Three-dimensional Point Cloud Plane Segmentation in Both Structured andUnstructured Environments

Junhao Xiaoa,∗, Jianhua Zhangb, Benjamin Adlera, Houxiang Zhangc,∗∗, Jianwei Zhanga

aDepartment of Computer Science, University of Hamburg, Hamburg, GermanybCollege of Compute Science and Technology, Zhejiang University of Technology, Hangzhou, China

cFaculty of Maritime Technology and Operations, Alesund University College, Alesund, Norway

Abstract

This paper focuses on three-dimensional (3D) point cloud plane segmentation. Two complementary strategies areproposed for different environments, i.e., a subwindow based region growing (SBRG) algorithm for structured environ-ments, and a hybrid region growing (HRG) algorithm for unstructured environments. The point cloud is decomposed intosubwindows first, using the points’ neighborhood information when they are scanned by the laser range finder (LRF).Then, the subwindows are classified as planar or nonplanar based on their shape. Afterwards, only planar subwindowsare employed in the former algorithm, whereas both kinds of subwindows are used in the latter. In the growing phase, pla-nar subwindows are investigated directly (in both algorithms), while each point in nonplanar subwindows is investigatedseparately (only in HRG). During region growing, plane parameters are computed incrementally when a subwindow or apoint is added to the growing region. This incremental methodology makes the plane segmentation fast. The algorithmshave been evaluated using real-world datasets from both structured and unstructured environments. Furthermore, theyhave been benchmarked against a state-of-the-art point based region growing (PBRG) algorithm with regards to seg-mentation speed. According to the results, SBRG is 4 and 9 times faster than PBRG when the subwindow size is set to3 × 3 and 4 × 4 respectively; HRG is 4 times faster than PBRG when the subwindow size is set to 4 × 4. Open-sourcecode for this paper is available at https://github.com/junhaoxiao/TAMS-Planar-Surface-Based-Perception.git.

Keywords: 3D Point Cloud, Plane segmentation, Region Growing

1. Introduction

Range sensors, e.g., Laser Range Finders (LRFs), Time-of-Flight (ToF) cameras, stereo vision, active vision [1] andthe newly developed RGB-D style cameras are becomingmore and more popular in the application of mobile roboticsystems. Noisy range images from such sensors can be usedfor various kinds of tasks, such as navigation [2, 3], simul-taneous localization and mapping (SLAM) [4–7], semanticmapping [8, 9] and object recognition [10, 11]. Objectswith planar surfaces are prevalent in both indoor and ur-ban environments, such as floors, doors, walls, ceilings androads. If the surfaces are extracted as polygons, they canprovide a compressive representation of the point clouds,normally the data compression rate is higher than 90%[12, 13]. Furthermore, planar patch has been found to bea good geometric feature for scan registration since threeplanes with linearly independent normals can determine

∗Principle corresponding author∗∗Corresponding author

Email addresses: xiao@informatik.uni-hamburg.de (JunhaoXiao), zjh@zjut.edu.cn (Jianhua Zhang),adler@informatik.uni-hamburg.de (Benjamin Adler),hozh@hials.no (Houxiang Zhang),zhang@informatik.uni-hamburg.de (Jianwei Zhang)

the transformation between overlapping point clouds. Theplane-based registration algorithm MUMC in [14] has beenproven to be faster and more reliable than the classic it-erative corresponding point (ICP) [15, 16] algorithm andthe recently proposed normal distribution transformations(NDT) algorithm [17, 18]. In our previous work [19, 20],a novel plane-based registration algorithm has been pro-posed and applied to datasets obtained with different sen-sors from different scenarios. From field experiments, ouralgorithm has been found to be fast, accurate and robust,see [19, 20] for details.

Plane segmentation — which has been researched foryears and is still a very hot and complex task in robotics —is the first step of plane-based mapping systems. Althoughthere are available algorithms in the graphics community[21, 22], they cannot be adopted into robotic systems di-rectly due to relying on more exact depth information thanthat which robotic sensors can provide. Therefore, vari-ous algorithms on this topic [12, 23–34] to deal with noisydatasets have recently been proposed by researchers fromthe robotics community. In this paper, two complementaryplane segmentation algorithms are presented for differentenvironments, i.e., a subwindow based region growing al-gorithm for structured environments, and a hybrid regiongrowing algorithm for unstructured environments.

Preprint submitted to Robotics and Autonomous Systems June 24, 2013

Fig. 1: The custom-built 3D laser range finder which is installedon a Pioneer 3-AT robot. The laser range finder is connected tothe on-board computer through the inside slip-ring, which enables360 continuous pan motion without making the cables entanglementtogether. The UTM-30LX LRF is installed horizontally on the topbracket of D48E with its “Sensor Front” point upwards.

A light weight Pan-Tilt Unit (PTU) and an inexpen-sive two-dimensional (2D) LRF were integrated for three-dimensional (3D) point cloud gathering in our research,namely the FLIR R© PTU-D48E and the Hokuyo R© UTM-30LX; the setup is shown in Fig. 1. PTU-D48E is a high-performance real-time positioning system which offers highprecision positioning and speed control. We only make useof its pan motion which rotates the LRF to make an actu-ated LRF (aLRF). It has 360-continuous pan motion dueto an inside slip-ring. The finest pan resolution is 0.006

and the highest pan speed is 100/s. The 2D LRF wasdesigned for both indoor and outdoor environments whichhas a 270 field of view in its sensing plane. It is rotatedby 180 on the PTU to obtain a 3D scan with a field ofview 360 × 135. In this work, the pan resolution andthe laser beam resolution have been set to 0.5 and 0.25

respectively. Therefore, the resulting point clouds have541 × 720 points. A typical point cloud gathered indoorby the sensor is illustrated in Fig. 2. Besides point cloudsfrom the custom-built scanner, we also make use of otherpublicly available datasets to analysis the performance ofthe proposed algorithms.

The work presented in this paper is partially based onpreviously published results [31], with the main additionsbeing the novel hybrid region growing algorithm and moreexperimental data. The paper is laid out as follows. Re-lated work on plane segmentation come in Section 2. InSection 3, we detail our plane segmentation approaches.Then the mathematical machinery for incremental planeparameter calculation whenever a subwindow or a singlepoint is added in the region growing is given in Section4. Afterwards, we present the experiments and results inSection 5. Finally, the paper is summarized in Section 6,which also states our conclusions, as well as future researchdirections.

Fig. 2: A typical point cloud gathered by our customized 3D laserrange finder. The data was sampled in the authors’ robot laboratory.The points are colored by height, and the color map is shown underthe point cloud.

2. Related work

The Expectation Maximization (EM) algorithm is aniterative method for finding maximum likelihood estimatesof model parameters. Lakaemper and Latecki [25] em-ployed an extended EM algorithm to fit planar patches in3D range data. The algorithm is called Split and MergeExpectation Maximization Patch Fitting (SMEMPF), italternated the following E-steps and M-steps until reach aconvergence. The E-step is performed for a current givenset of planes (an initialization set of plane parameters isneeded at the beginning), the probabilities for each pointof its correspondence to all planes are calculated basedon its distance to the planes. Given the probabilities com-puted in the E-step, the new positions of the planes are es-timated in the M-step. It works on arbitrary point clouds,but is not feasible for real time operation. This is due tothe iterative nature of EM, including a costly plane-pointcorrespondence check in its core.

Hough transform is a classic feature extraction methodwhich has been used in image processing for the detectionof lines or circles. In order to use it for plane detection in3D point cloud segmentation, Borrmann et al. [30] eval-uated different variants of the Hough Transform. It wasfound that the main problem is the representation of theaccumulator besides computational costs. To deal withthis, they proposed the accumulator ball as an accumu-lator design. The evaluation of different Hough methodsrecommended the Randomized Hough Transform for deal-ing with plane detection in 3D point clouds. However, itstill has a severe disadvantage, i.e., the processing timeincreases with the number of planes but not the numberof points in the point cloud. As it was reported in the pa-per, the segmentation time using the Randomized HoughTransform would be significant larger than region growingwhen more than 15 planes presented in the data. Similarly,Dube and Zell [35] also proposed to use the Randomized

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Hough Transform for plane detection from depth images.They concentrated on the Microsoft Kinect cameras andmade use of the sensor noise model to find proper param-eter metrics for the Randomized Hough Transform. Theyevaluated the influence of local sampling and found localsampling made the result better. However, their test en-vironment was a clean corridor with only walls, windows,roof and ground which means about 4 or even fewer planespresented in each frame. Furthermore, the high detect rateof the walls is 82.2% which is not satisfied for detail mapgeneration.

The RANdom SAmple Consensus (RANSAC) [36] isanother route for estimating parameters of a model froma dataset which may have outliers. When being appliedto 3D point cloud plane segmentation, dealing with mul-tiple models in one dataset should be considered. In [24],the cloud is decomposed into equal sized 3D cubes, andthen one model is fitted to each cube using RANSAC, thusavoiding the multiple model problems. Afterwards, smallplanar patches are merged using the cube neighborhood in-formation. However, the algorithm is still time consumingdue to the iterative nature of RANSAC. Trevor et al. [34]utilized the RANSAC algorithm for plane segmentation inanother way. In their work, the plane with the most inliersin the dataset is searched each time, then all inliers for thisplane are removed from the point cloud; the RANSAC isperformed again to find the next largest plane. This pro-cess terminates when no plane with a sufficient number ofpoints could be found. However, the segmentation timehas not been reported in these two papers.

The concept of organized point cloud (also known asrange image or structured point cloud) should be men-tioned before introducing the region growing approach. Anorganized point cloud resembles an organized image-likestructure, where the data is split into rows and columns.Examples of such point clouds include data delivered bystereo- and ToF-cameras. The advantage of such a pointcloud is that the relationship between adjacent points (likepixels in an image) is known, making nearest neighbor op-erations much more time-efficient. Some aLRFs can alsoproduce organized point cloud datasets, such as ours inFig. 1. The adjacency information is denoted as pixel-neighborhood information later on in this paper. Note thatthe nearest neighbor search is an important issue for re-gion growing, since it has to be performed at each step ofthe growth.

Region growing was proposed for image segmentationbased on color information between neighbor pixels. It wasextended to plane segmentation in [23]. In their system,planar segments were employed for smoothing the resultedmap, but not as features in the mapping phase, whichmeans the segmentation speed was not a critical point.To embed the plane segmentation into plane-based on-linemapping systems, Poppinga et al. [27] sped it up with thefollowing two improvements: the first step is to use pixel-neighborhood information for nearest neighbor search andthe second step is an incremental version for plane param-

eters computation. It is further accelerated without losingany precision via an efficient plane fitting error compu-tation in our previous work, see [31] for details. For un-organized point cloud, we have proposed a cached octreeregion growing algorithm in [20], which makes an compro-mise between time and memory. For the input point cloud,an octree is built first and the indices of nearest neighborsfor each point is cached for the region growing phase. Asa result, the algorithm provides an efficient plane segmen-tation solution for unorganized point clouds, see [20] fordetails.

It is noted that the growth unit is a single point inthe above region growing approaches. As alternatives, thegrowth unit can also be a line segment or a subwindow.In Harati et al. [26], the so-called bearing angle is com-puted for each point as a measure of the flatness of its localarea, using the pixel-neighborhood information. Based onthis measure, a line-based region growing algorithm is pro-posed. However, since bearing angle is the incident anglebetween the laser beam and edges of the scanned polygonin the selected direction, it can not be properly calculatedin cluttered environments. Georgiev et al. [32] started byextracting 2D line segments from each 2D scan slice (eachrow or column in an organized point cloud), where con-nected line segments represent candidate sets of coplanarsegments. Then, a region growing algorithm is utilizedto find coplanar segments and their least squares fitting(infinite) plane.

In Kaushik et al. [29], the point cloud is divided intosubwindows (named patches in their paper), and plane pa-rameters are computed for each subwindow. The resultingsubwindows are clustered into large surfaces by a Breadth-First search algorithm. One drawback of this approach isthat there are some subwindows whose appearance couldnot be approximated by a plane. Then it was extendedand published in Kaushik and Xiao [12], where the planarpatches and nonplanar patches are distinguished. How-ever, the plane parameters are not updated when newdata points are added; instead, the plane parameters ofthe selected seed patch are treated as the plane model.Therefore the resulting plane parameter — a fundamentalissue for plane-based registration and SLAM — would beinaccurate. To deal with this problem, in this paper wepresent an incremental version for plane parameter calcu-lation whenever a new subwindow (i.e., a patch in [12]) isadded to the growing region.

3. 3D point cloud plane segmentation

The proposed plane segmentation approaches are de-scribed in detail in this section. The first part is namedthe subwindow based region growing (SBRG) algorithm,since one subwindow is added to the region at each stepduring its growth. The second part is called the hybridregion growing (HRG) algorithm as there are two kindsof growth unit, i.e., a single point or a subwindow. Theapproaches are proposed for organized point clouds. For a

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Fig. 3: The subwindow classification result of the point cloud shownin Fig. 2. The size of subwindow is 3 × 3. Sparse, planar andnon-planar subwindows are colored in red, green and blue.

detailed discussion on different point cloud formats, pleaserefer to the point cloud library [37] which is under rapidlydeveloping. The feasibility of utilizing subwindows is eval-uated first since it is necessary in both algorithms, followedby detailing the two algorithms.

In this paper, the following notations are used:x or X, x, x a scaler, a vector, an unit vectorx · y vector dot productX a matrixX a set

3.1. Feasibility of using subwindows in plane segmentation

Subwindow is suitable for plane segmentation if thefollowing two assumptions are tenable. First, most sub-windows which locate on planar surfaces have a planarappearance. Second, subwindows from the same physi-cal surface have similar plane parameters. To confirm thefirst assumption, the point cloud is decomposed into smallsubwindows first, then the subwindows are classified intotwo categories based on their shape appearances, namelyplanar or non-planar.

To determine the appearance of a subwindow ω whichcontains valid points p0,p1, ...psw, the scatter matrix Cof the points is computed as in Eq. (3) (see Section 4),where m is the geometric center. Note that invalid pointsmay also appear in ω, for example when there is no objectin certain laser beam directions. Clearly, C is a positive-definite matrix. In other words, C has three positive eigen-values. Given its sorted eigenvalues λ1 < λ2 < λ3, theshape of ω is decided by the below criteria:

ω ∈

sparse, if sw < µsize(ω),

planar, if λ1 ≤ ηλ2,

non-planar, otherwise,

(1)

where µ, η ∈ (0, 1), and size(ω) is the total number ofvalid and invalid points. The subwindow is marked assparse when the number of valid points is smaller than agiven threshold. A parameter tuning step is needed for η

−1

−0.5

0

0.5

1

−1−0.5

00.5

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

z

Fig. 4: Distribution of unit normals for all planar subwindows inFig. 3. Note that it does not correspond to the view point of Fig.3. Although normals are present almost everywhere on the sphere,it is still easy to find six dense clusters corresponding to the six bigplanar surfaces in the point cloud.

in order to yield satisfactory classification results. Fromthe experiments, it is only related to the employed rangesensor, i.e., η is a fixed value for each sensor.

In this paper, the subwindow is set to be square, withsize larger than 2× 2, since 4 points are not adequate forshape analysis. For better understanding, the subwindowclassification result of Fig. 2 is illustrated in Fig. 3. Thesize of the subwindow is set to 3×3; µ and η are set to 0.7and 0.3 respectively. The point cloud library is utilized forvisualization, where sparse, planar and non-planar sub-windows are colored as light gray, green and blue. It isapparent that most subwindows which were scanned fromplanar surfaces have planar appearance. Similar resultshave been found for other scans, thus the first assumptionis confirmed.

Now we deal with the second assumption, it meansthat the normal of a subwindow is a good estimation ofthe surface. Hahnel et al. pointed out that local surfacenormals from a planar surface in the real world are almostuniformly distributed in [23]. However, two orthogonal2D laser scanners on a mobile robot were used for datacollection in their robotic system. A horizontal laser wasemployed to perform 2D SLAM to localize the robot. Atthe same time, a vertical upward pointing laser scannedthe 3D structure of the environment. Therefore, both lo-calization error and measurement noise exist in their 3Dpoint clouds. The noise level should be higher than thatof a point cloud sampled with the so-called stop-scan-go(also known as stop-and-scan) style by an aLRF, whichonly contains measurement noise. Considering the planarsubwindows of Fig. 3, their unit normals have been vi-sualized in Fig. 4. Although some random normals arestill present, several dense clusters are apparent accordingto the planar surfaces in Fig. 3, thus the second assump-tion has been verified. Therefore, it can be concluded that

4

Algorithm 1: subwindow based region growing

Input: Ψ: an organized point cloudOutput: R: planar segments, R′: uncertain pointsR ← ∅, R′ ← ∅,G ← ∅,Q ← ∅,Ω← ∅ ;1

Ω=planarSubwindows(Ψ) ;2

while Ω \ (R) ∪R′) 6= ∅ do3

select ω with minimum e in Ω \ (R∪R′) ;4

G ← ω, Q ←NN(ω) ;5

while Q 6= ∅ do6

ωc = Q.pop() ;7

if |nG · (mG −mωc)| < γ && nG · nωc

>8

δ && mse(G ∪ ωc) < ε thenG ← G ∪ ωc ;9

Q ← Q ∪ NN(ωc) ;10

end11

end12

if size(G)≥ θ then13

R ← R∪ G ;14

else15

R′ ← R′ ∪ G ;16

end17

end18

subwindow can be used in plane segmentation.

3.2. Subwindow based region growing (SBRG)

The proposed SBRG algorithm proceeds as follows: Foran input organized point cloud Ψ, it is decomposed intosubwindows first using the image-like structure. The sub-windows are classified as planar and non-planar based onthe method presented in Section 3.1, then only the pla-nar subwindows will be kept for plane segmentation (Al-gorithm 1, line 2). At the same time, local plane param-eters for each planar subwindow are computed as well asthe mean square error (MSE); MSE is denoted by e in thispaper. Afterwards, the subwindow ω with the minimumMSE among all unidentified subwindows is chosen as anew seed (Algorithm 1, line 4). A growing region G is ini-tialized by ω, and its unidentified neighbors are put into aFirst-In-Last-Out (FILO) queue Q which keeps G’s near-est neighbors (Algorithm 1, line 5). Then, G is extendedby investigating its neighbors in Q. Suppose that ωc is theneighbor subwindow being considered, it is assigned to Giff it meets the following criteria (Algorithm 1, Line 8 –11).

1. The dot product between the normal vectors of ωcand G is greater than δ. Actually arccos(nG · nωc

)is the angle between G and ωc, so this criterion is toensure that the investigated subwindow has a similarsurface normal direction to G.

2. To avoid adding a subwindow which is parallel butnot coplanar to G, the distance from the mass centerof ωc to the optimal plane of G should be less thanγ.

(a)

(b)

(c)

(d)

Fig. 5: A segmentation quality comparison between the subwindowbased- and hybrid-region growing when applied to unstructured en-vironments. (a): One scan from the on-line dataset Collapsed CarParking Lot. (b): A close-up view of one part in (a) which hasabundant planar surfaces. (c) and (d): Close-up views of the planesegmentation results.

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3. To guarantee an acceptable flatness of the resultingsegment, the plane fitting error e of G ∪ωc should beless than ε.

This process terminates when no more neighbors canbe added to G, i.e., when Q is empty (Algorithm 1, Line 6– 12). Since our goal is to extract large planes in the scene,only a G with more than θ subwindows is regarded as aplanar segment and added to the plane set R, otherwiseit is added to the uncertain region set R′ (Algorithm 1,Line 13 – 17). The algorithm ends when every planar sub-window is assigned either to R or R′. The aforementionedparameters (δ, γ, ε, θ) are pre-set thresholds which need tobe tuned.

3.3. Hybrid region growing (HRG)

The SBRG algorithm has good performance in struc-tured environments. However, the result is unsatisfactorywhen applied to unstructured environments — see Fig. 5for an example. In such cases, the segmentation resultfrom pure subwindow based region growing is not satisfiedsince the surface edges may be classified as non-planar orsparse. This also happens when there are trees in front ofa wall, as part of the wall will be occluded by the leavesand branches, making the corresponding subwindow non-planar. The HRG algorithm is proposed to cope with thisproblem by considering all subwindows in the region grow-ing process.

The procedure used in the HRG algorithm is quite sim-ilar to that of the SBRG algorithm. For a point cloud Ψ, itis decomposed into subwindows first and the subwindowsare classified into planar, non-planar, and sparse based onthe method presented in Section 3.1, the planar subwin-dows are put into a list Ω and the other subwindows areput into another list Θ (Algorithm 2, line 2). When thereare still unidentified planar subwindows, the subwindowω with the minimum MSE among all unidentified planarsubwindows is chosen as a new seed (Algorithm 2, line 4).A growing region G is initialized by ω, and its unidentifiedneighbors are put into a FILO queue Q (Algorithm 2, line5). Then G is extended by investigating its neighbors. Ifthe considering subwindow is planar, the criteria for deter-mining whether to add it into G are the same as those inSBRG (Algorithm 2, line 8 – 13). Otherwise, each pointin the subwindow will be investigated separately, and apoint pc will be added to G iff it passes the following tests(Algorithm 2, line 15 – 18).

1. The distance from the point to the optimal planefitted to G is smaller than γ, making sure pc is acoplanar point of G.

2. The plane fitting error of G ∪ pc should be less thanε, this ensures the flatness of the segment acceptable.

This process will continue until no new neighbor of Gcan be found. Afterwards, if G has more points than athreshold θ, it will be viewed as a planar segment; oth-erwise, it will be marked as uncertain area (Algorithm 2,

Algorithm 2: hybrid region growing

Input: Ψ: an organized point cloudOutput: R: planar segments, R′: uncertain pointsR ← ∅, R′ ← ∅,G ← ∅,Q ← ∅,Ω← ∅,Θ← ∅ ;1

(Ω,Θ)=subwindows(Ψ) ;2

while Ω \ (R) ∪R′) 6= ∅ do3

select ω with minimum e in Ω \ (R∪R′) ;4

G ← ω, Q ←NN(ω) ;5

while Q 6= ∅ do6

ω = Q.pop() ;7

if isPlanar(ω)==true then8

if mse(G ∪ ωc)< ε && nG · nωc>9

δ && |nG · (mG −mωc)| < γ thenG ← G ∪ ωc ;10

Q ← Q ∪ NN(ωc) ;11

end12

else13

for each point pc in ω do14

if mse(G ∪ pc) < ε &&15

nG · (mG − pc) < γ thenG ← G ∪ pc ;16

Q ← Q ∪ NN(ωc) ;17

end18

end19

end20

end21

if size(G)≥ θ then22

R ← R∪ G ;23

else24

R′ ← R′ ∪ G ;25

end26

end27

line 22 – 26). The algorithm ends when every point hasbeen assigned to R or R′. The aforementioned parameters(δ, γ, ε, θ) are pre-set thresholds. One parameter tuningstep is needed for a specific range sensor.

4. Incremental plane parameter calculation

As depicted in both algorithms, a plane should be fit-ted to the growing region whenever a new subwindow ora single point is added, the time cost of it is critical fora plane-based mapping system. In order to make it fast,the consequent incremental version is proposed. Differentforms of plane equations exist in the literature. A com-parison of them can be found in [38]. The Hessian formequation is chosen to represent planes in this work, for thereason that it can be obtained straightforward when thesurface normal vector and an arbitrary point on the planeare known. It is described as

n · p = d, (2)

where n is the unit normal vector of the plane, p is anarbitrary point on the plane and d is the distance from

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the origin to the plane. To make the equation of eachplane unique, we choose n in the direction which makesd > 0.

In order to determine the shape appearance of a sub-window ω, the scatter matrix of the valid points in it iscomputed as

C =1

K

K∑i=1

(pi −m)(pi −m)T (3)

where K is the number of valid points and

m =1

K

K∑i=1

pi (4)

is the geometric center. Then the eigenvalues of C arecomputed through eigenvalue decomposition and the sub-window classification is carried out as in Section 3.1.

If a subwindow is classified as planar, a plane will befitted to it using the least squares. According to the leastsquare fitting, the geometric center m is on the optimalplane, and the eigenvector corresponding to the smallesteigenvalue of C is the plane normal. As a result, d can beeasily computed as:

d = n ·m. (5)

When shape classification is finished, the following quan-tities are tracked for each planar subwindow besides C, m,n and d.

e = 1Kλmin(C)

J =

K∑i=1

pipTi

(6)

where e is the plane fitting error, J is the second order mo-ment about the origin, and λmin(C) stands for the mini-mum eigenvalue of C.

The above quantities are also tracked for the growingregion; they are used to derive the new plane parameterstogether with that of the subwindow to be added. At thebeginning of the region growing, i.e., when a new seed isselected, the quantities of the growing region are simplyassigned to those of the seed. Then the quantities of thegrowing region is updated incrementally which is explainedbelow. Suppose there are KG points in the current growingregion; the subwindow ω has passed the coplanar testsand is going to be added, which has Kω points. In Eq.(7), G and ω are used as subscripts to denote the growingregion and the investigated subwindow respectively, whilethere is no subscript for the combined region. Note that Gmay contain just one subwindow, i.e., when only the seedsubwindow is added. Eq. (7) is developed to calculatethe plane parameters of the combined region using theabove tracked quantities. Obviously, it can also be usedfor computing the plane parameters when merging two

coplanar segments.

s = mGKG + mωKω

m = s/(KG +Kω)

J = JG + Jω

C = 1KG+Kω

KG+Kω∑i=1

(pi −m)(pi −m)T

n = emin(C)

d = n ·m

e = λmin(C)KG+Kω

(7)

In Eq. (7), emin(C) stands for the normalized eigen-vector corresponding to λmin(C). All the other equationsneed constant time except the computation of C. Thetime complexity to find the eigenvalues of a n × n ma-trix is O(n3), so the eigenvalue decomposition of C isthe most time consuming part when KG is small at thestarting stage of a growing region. Then KG increases asthe region grows, and the calculation of C becomes themost time consuming part when there are many points(KG > n3, n = 3) in the growing region. In order to makethe algorithm fast, we calculate C using other trackedquantities. After some algebra, it can be simplified as:

C = J− smT. (8)

Eq. (7) and (8) yield an incremental version for com-puting the plane parameters in the SBRG algorithm. How-ever it is not sufficient for the HRG algorithm, because theplane parameters should be also updated when adding asingle point. Assume the point p has passed the coplanartests and is going to be added. The new plane parameterswill be calculated as:

s = mGKG + p

m = s/(KG + 1)

J = JG + ppT

C = J− smT

n = emin(C)

d = n ·m

e = λmin(C)KG+1

(9)

It can be seen that Eq. (9) is a special case of Eq. (7),i.e., when Kω = 1. To conclude this section, the plane pa-rameters are computed incrementally when a subwindowor a single point is added to the segment, which makesSBRG and HRG fast.

4.1. Computational complexity analysis

Suppose an organized point cloud from structured en-vironment contains n points, which is to be segmented

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(a) (b)

(c) (d)

Fig. 6: subwindow based region growing plane segmentation for indoor structured environment, (a) and (c) are two different viewpoints fromthe segmentation result of one point cloud, while (b) and (d) are two different viewpoints for another point cloud.

using the SBRG algorithm. The subwindow size is set tok, which means there are m = bn/kc subwindows. Shapeclassification needs constant time for each subwindow witha time complexity O(m). In addition, the neighbor searchexecutes with a time complexity at most O(m logm), forall subwindows belonging to one segment. Computing theplane parameter when a new subwindow is added to theregion also needs constant time with time complexity ofO(m). To sum up, the overall time complexity of theSBRG algorithm is O(m logm). For each subwindow andsegment, at most seven variables are tracked which yieldsthe memory complexity O(m).

There are two growth units in HRG, namely a singlepoint or a subwindow. As a result, the time complexity ofHRG is between the point based algorithm and the sub-window based algorithm. Its time complexity is equal toSBRG when there are no non-planar/sparse subwindows,and is equal to point based region growing when thereare no planar subwindows in the given point cloud. For apoint cloud with n points, the time complexity of SBRGis O(nk log n

k ), where k is the subwindow size. The timecomplexity of point based region growing is O(n log n) asreported in [27]. Consequently, the time complexity of

HRG is in the range [O(nk log nk ), O(n log n)], it increases

with the clutter level of the environment and vice versa.

5. Experiments and results

Both the proposed algorithms have been implementedin C++; the code has been published online, and can beaccessed from the following URL: https://github.com/junhaoxiao/TAMS-Planar-Surface-Based-Perception.

git. All experiments were carried out on a standard desk-top computer under Ubuntu 12.04. The efficiency of linearalgebra is crucial in the approaches, especially for calculat-ing the eigenvalues and eigenvectors of a square matrix, asit has to be performed whenever a point or a subwindowis investigated during the region growing phase. ThereforeEigen [39], a C++ template library for linear algebra, hasbeen employed. The point cloud library has been utilizedfor point cloud reading and writing, as well as 3D visual-ization. For figures illustrating segmentation results, thesegments have been colored randomly; therefore, one colormay be patched to multiple segments. Due to space lim-itation, we cannot present all the segmentation results.

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Fig. 7: One typical segmentation result using the subwindow based region growing algorithm in the second indoor dataset. Two differentviewpoints have been given.

Instead, some typical results have been selected for expla-nation.

As mentioned in Sections 3.2 and 3.3, parameter tun-ing is necessary for both algorithms. Experimental tuningwas performed offline in this work. Obviously, the parame-ters depend on the measurement model of the 3D scanner.Actually the parameters can be intuitively selected — thethresholds can be chosen by manually identifying the pla-nar surfaces in a given scan.

Furthermore, Kaustubh Pathak has kindly provided usthe access to their code used in [27]. Their algorithm servesas a baseline for comparison and is denoted as PBRG(Point Based Region Growing). It should be noted thatplane parameter uncertainties have also been computedduring the region growing procedure in PBRG, which hasnot been considered in the proposed algorithms. There-fore, the comparison is not fully impartial; however, theuncertainty computation does not induce much time com-plexity compared to the coplanar points detection, see [27]for details.

5.1. Structured environment datasetsTwo indoor datasets have been employed for evaluat-

ing the proposed SBRG algorithm. The first dataset isnamed TAMS Office. It was gathered using our customized3D scanner at different positions on a floor that has akitchen, a robot laboratory and several offices, including40 point clouds. One typical scan in the laboratory is il-lustrated in Fig. 2. The dataset has been made publiclyavailable at http://tams.informatik.uni-hamburg.de/research/datasets/index.php. The second dataset isfrom [29]; it was obtained by spinning a Hokuyo R© URG-04LX which is designed for indoor use only, it is denotedas Indoor Hokuyo in this paper.

Two segmentation results from the first dataset usingthe SBRG algorithm are depicted in Fig. 6, and one from

the second dataset is depicted in Fig. 7. Two differentviewpoints have been given for each result.

5.2. Unstructured environment datasets

Two datasets have been employed to evaluate theproposed HRG algorithm. Both of them are pub-licly available. The first dataset, Collapsed Car Park-ing Lot, was gathered in an unstructured environ-ment during NIST Response Robot Evaluation Exer-cise 2008 at Disaster City, Texas. The dataset isaccessed at http://www.robotics.jacobs-university.

de/datasets/RAW/RREE08/crashedCarPark/. The scanswere gathered by a track robot equipped with an aLRF;the aLRF is based on a SICK S 300 which has a FoV of270 of 541 beams. The sensor is pitched from −90 to+90 at a spacing of 0.5, which leads to an organizedpoint cloud of 541×361 = 195, 301 points per sample. See[7] for details.

The second dataset, Barcelona Robot Lab, covers10,000 square meters of the UPC Nord Campus inBarcelona, including 400 dense 3D point clouds. Thereare about 380,000 points in each scan. Since onlyunorganized point clouds are provided in this dataset,the points have been ordered using their spherical co-ordinates, see [20]. The original dataset is availableat http://www.iri.upc.edu/research/webprojects/

pau/datasets/BRL/php/dataset_data_access.php; inaddition, the organized point clouds for it can be down-loaded at http://tams.informatik.uni-hamburg.de/

research/datasets/index.php. Typical segmentationresults have been drawn in Fig. 8 and Fig. 9.

5.3. Discussion

As seen in Fig. 5, Fig. 7, and Fig. 6, SBRG has goodperformance in structured indoor environments while pro-ducing unsatisfactory segmentation results in unstructured

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(a)

(b)

Fig. 8: Two typical segmentation results using the HRG algorithm for the Collapsed Car Parking Lot dataset.

environments. HRG can deal with the unstructured envi-ronments which can be seen from Fig. 8 and Fig. 9. It hasalso been applied to the indoor datasets and is proven tohave no advantage when compared to the SBRG algorithmin such environments.

To analyze their speeds, the algorithms are comparedto the PBRG algorithm. We do not compare the algo-rithms to the Hough Transform method [30], because eachpoint cloud in our experiment contains much more than15 planar surfaces, and Borrmann et al. already reportedthat the Hough Transform is much slower than PBRG insuch datasets. We do not compare our algorithms to thatof [29] either, as the plane parameters are not updatedduring the patch clustering procedure in their algorithm,see Section 2.

For comparison, PBRG, SBRG and HRG have been ap-plied to the indoor environments, while PBRG and HRGhave been applied to the unstructured environments. Thedatasets Indoor Hokuyo and Barcelona Robot Lab are cho-

sen for benchmarking the speed since they contain morepoint clouds than the other two datasets, and are thereforebetter for statistical analysis. The segmentation time foreach point cloud with different algorithms is depicted inFig. 10 and Fig. 11. In Fig. 10, the subwindow size of3×3 and 4×4 are used for both SBRG and HRG, and thepoint cloud size corresponds to the number of valid pointsin each point cloud. Since most subwindows have a pla-nar appearance, SBRG and HRG have approximately thesame speed when using equal subwindow sizes. The aver-age segmentation time has been illustrated in Tab. 1, itcan be seen that SBRG is about 4 times faster than PBRGwhen the subwindow size is set to 3× 3 and 9 times fasterfor subwindow size 4 × 4. For a point cloud with about135, 000 points, SBRG needs only 0.1 second for plane seg-mentation. For aLRFs, the plane segmentation time usingSBRG is much shorter than obtaining a point cloud whichis usually tens of seconds.

In Fig. 11, subwindow size 3×3 and 4×4 were utilized

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(a)

(b)

Fig. 9: Two typical segmentation results using the HRG algorithm for the Barcelona Robot Lab dataset.

in the HRG algorithm. Even larger subwindow sizes havealso been tested, however, they made the narrow planarsurfaces unidentified. Note the approximately linear rela-tion between the processing time and the point cloud sizes.The same as Fig. 10, the point cloud size corresponds tothe number of valid points. The linearity of HRG is worsethan PBRG since two kinds of growth units are used init, and the time complexity depends on the clutter level ofthe environment which is analyzed in Section 4.1. Whenthe subwindow size is increased, the segmentation speed isfaster. For a point cloud with as much as 380,000 points,

HRG needs less than 0.4 second for plane segmentationwhen the subwindow size is set to 4× 4. Again, the planesegmentation time is much smaller than the data gatheringtime which means real-time data processing is promised.The average plane segmentation time is illustrated in Tab.2. It shows that HRG is 4 times faster compared to PBRGwhen using subwindow size of 4× 4.

In general, the speed of both algorithms is faster whencompared to the PBRG algorithm. The results are promis-ing for plane-based mapping systems in both structuredor unstructured environments, thus have attracted appli-

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1.24 1.26 1.28 1.3 1.32 1.34 1.36

x 105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

point cloud size

segm

enta

tion

time

[s]

PBRG

HRG, subwindow size 3×3

HRG, subwindow size 4×4

SWRG, subwindow size 3×3

SWRG, subwindow size 4×4

Fig. 10: Segmentation speed benchmarking result for a structuredenvironment, all point clouds in the Indoor Hokuyo dataset havebeen employed.

1 1.5 2 2.5 3 3.5 4

x 105

0

0.2

0.4

0.6

0.8

1

1.2

1.4

point cloud size

segm

enta

tion

time

[s]

PBRG

HRG, subwindow size 3×3

HRG, subwindow size 4×4

Fig. 11: Segmentation speed benchmarking result for an unstruc-tured environment, all point clouds in the Barcelona Robot Labdataset have been employed.

Table 1: Average plane segmentation time for the Indoor Hokuyodataset. The time for SBRG and HRG is specified to each underlyingsubwindow size.

algorithm PBRGSBRG HRG

3× 3 4× 4 3× 3 4× 4time [s] 0.418 0.109 0.048 0.124 0.061

Table 2: Average plane segmentation time for the Barcelona RobotLab dataset. The time for HRG is specified to each underlying sub-window size.

algorithm PBRGHRG

3× 3 4× 4time [s] 0.864 0.408 0.198

cation areas such as home service robots and field robots.For indoor structured environments, the SBRG algorithmis suggested while the HRG algorithm is recommended forunstructured environments.

However, the algorithms have their limitations. First,they can only be employed when dealing with organizedpoint clouds due to they need the image-like structure forgenerating subwindows. Second, an optional subwindowsize should be tuned for the working environment. If thesize if too small, the plane parameter estimated for eachsubwindow is disturbed by the sensor noise, and the seg-mentation time is longer compared to a larger subwindowsize. If the size is too large, many subwindows which lo-cate on the surface boundaries are classified as non-planar,making surface details missing in the SBRG algorithm.Furthermore, the long narrow planar surfaces will be lostin the segmentation result for both algorithms. From theexperiments, we suggest to use subwindow size not smallerthan 3 × 3 and not larger than 10 × 10. Third, the algo-rithms are limited to segmenting planar segments, whilehigher order surfaces are common and should be consid-ered for more general environments.

6. Conclusion and future work

We presented two plane segmentation approaches inthis paper, one for structured environments and the otherfor unstructured environments. The main idea is to use arelatively larger growth unit in the region growing pro-cedure, i.e., using subwindows. It was found that thepure subwindow based region growing algorithm is suit-able for structured environments and has unsatisfactoryperformance when applied to unstructured environments.In order to deal with this problem, we proposed to use bothsubwindow and single point as alternative growth unit inthe hybrid region growing algorithm.

Both algorithms were evaluated using real world datasets,which gives promising results. For structured environ-ments, the subwindow based region growing algorithm canextract the planar surfaces from a point cloud with about135, 000 points within 0.1 second. For unstructured envi-ronments, the hybrid region growing algorithm needs lessthan 0.4 second for segmenting a point cloud with as manyas 380, 000 points. From the experiments, the algorithmsare about 4 times faster than the point based region grow-ing when a proper size is set.

After segmentation, each point cloud can be repre-sented as a set of planar segments. The area of eachsegment can be calculated using the range-image basedmethod proposed in our previous publication [20]. Ad-ditionally, the resulting area-attributed can be used fordetermining corresponding segments between overlappingscans; based on the correspondences, the scans can be reg-istered into a common coordinate system, the reader isreferred to [20] for details of the registration approach.

In the future, we will focus on learning other attributesof the resulted segments, such as plane parameter uncer-tainties and 2D outlines. Furthermore, since our custom-built 3D scanner can provide intensity information for eachpoint, we will also try to find intensity features inspiredby current image processing techniques such as gray-scale

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histograms. These attributes can help to improve the ro-bustness of our registration algorithm proposed in [20].An even further step of our research could be embeddingthe plane-based registration approach into a pose-graphSLAM system.

Acknowledgments

Junhao Xiao is funded by the China Scholarship Coun-cil (CSC), which is gratefully acknowledged. We wouldalso like to thank Prof. Kaustubh Pathak from the JacobsRobotics group at Jacobs University Bremen for providingus an access to their code (PBRG).

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Junhao Xiao (M’12) is a Ph.D. stu-dent in the Institute of Technical As-pects of Multimodal Systems (TAMS),Department Informatics, University ofHamburg. He received his bachelordegree in Automation from the Na-tional University of Defense Technology(NUDT), 2007. He got a scholarship

from the China Scholarship Council (CSC) and joinedTAMS in Sep. 2009. His research interest lies on mobilerobotics, sensor fusion and especially 3D robotic mapping.

Jianhua Zhang (M’11) received theMSc degree at Zhejiang University ofTechnology in 2009 and the Ph.D. de-gree at the University of Hamburg in2012. He joined Zhejiang University ofTechnology, China, in December 2012.His research interests include categorydiscovery, object detection, image seg-mentation, medical image analysis.

Benjamin Adler is a scientific assis-tant at the Institute of TAMS, Depart-ment Informatics, University of Ham-burg. He received his diploma degree inComputer Science at the University ofHamburg in 2008 and is currently work-ing on his Ph.D. thesis. His research in-terest focuses on mobile robotics, GNSSsystems and multi-sensor fusion.

Houxiang Zhang (M’04 – SM’12)received Ph.D. degree in Mechanicaland Electronic Engineering from Bei-jing University of Aeronautics and As-tronautics, China, in 2003. From2004, he worked as Postdoctoral Fel-low at TAMS, Department of Informat-ics, University of Hamburg, Germany.Then he joined the Faculty of MaritimeTechnology and Operations, Alesund

University College, Norway in April 2011 where he is afull Professor on Robotics and Cybernetics. The focus ofhis research lies on mobile robotics, especially on climb-ing robots and urban search and rescue robots, modular

robotics, and nonlinear control algorithms.

Jianwei Zhang (M’92) received bothhis Bachelor of Engineering (1986, withdistinction) and Master of Engineering(1989) from the Department of Com-puter Science of Tsinghua University,Beijing, China, and his PhD (1994) atthe Institute of Real-Time ComputerSystems and Robotics, Department ofComputer Science, University of Karl-

sruhe, Germany. Dr. Jianwei Zhang is professor and headof TAMS, Department of Informatics, University of Ham-burg, Germany.

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