EuroCG 2014, Ein-Gedi, Israel, March 3–5, 2014

Robust Normal Estimation using Order-k Voronoi Covariance∗

Louis Cuel† Jacques-Olivier Lachaud‡ Quentin Merigot§ Boris Thibert¶

Abstract

We present a robust method to estimate normals, cur-vature directions and sharp features from an unorga-nized point cloud approximating an hypersurface in Rn.We define the k-Voronoi Covariance Measure of a pointcloud, which exploits the robust geometric informationcontained in order-k Voronoi cells. Our main theoremasserts that the k-VCM is resilient to noise and even tooutliers. Numerical experiments show that the k-VCMcan be used to reliably estimate normals, curvature di-rections and sharp features from point cloud approxi-mation of surfaces in R3.

1 Introduction

Differential quantities estimation, surface reconstruc-tion and sharp feature detection are motivated by alarge number of applications in computer graphics, ge-ometry processing or computational geometry. Theoutput of many 3D acquisition devices is a point cloudsampling of an unknown underlying shape. It is there-fore crucial to extract robust geometric informationsdirectly from the point cloud. Here, we address theproblem of robust estimation of normals, curvature, andsharp features in the presence of noise and even outliers.

Voronoi-based normal estimation Classical principalcomponent analysis methods try to estimate normalsby fitting a tangent plane. In contrast, Voronoi-basedmethods try to fit the normal cones to the underlyingshape, either geometrically [1] or more recently usingthe covariance of the Voronoi cells [2, 3]. In [2], the co-variance matrices of the Voronoi cells are defined withrespect to the center of mass of the cells. This approachhas been improved in [3] by changing the domain of in-tegration and the averaging process. The authors de-fine the Voronoi Covariance Measure (VCM) of anycompact sets, and show that this notion is stable un-der Hausdorff perturbation. Moreover, the VCM of asmooth surface encodes some differential quantities ofthis surfaces, such as its normals and curvatures. Withthe stability result, one can therefore use the VCM to

∗This research has been supported in part by the ANR grantsDigitalSnow ANR-11-BS02-009, KIDICO ANR-2010-BLAN-0205and TopData ANR-13-BS01-0008†Laboratoire Jean Kuntzman, Universite Joseph Fourier‡LAMA, Universite de Savoie§Laboratoire Jean Kuntzman, CNRS¶Laboratoire Jean Kuntzman, Universite Joseph Fourier

estimate differential quantites of a surface from a Haus-dorff approximation.

Distance to a measure Methods mentioned above arebased on the classical distance fonction. The distancefunction to a compact set is stable under Hausdorff per-turbation but is sensitive to outliers. In contrast, thedistance to a measure (here the k-distance) has beenintroduced in [4] in the context of geometric inferenceand is robust to outliers.

Contribution We introduce the notion of k-VCM of apoint cloud, which is a generalization of the VCM. In-stead of relying on the classical distance function, thek-VCM is based on the k-distance. We show a stabilityresult that implies that normals and curvature direc-tions can be estimated from a point cloud, using thek-VCM, even in presence of outliers.

2 Background

Our notion of k-Voronoi Covariance Measure combinesthe notions of distance function to a measure [4], withthe Voronoi Covariance Measure [3]. We recall the rel-evant definitions and properties of these objects.

2.1 Voronoi covariance measure

The Voronoi covariance measure (VCM) has been in-troduced in [3] for normals and curvature estimations.Let K be a compact subset of Rn and dK the distancefunction to K, i.e. the map dK(x) := minp∈K ‖p− x‖.A point p where the previous minimum is reached iscalled a projection of x on K. Almost every pointadmits a single projection on K, thus definining amap pK : Rn → K almost everywhere. The R-offset of K is the R-sublevel set of dK , i.e. the setKR := d−1K (]−∞, R[). The VCM maps any integrablefunction χ : Rn → R+ to the matrix

VK,R(χ) :=

∫KR

(x− pK(x))(x− pK(x))tχ(pK(x))dx.

Remark that this definition matches the definition in-troduced in [3]: when χ is the indicatrix of a ball, onerecovers a notion similar to the convolved VCM.

Point cloud In the specific case where the compact setK is a point cloud P , the VCM can be redefined usingthe Voronoi diagram of P . The Voronoi cell of a point

This is an extended abstract of a presentation given at EuroCG 2014. It has been made public for the benefit of the community and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.

30th European Workshop on Computational Geometry, 2014

p in P is the set VorP (p) := {x ∈ R, ∀q ∈ P ‖x− p‖ ≤‖x − q‖}, and pK maps (almost) every point x to thecenter of its Voronoi cell. Consequently,

VK,R(χ) =∑p∈P

χ(p)

∫VorP (p)∩PR

(x− p)(x− p)tdx

Smooth surface The VCM of a smooth compact sur-face S of R3 contains information on the normals nand on the principal curvatures κi and directions ei [3].Applying the VCM to the indicatrix of a ball B(p, r)centered at a point p of the surface gives, as r → 0,

VK,R(χ) ≈ 2π

3R3r2

[n(p)n(p)t +

r2

4

2∑i=1

κ2i (p)ei(p)ei(p)t

].

Stability The following stability result [3] implies thatinformation extracted from the covariance matrix suchas normals or principal directions are stable with re-spect to Hausdorff perturbation. We denote by ‖.‖opthe matrix norm induced by the Euclidean metric.Given a function f : Rn → R, we let ‖f‖∞ =maxx∈Rn |f(x)| and denote Lip f = maxx 6=y |f(x) −f(y)|/ ‖x− y‖ its Lipschitz constant, which can beinfinite. We define the bounded-Lipschitz norm by‖f‖BL = ‖f‖∞ + Lip(f).

Theorem 1 [3] Let χ be a bounded Lipschitz function,K a compact set of Rd and R > 0. Then, for everycompact set K ′, one has

‖VK,R(χ)− VK′,R(χ)‖op ≤ C0 ‖χ‖BL dH(K,K ′)12 ,

where the constant C0 depends only on K and R.

However, the Hausdorff distance dH becomes very largein the presence of outliers. We therefore introduce thek-distance, which is known to be resilient to outliers.

2.2 k-distance

The distance to a measure has been introduced in [4]and is defined for any probability measure on Rd. Weuse its definition when the measure is uniform over afinite point cloud, and call it k-distance, following [5].

Definition 1 Let P be a point cloud and k an integer.The k-distance dP,k is defined for any point x in Rn by

d2P,k(x) =1

k

∑pi∈NP,k(x)

‖x− pi‖2 ,

where NP,k(x) are the k nearest neighbors of x in P .

Theorem 3.5 in [4] shows that the distance to a mea-sure is resilient to outliers. In the case of two pointclouds P and P ′ this theorem asserts that

‖dP,k − dP ′,k‖∞ ≤1√k

W2(µP , µP ′),

where µP and µP ′ denote the uniform probability mea-sures on P and P ′, and W2 is the Wasserstein distance.In particular, when P and P ′ have the same cardinal,the distance W2(µP , µP ′) is the square root of the costof a least square assignment between the point clouds.

Power Diagram Let x be a point in Rn and b denotethe barycenter of the k-nearest neighbors of x in P . Asimple computation [5, Proposition 3.1] shows that

d2P,k(x) = ‖x− b‖2 − ωb, (1)

where ωb = − 1k

∑pi∈NP,k(x)

‖b − pi‖2. The k-distanceis therefore encoded by the power diagram associatedto the set B of barycenters of k points of P and theweights (ωb) defined above. Its cells are defined by

PowωB(b) = {x ∈ Rn;∀a ∈ B, ‖x− b‖2 − ωb

≤ ‖x− a‖2 − ωa}.

These cells coincide with the order-k Voronoi cells of P .

3 k-Voronoi Covariance Measure

We generalise the VCM by replacing the distance func-tion in the VCM by the k-distance. Let P be a pointcloud, and k an integer. For all point x in Rn, we denoteby pP,k(x) the barycenter of the k-nearest neighbors ofthe point x in P . This function is well-defined almosteverywhere. We also put PRk = d−1P,k(]−∞, R[).

3.1 Definition

The k-Voronoi Covariance Measure (k-VCM) of P is amatrix-valued measure. It associates to any integrablefunction χ : Rn → R+ a matrix defined by

VP,k,R(χ) :=

∫PR

k

(x−pP,k(x))(x−pP,k(x))tχ(pP,k(x))dx.

Decomposing this integral on order-k Voronoi cells andusing Equation (1) we get:

VP,k,R(χ) =∑b∈B

χ(b)

∫Powω

B(b)∩PRk

(x−b)(x−b)tdx. (2)

Moreover, as in the case of Voronoi cells, the sets overwhich the integrals are evaluated can be written as theintersection of power cells with a single ball, i.e.

PowωB(b) ∩ PRk = Powω

B(b) ∩ B(b, (R2 + ωb)

1/2).

3.2 Stability of the k-VCM

We state here the main result, that generalises the sta-bility theorem for the VCM [3]. This theorem assertsthat if the distance function to a compact set K is wellapproximated by the k-distance of a point cloud P , thenthe VCM of K is also well approximated by the k-VCMof P . The hypothesis of this theorem is satisfied for in-stance under the sampling model used in [5].

EuroCG 2014, Ein-Gedi, Israel, March 3–5, 2014

Theorem 2 Let P be a point cloud and K a compactset. If χ is a bounded Lipschitz function on Rn, then

‖VP,k,R(χ)− VK,R(χ)‖op ≤ C1 ‖χ‖BL ‖dP,k − dK‖12∞ ,

where the constant C1 depends on K and R only.

Note that the lipchitz function χ can be used to lo-calise the k-VCM. This result can even be generalisedto VCM involving distance-like functions. We prove itin the remaining of this section. Section 3.3 gives anintermediate result about the stability of gradients.

3.3 Stability of gradients of distance-like functions

We first need to recall the notion of distance-like func-tion, which is a generalisation of both the distance func-tion and the distance to a measure. A non-negativefunction δ on Rn is called distance-like function if thefunction ψδ(x) := 1

2 (‖x‖2 − δ2(x)) is convex. Thefollowing result guarantees the L1-stability of gradi-ents of distance-like functions, and follows from The-orem 3.5 of [6]. For any set E of Rn, we denote‖f‖L1(E) =

∫E‖f(x)‖dx and ‖f‖∞,E = supx∈E ‖f(x)‖.

Theorem 3 Let E be a set of Rn with rectifiableboundary. For any distance-like functions δ and δ′

bounded on E, one has

‖Oψδ − Oψδ′‖L1(E) ≤ C2‖δ − δ′‖12

∞,E ,

where C2 is a constant that depends on n, Hn(E),Hn−1(∂E), diam(E) and R = max(‖δ‖∞,E , ‖δ′‖∞,E).

The proof of this theorem relies on the following lemma.We skip the proof of this lemma, which is just a simplecalculation based on triangle inequalities.

Lemma 4 Under the assumptions of Theorem 3:

(i) ∀x ∈ E, |ψδ(x)− ψδ′(x)| ≤ 2R‖δ − δ′‖12

∞,E ;

(ii) diam(Oψ(E) ∪ Oψ′(E)) ≤ 2diam(E) + 4R.

This lemma allows us to apply Theorem 3.5 of [6] tothe functions ψδ and ψδ′ , thus implying Theorem 3.

3.4 Proof of Theorem 2

This proof follows the one of the VCM stability theoremin [3]. The idea is to compare the two integrals on thecommon set E = KR−ε with ε = ‖dP,k − dK‖∞ and toshow that remaining parts are negligeable. We have

VK,R(χ) =

∫E

∆K(x)∆K(x)tχ(pK(x))dx

+

∫KR\E

∆K(x)∆K(x)tχ(pK(x))dx,

where ∆K(x) = x − pK(x). For every x ∈ KR, onehas by definition ‖∆K(x)‖ ≤ R. Furthermore, the sec-ond term in the integral above is bounded by R2 ·Mtimes the d-volume ofKR\E, whereM = ‖χ‖∞. Corol-lary 4.4 of [6] implies that this volume is bounded bycst(K,R) · ε. Thus, the second term in the integral isbounded by M · cst(K,R) · ε.

We proceed similarly for the k-VCM, and introducethe quantity ∆P,k(x) := x−pP,k(x). Equation (1) givesus ‖∆P,k(x)‖ ≤ |dP,k(x)| ≤ R. Moreover, the definitionof ε gives the inclusion d−1P,k(]−∞, R[) ⊆ KR+ε\KR−ε.The volume of this set is bounded by cst(K,R) · ε, and

VP,k,R(χ) '∫E

∆P,k(x)∆P,k(x)tχ(pP,k(x))dx,

up to an error term of size M · cst(K,R) · ε.We now need to bound the operator norm of the in-

tegral∫E

[AK(x)−AP (x)]dx, where

AK(x) := ∆K(x)∆K(x)tχ(pK(x)),

and AP is defined similarly. The difference of these twoterms can be decomposed as:

AK(x)−AP (x)

= χ(pK(x))(∆K(x)∆K(x)t −∆P,k(x)∆P,k(x)t)

+ [χ(pK(x))− χ(pP,k(x))]∆P,k(x)∆P,k(x)t

One can factorise the first term of the sum by :

χ(pK(x))·[∆K(x)(∆K(x)−∆P,k(x))t

+ (∆P,k(x)−∆K(x))∆P,k(x)t]

Using the bound of ∆K and ∆P,k above, this first termis bounded by ‖pK(x) − pP,k(x)‖ · 2RM . The secondterm is bounded by L‖pK(x) − pP,k(x)‖ · R2, where Lis the Lipschitz constant of χ. Combining these twobounds, we get

‖AK(x)−AP (x)‖op ≤ (LR2 +2MR)‖pK(x)−pP,k(x)‖,

which by integration leads to∥∥∥∥∫E

[AK(x)−AP (x)]dx

∥∥∥∥op

≤ (LR2+2MR) ‖pK−pP,k‖L1(E).

The relations∇ψdP,k= pP,k and∇ψdK = pK , and The-

orem 3 imply that the norm ‖pK−pP,k‖L1(E) is bounded

by ε12 times a multiplicative constant that depends on

K and R only. This concludes the proof.

4 Experiments

Implementation. It is not practically feasible to com-pute Eq. (2) directly due to the very high number oforder-k Voronoi cells. To overcome this problem, wereplace the k-distance by the witnessed k-distance [5].A barycenter in B is called a witnessed barycenter if it

30th European Workshop on Computational Geometry, 2014

20K pts 40K pts 60K pts 80K ptsellipsoid 3.38 s 8.31s 14.7 s 22.51 sbimba 3.55 s 9.06 s 15.35 s 23.08 s

Table 1: Time calculation of the k-VCM with parame-ters : k=30, R=0.1, r= 0.2 .

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6

angle

en d

egre

noise

normals deviation on the ellipsoid. Hausdorf noise

convolved-kvcm-r=0.2-R=0.1Jet-fitting k=50Jet-fitting k=80

convolved-vcm-r=0.1-R=0.1convolved-vcm-r=0.1-R=0.2convolved-vcm-r=0.1-R=0.3convolved-vcm-r=0.2-R=0.1convolved-vcm-r=0.2-R=0.2convolved-vcm-r=0.2-R=0.3

Figure 1: Normal deviation for k-VCM, Jet fitting andVCM methods on a noisy ellipsoid.

Figure 2: Minimal principal direction estimation on thebimba point cloud of diameter 1 with 30% of outliers :dH(K,P ) = 0.2, R = 0.1, r = 0.15, k = 30

is the barycenter of k points p1, . . . , pk, where p1 lies inP and p2, . . . , pk−1 are the (k−1)-nearest neightbors ofp1 in P \{p1}. The witnessed k-distance is then definedby the following formula:

dwP,k(x) =

(minbw‖x− bw‖2 − ωbw

)1/2

,

where the minimum is taken over the witnessedbarycenters of P . Practically, we replace in Equation(2) the order-k Voronoi cells of the point cloud P bythe power cells associated to the witnessed barycenters.Note that the number of cells in this power diagram isbounded by the cardinal of the point cloud P , and cantherefore be computed efficiently using e.g. CGAL [7].The stability properties of the k-VCM of Theorem 2remain true for this relaxation. Indeed, Theorem 2 canbe generalised to VCM involving distance-like functionsand the witnessed k-distance is a distance-like function.Moreover, Theorem 4 of [5] allows to control the ap-proximation error ‖dK − dwP,k‖∞.

Normal estimation Given a point cloud P approxi-mating a smooth surface, we take as a normal estima-

Figure 3: Edge detection on a noisy fandisk point cloudof diameter 1 with : dH(K,P ) = 0.05, R = 0.03, r =0.5, k = 30

tion the eigenvector associated with the largest eigen-value of the k-VCM. The results obtained are comparedto the VCM [3] and jet fitting[8]. Figure 1 displays theaverage angular deviation between the estimated nor-mal and the ground truth in the case of a noisy ellipsoidwith outliers. Our method improves the mean deviationof the normals by at least 30%, and this improvementincreases with noise.

Estimation of curvature directions We estimate min-imal and maximal principal curvature directions usingthe second and the third eigenvectors of the k-VCM.Figure 2 shows an example of minimal curvature direc-tion estimation in the presence of many outliers.

Edge detection. One can also detect edges with the k-VCM matrix by considering the ratio of the two lowesteigenvalues. This ratio measures the anisotropy of theunion of the order-k Voronoi cells. Figure 3 displays inred points that are selected by thresholding accordingto this ratio (input is fandisk perturbated with noise).

References

[1] Nina Amenta and Marshall Bern. Surface reconstructionby voronoi filtering. Discrete & Computational Geometry,22(4):481–504, 1999.

[2] Pierre Alliez, David Cohen-Steiner, Yiying Tong, and Math-ieu Desbrun. Voronoi-based variational reconstruction of un-oriented point sets. In Symposium on Geometry processing,volume 7, page 3948, 2007.

[3] Quentin Merigot, Maks Ovsjanikov, and Leonidas Guibas.Voronoi-based curvature and feature estimation from pointclouds. IEEE Transactions on Visualization and ComputerGraphics, 17(6):743–756, 2011.

[4] Frederic Chazal, David Cohen-Steiner, and Quentin Merigot.Geometric inference for probability measures. Foundations ofComputational Mathematics, 11(6):733–751, December 2011.

[5] Leonidas Guibas, Dmitriy Morozov, and Quentin Merigot.Witnessed k-distance. Discrete & Computational Geometry,49(1):22–45, January 2013.

[6] Frederic Chazal, David Cohen-Steiner, and Quentin Merigot.Boundary measures for geometric inference. Foundations ofComputational Mathematics, 10(2):221–240, April 2010.

[7] Cgal, Computational Geometry Algorithms Library.http://www.cgal.org.

[8] Frederic Cazals and Marc Pouget. Estimating differen-tial quantities using polynomial fitting of osculating jets.Computer Aided Geometric Design, 22(2):121–146, February2005.