Fast Geometric Point Labeling using Conditional Random Fields

Radu Bogdan Rusu, Andreas Holzbach, Nico Blodow, Michael BeetzIntelligent Autonomous Systems, Computer Science Department, Technische Universitat Munchen

Boltzmannstr. 3, Garching bei Munchen, 85748, Germany{rusu,holzbach,blodow,beetz}@cs.tum.edu

Abstract— In this paper we present a new approach forlabeling 3D points with different geometric surface primitivesusing a novel feature descriptor – the Fast Point FeatureHistograms, and discriminative graphical models. To buildinformative and robust 3D feature point representations, ourdescriptors encode the underlying surface geometry around apoint p using multi-value histograms. This highly dimensionalfeature space copes well with noisy sensor data and is notdependent on pose or sampling density. By defining classes of 3Dgeometric surfaces and making use of contextual informationusing Conditional Random Fields (CRFs), our system is ableto successfully segment and label 3D point clouds, based onthe type of surfaces the points are lying on. We validate anddemonstrate the method’s efficiency by comparing it againstsimilar initiatives as well as present results for table settingdatasets acquired in indoor environments.

I. INTRODUCTION

Segmenting and interpreting the surrounding environmentthat a personal robot operates in from sensed 3D data isan important research topic. Besides recognizing a certainlocation on the map for localization or map refinementpurposes, obtaining accurate and informative object modelsis essential for precise manipulation and grasping. Addition-ally, although the acquired data is discrete and representsonly a few samples of the underlying scanned surfaces, itquickly becomes expensive to store and work with. It istherefore imperative to find ways to address this dimension-ality problem and group clusters of points sampled fromsurfaces with similar geometrical properties together, or insome sense try to “classify the world”. Achieving the latterannotates sensed 3D points and geometric structures withhigher level semantics and will greatly simplify research inthe aforementioned topics (e.g. manipulation and grasping).

In general, at the 3D point level, there are two basicalternatives for acquiring these annotations:

1) use a powerful, discriminative 3D feature descriptor,and learn different classes of surface or object typeseither by supervised or unsupervised learning, and thenuse the resultant model to classify newly acquired data;

2) use geometric reasoning techniques such as point cloudsegmentation and region growing, in combination withrobust estimators and non-linear optimization tech-niques to fit linear (e.g. planes, lines) and non-linear(e.g. cylinders, spheres) models to the data.

Both approaches have their advantages and disadvantages,but in most situations machine learning techniques – andthus the first approach, will outperform techniques purely

based on geometric reasoning. The reason is that while linearmodels such as planes can be successfully found, fitting morecomplicated geometric primitives like cones becomes verydifficult, due to noise, occlusions, and irregular density, butalso due to the higher number of shape parameters that needto be found (increased complexity). To deal with such largesolution spaces, heuristic hypotheses generators, while beingunable to provide guarantees regarding the global optimum,can provide candidates which drastically reduce the numberof models that need to be verified.

Fig. 1. An ideal 3D point based classification system providing twodifferent point labels: geometry (L1) and object class (L2).

Additionally, we define a system able to provide twodifferent hierarchical point annotation levels as an idealpoint classification system. Because labeling objects lyingon a table just with a class label, say mug versus cerealbox, is not enough for manipulation and grasping (becausedifferent mugs have different shapes and sizes and need to beapproached differently by the grasp planner), we require theclassification system to annotate the local geometry aroundeach point with classes of 3D geometric primitive shapes.This ensures that besides the object class, we are able toreconstruct the original object geometry and also plan bettergrasping points at the same time. Figure 1 presents the tworequired classification levels for 3 points sampled on separateobjects with different local geometry.

This paper concentrates on the first annotation approach,and introduces a novel 3D feature descriptor for point clouds:the Fast Point Feature Histograms (FPFH). Our descriptorsare based on the recently proposed Point Feature Histograms(PFH) [1] and robustly encode the local surface geometryaround a point p using multi-value histograms, but reducethe original PFH computational complexity from O(N2) to

O(N) while retaining most of their discriminative power.The FPFH descriptors cope well with noisy sensor dataacquired from laser sensors and are independent of pose orsampling density. We present an in-depth analysis on their us-age for accurate and reliable point cloud classification, whichprovides candidate region clusters for further parameterizedgeometric primitive fitting.

By defining classes of 3D geometric surfaces, such as:planes, edges, corners, cylinders, spheres, etc, and makinguse of contextual information using discriminative undirectedgraphical models (Conditional Random Fields), our systemis able to successfully segment and label 3D points basedon the type of surfaces the points are lying on. We viewthis is as an important building block which speeds up andimproves the recognition and segmentation of objects in realworld scenes.

As an application scenario, we will demonstrate the pa-rameterization and usage of our system for the purpose ofpoint classification for objects lying on tables in indoorenvironments. Figure 2 presents a classification snapshotfor a scanned dataset representing a table with objects ina kitchen environment. The system outputs point labels forall objects found in the scene.

Fig. 2. Top: raw point cloud dataset acquired using the platform in Figure 7containing approximatively 22000 points; bottom: classification results forpoints representing the table and the objects on it. The color legend is:dark green for planar, mid green for convex cylinders, yellow for concavecylinders, light green for corners, pink for convex edges and brown forconvex tori. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

The remainder of the paper is organized as follows. Weaddress related work on similar initiatives in the next section,and describe the Fast Point Feature Histograms (FPFH) inSection III. The classes of 3D primitive geometric shapesused for learning the model are presented and analyzed inSection IV. We introduce our variant of Conditional RandomFields in Section V and discuss experimental results inSection VI. Finally we conclude and give insight into our

future work in Section VII.

II. RELATED WORK

The area of 3D point-based surface classification has seensignificant progress over time, but more in recent years dueto advances in electronics which fostered both better sensingdevices as well as personal computers able to process the datafaster. Suddenly algorithms for 3D feature estimation whichwere intractable for real world applications a few years agobecame viable solutions for certain well-defined problems.Even so however, most of the work in 3D feature estimationhas been done by the computer graphics community, ondatasets usually acquired from high-end range scanners.

Two widely used geometric point features are the under-lying surface’s estimated curvature and normal [2], whichhave been also investigated for noisier point datasets [3].Other proposals for point features include moment invariantsand spherical harmonic invariants [4], and integral volumedescriptors [5]. They have been successfully used in ap-plications such as point cloud registration, and are knownto be translation and rotation invariant, but they are notdiscriminative enough for determining the underlying surfacetype. Specialized point features that detect peaks, pits andsaddle regions from Gaussian curvature thresholding for theproblem of face detection and pose estimation are presentedin [6]. Conformal factors are introduced by [7], as a measureof the total curvature of a shape at a given triangle vertex.

In general, descriptors that characterize points with asingle value are not expressive enough to make the necessarydistinctions for point-surface classification. As a direct conse-quence, most scenes will contain many points with the sameor very similar feature values, thus reducing their informativecharacteristics. Alternatively, multiple-value point featuressuch as curvature maps [8], or spin images [9], are someof the better local characterizations proposed for 3D mesheswhich got adopted for point cloud data. However, theserepresentations require densely sampled data and are not ableto deal with the amount of noise usually present in 2.5Dscans.

Following the work done by the computer vision commu-nity which developed reliable 2D feature descriptors, somepreliminary work has been done recently in extending thesedescriptors to the 3D domain. In [10], a keypoint detectorcalled THRIFT, based on 3D extensions of SURF [11] andSIFT [12], is used to detect repeated 3D structure in rangedata of building facades, however the authors do not addresscomputational or scalability issues, and fail to compare theirresults with the state-of-the art. In [13], the authors proposeRIFT, a rotation invariant 3D feature descriptor utilizingtechniques from SIFT, and perform a comparison of theirwork with spin images, showing promising results for theproblem of identifying corresponding points in 3D datasets.

We extend our previous work presented in [1] by com-puting fast local point feature histograms for each pointin the cloud. We make an in-depth analysis of the points’signatures for different geometric primitives (i.e. planes,cylinders, toruses, etc) and show classification results using

two different machine learning methods: Support VectorMachines (SVM) and Conditional Random Fields (CRF).

The work in [14], [15] is similar to our labeling approach,as they too use probabilistic graphical models for the problemof point-cloud classification. In both, sets of simple point-based 3D features are defined, and a model is trained toclassify points with respect to object classes such as: chairs,tables, screens, fans, and trash cans [15], respectively: wires,poles, ground, and scatter [14]. The main difference of ourframework is that we do not craft sets of simpler 3D featuresfor individual problems to solve, but propose the usage of asingle well defined 3D descriptor for all.

III. FAST POINT FEATURE HISTOGRAMS (FPFH)The computation of a Fast Point Feature Histogram

(FPFH) at a point p relies on the point’s 3D coordinatesand the estimated surface normal at that point. As shownby [1], the PFH formulation is sufficient for capturing thelocal geometry around the point of interest, but if needed, isextensible to the use of other local properties such as color,curvature estimates, 2nd order moment invariants, etc.

The main purpose of the FPFH formulation is to createa feature space in which 3D points lying on primitivegeometric surfaces (e.g. planes, cylinders, spheres, etc) canbe easily identified and labeled. As a requirement then, thediscriminating power of the feature space has to be highenough so that points on the same surface type can begrouped in the same class, while points sampled on differentsurface types are assigned to different classes. In addition,the proposed feature space needs to be invariant to rigidtransformations, and insensitive to point cloud density andnoise to a certain degree.

To formulate the FPFH computational model, we introducethe following notations:• pi is a 3D point with {xi, yi, zi} geometric coordinates;• ni is a surface normal estimate at point pi having a{nxi, nyi, nzi} direction;

• Pn = {p1,p2, · · · } is a set of nD points (also repre-sented by P for simplicity);

• Pk is the set of points pj , (j ≤ k), located in the k-neighborhood of a query point pi;

• ‖ · ‖x is the Lx norm (e.g. ‖ · ‖1 is the Manhattan or L1

norm, ‖ · ‖2 is the Euclidean or L2 norm).Using the above formulation, the estimation of a FPFH

descriptor includes the following steps: i) for each point pi,the set of Pk neighbors enclosed in the sphere with a givenradius r are selected (k-neighborhood); ii) for every pair ofpoints pi and pj (i 6= j) in Pk, and their estimated normalsni and nj (pi being the point with a smaller angle betweenits associated normal and the line connecting the points), wedefine a Darboux uvn frame (u = ni, v = (pj − pi) ×u, w = u×v) and compute the angular variations of ni andnj as follows:

α = v · nj

φ = (u · (pj − pi))/||pj − pi||2θ = arctan(w · nj , u · nj)

(1)

Then, for all sets of computed < α, φ, θ > values in Pk,a multi-value histogram is created as described in [1]. Wecall this the SPFH (Simplified Point Feature Histogram) asit only describes the direct relationships between a querypoint pi and its k neighbors. To capture a more detailedrepresentation of the underlying surface geometry at Pk, ina second step we refine the final histogram (FPFH) of pi byweighting all the neighboring SPFH values:

FPFH(pi) = SPFH(pi) +1k

k∑i=1

1ωk· SPFH(pk) (2)

where the weight ωk is chosen as the distance between thequery point pi and a neighboring point pk in some metricspace (Euclidean in our formulation), but could just as wellbe selected as a different measure if necessary. To understandthe importance of this weighting scheme, Figure 3 presentsthe influence region diagram for a Pk set centered at pq . TheSPFH connections are illustrated using red lines, linking pi

with its direct neighbors, while the extra FPFH connections(due to the additional weighting scheme) are shown in black.

p7

p6

p8

p9

p10

p11

pk5

pk1

pk2

pk3p

k4

pq

p12

p14

p15

p16

p13

p17

Fig. 3. The influence region diagram for a Fast Point Feature Histogram.Each query point (red) is connected only to its direct k-neighbors (enclosedby the gray circle). Each direct neighbor is connected to its own neighborsand the resulted histograms are weighted together with the histogram of thequery point to form the FPFH. The thicker connections contribute to theFPFH twice.

The number of histogram bins that can be formed in a fullycorrelated FPFH space is given by qd, where q is the numberof quantums (i.e. subdivision intervals in a feature’s valuerange) and d the number of features selected (e.g. in ourcase dividing each feature interval into 5 values will resultin 53 = 125 bins). In this space, a histogram bin incrementcorresponds to a point having certain values for all its 3features. Because of this however, the resulting histogramwill contain a lot of zero bins, and can thus contribute to acertain degree of information redundancy. A simplificationof the above is to decorrelate the values, that is to simplycreate d separate histograms, one for each feature dimensionin Equation 1, and then concatenate them together to createthe FPFH.

In contrast to our PFH formulation [1], the FPFH de-scriptors reduce the computational complexity from O(N2)

to O(N). In the next section we analyze if the FPFHdescriptors are informative enough to differentiate betweenpoints sampled on different 3D geometric shapes.

IV. 3D GEOMETRIC PRIMITIVES AS CLASS LABELS

A straightforward analysis of the FPFH discriminatingpower can be performed by looking at how features com-puted for different geometric surfaces resemble or differ fromeach other. Since the FPFH computation includes estimatedsurface normals, the features can be separately computed andgrouped for both two cases of convex and concave shapes.

Figure 4 presents examples of FPFH signatures for pointslying on 5 different convex surfaces, namely a cylinder, anedge, a corner, a torus, and finally a plane. To illustrate thatthe features are discriminative, in the left part of the figure weassembled a confusion matrix with gray values representingthe distances between the mean histograms of the differentshapes (a low distance is represented with white, while ahigh distance with black), obtained using the HistogramIntersection Kernel [16]:

d(FPFH1, FPFH2) =N∑

i=1

min(FPFHi1, FPFH

i2) (3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27Bins

0

50

100

150

200

250

300

350

400

Rat

ioof

poin

tsin

one

bin

(%)

FPFH for different geometric scanned surfaces

planecornercylinder (r=3cm)edgetorus (r=3cm)

torus edge cylinder corner plane

toru

sed

gecy

linde

rco

rner

plan

e

Histogram Intersection Kernel

Fig. 4. Example of Fast Point Feature Histograms for points lying onprimitive 3D geometric surfaces.

The results show that if the computational parameters arechosen carefully (i.e. the scale), the features are informativeenough to differentiate between points lying on differentgeometric surfaces.

V. CONDITIONAL RANDOM FIELDS

Generative graphical models, like Naive Bayes or HiddenMarkov Models, represent a joint probability distributionp(x, y), where x expresses the observations and y the classi-fication label. Due to the inference problem this approach isnot well applicable for fast labeling of multiple interactingfeatures like in our case. A solution is to use discriminativemodels, such as Conditional Random Fields, which representa conditional probability distribution p(y|x). That meansthere is no need to model the observations and we canavoid making potentially erroneous independence assump-tions among these features [17]. The probability p(y|x) canbe expressed as:

p(y|x) =p(y, x)p(x)

=p(y, x)∑y′ p(y′, x)

=1Z

∏c∈C ψc(xc, yc)

1Z

∑y′

∏c∈C ψc(xc, y′c)

(4)Therefore we can formulate a CRF model as:

p(y|x) =1

Z(x)

∏c∈C

ψc(xc, yc) (5)

where Z(x) =∑

y′∏

c∈C ψc(xc, y′). The factors ψc are

potential functions of the random variables vC within aclique c ∈ C [18].

According to this mathematical definition, a CRF can berepresented as a factor graph (see Figure 5). By definingthe factors ψ(y) = p(y) and ψ(x, y) = p(x|y) we get anundirected graph with state and transition probabilities.

Fig. 5. Graphical representation of a CRF, as a factor graph (left), and asan undirected graph (right).

The potential functions ψc can be split into edge potentialsψij and node potentials ψi as follows:

p(y|x) =1

Z(x)

∏(i,j)∈C

ψij(yi, yj , xi, xj)N∏

i=1

ψi(yi, xi) (6)

where the node potentials are

ψi(yi, xi) = exp{∑L

(λLi xi)yL

i } (7)

and the edge potentials are

ψij(yi, yj , xi, xj) = exp{∑L

(λijxixj)yLi y

Lj } (8)

Learning in a Conditional Random Field is performedby estimating the node weights λi = {λ1

i , . . . , λLi } and

the edge weights λij = {λ1ij , . . . , λ

Lij}. The estimation is

done by maximizing the log-likelihood of p(y|x) [19]. Oneway for solving this nonlinear optimization problem is byapplying the Broyden-Fletcher-Goldfarb-Shannon (BFGS)method. Specially Limited-memory BFGS gains fast resultsby approximating the inverse Hessian Matrix.

The given problem of labeling neighbored laser-scannedpoints motivates the exploitation of the graphical structure ofCRFs. In our work, each conditioned node represents a laser-scanned point and each observation node a calculated feature.Figure 6 shows a simplified version of our model. The outputnodes are connected with their observation nodes and theobservation nodes of their surrounding neighbors, and eachobservation node is connected to its adjacent observationnodes.

y1

y2

x11

x12

x13

x21

x22

x23

CalculatedFeatures

Labeled laser points

Fig. 6. Simplified version of our Conditional Random Field model

VI. DISCUSSIONS AND EXPERIMENTAL RESULTS

To validate our framework, we have performed severalexperiments of point cloud classification for real worlddatasets representing table scenes. The 3D point clouds wereacquired using a SICK LMS400 laser scanner, mounted ona robot arm and sweeped in front of the table (see Figure 7).The average acquisition time was ≈1 s per complete 3D pointcloud, and each dataset comprises around 70000 − 80000points. In total we acquired 10 such datasets with differentobjects and different arrangements (see Table II).

Fig. 7. Left: the robot used in our experiments; right: an example of oneof the scanned table scenes.

Since we are only interested in classifying the points be-longing to the objects on the table, we prefilter the raw pointclouds by segmenting the table surface and retaining only theindividual clusters lying on it. To do this, we assumed that theobjects of interest are lying on the largest (biggest number ofinliers) horizontal planar component in front of the robot, andwe used a robust estimator to fit a plane model to it. Oncethe model was obtained, our method estimates the boundariesof the table and segments all object clusters supported by it(see Figure 8). Together with the table inliers, this constitutesthe input to our feature estimation and surface classificationexperiments, and accounts for approximatively 22000-23000points (≈ 30% of the original point cloud data).

Fig. 8. Left: raw point cloud dataset; right: the segmentation of the table(green) and the objects supported by it (random colors) from the rest of thedataset (black).

TABLE IFEATURE ESTIMATION, MODEL LEARNING, AND TESTING RESULTS FOR

THE DATASET IN FIGURE 2. THE METHOD INDICES REPRESENT THE

TYPE OF FEATURES USED: PFH (1) AND FPFH (2). ALL COMPUTATIONS

WERE PERFORMED ON A CORE2DUO @ 2 GHZ WITH 4 GB RAM.

Feature Model ModelMethod Estimation Training Testing Accuracy

(pts/s) (s) (pts/s)

CRF1 331.2 0.32 18926.17 89.58%SVM1 331.2 1.88 1807.11 90.13%CRF2 8173.4 0.091 76996.59 97.36%SVM2 8173.4 1.98 4704.90 89.67%

To learn a good model, we used object clusters from 3 ofthe data sets, and then tested the results on the rest. This is incontrast to our previous formulation in [1] where we trainedthe classifiers with synthetically noisified data. The trainingset was built by choosing primitive geometric surfaces out ofthe data sets, like corners, cylinders, edges, planes, or tori,both concave and convex. Our initial estimate is that theseprimitives account for over 95% data in scenes similar toours.

We used the training data to compute two different models,using Support Vector Machines and Conditional RandomFields. Since ground truth was unavailable at the data ac-quisition stage, we manually labeled two scenes to test theindividual accuracy of the two machine learning modelstrained. Additionally, on the same datasets, we estimatedand tested two extra SVM and CRF models based on PFHfeatures. Table I presents the computation time spent for boththe feature estimation, and model training and testing parts.

As shown, the FPFH algorithm easily outperforms thePFH algorithm in computation performance. It calculatespoint histograms over 20 times faster while still gainingan edge in CRF classification accuracy. The training errorcurves for the CRF models for both PFH and FPFH areshown in Figure 9. Since the error norm becomes very smallafter approximatively 50 iterations, we stopped the modeltraining there. Table II presents visual examples of someof the classified table scenes with both the SVM and CRFmodels.

0 10 20 30 40 50 600

5000

10000

15000

20000

25000

30000

35000

40000

Err

or

norm

Error norm and Learning Curve

0 10 20 30 40 50 60Training Iterations

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Acc

ura

cy

FPFHF Error NormFPFHF Test AccuracyPFH Error NormPFH Test Accuracy

Fig. 9. Test accuracy and training error curves for the two CRF models.

TABLE IIA SUBSET OF 4 DIFFERENT DATASETS USED TO TEST THE LEARNED MODELS. THE IMAGES (TOP TO BOTTOM) REPRESENT: GRAYSCALE INTENSITY

VALUES, CLASSIFICATION RESULTS OBTAINED WITH THE SVM MODEL, AND CLASSIFICATION RESULTS OBTAINED WITH THE CRF MODEL.

Scene 1 Scene 2 Scene 3 Scene 4

VII. CONCLUSIONS AND FUTURE WORK

In this paper we presented FPFH (Fast Point FeatureHistogram), a novel and discriminative 3D feature descriptor,useful for the problem of point classification with respectto the underlying geometric surface primitive the point islying on. By carefully defining 3D shape primitives, andmaking use of contextual information using probabilisticgraphical models such as CRFs, our framework can robustlysegment point cloud scenes into geometric surface classes.We validated our approach on multiple table setting datasetsacquired in indoor environments with a laser sensor. Thecomputational properties of our approach exhibit a favorableintegration for fast, real time 3D classification of laser data.

As future work, we plan to increase the robustness of theFPFH descriptor against noise to be able to use it for pointclassification in datasets acquired using stereo cameras orother sensors less precised than laser scanners.

Acknowledgments This work was supported by theCoTeSys (Cognition for Technical Systems) cluster of ex-cellence.

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