Hong and BH Lee [16] proposed to convert thereference point cloud into a disk-like distribution suitable forthe point cloud structure to improve the registration accuracyof the NDTalgorithm However there are few studies of NDTalgorithms in China Zhang Xiao [17] performed a featurepoint search based on the speeded up robust features (SURF)algorithm an improvement of the 3D-NDT algorithm HuXiuxiang [18] proposed the normal aligned radial feature(NARF) algorithmwhich improvedNDTChenQingyan [1920] study is based on curvature feature of NDT registrationmethod Zheng Fen [21] et al improved 3D scale invariantfeature transform (3DSIFT) algorithm and 3D-NDT algo-rithm The above methods all improve the original NDTalgorithm in different ways with various accuracy and timeadvantages However it has always been a challenge to obtaina balance between time and accuracy when registering large
Mathematical Problems in Engineering 3
amounts of point cloud data The registration algorithm inthis paper improves the normal distribution transformationINDT algorithm in the accurate registration stage improvesthe speed of the registration algorithm and ensures goodregistration accuracy
Existing initial and precise registration algorithms faceproblems when dealing with a large volume of point clouddata This paper proposes an algorithm that is based onfeatures andmatching which improves the registration speedwhile ensuring good registration accuracy
3 Rough Registration of Point Clouds
The main ideas of the initial algorithm are as followsFirst the feature points are extracted from the source pointcloud and the target point cloud respectively reducing thenumber of points involved in registration and removing theredundant points and thus greatly reducing the registrationtime Second the high dimensional descriptor FPFH is usednot only as a standard but also the Moorhouse multihusbanddistance to further determine the correspondence betweenpoints The RANSAC algorithm is then used to remove thecorresponding pairs of errors Finally we use singular valuedecomposition (SVD) to solve the coordinate transformationmatrix and apply it to the source point cloud to complete thecoarse registration Next we introduce the main links of therough registration algorithm
31 Feature Point Extraction The extraction of feature pointsreduces the number of point sets involved in registrationthereby improving the speed of registration With fewer pointcloud data points the introduction of feature points greatlyimproves the algorithms speed But a large volume of pointcloud data will increase the running time of feature pointextraction This can easily burden the entire registrationalgorithm which leads to long running time To solve thisproblem we judge the retention points before judging thefeature points The judgment of reserved points cannot onlyremove obvious outliers but can improve the efficiency of thewhole feature point extraction algorithmMoreover themorepoint cloud data the greater the time advantage of the featurepoint extraction algorithm
The method of extracting feature points in this paper isdivided into two steps (1) removing somepoints according tothe mean curvature of the point ie selecting the reservationpoint and (2) determining the feature points by the concaveand convex points as described below
The first step is the judgment of the reservation pointThemean curvature of each point in the point cloud is calculatedand the mean 120583 and variance 120590 of the point average meancurvature are calculated Second the points with averagecurvature greater than 120583+119901120590 are chosen as reservation pointsto remove the other points in the point cloud Among them119901 is the proportional coefficient whose value controls thenumber of reserved points When 119901 is large there are fewerreserved points and vice versa
Curvature generally includes the principal normalmean and Gaussian curvature We use the mean curvature
to filter feature points The normal curvature measures thedegree of curvature of a surface in a certain direction andthe principal curvature represents the extreme value of thenormal curvature Gaussian curvature is an intrinsic measureof curvature ie its value depends on how the distance on thesurface is measured rather than how the surface is embeddedin space Average curvature is an ldquoexternalrdquo bendingmeasure-ment standard which describes the curvature of a surfaceembedded in the surrounding space locally Therefore it canbetter reflect the degree of change and particularity of thecurrent point so in this paper average curvature is used asthe measure of screening feature points
The second step is the judgment of the concave andconvex points If a point on any surface is concave or convexit is taken as a feature point [22] Based on this principle wecalculate the value of 119878(119901)(119904(119901) isin (0 1)) using the followingequation
119878 (119901) = 12 minus 1120587 arctan1198961 (119901) + 1198962 (119901)1198961 (119901) minus 1198962 (119901) (1)
If 119878(119901119894) gt max(119878(1199011198941) 119878(1199011198942) 119878(119901119894119896)) then point 119901119894 isa local convex point If 119878(119901119894) lt min(119878(1199011198941) 119878(1199011198942) 119878(119901119894119896))then 119901119894 is a local concave point In this the 119878(1199011198941)119878(1199011198942) 119878(119901119894119896) are the 119878(119875) value of the point 119901119894 neigh-borhood points Points in point cloud data that are concavepoints or bumps are characteristic points
The steps to extract feature points from point cloud 119875 areas follows(1) Calculate the mean curvature of each point(2) Calculate the mean 120583 and variance 120590 of meancurvature and select points with mean curvature greater than120583+119901120590 as retention points (119901 is proportional to the coefficientand the 119901 is used to control the number of reserved points)(3)The concave points are judged by the reserved pointsand the feature points are extracted
32 Corresponding Point Pair Lookup Based on FPFH andHausdorff Distance To find the corresponding point is tofind the nearest point of the query point in another pointcloud If we calculate the Euclidean distance from eachpoint in the other point cloud to the query point wewill find the corresponding point pair but this requiresextensive calculation with little accuracy in the results Theaccuracy of the corresponding point pairs directly affects theregistration effect of point clouds In this paper the robustFPFH descriptor is added to the corresponding point pairsearch algorithm The fast point feature histogram (FPFH)is based on the relationship between the sampling pointand the neighborhood point normal which is representedby 33 dimensional descriptors In addition to find thecorresponding pointsmore accurately and reduce the effect oferror correspondences the Hausdorff distance is introducedto the algorithm
We use FPFH points and Hausdorff distances to findcorresponding points First the feature points set in thesource point cloud and target point cloud are describedby FPFH Next we find the point of minimum differencebetween the FPFH points of each point of the source point
4 Mathematical Problems in Engineering
Input source cloud 119875 target cloud119876Output Registered point cloud 1199001199061199051198621198971199001199061198891 Extract feature points from the source cloud and the target cloud separately2 for all points 119901119894 isin 119875 do3 if 119901119894 is a key point then4 119875119896119890119910 larr997888 1199011198945 end if6 end for7 for all 119901119894 isin 119875119896119890119910 do8 119875119889119890119904minus119896119890119910 larr997888 1199011198949 end for10 gets 119876119896119890119910 and 119876119889119890119904minus119896119890119910 in the same way11 for point 119902119895 isin 119876119889119890119904minus119896119890119910 do12 gets 11987911990213 end for14 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do15 119878 larr997888most similar point to 119901119894 in 11987911990216 119862119888119900119899 larr997888 11987811989417 end for18 getRemainingCorrespondences()19 estimateRigidTransformation(119875119896119890119910119876119896119890119910)20 transformPointCloud(119875119876transform matrix)21 Return 119877119878
Algorithm 1 Point cloud initial registration algorithm
cloud feature point through the nearest neighbor search inthe target point cloud feature point set Finally we calculatethe Hausdorff distance if it is less than the threshold valuethen the two are corresponding points
33 RANSAC Culling Error Corresponding Point Pair Thecorresponding error pairs will affect the estimation of thefinal rigid transformation matrix thus affecting the pointcloud registration Therefore wrong corresponding pointpairsmust be eliminated By finding the corresponding pointsbased on FPFH and the Hausdorff distance in the previoussection we obtained a corresponding point pair with a highermatching degree however with a certain gap in the actualapplication of the distance point cloud registration This ismainly due to noise in the gathering process of the pointcloud which leads to the topology of the point cloud data ofthe two times of obtaining the same area being not completelyconsistent so there are some wrong correspondences In thispaper the RANSAC (random sample consistency) algorithmis used to eliminate error correspondences
34 Description of the Algorithm The rough point cloudregistration algorithm for feature extraction and matchingmainly uses the FPFH description Hausdorff distance andRANSAC algorithm to perform pairwise registration of pointclouds aiming to provide their accurate registration of pointclouds and good initial position The specific algorithmdescription is shown as Algorithm 1
In lines 1-6 the feature points are extracted according tothe method mentioned in Section 31 The reserved points
are obtained in lines 2-4 and concave and convex points arejudged in lines 5 and 6 Finally the feature points of the sourcepoint cloud are obtained Line 7 performs similar operationson the target point cloud to get the feature points set Lines8-12 as mentioned in Section 32 perform a lookup basedon corresponding points Line 13 uses RANSAC to eliminatethe corresponding error points Line 14 solves the rigid bodytransformation matrix Line 15 transforms the source pointcloud to complete the initial registration
4 Accurate Registration Based on ImprovedNDT Algorithm
Weuse the improvedNDTalgorithm INDT for accurate reg-istration First only the feature point sets of the source cloudand the target cloud are involved in the algorithm during theregistration process Second the algorithm replaces a singleprobability density function with a mixed probability densityfunction Finally the algorithm updates the Newton iterativealgorithm through a linear searchWewill describe the INDTalgorithm in detail
41 Mixed Probability Density Function For noise pointsthe negative logarithm probability function of the normaldistribution will grow without limit which will greatlyaffect the result Therefore using a single probability densityfunction as shown in Formula (1) the registration effect isnot good for noisy point cloud data Like Lu Ju [23] we use
Mathematical Problems in Engineering 5
04
035
03
025
02
015
01
005
0
minus4 minus2 0 2 4
xminus4 minus2 0 2 4
x
p(x)
5
4
3
2
1
0
-log(p(x))(x)p -log( )(x)p
Figure 1 Comparison of single probability density function and mixed probability density function
the PDF of the PDF which consists of a normal distributionand a uniform distribution
119901 (119909) = 1198881 exp((119909 minus 120583)119879summinus1 (119909 minus 120583)2 ) + 11988821199010 (2)
1199010 is the expected ratio of noise and 1198881 1198882 are usedto determine the normalization of the probability densityfunction by the spatial span of voxelsThe effect of noise whenusing the mixed probability density function is bounded asshown in Figure 1
The form of 997888rarr119901(119909) = minus log(1198881 exp((119909minus120583)119879summinus1(119909minus120583)2)+1198882) is a logarithmic function to be optimized These termshave no simple first and second derivatives However thenegative logarithm probability function can be approximatedby a Gaussian function This form of function is similar tothe Gauss function 119901(119909) = 119889119894 exp(minus1198892119909221205902) The requiredfitting parameters should be consistent with x=0 x=120590 andx=infin as shown in Formula (3) Using the Gaussian functionto estimate the influence of a point in the current point cloudon the NDT fractional function is shown in Formula (4)The NDT fractional function is simpler than the derivativeof Formula (2) but it still has the same general properties inthe optimization Since 1198893 only adds a constant offset to thefractional function when registering with NDT and does notchange its shape or its optimized parameters 1198893 is roundedoff from Formula (4)
1198893 = minus log (1198882)1198891 = minus log (1198881 + 1198882) minus 1198893
119901 (119909119896) = minus1198891 exp(minus11988922 (119909119896 minus 120583119896)119879)minus1sum119896
(119909119896 minus 120583119896) (4)
In Formula (4)119909119896minus120583119896is the average number ofNDTcellssumminus1119896 (119909119896 minus 120583119896) is the covariance of the NDT cell
42 Newton Iteration Algorithm for Parameter SolutionGiven point set 119909 = 1199091 1199092 119909119899 pose 119901 and the transformfunction 119879(119901 119909) that transforms 119909 by 119901 for the currentparameter vector the NDT fractional function is shown in
119904 (119901) = minus 119899sum119896minus1
119901 (119879 (119901 minus 119909119896)) (5)
This corresponds to the possibility of point 119909119896 beinglocated on the surface of the target point cloud after 119901transformation
The probability function requires the inverse of thecovariance matrix summinus1 Where the points in a cell are com-pletely coplanar or collinear the covariance matrix is singularand cannot be inverted In the 3D case the covariance matrixcalculated from three or fewer points is singular ThereforePDFs only calculate cells that contain more than three pointsIn addition to prevent numerical problems as long as it isfound that the sum is almost singular sum is slightly enlarged Ifthe maximum eigenvalue of sum 1205823 is 100 times greater thanthe eigenvalue 1205821 or 1205822 the smaller eigenvalue 120582119895 will bereplaced by 120582119895 = 1205823100 Replace the covariance matrix sum
6 Mathematical Problems in Engineering
Input two point clouds 119875 and 119876 to be registeredOutput Point cloud 1198751015840 and 1198761015840 after accurate registration1 Initialization2 Divide the ell structure B3 for all points 119902119894 isin 119876 do4 find the cell 119887119894 containing the point 119902119894119887119894 isin 1198615 store 119902119894 in 1198871198946 end for7 for all cells 119887119894 isin 119861 do8 1199011015840 997888rarrall points in 1198871198949 119898119906119894 997888rarr 1119899 sum119898119896=1 119910101584011989610 sum119894 997888rarr 1119898 minus 1 sum119898119896=1(1199101015840119896 minus 120583)(1199101015840119896 minus 120583)11987911 end for12 Registration13 while not converged do14 119904119888119900119903119890 997888rarr 015 119892 997888rarr 016 119867 997888rarr 017 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do18 find the cell 119887119894 containing the conversion function 119879(119901 119909119896)19 119904119888119900119903119890 997888rarr 119904119888119900119903119890+119901(119879(119901 119909119896))20 update 11989221 update11986722 end for23 solve119867Δ119901 = minus11989224 119901 997888rarr 119901 + Δ11990125 end while
Algorithm 2 INDT algorithm
with a matrix sum1015840 = orand1015840or where or contains the eigenvalue ofsum and and1015840 is shown in
The Newton algorithm can be used to find the parameter119901 for optimization 119904(119901) The algorithm iteratively solves theequation 119867Δ119901 = minus119892 where119867 and 119892 are the Hessian and thegradient vector of the fractional function 119904(119901) respectivelyDuring each iteration119901 997888rarr 119901+Δ119901 (the increment isΔ119901) Let119909119896 equiv 119879(119901 119909119896)minus120583119896 in other words 119909119896) is the transformation ofpoint 119909119896) by the current pose parameter relative to the centerof the cell PDF to which it belongs The element 119892119894 of thegradient vector 119892 is as shown as
Regardless of the dimension of registration the gradientof the NDT fractional function and the expression of theHessian matrix are the same and are similarly independentof the transformation representation
43 Description of the Algorithm The algorithm uses thefeature point set of the source point cloud and the target pointcloud to register and replaces the single probability densityfunction with the mixed probability density function witha normal distribution and a uniform distribution and usesthe Newton algorithm to iteratively solve the transformationparameters The algorithm is shown as Algorithm 2
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 3
amounts of point cloud data The registration algorithm inthis paper improves the normal distribution transformationINDT algorithm in the accurate registration stage improvesthe speed of the registration algorithm and ensures goodregistration accuracy
Existing initial and precise registration algorithms faceproblems when dealing with a large volume of point clouddata This paper proposes an algorithm that is based onfeatures andmatching which improves the registration speedwhile ensuring good registration accuracy
3 Rough Registration of Point Clouds
The main ideas of the initial algorithm are as followsFirst the feature points are extracted from the source pointcloud and the target point cloud respectively reducing thenumber of points involved in registration and removing theredundant points and thus greatly reducing the registrationtime Second the high dimensional descriptor FPFH is usednot only as a standard but also the Moorhouse multihusbanddistance to further determine the correspondence betweenpoints The RANSAC algorithm is then used to remove thecorresponding pairs of errors Finally we use singular valuedecomposition (SVD) to solve the coordinate transformationmatrix and apply it to the source point cloud to complete thecoarse registration Next we introduce the main links of therough registration algorithm
31 Feature Point Extraction The extraction of feature pointsreduces the number of point sets involved in registrationthereby improving the speed of registration With fewer pointcloud data points the introduction of feature points greatlyimproves the algorithms speed But a large volume of pointcloud data will increase the running time of feature pointextraction This can easily burden the entire registrationalgorithm which leads to long running time To solve thisproblem we judge the retention points before judging thefeature points The judgment of reserved points cannot onlyremove obvious outliers but can improve the efficiency of thewhole feature point extraction algorithmMoreover themorepoint cloud data the greater the time advantage of the featurepoint extraction algorithm
The method of extracting feature points in this paper isdivided into two steps (1) removing somepoints according tothe mean curvature of the point ie selecting the reservationpoint and (2) determining the feature points by the concaveand convex points as described below
The first step is the judgment of the reservation pointThemean curvature of each point in the point cloud is calculatedand the mean 120583 and variance 120590 of the point average meancurvature are calculated Second the points with averagecurvature greater than 120583+119901120590 are chosen as reservation pointsto remove the other points in the point cloud Among them119901 is the proportional coefficient whose value controls thenumber of reserved points When 119901 is large there are fewerreserved points and vice versa
Curvature generally includes the principal normalmean and Gaussian curvature We use the mean curvature
to filter feature points The normal curvature measures thedegree of curvature of a surface in a certain direction andthe principal curvature represents the extreme value of thenormal curvature Gaussian curvature is an intrinsic measureof curvature ie its value depends on how the distance on thesurface is measured rather than how the surface is embeddedin space Average curvature is an ldquoexternalrdquo bendingmeasure-ment standard which describes the curvature of a surfaceembedded in the surrounding space locally Therefore it canbetter reflect the degree of change and particularity of thecurrent point so in this paper average curvature is used asthe measure of screening feature points
The second step is the judgment of the concave andconvex points If a point on any surface is concave or convexit is taken as a feature point [22] Based on this principle wecalculate the value of 119878(119901)(119904(119901) isin (0 1)) using the followingequation
119878 (119901) = 12 minus 1120587 arctan1198961 (119901) + 1198962 (119901)1198961 (119901) minus 1198962 (119901) (1)
If 119878(119901119894) gt max(119878(1199011198941) 119878(1199011198942) 119878(119901119894119896)) then point 119901119894 isa local convex point If 119878(119901119894) lt min(119878(1199011198941) 119878(1199011198942) 119878(119901119894119896))then 119901119894 is a local concave point In this the 119878(1199011198941)119878(1199011198942) 119878(119901119894119896) are the 119878(119875) value of the point 119901119894 neigh-borhood points Points in point cloud data that are concavepoints or bumps are characteristic points
The steps to extract feature points from point cloud 119875 areas follows(1) Calculate the mean curvature of each point(2) Calculate the mean 120583 and variance 120590 of meancurvature and select points with mean curvature greater than120583+119901120590 as retention points (119901 is proportional to the coefficientand the 119901 is used to control the number of reserved points)(3)The concave points are judged by the reserved pointsand the feature points are extracted
32 Corresponding Point Pair Lookup Based on FPFH andHausdorff Distance To find the corresponding point is tofind the nearest point of the query point in another pointcloud If we calculate the Euclidean distance from eachpoint in the other point cloud to the query point wewill find the corresponding point pair but this requiresextensive calculation with little accuracy in the results Theaccuracy of the corresponding point pairs directly affects theregistration effect of point clouds In this paper the robustFPFH descriptor is added to the corresponding point pairsearch algorithm The fast point feature histogram (FPFH)is based on the relationship between the sampling pointand the neighborhood point normal which is representedby 33 dimensional descriptors In addition to find thecorresponding pointsmore accurately and reduce the effect oferror correspondences the Hausdorff distance is introducedto the algorithm
We use FPFH points and Hausdorff distances to findcorresponding points First the feature points set in thesource point cloud and target point cloud are describedby FPFH Next we find the point of minimum differencebetween the FPFH points of each point of the source point
4 Mathematical Problems in Engineering
Input source cloud 119875 target cloud119876Output Registered point cloud 1199001199061199051198621198971199001199061198891 Extract feature points from the source cloud and the target cloud separately2 for all points 119901119894 isin 119875 do3 if 119901119894 is a key point then4 119875119896119890119910 larr997888 1199011198945 end if6 end for7 for all 119901119894 isin 119875119896119890119910 do8 119875119889119890119904minus119896119890119910 larr997888 1199011198949 end for10 gets 119876119896119890119910 and 119876119889119890119904minus119896119890119910 in the same way11 for point 119902119895 isin 119876119889119890119904minus119896119890119910 do12 gets 11987911990213 end for14 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do15 119878 larr997888most similar point to 119901119894 in 11987911990216 119862119888119900119899 larr997888 11987811989417 end for18 getRemainingCorrespondences()19 estimateRigidTransformation(119875119896119890119910119876119896119890119910)20 transformPointCloud(119875119876transform matrix)21 Return 119877119878
Algorithm 1 Point cloud initial registration algorithm
cloud feature point through the nearest neighbor search inthe target point cloud feature point set Finally we calculatethe Hausdorff distance if it is less than the threshold valuethen the two are corresponding points
33 RANSAC Culling Error Corresponding Point Pair Thecorresponding error pairs will affect the estimation of thefinal rigid transformation matrix thus affecting the pointcloud registration Therefore wrong corresponding pointpairsmust be eliminated By finding the corresponding pointsbased on FPFH and the Hausdorff distance in the previoussection we obtained a corresponding point pair with a highermatching degree however with a certain gap in the actualapplication of the distance point cloud registration This ismainly due to noise in the gathering process of the pointcloud which leads to the topology of the point cloud data ofthe two times of obtaining the same area being not completelyconsistent so there are some wrong correspondences In thispaper the RANSAC (random sample consistency) algorithmis used to eliminate error correspondences
34 Description of the Algorithm The rough point cloudregistration algorithm for feature extraction and matchingmainly uses the FPFH description Hausdorff distance andRANSAC algorithm to perform pairwise registration of pointclouds aiming to provide their accurate registration of pointclouds and good initial position The specific algorithmdescription is shown as Algorithm 1
In lines 1-6 the feature points are extracted according tothe method mentioned in Section 31 The reserved points
are obtained in lines 2-4 and concave and convex points arejudged in lines 5 and 6 Finally the feature points of the sourcepoint cloud are obtained Line 7 performs similar operationson the target point cloud to get the feature points set Lines8-12 as mentioned in Section 32 perform a lookup basedon corresponding points Line 13 uses RANSAC to eliminatethe corresponding error points Line 14 solves the rigid bodytransformation matrix Line 15 transforms the source pointcloud to complete the initial registration
4 Accurate Registration Based on ImprovedNDT Algorithm
Weuse the improvedNDTalgorithm INDT for accurate reg-istration First only the feature point sets of the source cloudand the target cloud are involved in the algorithm during theregistration process Second the algorithm replaces a singleprobability density function with a mixed probability densityfunction Finally the algorithm updates the Newton iterativealgorithm through a linear searchWewill describe the INDTalgorithm in detail
41 Mixed Probability Density Function For noise pointsthe negative logarithm probability function of the normaldistribution will grow without limit which will greatlyaffect the result Therefore using a single probability densityfunction as shown in Formula (1) the registration effect isnot good for noisy point cloud data Like Lu Ju [23] we use
Mathematical Problems in Engineering 5
04
035
03
025
02
015
01
005
0
minus4 minus2 0 2 4
xminus4 minus2 0 2 4
x
p(x)
5
4
3
2
1
0
-log(p(x))(x)p -log( )(x)p
Figure 1 Comparison of single probability density function and mixed probability density function
the PDF of the PDF which consists of a normal distributionand a uniform distribution
119901 (119909) = 1198881 exp((119909 minus 120583)119879summinus1 (119909 minus 120583)2 ) + 11988821199010 (2)
1199010 is the expected ratio of noise and 1198881 1198882 are usedto determine the normalization of the probability densityfunction by the spatial span of voxelsThe effect of noise whenusing the mixed probability density function is bounded asshown in Figure 1
The form of 997888rarr119901(119909) = minus log(1198881 exp((119909minus120583)119879summinus1(119909minus120583)2)+1198882) is a logarithmic function to be optimized These termshave no simple first and second derivatives However thenegative logarithm probability function can be approximatedby a Gaussian function This form of function is similar tothe Gauss function 119901(119909) = 119889119894 exp(minus1198892119909221205902) The requiredfitting parameters should be consistent with x=0 x=120590 andx=infin as shown in Formula (3) Using the Gaussian functionto estimate the influence of a point in the current point cloudon the NDT fractional function is shown in Formula (4)The NDT fractional function is simpler than the derivativeof Formula (2) but it still has the same general properties inthe optimization Since 1198893 only adds a constant offset to thefractional function when registering with NDT and does notchange its shape or its optimized parameters 1198893 is roundedoff from Formula (4)
1198893 = minus log (1198882)1198891 = minus log (1198881 + 1198882) minus 1198893
119901 (119909119896) = minus1198891 exp(minus11988922 (119909119896 minus 120583119896)119879)minus1sum119896
(119909119896 minus 120583119896) (4)
In Formula (4)119909119896minus120583119896is the average number ofNDTcellssumminus1119896 (119909119896 minus 120583119896) is the covariance of the NDT cell
42 Newton Iteration Algorithm for Parameter SolutionGiven point set 119909 = 1199091 1199092 119909119899 pose 119901 and the transformfunction 119879(119901 119909) that transforms 119909 by 119901 for the currentparameter vector the NDT fractional function is shown in
119904 (119901) = minus 119899sum119896minus1
119901 (119879 (119901 minus 119909119896)) (5)
This corresponds to the possibility of point 119909119896 beinglocated on the surface of the target point cloud after 119901transformation
The probability function requires the inverse of thecovariance matrix summinus1 Where the points in a cell are com-pletely coplanar or collinear the covariance matrix is singularand cannot be inverted In the 3D case the covariance matrixcalculated from three or fewer points is singular ThereforePDFs only calculate cells that contain more than three pointsIn addition to prevent numerical problems as long as it isfound that the sum is almost singular sum is slightly enlarged Ifthe maximum eigenvalue of sum 1205823 is 100 times greater thanthe eigenvalue 1205821 or 1205822 the smaller eigenvalue 120582119895 will bereplaced by 120582119895 = 1205823100 Replace the covariance matrix sum
6 Mathematical Problems in Engineering
Input two point clouds 119875 and 119876 to be registeredOutput Point cloud 1198751015840 and 1198761015840 after accurate registration1 Initialization2 Divide the ell structure B3 for all points 119902119894 isin 119876 do4 find the cell 119887119894 containing the point 119902119894119887119894 isin 1198615 store 119902119894 in 1198871198946 end for7 for all cells 119887119894 isin 119861 do8 1199011015840 997888rarrall points in 1198871198949 119898119906119894 997888rarr 1119899 sum119898119896=1 119910101584011989610 sum119894 997888rarr 1119898 minus 1 sum119898119896=1(1199101015840119896 minus 120583)(1199101015840119896 minus 120583)11987911 end for12 Registration13 while not converged do14 119904119888119900119903119890 997888rarr 015 119892 997888rarr 016 119867 997888rarr 017 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do18 find the cell 119887119894 containing the conversion function 119879(119901 119909119896)19 119904119888119900119903119890 997888rarr 119904119888119900119903119890+119901(119879(119901 119909119896))20 update 11989221 update11986722 end for23 solve119867Δ119901 = minus11989224 119901 997888rarr 119901 + Δ11990125 end while
Algorithm 2 INDT algorithm
with a matrix sum1015840 = orand1015840or where or contains the eigenvalue ofsum and and1015840 is shown in
The Newton algorithm can be used to find the parameter119901 for optimization 119904(119901) The algorithm iteratively solves theequation 119867Δ119901 = minus119892 where119867 and 119892 are the Hessian and thegradient vector of the fractional function 119904(119901) respectivelyDuring each iteration119901 997888rarr 119901+Δ119901 (the increment isΔ119901) Let119909119896 equiv 119879(119901 119909119896)minus120583119896 in other words 119909119896) is the transformation ofpoint 119909119896) by the current pose parameter relative to the centerof the cell PDF to which it belongs The element 119892119894 of thegradient vector 119892 is as shown as
Regardless of the dimension of registration the gradientof the NDT fractional function and the expression of theHessian matrix are the same and are similarly independentof the transformation representation
43 Description of the Algorithm The algorithm uses thefeature point set of the source point cloud and the target pointcloud to register and replaces the single probability densityfunction with the mixed probability density function witha normal distribution and a uniform distribution and usesthe Newton algorithm to iteratively solve the transformationparameters The algorithm is shown as Algorithm 2
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
4 Mathematical Problems in Engineering
Input source cloud 119875 target cloud119876Output Registered point cloud 1199001199061199051198621198971199001199061198891 Extract feature points from the source cloud and the target cloud separately2 for all points 119901119894 isin 119875 do3 if 119901119894 is a key point then4 119875119896119890119910 larr997888 1199011198945 end if6 end for7 for all 119901119894 isin 119875119896119890119910 do8 119875119889119890119904minus119896119890119910 larr997888 1199011198949 end for10 gets 119876119896119890119910 and 119876119889119890119904minus119896119890119910 in the same way11 for point 119902119895 isin 119876119889119890119904minus119896119890119910 do12 gets 11987911990213 end for14 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do15 119878 larr997888most similar point to 119901119894 in 11987911990216 119862119888119900119899 larr997888 11987811989417 end for18 getRemainingCorrespondences()19 estimateRigidTransformation(119875119896119890119910119876119896119890119910)20 transformPointCloud(119875119876transform matrix)21 Return 119877119878
Algorithm 1 Point cloud initial registration algorithm
cloud feature point through the nearest neighbor search inthe target point cloud feature point set Finally we calculatethe Hausdorff distance if it is less than the threshold valuethen the two are corresponding points
33 RANSAC Culling Error Corresponding Point Pair Thecorresponding error pairs will affect the estimation of thefinal rigid transformation matrix thus affecting the pointcloud registration Therefore wrong corresponding pointpairsmust be eliminated By finding the corresponding pointsbased on FPFH and the Hausdorff distance in the previoussection we obtained a corresponding point pair with a highermatching degree however with a certain gap in the actualapplication of the distance point cloud registration This ismainly due to noise in the gathering process of the pointcloud which leads to the topology of the point cloud data ofthe two times of obtaining the same area being not completelyconsistent so there are some wrong correspondences In thispaper the RANSAC (random sample consistency) algorithmis used to eliminate error correspondences
34 Description of the Algorithm The rough point cloudregistration algorithm for feature extraction and matchingmainly uses the FPFH description Hausdorff distance andRANSAC algorithm to perform pairwise registration of pointclouds aiming to provide their accurate registration of pointclouds and good initial position The specific algorithmdescription is shown as Algorithm 1
In lines 1-6 the feature points are extracted according tothe method mentioned in Section 31 The reserved points
are obtained in lines 2-4 and concave and convex points arejudged in lines 5 and 6 Finally the feature points of the sourcepoint cloud are obtained Line 7 performs similar operationson the target point cloud to get the feature points set Lines8-12 as mentioned in Section 32 perform a lookup basedon corresponding points Line 13 uses RANSAC to eliminatethe corresponding error points Line 14 solves the rigid bodytransformation matrix Line 15 transforms the source pointcloud to complete the initial registration
4 Accurate Registration Based on ImprovedNDT Algorithm
Weuse the improvedNDTalgorithm INDT for accurate reg-istration First only the feature point sets of the source cloudand the target cloud are involved in the algorithm during theregistration process Second the algorithm replaces a singleprobability density function with a mixed probability densityfunction Finally the algorithm updates the Newton iterativealgorithm through a linear searchWewill describe the INDTalgorithm in detail
41 Mixed Probability Density Function For noise pointsthe negative logarithm probability function of the normaldistribution will grow without limit which will greatlyaffect the result Therefore using a single probability densityfunction as shown in Formula (1) the registration effect isnot good for noisy point cloud data Like Lu Ju [23] we use
Mathematical Problems in Engineering 5
04
035
03
025
02
015
01
005
0
minus4 minus2 0 2 4
xminus4 minus2 0 2 4
x
p(x)
5
4
3
2
1
0
-log(p(x))(x)p -log( )(x)p
Figure 1 Comparison of single probability density function and mixed probability density function
the PDF of the PDF which consists of a normal distributionand a uniform distribution
119901 (119909) = 1198881 exp((119909 minus 120583)119879summinus1 (119909 minus 120583)2 ) + 11988821199010 (2)
1199010 is the expected ratio of noise and 1198881 1198882 are usedto determine the normalization of the probability densityfunction by the spatial span of voxelsThe effect of noise whenusing the mixed probability density function is bounded asshown in Figure 1
The form of 997888rarr119901(119909) = minus log(1198881 exp((119909minus120583)119879summinus1(119909minus120583)2)+1198882) is a logarithmic function to be optimized These termshave no simple first and second derivatives However thenegative logarithm probability function can be approximatedby a Gaussian function This form of function is similar tothe Gauss function 119901(119909) = 119889119894 exp(minus1198892119909221205902) The requiredfitting parameters should be consistent with x=0 x=120590 andx=infin as shown in Formula (3) Using the Gaussian functionto estimate the influence of a point in the current point cloudon the NDT fractional function is shown in Formula (4)The NDT fractional function is simpler than the derivativeof Formula (2) but it still has the same general properties inthe optimization Since 1198893 only adds a constant offset to thefractional function when registering with NDT and does notchange its shape or its optimized parameters 1198893 is roundedoff from Formula (4)
1198893 = minus log (1198882)1198891 = minus log (1198881 + 1198882) minus 1198893
119901 (119909119896) = minus1198891 exp(minus11988922 (119909119896 minus 120583119896)119879)minus1sum119896
(119909119896 minus 120583119896) (4)
In Formula (4)119909119896minus120583119896is the average number ofNDTcellssumminus1119896 (119909119896 minus 120583119896) is the covariance of the NDT cell
42 Newton Iteration Algorithm for Parameter SolutionGiven point set 119909 = 1199091 1199092 119909119899 pose 119901 and the transformfunction 119879(119901 119909) that transforms 119909 by 119901 for the currentparameter vector the NDT fractional function is shown in
119904 (119901) = minus 119899sum119896minus1
119901 (119879 (119901 minus 119909119896)) (5)
This corresponds to the possibility of point 119909119896 beinglocated on the surface of the target point cloud after 119901transformation
The probability function requires the inverse of thecovariance matrix summinus1 Where the points in a cell are com-pletely coplanar or collinear the covariance matrix is singularand cannot be inverted In the 3D case the covariance matrixcalculated from three or fewer points is singular ThereforePDFs only calculate cells that contain more than three pointsIn addition to prevent numerical problems as long as it isfound that the sum is almost singular sum is slightly enlarged Ifthe maximum eigenvalue of sum 1205823 is 100 times greater thanthe eigenvalue 1205821 or 1205822 the smaller eigenvalue 120582119895 will bereplaced by 120582119895 = 1205823100 Replace the covariance matrix sum
6 Mathematical Problems in Engineering
Input two point clouds 119875 and 119876 to be registeredOutput Point cloud 1198751015840 and 1198761015840 after accurate registration1 Initialization2 Divide the ell structure B3 for all points 119902119894 isin 119876 do4 find the cell 119887119894 containing the point 119902119894119887119894 isin 1198615 store 119902119894 in 1198871198946 end for7 for all cells 119887119894 isin 119861 do8 1199011015840 997888rarrall points in 1198871198949 119898119906119894 997888rarr 1119899 sum119898119896=1 119910101584011989610 sum119894 997888rarr 1119898 minus 1 sum119898119896=1(1199101015840119896 minus 120583)(1199101015840119896 minus 120583)11987911 end for12 Registration13 while not converged do14 119904119888119900119903119890 997888rarr 015 119892 997888rarr 016 119867 997888rarr 017 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do18 find the cell 119887119894 containing the conversion function 119879(119901 119909119896)19 119904119888119900119903119890 997888rarr 119904119888119900119903119890+119901(119879(119901 119909119896))20 update 11989221 update11986722 end for23 solve119867Δ119901 = minus11989224 119901 997888rarr 119901 + Δ11990125 end while
Algorithm 2 INDT algorithm
with a matrix sum1015840 = orand1015840or where or contains the eigenvalue ofsum and and1015840 is shown in
The Newton algorithm can be used to find the parameter119901 for optimization 119904(119901) The algorithm iteratively solves theequation 119867Δ119901 = minus119892 where119867 and 119892 are the Hessian and thegradient vector of the fractional function 119904(119901) respectivelyDuring each iteration119901 997888rarr 119901+Δ119901 (the increment isΔ119901) Let119909119896 equiv 119879(119901 119909119896)minus120583119896 in other words 119909119896) is the transformation ofpoint 119909119896) by the current pose parameter relative to the centerof the cell PDF to which it belongs The element 119892119894 of thegradient vector 119892 is as shown as
Regardless of the dimension of registration the gradientof the NDT fractional function and the expression of theHessian matrix are the same and are similarly independentof the transformation representation
43 Description of the Algorithm The algorithm uses thefeature point set of the source point cloud and the target pointcloud to register and replaces the single probability densityfunction with the mixed probability density function witha normal distribution and a uniform distribution and usesthe Newton algorithm to iteratively solve the transformationparameters The algorithm is shown as Algorithm 2
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 5
04
035
03
025
02
015
01
005
0
minus4 minus2 0 2 4
xminus4 minus2 0 2 4
x
p(x)
5
4
3
2
1
0
-log(p(x))(x)p -log( )(x)p
Figure 1 Comparison of single probability density function and mixed probability density function
the PDF of the PDF which consists of a normal distributionand a uniform distribution
119901 (119909) = 1198881 exp((119909 minus 120583)119879summinus1 (119909 minus 120583)2 ) + 11988821199010 (2)
1199010 is the expected ratio of noise and 1198881 1198882 are usedto determine the normalization of the probability densityfunction by the spatial span of voxelsThe effect of noise whenusing the mixed probability density function is bounded asshown in Figure 1
The form of 997888rarr119901(119909) = minus log(1198881 exp((119909minus120583)119879summinus1(119909minus120583)2)+1198882) is a logarithmic function to be optimized These termshave no simple first and second derivatives However thenegative logarithm probability function can be approximatedby a Gaussian function This form of function is similar tothe Gauss function 119901(119909) = 119889119894 exp(minus1198892119909221205902) The requiredfitting parameters should be consistent with x=0 x=120590 andx=infin as shown in Formula (3) Using the Gaussian functionto estimate the influence of a point in the current point cloudon the NDT fractional function is shown in Formula (4)The NDT fractional function is simpler than the derivativeof Formula (2) but it still has the same general properties inthe optimization Since 1198893 only adds a constant offset to thefractional function when registering with NDT and does notchange its shape or its optimized parameters 1198893 is roundedoff from Formula (4)
1198893 = minus log (1198882)1198891 = minus log (1198881 + 1198882) minus 1198893
119901 (119909119896) = minus1198891 exp(minus11988922 (119909119896 minus 120583119896)119879)minus1sum119896
(119909119896 minus 120583119896) (4)
In Formula (4)119909119896minus120583119896is the average number ofNDTcellssumminus1119896 (119909119896 minus 120583119896) is the covariance of the NDT cell
42 Newton Iteration Algorithm for Parameter SolutionGiven point set 119909 = 1199091 1199092 119909119899 pose 119901 and the transformfunction 119879(119901 119909) that transforms 119909 by 119901 for the currentparameter vector the NDT fractional function is shown in
119904 (119901) = minus 119899sum119896minus1
119901 (119879 (119901 minus 119909119896)) (5)
This corresponds to the possibility of point 119909119896 beinglocated on the surface of the target point cloud after 119901transformation
The probability function requires the inverse of thecovariance matrix summinus1 Where the points in a cell are com-pletely coplanar or collinear the covariance matrix is singularand cannot be inverted In the 3D case the covariance matrixcalculated from three or fewer points is singular ThereforePDFs only calculate cells that contain more than three pointsIn addition to prevent numerical problems as long as it isfound that the sum is almost singular sum is slightly enlarged Ifthe maximum eigenvalue of sum 1205823 is 100 times greater thanthe eigenvalue 1205821 or 1205822 the smaller eigenvalue 120582119895 will bereplaced by 120582119895 = 1205823100 Replace the covariance matrix sum
6 Mathematical Problems in Engineering
Input two point clouds 119875 and 119876 to be registeredOutput Point cloud 1198751015840 and 1198761015840 after accurate registration1 Initialization2 Divide the ell structure B3 for all points 119902119894 isin 119876 do4 find the cell 119887119894 containing the point 119902119894119887119894 isin 1198615 store 119902119894 in 1198871198946 end for7 for all cells 119887119894 isin 119861 do8 1199011015840 997888rarrall points in 1198871198949 119898119906119894 997888rarr 1119899 sum119898119896=1 119910101584011989610 sum119894 997888rarr 1119898 minus 1 sum119898119896=1(1199101015840119896 minus 120583)(1199101015840119896 minus 120583)11987911 end for12 Registration13 while not converged do14 119904119888119900119903119890 997888rarr 015 119892 997888rarr 016 119867 997888rarr 017 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do18 find the cell 119887119894 containing the conversion function 119879(119901 119909119896)19 119904119888119900119903119890 997888rarr 119904119888119900119903119890+119901(119879(119901 119909119896))20 update 11989221 update11986722 end for23 solve119867Δ119901 = minus11989224 119901 997888rarr 119901 + Δ11990125 end while
Algorithm 2 INDT algorithm
with a matrix sum1015840 = orand1015840or where or contains the eigenvalue ofsum and and1015840 is shown in
The Newton algorithm can be used to find the parameter119901 for optimization 119904(119901) The algorithm iteratively solves theequation 119867Δ119901 = minus119892 where119867 and 119892 are the Hessian and thegradient vector of the fractional function 119904(119901) respectivelyDuring each iteration119901 997888rarr 119901+Δ119901 (the increment isΔ119901) Let119909119896 equiv 119879(119901 119909119896)minus120583119896 in other words 119909119896) is the transformation ofpoint 119909119896) by the current pose parameter relative to the centerof the cell PDF to which it belongs The element 119892119894 of thegradient vector 119892 is as shown as
Regardless of the dimension of registration the gradientof the NDT fractional function and the expression of theHessian matrix are the same and are similarly independentof the transformation representation
43 Description of the Algorithm The algorithm uses thefeature point set of the source point cloud and the target pointcloud to register and replaces the single probability densityfunction with the mixed probability density function witha normal distribution and a uniform distribution and usesthe Newton algorithm to iteratively solve the transformationparameters The algorithm is shown as Algorithm 2
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Mathematical Problems in Engineering
Input two point clouds 119875 and 119876 to be registeredOutput Point cloud 1198751015840 and 1198761015840 after accurate registration1 Initialization2 Divide the ell structure B3 for all points 119902119894 isin 119876 do4 find the cell 119887119894 containing the point 119902119894119887119894 isin 1198615 store 119902119894 in 1198871198946 end for7 for all cells 119887119894 isin 119861 do8 1199011015840 997888rarrall points in 1198871198949 119898119906119894 997888rarr 1119899 sum119898119896=1 119910101584011989610 sum119894 997888rarr 1119898 minus 1 sum119898119896=1(1199101015840119896 minus 120583)(1199101015840119896 minus 120583)11987911 end for12 Registration13 while not converged do14 119904119888119900119903119890 997888rarr 015 119892 997888rarr 016 119867 997888rarr 017 for point 119901119894 isin 119875119889119890119904minus119896119890119910 do18 find the cell 119887119894 containing the conversion function 119879(119901 119909119896)19 119904119888119900119903119890 997888rarr 119904119888119900119903119890+119901(119879(119901 119909119896))20 update 11989221 update11986722 end for23 solve119867Δ119901 = minus11989224 119901 997888rarr 119901 + Δ11990125 end while
Algorithm 2 INDT algorithm
with a matrix sum1015840 = orand1015840or where or contains the eigenvalue ofsum and and1015840 is shown in
The Newton algorithm can be used to find the parameter119901 for optimization 119904(119901) The algorithm iteratively solves theequation 119867Δ119901 = minus119892 where119867 and 119892 are the Hessian and thegradient vector of the fractional function 119904(119901) respectivelyDuring each iteration119901 997888rarr 119901+Δ119901 (the increment isΔ119901) Let119909119896 equiv 119879(119901 119909119896)minus120583119896 in other words 119909119896) is the transformation ofpoint 119909119896) by the current pose parameter relative to the centerof the cell PDF to which it belongs The element 119892119894 of thegradient vector 119892 is as shown as
Regardless of the dimension of registration the gradientof the NDT fractional function and the expression of theHessian matrix are the same and are similarly independentof the transformation representation
43 Description of the Algorithm The algorithm uses thefeature point set of the source point cloud and the target pointcloud to register and replaces the single probability densityfunction with the mixed probability density function witha normal distribution and a uniform distribution and usesthe Newton algorithm to iteratively solve the transformationparameters The algorithm is shown as Algorithm 2
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 7
(a) Preregistrationpoint cloud
(b) Automatic regis-tration algorithm
(c) This article reg-istration algorithm
Figure 2 Result of point cloud registration
(a) Preregistrationpoint cloud
(b) Point cloudautomaticregistrationalgorithm
(c) This articleregistrationalgorithm
Figure 3 Results of point cloud chef registration
5 Experimental Results and Analysis
The algorithm was compiled on CMake371 and debuggedin Microsoft Visual Studio 2013 under the Win7 64-bitoperating system We ran the C++ development languageand combined the PCL point cloud library to develop andmaintain the algorithm Because there is no need for librariesin PCL version 16 we used PCL18 Version 18 has manymore cloud processing algorithms and some modules havedifferent usage
Experiment 1 the point cloud data 119903119886119887119887119894119905119901119888119889 and119903119886119887119887119894119905119905119901119888119889 of Stanford University consist of 35947 pointsA comparison between the algorithm and the point cloudautomatic registration algorithm based on feature extractionin reference 25 is shown in Figure 2 and the comparison ofregistration data is shown in Table 1
The running time of this algorithm was 76605 s (aver-age of 10 results) and the root mean square error was0000600173m The point cloud automatic registration algo-rithm time was 12957 s and the root mean square error was000367109m The time difference of the two methods wasabout 53 s but the registration errors were quite differentFrom Figure 2 and Table 1 we can see that our algorithmhas obvious advantages in both time and the accuracy ofregistration
Experiment 2 StanfordUniversityrsquos 119888ℎ119890119891119891119901119888119889 and 119888ℎ119890119891119891119905119901119888119889 point cloud data consist of 176920 points A com-parison of the registration algorithmrsquos registration results isshown in Figure 3 and the comparison of the registrationdata (including the running time and RMSE) is shown inTable 2
Figure 3(a) shows the preregistration point cloud andFigures 3(b) and 3(c) are respectively the registered pointcloud As can be seen from Figure 3 the registration effect ofthis algorithm is better than that of the point cloud automaticregistration algorithm The number of points in the Cheffmodel is 176920 which is about five times that of the pointcloud data in our first experiment It can be seen fromTable 2that the time advantage of the algorithm is more obvious Inaddition the accuracy of this algorithm is still better
Experiment 3 in this experiment the chicken pointcloud is used in initial registration and accurate registra-tion In the initial registration stage the SAC-IA (SampleConsensusndashInitial Alignment) algorithm [24] the Huangrsquosalgorithm [25] and our algorithm are compared and theresults are shown in Figure 4 and Table 3 In the accurate reg-istration stage NDT algorithm Curvature NDT algorithmand INDT algorithm are compared and the results are shownin Figure 5 and Table 4
In the initial registration stage in our algorithm thesource cloud and target cloud are almost overlapped whilethe RMSE is only 113234 Besides the running time of 95607seconds is much shorter than the other two comparisonalgorithms
In the accurate registration stage the difference betweenthe results of the three algorithms is almost indistinguishablefrom the naked eye However from the data in Table 4 theaccuracy of INDT algorithm is slightly behind other twocomparison algorithms and the difference is 30 and 43respectively while the running time is only 21 and 52of these two algorithms Thus the proposed algorithm hasobvious advantages in terms of time while ensuring accuracy
8 Mathematical Problems in Engineering
(a)Preregistrationpoint cloud
(b) SACminusIA (c) Huangrsquos algo-rithm
(d) This articlealgorithm
Figure 4 Results of point cloud chicken initial registration
(a)Preregistrationpoint cloud
(b) NDTalgorithm
(c) CurvatureNDT algorithm
(d) This articleNDT algorithm
Figure 5 Results of point cloud chicken accurate registration
Table 1 A comparison of the two registration methods
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
From the above experiments we can find that the greaterthe amount of point cloud data the more obvious the timeadvantage of the algorithm
6 Conclusions
In this paper a point cloud registration algorithm based onfeature extraction and matching is proposed The algorithmtreats the feature point of the registration point cloud usesFPFH and Hausdorff to find the corresponding point pairand uses the RANSAC algorithm to remove the error corre-sponding point pair thus completing the rough registrationof the point cloud In the stage of precise registration theimproved normal distribution transformation algorithm isusedThismethod has achieved remarkable results especiallywhen processing a large volume of point cloud data wherethe advantages in registration time and accuracy are obviousHowever some parts of the algorithm require further studyWe plan to carry out the following work (1) In the experi-ment the algorithm uses multiple experiments to determinethe threshold Next we can study the automatic selection ofvarious thresholds (2) In this paper the accuracy of the INDTalgorithm in the registration of a large amount of point clouddata is improved but there is still room for improvement andwe plan to study this
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the Natural Science Foundation ofHebei Province China under Grant no F2017203019
References
[1] R B Rusu N Blodow and M Beetz ldquoFast Point FeatureHistograms (FPFH) for 3D registrationrdquo IEEE InternationalConference on Robotics amp Automation IEEE Press pp 183ndash1892009
[2] M-PDubuisson andAK Jain ldquoAmodifiedHausdorff distancefor object matchingrdquo in Proceedings of the International Confer-ence on Pattern Recognition IEEE vol 1 pp 86ndash95 2002
[3] I Lysenkov andVRabaud ldquoPose estimationof rigid transparentobjects in transparent clutterrdquo in Proceedings of the 2013 IEEEInternational Conference on Robotics and Automation ICRA2013 pp 162ndash169 Germany May 2013
[4] S Ramalingam and Y Taguchi ldquoA theory of minimal 3D Pointto 3D Plane registration and its generalizationrdquo InternationalJournal of Computer Vision vol 102 no 1-3 pp 73ndash90 2013
[5] Y Meng and H Zhang ldquoRegistration of point clouds usingsample-sphere and adaptive distance restrictionrdquo The VisualComputer vol 27 no 6-8 pp 543ndash553 2011
[6] Y Zhang Research on Registration of Unorganized Point CloudChangan university 2015
[7] C S Chua F Han and Y K Ho ldquo3D human face recognitionusing point signaturerdquo in Proceedings of the Fourth IEEEInternational Conference pp 233ndash238 2000
[8] N Gelfand N J Mitra L J Guibas et al ldquoRobust global regis-trationrdquo in Proceedings of the Third Eurographics Symposium onGeometry Proceeding pp 197ndash206 2005
[9] Y Jianfeng and D Kezhong ldquoPoint Cloud Registration Algo-rithm Based on Extracting and Matching Feature Pointsrdquo inBulletin of Surveying and Mapping vol 9 pp 62ndash65 2013
[10] X Yang Study on Three-Dimensional Point Cloud RegistrationMethod Based onTheNormal Vector NorthUniversity of China2016
[11] P Biber and W Straszliger ldquoThe normal distributions transforma new approach to laser scan matchingrdquo in Proceedings of theIEEERSJ International Conference on Intelligent Robots andSystems (IROS rsquo03) vol 3 pp 2743ndash2748 IEEE October 2003
[12] MMagnusson 3D ScanMatching for Mobile Robots with Appli-cation to Mine Mapping Orebro University Orebro Sweden2006
[13] C Ulas and H Temeltas ldquoA 3D Scan Matching Method BasedOn Multi-Layered Normal Distribution Transformrdquo IFAC Pro-ceedings Volumes vol 44 no 1 pp 11602ndash11607 2011
[14] C Ulas and H Temeltas ldquo3D Multi-Layered Normal Distri-bution Transform for Fast and Long Range Scan Matchingrdquo
Journal of Intelligent ampRobotic Systems vol 71 no 1 pp 85ndash1082013
[15] H Hong and B H Lee ldquoKey-layered normal distributionstransform for point cloud registrationrdquo IEEE Electronics Lettersvol 51 no 24 pp 1986ndash1988 2015
[16] H Hong and B Lee ldquoA method of generating multi-scale disc-like distributions for NDTregistration algorithmrdquo InternationalJournal of Mechanical Engineering and Robotics Research vol 5no 1 pp 52ndash56 2016
[17] X Zhang A W Zhang and Z H Wang ldquoPoint Cloud Reg-istration Based on Improved Normal Distribution TransformAlgorithmrdquo Laser amp Optoelectronics Progress vol 51 no 4 pp96ndash105 2014
[18] H U Xiuxiang and L Zhang ldquoMulti-view Point Cloud Regis-tration Based on Improved 3D-NDTCombiningTheFeatureofNARFrdquo Journal of signal processing vol 13 no 12 pp 1674ndash16792015
[19] Y C Fang Q Y Chen N Sun et al Three-dimensional nor-mal distribution transformation point cloud registration methodbased on curvature feature CN 105069840 2015
[20] Q ChenThe craneworking environment oriented static scene 3Dreconstruction Nankai University 2015
[21] F Zheng RyadChellali M Y Dai et al ldquoResearch on NormalDistribution Transform Registration Method Based on devel-oped 3D SIFTrdquo Journal of Computer Applications vol 10 pp2875ndash2878 2017
[22] Y Wu W Wang K Lu Y Wei and Z Chen ldquoA New Methodfor Registration of 3D Point Sets with LowOverlapping RatiosrdquoProcedia CIRP vol 27 pp 202ndash206 2015
[23] J Lu W Liu D Dong and Q Shao ldquoPoint cloud registrationalgorithm based on NDT with variable size voxelrdquo in Proceed-ings of the 34th Chinese Control Conference CCC 2015 pp 3707ndash3712 China July 2015
[24] X Chen and Y Zhu ldquoPoint cloud registration technology basedon SAC-IA and improved ICP algorithmrdquo Journal of XianEngineering University vol 31 no 3 pp 395ndash401 2017
[25] H Yuan F Da and H Tao ldquoAn Automatic Registration Algo-rithm for Point Cloud Based on Feature Extractionrdquo ChineseJournal of Lasers vol 42 no 3 pp 242ndash248 2015
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
