Geometric Integrability and Consistency of 3D Point Clouds

George KamberovStevens Institute of Technology

Hoboken, NJ 07030gkambero@stevens.edu

Gerda KamberovaHofstra University

Hempstead , NY 11549cscglk@hofstra.edu

Abstract

Numerous applications processing 3D point data willgain from the ability to estimate reliably normals and dif-ferential geometric properties. Normal estimates are noto-riously noisy, the errors propagate and may lead to flawed,inaccurate, and inconsistent curvature estimates. Frankot-Chellappa introduced the use of integrability constraintsin normal estimation. Their approach deals with graphsz = f(x, y).

We present a newly discovered General OrientabilityConstraint (GOC) for 3D point clouds sampled from gen-eral surfaces, not just graphs. It provides a tool to quan-tify the confidence in the estimation of normals, topology,and geometry from a point cloud. Furthermore, similarly tothe Frankot-Chellappa constraint, the GOC can be used di-rectly to extract the topology and the geometry of the man-ifolds underlying 3D point clouds. As an illustration wedescribe an automatic Cloud-to-Geometry pipeline whichexploits the GOC.

1. IntroductionPresently it is possible to extract 3D point clouds from

complex 3D scenes, often at real-time video rates. What isneeded are automatic and computationally stable methodsto extract geometry from noisy non uniform clouds. Thesemethods must handle automatically under-sampled regions,for example, they have to deal with surface boundaries andresolve the ambiguities imposed by objects that are or ap-pear to be tangent to each other. Furthermore, given theexpected low quality of the data it is crucial to have meth-ods to evaluate the consistency of the recovered geometryeven in the absence of ground truth.

We present a novel orientation integrability condition,the General Orientability Constraint (GOC), which willserve as a foundation to provide an approach to quantify theperformance of geometry reconstruction techniques from3D point clouds and meshes. Furthermore, the GOC canbe used to enhance different reconstruction methods, e.g.,

[12, 17, 24, 28]. As an example, we use the integrabil-ity constraint to modify the automatic Cloud to GeometryPipeline from [17]. This leads to a significant speed up inthe geometry computation stage of the pipeline.

1.1. Related work

Estimating normals and geometry There is a vast lit-erature on geometry extraction from point clouds. It in-cludes local polynomial fits, global smooth approximations,Voronoi/Delaunay-based methods, and level set methods,for example, [1, 2, 7, 11, 25, 30]. Earlier heuristic ap-proaches gave no guarantees, but new powerful methods dogive guarantees of faithfulness to the underlying continu-ous shape in the limit of infinitely dense data [5, 6]—undersomewhat restrictive assumptions on object smoothness andthe evenness of the sampling.

The estimation of the normals is a key step in geometryreconstruction – it is also notoriously prone to errors. Theseerrors propagate and lead to unreliable estimates of localdifferential geometric properties. There are other factorscontributing to the errors in the geometry estimates, e.g.,discretization. As a result, many authors consider local dif-ferential geometric estimates unreliable [8, 9].

A number of authors have studied the reliability of ori-entation estimates. Dey et al. [4] have performed empiricalcomparisons of several approaches given ground truth data.Theoretical error bounds for a number of methods are de-rived in [22, 23]. The bounds derived by Mitra et al. [23]are only for the method for estimating normals presentedin [23] (thus it works well if the data is properly sampledand one has sensor noise model, but the method heuristicsfail on data sets like the one in Figure 2). [22] deals withsurfaces of the form z = f(x, y).

Integrability constraints for normals on graphsFrankot and Chellappa [10] popularized the integra-bility condition that must be satisfied by a vector field(N1(x, y), N2(x, y), N3(x, y)) to be the orientation

To Appear in Proc. ICCV 2007, Rio DeJaneiro, Oct 14-20, 2007.

(normal vector field) to a graph z = f(x, y),

∂

∂y

N1

N3=

∂

∂x

N2

N3. (1)

Many authors have used integrability on graph surfaces tohelp reconstruction and resolve ambiguities, for example,[9, 10, 27]. Using the conformal geometry approach to sur-face theory from [18] we look for integrability constraintssatisfied by the orientation of a smooth surface in R3, notnecessarily a graph.

1.1.1 Conformal geometry of surfaces in R3: back-ground

Conformal surface immersions provide a convenient ma-chine to treat general surfaces in R3, including surfaceswith self-intersections. Following [16, 18] we think of asmooth surface, S ⊂ R3, as a vector-valued map, f , fromsome orientable two-dimensional manifold M2 into Eu-clidean three space such that S = f(M2), the map f is animmersion with Gauss map N. The Gauss map assigns toeach point on M2 a unit vector orthogonal to S. The GOCand the computations underlying the framework in [16] arebased on analysis of the differentials df and dN of the im-mersion and the Gauss map. A differential of a map is just aone-form whose inputs are tangent vectors and whose out-puts are the corresponding directional derivatives. Recallthat the vector-valued map f : M2 → R3 is an immer-sion if and only if its differential has trivial kernel, that is,df(v) 6= 0 if v 6= 0. The immersion and the Gauss map in-duce a complex structure J on M2 defined as follows: let vbe a vector tangent to M2, then J(v) is the unique tangentvector satisfying

df (J(v)) = N× df (v) . (2)

Here × denotes the cross-product in R3. Essentially J isa counter-clockwise rotation in the tangent plane by 90o.The key to the conformal method for computing differentialgeometric properties [14] and to the orientability constraintis to look at two R3-valued 1-forms τ and ω defined in [18]by

τ(v) :=12

(dN(v)−N× dN(J(v))) (3)

ω(v) := dN(v)− τ(v). (4)

The fundamental identity

−Hdf = τ, (5)

where H is the mean curvature function implies that for ev-ery nonzero tangent vector v we have

H = −τ(v) · df(v)|df(v)|2 (6)

and

K = H2 − (ω(v) · df(v))2

|df(v)|4 − ((ω(v) · df(J(v))2

|df(v)|4 , (7)

where K is the Gauss curvature (of f ). Furthermore theprincipal curvature directions are expressed in closed formformulae involving df(v) and the dot products ω(v) ·df(v)and ω(v) · df(J(v)).

2. GOC: The General Orientability Constraint2.1. Smooth surfaces

Similarly to the Frankot-Chellappa we will search for thedifferential equations satisfied by the Gauss map N for theintegrability constraint. In fact the constraint involves theform τ and it follows from the identity (5) and its deriva-tives.

To express the constraint as partial differential equationswe will use isothermal coordinate systems (x, y) with re-spect to f (i.e., conformal parameters with respect to thecomplex structure J). A coordinate system (x, y) is isother-mal with respect to f iff:

∂f∂y

= N× ∂f∂x

. (8)

Such coordinate systems are defined in a neighborhood ofevery point. The condition of being conformal parameterscan be expressed in terms of the complex structure as

∂

∂y= J

(∂

∂x

). (9)

Denote by τx and τy the components of τ with respectto the conformal parameters (x, y) so

τ = τxdx + τydy, (10)

where

τx =12

(∂N∂x

−N× ∂N∂y

)(11)

τy =12

(∂N∂y

+ N× ∂N∂x

)(12)

Notice that in regions where the mean curvature is identi-cally zero (5) leads to the constraint

τ = 0 (13)

or equivalently∂N∂x

= N× ∂N∂y

. (14)

In regions where the mean curvature is not identically zerowe must differentiate (5) to obtain the constraint. By com-paring the coefficients of

dτ = dH ∧ d f , (15)

To Appear in Proc. ICCV 2007, Rio DeJaneiro, Oct 14-20, 2007.

where ∧ denotes the usual edge product of 1-forms, we ob-tain the constraint

∂δx

∂y= −∂δy

∂x, (16)

where δx and δy are uniquely defined by

dτ = (δxτx + δyτy) dx ∧ dy. (17)

Equation (16) is a third order partial differential equationfor the components of N.

Thus we can summarize the GOC:

Theorem 1 If N is the orientation of a C4 conformal im-mersion in R3 of a surface equipped with a complex struc-ture J , then in a neighborhood of every point, N satisfieseither (14) or (16). Here (x, y) are conformal with respectto J .

Remark The GOC does encompass the Frankot-Chellappa conditions, but one has to do a change of vari-ables first. It is easy to see that in general the (x, y) param-eters used to parameterize a graph f(x, y) = (x, y, z(x, y))are not isothermal.

2.2. Discrete GOC on 3D point clouds

We propose an approach to address orientation integra-bility of discrete oriented, topologized clouds in R3 withoutrequiring such high order smoothness of the underlying sur-faces as in Theorem 1.

Definition 1 An oriented point cloud in R3 is a set{(p, N) ∈ R3 × S2

}of 3D points with a ”normal” vector

attached to each point; the 3D points form the underlyingpoint cloud. The cloud is topologized if it is equipped witha connectivity graph which specifies the neighbors of eachpoint.

Integrability constraints are expressed in terms of deriva-tives or path integrals. Thus we will measure integrabilityin the context of a chosen discrete calculus approach.

Discrete GOC assumptions: Assume that we are givena topologized oriented point cloud and that a method forcomputing first order directional derivatives ( along tangentvectors) has been selected. So for every tangent vector u wecan estimate the directional derivatives dN(u) and df(u)from the oriented point cloud. The derivatives estimates aretangent-valued, i.e., orthogonal to N (in agreement with dif-ferential geometry).

There are many approaches to building a topologized ori-ented point cloud from a cloud of 3D points, e.g., one coulduse one of the many polyhedral reconstruction methods dis-cussed earlier, or say the methods in [13, 15, 20, 21, 23]. For

a given topologized oriented cloud, one can select amongmany possible discrete difference schemes to compute di-rectional derivatives. In Section 3 we apply the proposedapproach to GOC for a particular orientation-topology-discrete calculus scheme.

To derive the GOC we look at the fundamental identity(5). It implies that in integrable case when the normals areproperly chosen and there is no noise due to discretization:

τ(v)× df(v) = 0, for every tanget vector v (18)

and

h(v) :=τ(v) · df(v)|df(v)|2 is independent on v (19)

Thus, the extent to which integrability is violated is mea-sured by the rate of failure of (18) and (19). At each pointP = (p,N) in the oriented cloud we define the penaltyterms

α(P ) =∮

A(P )

| sin (] (τ(v), df(v))) |dv (20)

andχ(P ) = Var(h(v) |v ∈ A(P )), (21)

here A(P ) is the set of unit tangent vectors based at p inwhich we can compute cross products τ(v)× df(v) and thedot products h(v), (19). In this paper we use A(P ) to be theset of unit vectors in direction of the tangential projectionsof the vectors

→pqi where (qi,Ni) are the neighbors of P .

So the integral (20) is just a finite sum over all neighborsof the point. We measure the orientation integrability of anoriented point cloud C = {P = (p,N)} by

O(C) =∑

P∈C(α(P ) + χ(P )). (22)

3. The Cloud to Geometry Pipeline C2GUsing O(C) we modified the automatic pipeline intro-

duced in [17].

3.1. Topology and Calculus

To define a topology on a oriented point cloud we usethe proximity likelihood introduced in [15] and follow theframework in [16]. Given a choice of geometric scale ρ, theproximity likelihood 0 ≤ ∆P (Q) ≤ 1 defined in [15] mea-sures the likelihood that the oriented point Q is a neighborof the oriented point P on a surface patch. The likelihoodincorporates three estimators of surface distance: a linearestimator based on Euclidean distance, a quadratic estima-tor based on the cosine between normals, and a third or-der estimator δP which is crucial in distinguishing pointswhich are far on the surface but close in Euclidean distance

To Appear in Proc. ICCV 2007, Rio DeJaneiro, Oct 14-20, 2007.

(δP (Q) = O(s3) where s is the surface distance betweenthe points). Thus, neighborhoods built using the proximitylikelihood based on ∆P (Q) are in general tighter than theusual approach using k-nearest Euclidean neighbors.

We use the proximity likelihood to build neighborhoodsaround each oriented point P that have the smallest pos-sible radius in terms of surface distance (for the givenscale choice) and the smallest number of neighbors thatstill provide all the available information about the nor-mal distribution around P . We call these neighborhoods∆-neighborhoods. The ∆-neighborhoods provide a tool toclassify the points of an oriented cloud as isolated points,curve points, boundary surface points, or interior surfacepoints [16]. Isolated, curve and surface boundary pointsare called cloud boundary points. A surface point is dis-tinguished by the property that one can use the ∆ neigh-borhood to compute the orientation in at least two orthog-onal directions tangential directions at the point, i.e., twoorthogonal directions emanating from the point and perpen-dicular to the normal at the point. The point is an interiorsurface point if one can use the ∆-neighborhood to computethe orientation in a full circle of tangential directions, oth-erwise the surface point is a surface boundary point. Thesurface points are samples of 2D manifolds and we cancompute surface differential properties following the con-formal method [14]. Finding a ∆-neighborhood of a pointis a pretty straightforward region growing procedure. Theneighbors of a point are determined by starting at the pointand following the gradient of the proximity likelihood in-side the voxel determined by the geometry scale. The pro-cess stops if the point is identified as an interior surfacepoint or upon exiting the voxel. In the later case the pointis automatically classified as a cloud boundary point. Withthis approach there is no extra cost to treat automaticallyscenes with multiple objects and surfaces with boundary.

To define directional derivatives we identify surface im-mersions with their images so the tangent vectors are ac-tually vectors in R3, i.e., a tangent vector v at a orientedpoint (p,N) is just a 3D vector perpendicular to N andJ(v) = N × v. Each directional derivative df(v) is justv. A unit tangent vector v at P = (p,N) is called radialat P if it is parallel with the tangent projection of an edgefrom a point to one of its ∆- neighbors in the cloud. If vis a radial vector, then the directional derivative dN(v) iscomputed by a first order difference along the edge fromP to its neighbor and dN(J(v)) is computed by a linearinterpolation of the directional derivatives along the radialdirections closest to J(v). This setup enables us to com-pute the penalty terms α(P ) (see (20)) and χ(P ) (see (21))and therefore the integrability deviation measure O(C) ofthe topologized oriented point cloud (see (22) ).

Figure 1. The C2G pipeline.

3.2. The Pipeline Outline

The stages and the data flow in the Cloud-to-Geometry(C2G) pipeline are laid out in Figure 1. The pipeline isbased on the ideas introduced in [17]. The pipeline takes asinput a 3D point cloud and outputs:

1. A point-based reconstruction of the scene geometry:each point is equipped with a normal and a neighbor-hood of points closest to it in terms of surface distance;the points are classified as isolated, curve, and surfacepoints, and the surface points are divided into interiorand boundary points; the connected components of thetopologized cloud are identified thus segmenting thescene into geometric objects represented by 0D, 1D,and 2D submanifolds.

2. Estimates of differential geometric properties (curva-tures and principal curvature directions) at each sur-face point.

The path from 3D point samples to a complete descrip-tion of scene geometry leads through several stages. The 3Dpoint cloud is a sample of the underlying scene with noisearising from sensor errors or uncertainty in the environment.In the first stage we use a clustering method of [29] to groupthese noisy samples into local primitives which incorporatelocalized scene measurements plus uncertainties. The scaleat which this is done, we call the clustering/localizationscale. The primitives are fish scales [29] and consist of a lo-cal covariance ellipsoid approximating the 3D spatial den-sity, with the smallest eccentricity direction encoding the(unoriented) surface normal line. The clustering scales areestablished automatically and adaptively [3].

The next stage uses the unorganized fish-scales to es-tablish orientation and surface topology (i.e., connectivity),followed by the geometry. The key parameter is the geome-try/voxelization scale—the scale at which local shape “tex-ture” can be distinguished. Here we will assume that thegeometry details scale is chosen. We have developed meth-ods to learn this scale as we build the orientation and topol-ogy, but will discuss them in a separate paper due to lack

To Appear in Proc. ICCV 2007, Rio DeJaneiro, Oct 14-20, 2007.

of space and to keep the discussion focussed. To computeorientation and topology, we follow [16, 17]. This approachgives rapid orientation of the cloud starting from local es-timates of the unoriented surface normals. A prime advan-tage is that, unlike previous methods based on triangulationsand parameterizations, our method does not commit prema-turely to assignments of surface patch orientation based onlocal information.

The topology is computed by finding the ∆-neighborhood of each oriented point, then the orientationis iteratively re-evaluated by comparing orientations andpossibly flipping the orientations of adjacent connectedcomponents. The process continues while the integrabilitydeviation measure O(C) continues to decrease.

Note Thus, the topology-orientation stage minimizes theintegrability deviation measure.

The orientation-topology iteration is designed to analyzeautomatically the ambiguities occurring in regions wheredifferent surfaces meet [17]. The hardest case—tangentsurfaces—is handled by analyzing the proximity likelihooddistributions in the different candidate neighborhoods (com-ing from different bifurcation branches). This approach al-lows us to resolve automatically cases (See Figures 2 and3) where polygonal-based approaches and competing point-based approaches (e.g., [23]) need human intervention.

To compute the surface differential properties estimateswe modify the conformal method technique [19, 14]. Weuse the formulae (6), (7), and the closed form expressionsfor the principal curvature directions. Instead of com-puting redundant estimates along all possible radial direc-tions as in [19, 14] and then averaging, we use the esti-mates along the radial direction v for which the penalty| sin (] (τ(v), df(v))) | is minimal. Thus C2G needs a sin-gle pass through all neighbors of a point instead of the threepasses needed in [17]. As a result C2G, is considerablyfaster than the reference pipeline [17].

Note Essentially we replace average estimates withmost likely estimates, as measured using the GOC.

Figure 2. Two tangent surfaces: (Left) The original point cloudsampled from a sphere touching a plane. (Right) The two largestconnected components extracted by the C2G pipeline: part of thesphere and part of the plane.

Figure 3. (Left) A cloud 3D point cloud obtained from stereo [3];(Middle and Right) the two largest 2D components extracted fromthe 3D point cloud. The quality of the cloud segmentation intomanifolds matches the one reported in [17].

4. SummaryWe presented an orientation integrability constraint ap-

plicable to general 3D clouds. It provides a method to op-timize the reconstruction of topology and orientation froma 3D point cloud. The orientation integrability constraintand the underlying approach to estimating the consistencyof the topology, orientation, and geometry reconstructionsprovide methods to speed up and improve the reliability ofthe the computation of differential geometric properties bytaking the most likely estimates. The proposed approachis implemented in a pipeline that deals successfully withclouds in which previous methods fail, and provides geo-metric quality of the geometry estimates comparable to theoriginal conformal method.

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To Appear in Proc. ICCV 2007, Rio DeJaneiro, Oct 14-20, 2007.