(2007) robust smooth feature extraction from point clouds

8/6/2019 (2007) Robust Smooth Feature Extraction From Point Clouds

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Figure 1. Overview of our feature extraction pipeline; from left to right: original point cloud; candidatepoints identified to be close to possible features; candidate points projected onto the intersection ofRMLS surfaces; smoothed projected points; reconstructed polylines that identify the sharp features (withnormal vectors for the defining surfaces).

loops, allows us to extract surface patches that are en-

closed by the polylines. Furthermore, we show thatthis information can be used to improve previously de-veloped techniques that operate on point-sampled ge-ometry, e.g., surface meshing and compression.

In Section 2, we review related work in the field of fea-ture line extraction on mesh- and point-based models.Section 3 describes our feature extraction pipeline. InSection 4, we apply our results to improve surface re-construction and the compression of point-based mod-els. Section 5 presents experimental results, and weconclude the paper with final remarks and future workin Section 6.

2. Related Work

2.1 Mesh-Based Feature Extraction

Multiple techniques have investigated the identificationof feature edges on polygonal models. Hubeli et al. [11]create a multi-resolution framework and normal-basedclassification operators to define a set of edges on whicha thinning process extracts feature lines. Watanabeet al. [19] use discrete differential geometry operatorsto compute approximations of the mean and Gaus-

sian curvatures. Whereas, Hildebrandt et al. [10] useanisotropic filtering on discrete differential geometricapproximations of third order derivatives of the sur-face mesh. Both techniques [10, 19] build a set of fea-ture edges from the extrema triangles. Attene et al. [4]identify chamfer triangles to reconstruct sharp featureson meshes. They insert new vertices on the edges andfaces of these triangles and project the points to theintersection of planes fit to the surrounding surfaces.

The underlying assumption of connectivity and nor-

mals associated with the vertices of the mesh is notavailable for point-based models. In order to extractfeature lines from point clouds using these techniques, aconnectivity construction method (surface reconstruc-tion) must be applied in a preprocessing step. Theconstruction of connectivity is non-trivial, computa-tionally expensive, and moreover, the success of featureextraction relies on the ability of the polygonal meshingprocedure to accurately build the sharp edges.

2.2 Point-Based Feature Extraction

Feature line extraction from point-based models is not

straightforward in the absence of connectivity and nor-mal information. Pauly et al. [16] use covariance anal-ysis of the distance-driven local neighborhoods to flagpotential feature points. By varying the radius of theneighborhoods, they develop a multi-resolution schemecapable of processing noisy input data. Gumhold etal. [9] construct a Riemann graph over local neighbor-hoods and use covariance analysis to compute weightsthat flag points as potential creases, boundaries, or cor-ners. Both techniques [9,16] connect the flagged pointsusing a minimum spanning tree and fit curves to ap-proximate sharp edges. Demarsin et al. [5] computepoint normals using principal component analysis andsegment the points into groups based on the normalvariation in local neighborhoods. A minimum span-ning tree is constructed between the boundary pointsof the assorted clusters, which is used to build the finalfeature curves.

These techniques are capable of extracting features onpoint clouds by connecting existing points; however,their accuracy depends on the sampling quality of the


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Figure 3. The residual r(p) is computed as the max-imum distance between the points in the neighbor-hood around p and the polynomial fit f, whereasthe distances are considered in direction of theplane normal n.

dependently for all points, which may lead to situ-ations where nearby points do not grow to identicalsub-neighborhoods, resulting in jagged edges. Figures2a and 2b illustrate the occurrence of jagged edges,created by the RMLS projection. Figures 2c and 2d

show two different sub-neighborhood constructions forone set of points that leads to different configurationsof the feature region.

To overcome this problem, we propose to apply asmoothing procedure to the RMLS projected points.The goal of the following five-step algorithm is to ex-tract smooth and complete curves that approximatethe features well:

(1) Extract points from P that are near potential fea-tures,

(2) Use RMLS to fit multiple surfaces to the neighbor-

hoods of these points and project each point to itsnearest intersection between the surfaces,

(3) Smooth the projected points using covarianceanalysis of adaptively grown neighborhoods,

(4) Create an initial set of feature lines by grow-ing polylines that approximate the smoothed pro-

jected points,

(5) Analyze the end points of the feature polylines tocomplete gaps and connect to corner points.

The following subsections discuss the algorithmic de-tails of each of the five steps.

3.2. Identifying Potential Edge Regions

To identify potential edge regions, we adopt themethod of Fleishman et al. [7]. Specifically, given apoint p P, they consider the neighborhood N(p) Pthat is used for the polynomial fit of the MLS pro-

jection. Considering the polynomial f over the refer-ence plane H that is defined by point q and normal n,

(a) (b)

Figure 4. Identification of potential feature pointsfor the Fandisk model; the threshold is used toadjust the number of points retained near poten-tial features; the automatically computed thresh-old = 0.057 (a) and a manually adjusted threshold = 0.014 (b).

they evaluate the maximum polynomial residual r in aneighborhood around p,

r(p) = maxxN(p)

x xf,

where xf corresponds to the projection of x onto f indirection of the plane normal n, see Fig. 3.

For sharp features, r becomes reasonably large becausemany point sampled along the sharp edge lift awayfrom the polynomial. Based on this observation, theystraightforwardly define any point p P to be a po-tential feature point, if r(p) > for some user definedthreshold . Note that strong outliers will also recordlarge r; however, they are eliminated during the RMLS

projection stage.

This approach relies on an appropriate selection of thethreshold . We extend the pipeline in [7] by usingan automatic computation method, which produces anadaptive threshold. In particular, we found that setting to the mean residual value taken over all points in P, = 1|P|

pP r(p), leads to satisfying results. More-

over, we allow the user to modify manually, using aninteractive tool. Figure 4 shows the Fandisk model af-ter identifying potential feature points using differentvalues for the threshold . The output of the identi-fication procedure is a set of potential feature points,

i.e. F = {p P | r(p) > }.

3.3. Projecting Points to Edges

We use the RMLS procedure to project the pointsp F towards the features. The output of the pro-cedure is the definition of a number of surfaces fit tothe points in the neighborhood around p. This numberdescribes the feature types in the region.


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If RMLS returns a single surface, then this implies thatno feature was detected and the high residual was pro-duced by outliers. Two surfaces indicate the presenceof a sharp feature. We now use Newtons method, asin [16], to project p onto the intersection of the two sur-faces closest to p, yielding an edge point e. Note that

e does not correspond to the RMLS projected point,it is rather the result of the Newton projection seededat the RMLS projection of p. If three or more surfacesare found, then there are multiple sharp features and acorner point c is produced in addition to an edge pointe. At this stage, the output of the projection procedureis a feature point cloud F(p) that we compose of edgepoints E(p) and corner points C(p), F(p) = E(p) C(p).

The RMLS projection stage results in multiple cornerpoints projected to each region formed by three or moreintersecting surfaces. This is due to the occurrence of

jagged edges as described in section 3.1. The goal is to

group close corner points to retrieve a single point iden-tifying the feature corner. We achieve this by weightingthe corner points to prioritize their importance duringthe projection stage.

Specifically, for each corner point ci C(p) that wasprojected from pi F(p), we consider the correspond-ing projection to the two closest surfaces, ei E(p).We now assign a weight wi to the point ci being in-versely proportional to the distance between ci and ei,wi = 1/ci ei. The figure shows two close cornerpoints ci and cj with different weights. We now group

the corner points C(p)k C

(p) by considering neigh-

borhoods of a predefined feature radius and selectc C(p)k that is the clos-est to the weighted averageof the corner points in thegroup. All other points in

C(p)k are defined to be edgepoints and are moved fromC(p) to E(p).

3.4. Smoothing the Feature Points

The output of the projection procedure is a set of points

F

(p)

that are expected to identify the sharp features.However, the points in F(p) are affected by noise, sincethey were obtained using the RMLS operator, whichtends to produce jagged edges. Inspired by [13], weimplement a smoothing filter based on the covarianceanalysis of points in local neighborhoods.

Specifically, given p F(p), we consider the set of kfeature points in the -neighborhood of p identified byN(p) = {pi F(p) | pi p }, i = 1, . . . , k. We

(a)

(b)

Figure 5. Feature cloud smoothing using differentneighborhood sizes .

now compute the 3 3 covariance matrix

C = (p1 p , . . . , pk p) (p1 p , . . . , pk p)T,

where p is the mean, p = 1kk

i=1 pi. We solve theeigensystem Cxi = ixi, i {0, 1, 2} and assume(without loss of generality) that 0 1 2. Theeigenvector x0 that corresponds to the largest eigen-value 0 is expected to be aligned with the feature linein the neighborhood N(p). We achieve a smoothing ef-fect by projecting feature points p onto the line definedby the mean p and the eigenvector x0, yielding p.

The overall smoothing performance is determined bythe neighborhood size , see Fig. 5. In order to ob-

tain stable smoothing in the presence of noise and highcurvature, we adopt the adaptive neighborhood growthscheme in [13]. This method is based on a statisticalanalysis of point distributions on the plane.

In particular, given a point p, the algorithm fits a lo-cal plane to points pi in the neighborhood N(p) andprojects them onto the plane that is defined by thetwo eigenvectors, which correspond to the two largesteigenvalues. The resulting point set (in R2) is theninterpreted in terms of a distribution of two randomvariables, X and Y, which can be analyzed by statisti-cal methods. The cross-correlation coefficient ofX andY is defined by (X, Y) = cov(X,Y)

(X)(Y), where cov(X, Y) is

the covariance ofX and Y, and () is the standard de-viation. Large absolute values for indicate that thereis a linear relationship between X and Y, meaning thatthe points in N(p) are distributed along a feature line.

Given p, the adaptive neighborhood growth algorithmincreases the neighborhood N(p) by iteratively addingpoints closest to p from F(p) until a sufficiently largecorrelation coefficient is found. In practice we found


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(a)

(c) (b)

Figure 8. Feature completion requires an analysisof the endpoints of each feature curve to determineif it must fill a gap between multiple features (a),split a single feature into two thus forming a new

corner point (b), or be left alone unattached (c).

points of each end of the feature polyline. An alter-native approach is to evaluate the direction of the lastsegment of the feature curve; however, we found thatthe cubic fitting leads to better results.

For each end point p, we search for other feature pointsthat are within the cone formed by the tangent vectorat p and a predefined aperture angle. We search for thethree cases in Fig. 8: gap completion (a), corner cre-ation (b), and endpoint identification (c). The searchalgorithm finds the closest vertex of the feature poly-lines that exists within the volume of the search conethat is within the distance smax. The scalar shouldbe adjusted based on the sampling quality along thefeatures of the cloud. In practice we found that = 7accommodates well for the sampling of the models inthis paper.

Figure 8(a) shows an example for the case of gap com-pletion, which occurs due to poor sampling quality,i.e. the distance between two end points is larger thansmax. To resolve this problem, we merge any two poly-lines with endpoints that are within smax distanceto each other and corresponding tangent vectors thatpoint to opposing directions within the aperture angle.

Corner creation occurs if the end point of a polylineprojects to an interior point p of another feature line,Fig. 8(b). To prevent the production of two cornerpoints, we split the corresponding line segment, inserta new corner point at p, and reconnect the involvedline segments accordingly. This case resolves situa-tions where a single polyline missed a corner duringthe growing process.

The last case, Fig. 8c, occurs if a feature line dissipatesas detected on the Fandisk model. If no vertices offeature lines are found during the directional search,then we declare the end point to belong to a dissipatingfeature edge, and no connection operation is performed.

After the endpoints have been resolved the featurecurves are considered complete. Perturbations aresmoothed from the feature polylines by fixing the end-points and applying Laplacian smoothing on the inter-nal vertices weighted by inverse distances to the targetpoint. This process is limited to 1 or 2 passes to min-

imize shrinking on the features. Finally, we considerthe feature polylines to be complete and smooth.

4. Applications

The output of our method is a set of connected andsmoothed polylines that identify the sharp features ofa point-sampled surface. We are also able to extractsurface patches that are bounded by the feature lines.In this section, we will show the benefits of applying thefeature line information as input data for methods thatrely on point sets. We focus on surface reconstruction

and shape compression.We demonstrate the effectiveness of our feature ex-traction technique using point models that representmachine parts with sharp edges, such as the QuarterPiston model, the Ra model, the Scallop model, andthe well-known Fandisk model. We emphasize that alldata sets consist only of a set of 3D points. We do notassume associated normal vectors for the geometry.

4.1. Surface Reconstruction

Triangle meshes are a popular representational form

for models that are commonly used in geometry pro-cessing. The algorithms leverage additional informa-tion inherent in mesh models versus point set surfaces,mainly connectivity, to compute discrete differentialoperations. Often, the robustness and efficiency of thealgorithms are dependent on the quality of the inputmesh. This demand for high quality mesh structureshas motivated much of the development in surface con-struction and remeshing algorithms.

Schreiner et al. [18] extend the approach in [17] andproposed an advancing-front algorithm that produceshigh quality meshes from many different types of source

models, including point-set surfaces. Their approach isbased on defining a guidance field, based on local curva-ture information to determine the size of the triangles.Precomputing a specific guidance field allows the tri-angulation scheme to adapt to areas of high curvature,shrinking the allowable triangle size as the advancingfronts grow towards such regions.

The algorithm uses MLS for projection and curva-ture computations on point clouds. Consequently,


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Figure 9. Results obtained by applying the mesh-ing technique in [18] to Scallop (top), Quarter Pis-ton (middle), and Ra (bottom); the left half of eachcloseup view shows meshes obtained by the tra-ditional method; the corresponding right halvesshow the meshes after initializing the input of [18]with feature lines extracted with our method.

their method adaptively shrinks the triangle sizes nearsharp features and produces rounded edges. Extendingtheir projection and guidance field computations by us-ing RMLS enables reconstructions with sharp features;however, as previously discussed, this creates jaggededges and additional computational costs.

In response to this problem, we guide the method byincluding the extracted feature curves with evaluatednormals for the two meeting surfaces. Using these fea-ture lines as the initial polylines of the advancing front,

the remeshing system is able to generate a trianglemesh with sharp features without changing the MLSmethods. Consequently, the projection computationis not effected, while the algorithm automatically pro-duces models with smooth sharp features, Fig. 9.

Moreover, the feature lines aid the computation of abetter guidance field. Previously, sharp features com-pute large curvature values such that the guidance field

(a) (b)

Figure 10. The partition of Quarter Piston usingthe traditional patch composition method in [15](a); partition with initial segmentation (b).

scales the triangles to a very small size to capture anMLS smoothed region. By initializing the advancingfront to the feature edges, it is known that curvaturevalues computed here are misleading; therefore, largertriangles can be used to approximate the surface grownfrom the sharp edges. The mesh reconstructed usingthe feature polylines has a significantly reduced finaltriangle count, Fig. 9.

4.2. Surface Compression

Surface compression is an elementary problem in ge-ometry processing, the goal of which is to represent3D models compactly for the purpose of space efficientstorage and fast transmission. The task of the encoderis to transform a specific surface representation into acompact bit stream, which can be decoded at the re-

ceiver side in order to reobtain the original model oran approximation of it, if the encoding is lossy.

In [15] it was shown that a point model can be com-pressed by decomposing the input surface into a num-ber of patches, which are parameterized as height fieldsover planar domains and resampled on regular grids.The resulting images with arbitrary regions of supportare encoded using state-of-the-art wavelet compressionof shaped images. One key feature in their method isthe surface parameterization that is based on a split-merge partitioning procedure. In the first step, thesurface is recursively subdivided until all patches meet

some prescribed surface flatness constraint. In the con-sequent phase, adjacent patches are merged as long asthe resulting patches show height field properties.

Although their method yields a high rate-distortionperformance, the approximation quality depends heav-ily on the structure of the surface partition. Especiallyfor models with sharp features, the constructed par-titions show patches that contain the features, ratherthan identifying the features as patch boundaries. The


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0

1

2

3

0 0.5 1 1.5 2

RMS

bit rate [bpp]

traditionalguided

original (2 105 points) 0.08 bpp, Erms = 12.1 0.11 bpp, Erms = 4.53

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

RMS

bit rate [bpp]

traditionalguided

original (2 105 points) 0.26 bpp, Erms = 10.1 0.48 bpp, Erms = 2.45

Figure 11. Rate distortion performance for Fandisk (top row) and Quarter Piston (bottom row); the left

figure shows the original model, followed by two decodings for different rates; the rate-distortion curvesshow that the compression performance is significantly increased by applying the partitioning in [15]based on feature aware segmentation in this work (right); the error is the root-mean-square error in [15].

major drawback of this behavior is that sharp featureswithin patches correspond to high frequency compo-nents in the corresponding wavelet signal, which areconsequently hard to encode.

To overcome this problem, we propose to guide the par-titioning procedure in [15] by the sharp features that

were constructed with our method. In particular, theencoder receives the input model with attached pre-partition, which is extracted from the closed curves,see section 3.6. The splitting operations in the parti-tioning step are performed on each individual patch inthe pre-partition, while the merging is performed as inthe usual setting.

The advantage of this approach is that the featuresgive the encoder a first guess of the surface structureto build the patch layout, Fig. 10b. The individualpatches in the resulting partition show an improvedstructure as well as more regular boundary shapes.

In contrast, the traditional partitioning method doesnot recognize edges sufficiently, leading to patches thatwrap around sharp features, Fig. 10a.

Figure 11 shows compression results for Fandisk (toprow) and Quarter Piston (bottom row) at 2 105 pointseach. The left images show the original models followedby two decoded models at different bit rates. For theFandisk model we observe heavy compression artifacts

only at extremely low bit rates, e.g., 0.08 bits per point(bpp). Slightly increasing the bit rate leads to a signifi-cantly better geometric reconstruction quality. For theQuarter Piston model (Fig. 11 bottom row), however,we observe that more bits need are needed (0.26 bpp)to achieve an adequate reconstruction quality. We ex-plain this behavior by the fact that this model showsmore complex geometric properties that the Fandisk.Further increasing the bit rate, e.g., to 0.48 bpp, yieldsreconstructions that are close to the original model.

5. Experimental Results

We discuss the robustness of our method to noise andits run time efficiency. Figure 12 shows extracted fea-ture lines for the Quarter Piston model at differentnoise levels. We applied uniform noise to the origi-nal model by shifting each point p by a random vector(p) of restricted length, (p) l. The length l is thenoise level, which is expressed in units of the bound-ing box diagonal length dB of the original model. Theresults in Fig. 12 show that our method is robust tonoise and consequently extracts smooth feature lines.Only for high noise levels, we observe that the featureline completion step fails to connect all feature lines.

The total time required to construct the polylines isdetermined by the number of potential feature points


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(a) (b) (c) (d)

Figure 12. Feature extraction from the Quarter Piston model; original point cloud (a); features extractedfrom the original model (b); features extracted from the original model after applying uniform noise(p) 0.01dB (c), and (p) 0.05dB (d), where dB is the length of the model bounding box diagonal.

in data set, since only the feature point identificationstage operates on the complete point set. Table 1 showsrun times for all of the five stages in our algorithmfor different models at varying noise levels. We foundthat an increase in the noise level leads to a slight de-crease in computational costs, which can be explained

by the fact that the threshold becomes larger fornoisy models, which consequently results in a reducednumber of potential feature points. Carried out on aPentium4 2 GHz platform, our implementation needsfour to ten minutes to process models of half a millionpoints, where the major part of computational costs isrequired for the RMLS projection (stage 2).

The robust feature extraction produces smooth results,but can be limited by an extremely poor sampling qual-ity and by time constraints. High noise levels or sparsesampling worsens the performance of our method, sincethe distinction between sharp features and noisy re-gions becomes ambiguous. Because RMLS is computa-tionally expensive, the algorithm is run as a pre-processand its output improves the performance of other in-teractive methods.

Figure 13 illustrates the algorithm output on severaldifferent models with varying feature types. The re-finement and smoothing that occurs over the multi-ple stages removes perturbations, while extracting anaccurate representation of the sharp features. Addi-tionally, note that the extracted feature polylines arecoupled with normal vectors. We couple the projectedpoints with the RMLS evaluated normals during theprojection phase that identifies the edges. The feature

growing method groups normals of neighboring poly-line vertices to establish consistent vector orientation.

6 Conclusions and Future Work

We presented a method for extraction of feature curveson point-sampled surfaces. Our algorithm leverages therobust statistical methods of RMLS to project points

model noise |F| run time [seconds][102] (1) (2) (3) (4) (5) total

Quarter 0.0 0.5 66k 177 348 17 71 0.1 613Piston 0.01 1.2 62k 179 342 16 60 0.1 598

0.05 4.7 30k 182 365 10 12 1 570

Fandisk 0.0 0.7 39k 178 198 10 22 0.1 4070.01 1.1 37k 180 226 10 19 0.1 435

0.05 4.8 8k 186 83 2 0.6 2 274Ra 0.0 0.8 31k 176 164 7 12 0.0 359

0.01 1.4 28k 176 153 7 10 0.0 3470.05 4.7 11k 182 133 3 0.9 0.3 319

Table 1. Run time performance of our feature ex-traction implementation for models that contain500000 points each; the stages in our pipeline cor-respond to point identification (1), point projection(2), point smoothing (3), curve propagation (4), andcurve completion (5).

to all possible features such that they do not need tobe exactly sampled by the input point cloud. The out-put curves are described as complete and smooth andprove valuable as inputs to existing geometric process-ing applications for point clouds. We were able to sig-nificantly improve the performance of a previously pro-posed surface reconstruction and a point-compressionmethod without any modifications to their algorithms.

In future work, we aim at improving initial neighbor-hood selection methods to speed up the RMLS method,improving data segmentation for CAD purposes, andinvestigating smoothing and resampling methods tomaintain the sharp features.

Acknowledgments. This work was partially sup-

ported by NSF grants CCR-0310705, CCF-0528201, OISE-

0405402, CCF-0401498, EIA-0323604, IIS-0513692, and

OCE-0424602, NIH grant 573996, the Department of En-

ergy, an IBM Faculty Award, the Army Research Office,

and the DFG Graduiertenkolleg 1042 Explorative Analy-

sis and Visualization of Large Information Spaces. L. Ha is

supported by a Vietnam Education Foundation fellowship.


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Figure 13. Features extracted with our method; displaying the original point sets without normal informa-

tion does not show surface characteristics; our method identifies sharp features and can define orientedsurface patches without relying on input normal vectors.

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(2007) robust smooth feature extraction from point clouds

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