Think Globally, Cluster Locally:A Unified Framework for Range Segmentation

Gene Yu Michael Grossberg George Wolberg Ioannis Stamos���

Dept. of Computer ScienceCity College of New York / CUNY

New York, NY 10031

�Dept. of Computer ScienceHunter College / CUNYNew York, NY 10065

(a) acquisition (b) clustering (c) refinementFigure 1. Segmentation pipeline. (a) In this simulated LIDAR setup, the frustum represents a scannerprojecting a beam onto a 3D model. The beam strikes the nearest surface and measures the distance,rendered here in false color. (b) Similarity based on local plane fitting drives a hierarchical clusteringprocess. (c) Planar components are refined and merged using a variant of the � -means algorithm.

Abstract

Modern range scanners can capture the geometry oflarge urban scenes on an unprecedented scale. While thevolume of data is overwhelming, urban scenes can be ap-proximated well by parametric surfaces such as planes.Piecewise planar representation can reduce the size of thedata dramatically. Furthermore, it is ideal for renderingand other high-level applications. We present a segmen-tation algorithm that extracts a piecewise planar functionfrom a large range image. Many existing algorithms forlarge datasets apply planar criteria locally to achieve effi-cient segmentations. Our novel framework combines localand global approximants to guarantee truly planar compo-nents in the output. To demonstrate the effectiveness of ourapproach, we present an evaluation method for piecewiseplanar segmentation results based on the minimum descrip-tion length principle. We compare our method to regiongrowing on simulated and actual data. Finally, we presentresults on large scale range images acquired at New York’sGrand Central Terminal.

1. IntroductionMany papers in the range segmentation literature present

algorithms that have advantages when applied to smallmeshes or to simple closed objects. Sophisticated ap-proaches are possible when the number of points is limitedor the topology is known, but the large, complex datasetsproduced by modern range scanners favor greedy local ap-proaches such as region growing. The problem with suchmethods is that they do not produce a truly planar segmenta-tion. Locally planar criteria drive the segmentation process,but they cannot guarantee output components that are verifi-ably planar in a global sense. We present a novel frameworkfor enforcing planarity both locally and globally to producea segmentation composed of output components that can bereplaced by polygons with small error. Figure 1 illustratesour segmentation pipeline. First, a range image is acquiredby a scanner. A local clustering phase groups points to-gether if they are likely to belong to the same plane. Then,we refine the set of planes in a global setting using a vari-ant of the � -means algorithm. Our approach shows that a�email: � yu,grossberg,wolberg � @cs.ccny.cuny.edu,

istamos@hunter.cuny.edu

Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA

combination of simple techniques can be used to producehigh-fidelity segmentation results at a manageable cost.

We borrow liberally from the image segmentation litera-ture to help us manage the cost of our algorithm. Image seg-mentation and range segmentation, while closely related,are not interchangeable. Image segmentation methods typ-ically use the graph topology of the image grid. In cur-rent range segmentation work, such as moving least squaresand other point-based methods, the image topology is oftendisregarded. The usual range data processing pipeline pro-ceeds by performing registration and combining the scansfirst. Segmentation follows in unorganized point clouds.We feel strongly that segmentation should come first. Thus,we perform segmentation on individual range images andpostpone the integration of multiple scans.

Since our goal is to replace planar components withpolygonal approximants, we evaluate our results by mea-suring the fidelity of the output function to the original data.We develop a novel method for comparing piecewise pla-nar segmentations based on the minimum description lengthprinciple. By encoding our output for compression, wecan compare our results directly to the original dataset andto other segmentations by computing compression ratios.Tests on simulated range images show that a piecewise pla-nar representation can approach the ground truth segmenta-tion, while tests on actual range images show a correspond-ing empirical improvement over region growing.

2. Related Work

Region growing is commonly applied to large rangedatasets because it is fast and easy to implement [2, 3, 9].Several methods to augment basic region growing useglobal techniques in a similar spirit to our work. Gotardo etal. [8] employ robust statistics to extract seed regions, whileBab-Hadiashar and Gheissari [1] group inliers by analyz-ing residuals from parametric fitting. We also perform sta-tistical clustering, but instead of shrinking clusters to seedregions, we use the clusters directly, similar to the graph-based method of Felzenszwalb and Huttenlocher [7]. Theirgreedy similarity measure is based on minimum spanningtrees, while our iterative refinement, though not as fast,gives us greater control over the output.

Our refinement phase, like all iterative methods, con-verges more quickly the closer the initial guess is to the finalresult. Therefore, we use an anisotropic filter to improve thelocal plane fit that drives the clustering process. The bilat-eral filter introduced by Tomasi and Manduchi [13] com-bines weighting functions in orthogonal domains to createan edge-aware convolution kernel. Choudhury and Tum-blin [4] use geometric information in the form of piecewiselinear approximation to improve the bilateral filter near edgediscontinuities. In our purely geometric context, we apply

piecewise linear approximation to range images using a lo-cal plane fit based on moving least squares [12].

After clustering the points into connected components,we refine the components using a variant of the � -means al-gorithm. A form of � -means iteration was used by Cohen-Steiner et al. [5] to simplify piecewise planar manifolds.Because the topology is known, they are able to choose thenumber of components � arbitrarily. Our refinement phaseexpands the applicability of such an approach to the topo-logically complex surfaces present in raw range images byfixing � automatically during the clustering process.

To evaluate the final result, we measure the space sav-ings and the fidelity of the segmentation using a minimumdescription length (MDL) criterion. Darrell et al. [6] usedMDL locally to allow neighboring components to competefor boundary points. Our method also allows competitionbetween neighboring components, but using global planefitting directly ensures that a point belongs to a given com-ponent only if it fits the output representation. Once weknow that the output truly matches the desired type, we ap-ply MDL globally to evaluate the final representation.

3. Segmentation AlgorithmOur algorithm proceeds in two distinct phases. The first

is based on local surface fitting, while the second is based onglobal surface fitting. A block diagram illustrating our ap-proach is presented in figure 2. By allowing local and global

Are the pointscoplanar?

Clustering

Compute thesimilarity measure

Cut the graph by thresholding edges

no

yes

initial labeling

input image

Convergence?

Refinement

Relabel the pointsby plane projection

Refit planes to thenew components

no

yes

final labeling

weightedgraph

connectedcomponent

recurse

intermediatelabeling

new planeparameters

iterate

Figure 2. Block diagram. On the left, graph-based clustering based on local surface fit-ting generates a hierarchy of connected com-ponents. Then, on the right, we traverse thehierarchy bottom-up, refining the subtree ateach internal node by fitting global surfacesto the individual components.

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Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA

notions of planarity to communicate, we expand the rangeof evidence available at each point to determine its member-ship in a component. Using only local relations, boundariescannot be resolved cleanly due to ambiguities at corners anddepth discontinuities. In addition, local variations in scaleand sampling rate make suitable thresholds difficult to de-termine, since no single set of parameters can be optimal inall areas of the image. By contrast, purely global methodsrequire additional information beyond the input data points,such as the number of components we expect to find. Wemust bootstrap global methods by aggregating local surfacefeatures or by applying some form of domain knowledge.Our method is motivated by the observation that local meth-ods can achieve a partial solution to the planar segmentationproblem while providing exactly the information needed toinitiate a global process of refinement.

The purpose of the first phase of our algorithm is tobuild a hierarchical subdivision of the input image suchthat the leaf nodes of the resulting tree represent planarconnected components. The second phase then traversesthe tree bottom-up, refining the plane parameters for eachcomponent and merging components where possible. Af-terward, the set of discovered planes is removed from thedataset, and the process is repeated on the remaining points.Since the distribution of similarity weights changes at eachiteration, repeated passes target surfaces with different errorcharacteristics. Thus, we can distinguish planes at differentdepths and sampling rates, and made of different materials.������ ������������������

Computing the similarity measure. Local surfacefitting in actual range data is typically performed usingsmoothing operators to combat noise. We use a form ofmoving least squares (MLS) approximation to fit planes insmall neighborhoods around each point. Because MLS ap-proximants are weighted operators that decay with distance,we can achieve a degree of anisotropy that improves the ini-tial clustering over simpler isotropic convolution operators.In our formulation, a local frame is represented by a tuple� �"! #%$'&)(

with origin#

and normal vector&

. MLS pro-jection maps any point

#within a radius * of the input data

to a point#

on the MLS surface (fig. 3a). To determine theradius * , we choose a window size + and use the points ina +-,.+ neighborhood in the image grid. This choice hastwo practical advantages:

1. There is no need to build either explicit connectivity orsearch data structures such as � d-trees.

2. Neighborhoods scale automatically to the samplingrate of the scanner.

The global threshold + is more robust than choosing * di-rectly would be, because the derived radius scales grace-fully to adjust for the spread of the scanner beam at different

// 0132

(a) MLS projection.

4�5 4�6

7 5

7 6

8:9 4 5<; 7 6>= 8:9 4?6 ; 7 5 =

(b) Point-to-point distance.

Figure 3. Similarity. (a) Project#

to the lo-cal frame

�@�A! #$B&C(induced by the neigh-

borhood of radius * around#

. The origin is#, and the normal is

&. (b) Distance D�EGF be-

tween# E and

# F is the sum of point-to-planedistances D !H# E $B� F ( and D !<# F $I� E ( . Similarity isthe probability, given D EGF , that

# E and# F be-

long to the same plane.

depths. It is only necessary to determine the general level ofnoise in the image, which is related to the error character-istics of the scanner. For the Leica HDS 2500 [11] scannerused in our study, a small JK,LJ window is generally satis-factory, with MN,OM or P�,�P windows being useful sometimesif the surfaces in the image are bumpy or if they are madeof absorptive materials that partially degrade the signal.

The local frame at#

is determined by minimizing theweighted least squares functional

Q �SRNTURVEXWZY[ !H# E]\ #�(%^_&�`_acbd!Bef# E�\ #Neg(f$ (1)

whereb

is the weighting function. We defineb

using thetail probability of the normal distribution

bh!<ij(N�lk \nm !Hij(f$ (2)

where m !<ij(o�p!HqZr s�t�(fu Y]v)wu]x y_z|{ [ \ !<} \�~ ( a:� !�s�q a (�` D }is the cumulative distribution function with mean ~ andstandard deviation

q. The tail probability is the probability

that a normally distributed random variable exceedsi. We

prefer the tail probability to simpler weighting functions be-cause it never decays so rapidly that all of the weights van-ish. Parameters ~ and

bare determined by fitting a normal

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Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA

distribution to the distance valuesef# E�\ #Ne . We minimize

the MLS functional (eq. 1) at#

by computing the singularvalue decomposition (SVD) of the weighted covariance ma-trix of the points

# E in the neighborhood of#

, centered attheir mean position

#. The SVD produces the matrix �.�

of right eigenvectors containing the normal vector&

cor-responding to the smallest eigenvalue. In addition, since� � encodes the rotation from world coordinates to the lo-cal frame, it is convenient to compute the perpendicular dis-tance from a point

#to a plane

�as

D !<#%$B��(�� � � !H# \ #Z(C^�&N� (3)

To compute the distance between two points# E and

# F(fig. 3b), we sum two perpendicular projections

D�E�F ��� D !<# E $B� F (g�_��� D !<# F $I� E (_��� (4)

Cut the graph by thresholding edges. The search forplanar components begins by representing the image as aweighted graph, then removing edges from the graph byhistogram thresholding. We compute point-to-point dis-tance (eq. 4) for each edge between a pair of � -connectedvertices in the image. Then we generate similarity weightsby fitting a normal distribution to the distance values andscoring edges using the tail probability (eq. 2). The chang-ing distribution of weights makes it possible for repeatedapplication of the clustering method to find different setsof planes. By contrast, region growing would produce thesame components on each pass no matter how many pointswere removed from the dataset, unless some mechanism forautomatically adjusting the parameters were devised.

By generating a histogram, it is easy to determine a suit-able threshold automatically by the midpoint method. First,we initialize the threshold to the mean weight. Then, wepartition the weights into two groups above and below thethreshold, and compute the mean of each group. The meanof the two means becomes the new threshold, and the pro-cess iterates until convergence. Implicitly, we are positingthat the distribution is bimodal. One mode corresponds toall edges connecting coplanar points, while the other modecorresponds to all edges that cross discontinuities.

Once the threshold is chosen, removing edges withweight below the threshold cuts the graph into connectedcomponents. Then, we filter the components, searching forplanar pieces. A component is rejected immediately if ithas fewer points than some minimum number � . Other-wise, we treat each component as a candidate plane by fit-ting plane parameters to its constituent points. To estimatethe parameters, we construct an unweighted covariance ma-trix in homogeneous coordinates, and solve for the eigen-vector corresponding to the smallest eigenvalue using SVD.The eigenvector is a plane equation in point-normal form� ��!�&N$���(

, where&

is the surface normal and�

is the per-pendicular distance to the origin. The distance between a

Figure 4. Three levels of hierarchical clus-tering from top to bottom. Left: Initially,all points are nonplanar (red) and connectedby weighted edges. Removing low-weightedges cuts the image into components. Pla-nar components (white) are removed, whilethe remaining components are processed re-cursively. Right: hierarchical subdivision ofregions shown as a tree structure.

point#

in space and a global plane�

is� !<#%$ � (���&.^_#��.���

(5)

If the standard error of the residuals is smaller than a pre-determined tolerance � , then the candidate is planar. Werecurse on the nonplanar candidates, creating a hierarchyof nested components with the planar components stored inthe leaves (fig. 4). The global threshold � limits the depthof the component hierarchy. It can be determined by cur-sory inspection of the data, such that � corresponds to thesmallest planar component desired. The threshold � can beset according to the manufacturer’s error estimate for thescanner used to create the image. For the HDS 2500, weused � ��� mm for all of the examples shown, based on thepositional accuracy provided by Leica.��H�%���n�h�����������C�

After constructing the hierarchy, refinement proceedsbottom-up by the � -means method using objective function

� �  ¢¡X£¤'¥h¦�§B¨¦ª©�«¬VEXWCYV­¯® ¥�¦

� !<#%$ � E (c$ (6)

where the °�E are disjoint components. The children of eachnode of the component hierarchy partition its points into

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Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA

connected subspaces, each containing a set of plane param-eters that are fitted to the constituent points. By alternatelyrelabeling the points, then updating the parameters, we it-eratively converge to the largest and most accurate planarcomponents we can find.

Relabel the points by plane projection. Duringbottom-up traversal, we update membership in the currentnode by projecting each point to the set of child planes andallowing the point to join the closest child. Most points re-join the same child, but adjacent components compete alongtheir boundary, which makes our algorithm particulary ef-fective at corners. Planarity is enforced using a threshold�U± �² ¢¡ª£�! � $B³�q E ( where

q E is the standard error of the ´ thcomponent. Using a multiple of the standard error tends toclip peaks, which we repair in a post-processing step by di-lating components subject the threshold �h± ± �µ!<¶·��k:(>q Ewhere

¶is the number of neighbors that belong to compo-

nent ´ . This constraint fills small holes liberally, but restrictsgrowth along the boundaries.

Refit planes to the new components. After relabel-ing the points, we recompute the plane parameters for eachchild to reflect the change in membership, repeating the pro-cess until the membership stabilizes. Then, we remove theparent, promote the child segments one level, and continuetraversing upward to the root. As we travel up the tree, wefit our candidate planes to larger and larger sets of points,and our confidence in the accuracy of the fit grows.

Overfitting occurs when adjacent, coplanar componentsare created in different branches of the hierarchy. Redun-dant planes are also removed during post-processing by test-ing the fit of each plane to the points in adjacent compo-nents. We use global point-to-plane distance (5) to projectthe points from one component into the plane of the other,merging components if the error distributions agree.

��<�%�� ¹¸)��º��»�h¼N�H��½�¾¿�NÀZ�H½�Á���In the best case, our algorithm is a typical recursive

divide-and-conquer algorithm, which runs in  !H¶OÃXĶ�(time. Unfortunately, we usually cannot label all of thepoints on the first pass. Consider two planes meeting ata corner. Local neighborhoods degrade in quality as theyapproach the corner because outliers cross the edge discon-tinuity. However, if one of the surfaces can be extracted, itspoints no longer pollute the distribution of the other. There-fore, repeated passes often reveal finer levels of detail.

In the worst case, one component containing the mini-mum number of points is extracted on each pass, and totalthe running time is Å !<¶ a ( . To investigate the average case,we segmented

s J actual range scans from Grand CentralTerminal at their full size of P�PÁP., PÁP�P pixels. We alsotiled the images into

k , s , s , s , ³ , ³ , �K,Æ� , J¢,.J , andÇ , Ç regions. Results appear in figure 5. For tiled images,we recorded only the worst running time encountered. Error

0.01

0.1

1

10

100

1000

10 100 1000

runn

ing

time

(min

utes

)

thousands of points

Figure 5. Timing. Log-log plot of number ofpoints vs. mean running time for 25 scans ona 3GHz Intel Xeon 5160 processor with + � 5,� � 6mm and � ranging from 400 points atfull size down to 40 for the smallest tile. Errorbars indicate minimum and maximum times.The best-fitting line has a slope of 1.3.

bars indicate minimum and maximum times—the spread re-flects dependence on image content. A straight line throughthe mean values in the log-log graph suggests an averagecase power law relationship with a slope of

k�� ³. The aver-

age case running time appears to fall closer to the best case.Our algorithm is implemented in C++ on a Linux sys-

tem with data structures and procedures from the StandardTemplate Library and Boost. For linear algebra, we use thehardware-optimized ATLAS library integrated with CLA-PACK. We compute the right tail probability (eq. 2) usingthe GNU Scientific Library.

Finding the major planes ins J range images of Grand

Central Terminal took an average of� P minutes on a 3GHz

Intel Xeon 5160 processor. The complex example in fig-ure 6 took three hours to find major planes covering

�ÁÈ%

in the image (see inset). On subsequent runs, we increasedthe window size to approximate the curved ceiling, and wedecreased the window size to capture fragments formed bypartial occlusion. A large � �µs cm captured the fenes-trated panels at either end of the Main Concourse. Since� affects the depth of the component hierarchy, it can havea great impact on running time. Nevertheless, the GrandCentral Terminal dataset is highly complex, so we included� � J È , which took over M hours to run on average. Evenwith such a small number of points per component, how-ever, our method was able to avoid nonplanar areas such asthe hanging lamps in figure 6 robustly.

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Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA

Figure 6. Segmentation results for a 999 x 999range image of Grand Central Terminal.Black denotes holes where the scanner failedto return a value. Red denotes points clas-sified as nonplanar by our algorithm. Eightother colors are assigned randomly to differ-entiate the planar regions. The inset showsthe major planes found in the first of fourruns ( + � 5, � � 6mm, and � � 400).

4. Experimental Results

Hoover et al. [9] first compared planar segmentations us-ing a collection of heuristic counts. More recently, Jiang etal. [10] examined statistical distance measures, includingan information-theoretic distance derived from mutual in-formation. In the same vein, we designed a practicalinformation-theoretic evaluation procedure based on MDL,using the byte coder bzip2 as a proxy for entropy. We pa-rameterize each component in a coordinate frame that min-imizes the variance in É . If the constituent points are trulycoplanar, the É -coordinates of all such points will be closeto zero in parameter space. After quantization, the greaterproportion of zero bytes in the data stream leads to a bettercompression ratio. For each component,

1. Fit a least squares plane.

2. Transform each point to the local coordinate frame.

3. Quantize the transformed values.

After transforming each component, we write the quantizedvalues to a flat file, and compress the file using bzip2.

2.2

2.4

2.6

2.8

3

3.2

0 3 6 9 12 15

com

pres

sion

ratio

noise (mm)

no labelingground truth

region growingour method

Figure 7. Evaluation. Minimum descriptionlength comparison between region growingand our method on simulated LIDAR imageswith fixed amounts of white noise added.

(a) region growing (b) our method

Figure 8. Failure modes. Region growingtends to underfit, while our method is morelikely to overfit. (a) One component spanstwo faces of the icosahedron. (b) One faceof the cube is described by two components.

We compared our results to the region growing methodof Chen and Stamos [3]. Figure 7 shows the results of di-rect comparison on simulated LIDAR images. Since thesimulated examples are constructed to match our assump-tions about the surfaces in a range image, our algorithmpredictably approaches the ground truth compression ratio.Cases in which our method seem to beat ground truth cor-respond to overfitting. Region growing, on the other hand,consistently undersegments. A comparison of failure modesis presented in figure 8.

Results on actual data are more difficult to interpret, be-cause region-growing treats smoothly curving surfaces suchas the ceiling of the Main Concourse as one connected com-ponent, while our algorithm cuts curved surfaces into planarpieces. Nevertheless, our results were consistent with the

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Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA

(a) region growing (b) our method

Figure 9. Actual data. 3D point renderingof two detail comparisons between regiongrowing and our method on range imagesfrom Grand Central Terminal. Our methodproduces tight corners, and it correctly dis-tinguishes parallel planes that are separatedby small depth discontinuities.

simulated results ink:s

trials on actual data. On simulateddata with

�mm of noise, our method achieved an average

compression ratio ofsU� M k vs.

sU� �Á�for region-growing. On

actual data, our method achieved a ratio ofs|� M s vs.

s|� �¯M forregion-growing. Two examples from Grand Central Termi-nal are rendered using 3D points in figure 9. These areascontain both planes and sculptural detail. Our method ex-tracts planes with great precision, especially in corners, andit is able to distinguish difficult cases such as parallel planesthat are separated by small depth discontinuities. Additionalexamples on full-size scans are presented in figure 10.

5. ConclusionThe main contribution of our work is a unified frame-

work for range segmentation utilizing local and globalmethods driven by a single model. In our novel framework,a simple hypothesis test negotiates with a simple iterativerefinement step to narrow down a huge range of possiblescales, orientations, and noise levels.

Many previous segmentation methods are not directlycomparable to our algorithm because they do not yield aplanar segmentation. They do not distinguish between flatand curved surfaces, they group non-coplanar regions, andthey perform poorly on edges. The remedy we propose is

to evaluate the output components dynamically. By usinga single model both to select components locally, and toevaluate those components globally, we can resolve highlyambiguous configurations of points. It is this unification oflocal and global methods around a single model that allowsus to produce a truly planar segmentation.

AcknowledgementsThe authors thank Cecilia Chao Chen for generating the

region growing results, and Irina Gladkova for suggestingthe MDL criterion. This work was supported in part bygrant DOE DE-FG52-06NA27503.

References[1] A. Bab-Hadiashar and N. Gheissari. Range image segmen-

tation using surface selection criterion. IEEE Trans. ImageProcess., 15(7):2006–2018, 2006.

[2] P. J. Besl and R. C. Jain. Segmentation through variable-order surface fitting. IEEE Trans. Pattern Anal. Mach. In-tell., 10(2):167–192, 1988.

[3] C. Chen and I. Stamos. Range image segmentation for mod-eling and object detection in urban scenes. In Intl. Conf.3-D Digital Imaging and Modeling (3DIM), pages 185–192,Montreal, Canada, 2007.

[4] P. Choudhury and J. Tumblin. The trilateral filter for highcontrast images and meshes. In Proc. Eurographics Symp.Rendering, pages 186–196, Leuven, Belgium, 2003.

[5] D. Cohen-Steiner, P. Alliez, and M. Desbrun. Variationalshape approximation. In Proc. SIGGRAPH, pages 905–914,Los Angeles, CA, 2004.

[6] T. Darrell, S. Sclaroff, and A. Pentland. Segmentation byminimal description. In Proc. Intl. Conf. Computer Vision(ICCV), pages 112–116, Osaka, Japan, 1990.

[7] P. Felzenszwalb and D. Huttenlocher. Efficient graph-basedimage segmentation. Intl. J. Computer Vision, 59(2):167–181, 2004.

[8] P. F. U. Gotardo, K. L. Boyer, O. R. P. Bellon, and L. Silva.Robust extraction of planar and quadric surfaces from rangeimages. In Proc. Intl. Conf. Pattern Recognition (ICPR),pages 216–219, Cambridge, England, 2004.

[9] A. Hoover, G. Jean-Baptiste, X. Jiang, P. J. Flynn, H. Bunke,D. B. Goldgof, K. Bowyer, D. W. Eggert, A. Fitzgibbon, andR. B. Fisher. An experimental comparison of range imagesegmentation algorithms. IEEE Trans. Pattern Anal. Mach.Intell., 18(7):673–689, 1996.

[10] X. Jiang, C. Marti, C. Irniger, and H. Bunke. Distance mea-sures for image segmentation evaluation. EURASIP J. Ap-plied Signal Processing, 2006:1–10, 2006.

[11] Leica Geosystems. http://www.leica-geosystems.com.[12] D. Levin. Mesh-independent surface interpolation. In

G. Brunnett, B. Hamann, and K. Mueller, editors, Geo-metric Modeling for Scientific Visualization, pages 37–49.Springer-Verlag, Berlin, 2003.

[13] C. Tomasi and R. Manduchi. Bilateral filtering for gray andcolor images. In Proc. Intl. Conf. Computer Vision (ICCV),pages 839–846, 1998.

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DAV

IDIL

IFF

(a) Photograph of the Grand Concourse. (b) Southwest corner.

(d) East end / Entrance to 42nd St. passage. (c) South side / Ticket booths.

Figure 10. Segmentation results for 999 x 999 range images taken inside Grand Central Terminal.Black denotes holes where the scanner failed to return a value. Red denotes points classified asnonplanar by our algorithm. Eight other colors are assigned randomly to the planar components todifferentiate the regions. (a) A photograph of the Grand Concourse, facing east. (b) Planar areasyield high fidelity components with sharp corners. Our method can also handle noisy componentswith complex topology, such as the large fenestrated panels. (c) Piecewise planar approximationof smooth curved surfaces such as the vaulted ceiling are easy to generate. (d) Highly nonplanarareas such as ticket windows that cannot be approximated well by piecewise planar components arecorrectly identified and excluded.

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Proceedings of 3DPVT'08 - the Fourth International Symposium on 3D Data Processing, Visualization and Transmission

June 18 - 20, 2008, Georgia Institute of Technology, Atlanta, GA, USA