Jacobson et al. [JKSH13] hypothesized that the local coherency of the
generalized winding number function could be used to correctly determine
consistent facet orientations in polygon meshes. We report on an approach to
consistently orienting facets in polygon meshes by minimizing the Dirichlet
energy of generalized winding numbers. While the energy can be concisely
formulated and efficiently computed, we found that this approach is
fundamentally flawed and is unfortunately not applicable for most handmade
meshes shared on popular mesh repositories such as Google 3D Warehouse.Comment: 6 pages, 4 figure

Jacobson, Alec

Kavan, Ladislav

Sorkine-Hornung, Olga

Takayama, Kenshi

English

arXiv

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Consistently Orienting Facets in Polygon Meshes by Minimizing the Dirichlet Energy of Generalized Winding Numbers

Jacobson et al. [JKSH13] hypothesized that the local coherency of the generalized winding number function proposed in that work could be used to correctly determine consistent facet orientations in polygon meshes. We report on an approach to consistently orienting facets in polygon meshes by minimizing the Dirichlet energy of generalized winding numbers. While the energy can be concisely formulated and efficiently computed, we found that this approach is fundamentally flawed and is unfortunately not applicable for most handmade meshes shared on popular mesh repositories such as Google 3D Warehouse

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsCenter for Human Modeling and Simulation Department of Computer & Information Science2014Consistently Orienting Facets in Polygon Meshesby Minimizing the Dirichlet Energy of GeneralizedWinding NumbersKenshi TakayamaAlec JacobsonLadislav KavanUniversity of Pennsylvania, ladislav.kavan@gmail.comOlga Sorkine-HornungFollow this and additional works at: http://repository.upenn.edu/hmsPart of the Engineering Commons, and the Graphics and Human Computer InterfacesCommonsThis paper can be found in arXiv at http://arxiv.org/abs/1406.5431.This paper is posted at ScholarlyCommons. http://repository.upenn.edu/hms/145For more information, please contact libraryrepository@pobox.upenn.edu.Recommended CitationTakayama, K., Jacobson, A., Kavan, L., & Sorkine-Hornung, O. (2014). Consistently Orienting Facets in Polygon Meshes byMinimizing the Dirichlet Energy of Generalized Winding Numbers. Retrieved from http://repository.upenn.edu/hms/145Consistently Orienting Facets in Polygon Meshes by Minimizing theDirichlet Energy of Generalized Winding NumbersAbstractJacobson et al. [ JKSH13] hypothesized that the local coherency of the generalized winding number functionproposed in that work could be used to correctly determine consistent facet orientations in polygon meshes.We report on an approach to consistently orienting facets in polygon meshes by minimizing the Dirichletenergy of generalized winding numbers. While the energy can be concisely formulated and efficientlycomputed, we found that this approach is fundamentally flawed and is unfortunately not applicable for mosthandmade meshes shared on popular mesh repositories such as Google 3D Warehouse.DisciplinesComputer Sciences | Engineering | Graphics and Human Computer InterfacesCommentsThis paper can be found in arXiv at http://arxiv.org/abs/1406.5431.This working paper is available at ScholarlyCommons: http://repository.upenn.edu/hms/145Consistently Orienting Facets in Polygon Meshes byMinimizing the Dirichlet Energy of Generalized Winding NumbersKenshi TakayamaETH ZurichAlec JacobsonETH ZurichLadislav KavanUniversity of PennsylvaniaOlga Sorkine-HornungETH ZurichAbstractJacobson et al. [JKSH13] hypothesized that the local coherency of the generalized winding numberfunction proposed in that work could be used to correctly determine consistent facet orientations in polygonmeshes. We report on an approach to consistently orienting facets in polygon meshes by minimizing theDirichlet energy of generalized winding numbers. While the energy can be concisely formulated and efficientlycomputed, we found that this approach is fundamentally flawed and is unfortunately not applicable for mosthandmade meshes shared on popular mesh repositories such as Google 3D Warehouse.1 IntroductionThe robust inside-outside segmentation method based on the generalized winding number proposed byJacobson et al. [JKSH13] assumes that the polygonal facets in the input mesh are consistently oriented.If this assumption does not hold, which is often the case for hand-made meshes shared on the Internet(e.g., Google 3D Warehouse), it is necessary to first correct the facet orientations. In this technical report,we present an approach to correcting inconsistent facet orientations based on the Dirichlet energy of thegeneralized winding number. Our motivation came from the fact that if the mesh is watertight and its facetsare consistently oriented, the generalized winding number is piecewise constant, and hence its Dirichletenergy is zero. Our intuition was that for a mesh with various imperfections such as holes and intersections,more consistently oriented facets would result in a smaller Dirichlet energy of the corresponding generalizedwinding number.2 Method2.1 Patch extractionAs a preprocessing step, facets that are topologically connected and forming a two-manifold surface aregrouped together as a patch, similar to Borodin et al.’s method [BZK04]. If the mesh has non-manifoldedges, the mesh is split around these edges, and multiple patches are generated. All facets belonging tothe same patch share the same decision about whether they should be flipped or not in the output. Givenn patches, our goal is to seek an assignment for n binary variables representing whether the i-th patch Pishould be flipped or not (1 ≤ i ≤ n).2.2 Generalized winding numbersLet us first review the concept of generalized winding numbers [JKSH13]. The generalized winding number(hereafter we drop the word “generalized”) at a point p ∈ R3 is defined asw(p) =14pi∑τ∈Mθτ (p) , (1)where M is the input mesh (denoted as a set of facets), τ ∈ M is a facet in M, and θτ is the signed solidangle subtended by τ viewed from p. Note that the sign of the angle is defined to be positive for points inthe half space corresponding to the back side of a facet. In particular, the winding number of points insideand outside a watertight mesh is 1 and 0, respectively.2.3 Dirichlet energy of the winding numberSince the winding number of a watertight and consistently oriented mesh is piecewise constant and henceits Dirichlet energy is zero, our intuition is that the Dirichlet energy of the winding number for an imperfectmesh should decrease as its facet orientations become more consistent. We give a mathematical formulationof the Dirichlet energy in the following.1For each i-th patch Pi ⊆ M (1 ≤ i ≤ n), we want to determine a binary variable si ∈ {+1,−1} calledsign, where si = −1 means that all facets in Pi should be flipped in the output. Now we can define thewinding number with respect to the given sign s = (s1, · · · , sn) asws(p) =14pin∑i=1si∑τ∈Piθτ (p) . (2)Note that the orientation of each triangle τ is fixed to the original configuration when computing θτ . TheDirichlet energy of ws denoted as E(s) is a function of s and is defined as the integral of the gradientmagnitude squared of ws over the volumetric domain Ω = R3 \M:E(s) =∫p∈Ω‖∇ws(p)‖2 dV . (3)Reorganizing the integrand as‖∇ws(p)‖2 =∥∥∥∥∥∥ 14pin∑i=1si∑τ∈Pi∇θτ (p)∥∥∥∥∥∥2(4)=116pi2 n∑i=1si∑τ∈Pi∇θτ (p) · n∑j=1sj∑µ∈Pj∇θµ(p) (5)=116pi2n∑i=1n∑j=1sisj∑τ∈Pi∇θτ (p) ·∑µ∈Pj∇θµ(p) , (6)the energy can be written concisely in quadratic form asE(s) = s>Q s , (7)where Q is an n-by-n matrix with elementsQij =116pi2∑τ∈Pi∑µ∈Pj∫p∈Ω∇θτ (p) · ∇θµ(p) dV . (8)Q is fixed per model independently of s. Our goal is to find a set of facet orientations s∗ that gives thesmallest Dirichlet energy:s∗ = arg mins∈{−1,+1}nE(s). (9)Since s2i = 1, the energy can be further reorganized asE(s) = 2n−1∑i=1n∑j=i+1sisjQij +n∑i=1Qii︸ ︷︷ ︸const.. (10)Ignoring the diagonals of Q, we can also considerE˜(s) =n−1∑i=1n∑j=i+1sisjQij (11)as the energy instead.2.4 Computing the Dirichlet energyWe need to compute Qij in order to evaluate E(s). A straightforward approach would be to tessellatethe finite volume around the input mesh with sufficiently large volume and fine resolution to capture thevariations of the winding numbers and then numerically compute the integral over this tessellation. However,this approach can be computationally expensive, especially when the mesh contains many small details anda fine tessellation is needed for achieving sufficient accuracy.Fortunately, since the winding number function is harmonic (i.e., ∆w(p) = 0) [JKSH13], we can convertthe volumetric integral over the domain Ω into a surface integral over the domain boundary ∂Ω using Green’sfirst identity. For a once differentiable scalar function ψ and a twice differentiable scalar function φ, Green’sfirst identity states ∫Ωψ∆φ dV +∫Ω∇ψ · ∇φ dV =∫∂Ωψ (∇φ · n) dS (12)where n is an outgoing normal at the domain boundary (the positional variable p is omitted here for brevity).By substituting ψ = θτ and φ = θµ and recalling that winding numbers are harmonic, we get:0∫Ωθτ∆θµ dV +∫Ω∇θτ · ∇θµ dV =∫∂Ωθτ (∇θµ · n)dS , (13)2which shows that the volumetric integral in (8) is now converted to a surface integral.To compute this surface integral, we make the following notes on the domain and its boundary. Thedomain Ω consists of the entire 3D space R3 with a hole of infinitesimally thin slab carved out by each facetτ ∈M, and the domain boundary ∂Ω consists of such holes. Each hole corresponding to τ has its front andback sides denoted by τ+ and τ−, respectively. Note that the domain’s outgoing normals on τ+ and τ− arepointing toward the hole and are opposite to each other, see Figure 1. Now the domain boundary can bewritten as∂Ω =⋃τ∈Mτ+ ∪ τ−. (14)For a facet τ ∈ M and a point on it p ∈ τ , let us denote the corresponding point on the front and backsides as p+ ∈ τ+ and p− ∈ τ−, respectively. Then, it is easy to see that the following properties hold:θτ (p+) = −θτ (p−) = −2pi (15)∇θτ (p+) = ∇θτ (p−) , (16)where the latter is due to the derivative of the arctangent. Note that θτ and ∇θτ are undefined along theedges of τ ; we simply ignore the edge parts (i.e., parts connecting τ+ and τ−) in the surface integral, sincethe measure of all the edges vanishes (as opposed to the measure of the facets’ interiors).Original facet normal Normal on Normal on Figure 1: Notations for the domain boundary.For any other third facet γ 6= τ , θτ and ∇θµ take the same value on γ+ and γ−. Therefore,∫γ+∪γ−θτ (∇θµ · n)dS =∫γ+θτ (∇θµ · n)dS +∫γ−θτ (∇θµ · n)dS (17)=∫γ+θτ (∇θµ · n)dS +∫γ+θτ (∇θµ · (−n))dS (18)= 0 (19)where in (18) we replaced the integral over γ− with an integral over γ+ with a negated surface normal.Therefore, the integral over ∂Ω is reduced to an integral over τ+ ∪ τ−, i.e.,∫∂Ωθτ (∇θµ · n)dS =∑γ∈M∫γ+∪γ−θτ (∇θµ · n)dS (20)=∫τ+∪τ−θτ (∇θµ · n)dS . (21)Using the properties above, we arrive at:∫τ+∪τ−θτ (∇θµ · n)dS =∫τ+(−2pi)(∇θµ · n)dS +∫τ−(2pi)(∇θµ · n)dS (22)= −2pi∫τ+∇θµ · n dS + 2pi∫τ+∇θµ · (−n) dS (23)= −4pi∫τ+∇θµ · n dS . (24)In summary, Qij is computed asQij =∑τ∈Pi∑µ∈PjF (τ, µ) (25)whereF (τ, µ) = − 14pi∫p∈τ+∇θµ(p) · n(p) dS . (26)32.5 Convergence of the Dirichlet energyBy deriving an analytic form of F (τ, µ) using Maple, we found that it evaluates to infinity (either positiveor negative depending on relative orientation of τ and µ) when some edges of τ and µ coincide geometrically.An intuitive explanation for this is that the magnitude of ∇θµ(p) becomes infinitely large when p ∈ τ+ goesinfinitesimally close to the edges of τ that are geometrically coinciding with the edges of µ. In particular,the diagonal entries Qii always evaluate to positive infinity. Although we can ignore the diagonals using(11), off-diagonal entries Qij can still evaluate to infinity if some boundary edges of Pi and Pj coincidegeometrically. In our experiments, we numerically evaluate the energy with a sufficiently fine discretizationof ∂Ω where the finite entries in Q are converged.2.6 Finding the minimizer of the Dirichlet energyThe minimization problem in (9) is referred to as Quadratic Unconstrained Binary Optimization (QUBO).QUBO is NP-hard, and various heuristic methods exist [BHT07]. In our experiments, we take a bruteforce approach and exhaustively search for the optimal solution, so that we can examine the nature of ourapproach with small toy examples.3 ResultsWe implemented our approach in 3D as well as in 2D where the input consists of open or closed polylines(still referred to as “patches” in the following) and the same formulation applies. Although here we showour experiments only in 2D for the ease of visualization, the same argument also holds in 3D. Figure 2 showscases where the input corresponds to the boundary of a solid shape with some missing pieces. In such cases,our algorithm produces the expected results, i.e., it orients the facets consistently and correctly identifiesthe shape’s interior. On the other hand, Figure 3 shows a case where our approach is fundamentally flawed:the input has some small patches attached to larger patches, and those small patches are oriented inside-out.The fundamental reason for this is that the Dirichlet energy only measures the variation of the functionwhile ignoring the sign, and hence there is no penalty for orienting facets inside-out.Optimized Optimized 11.3 6.83 16.4 6.94 Figure 2: Examples where our approach produces reasonable results. The input polylines are represented bywhite lines with their orientations indicated by yellow arrows. The number below each image is the Dirichletenergy of the corresponding winding number function. The color maps represent the associated winding num-bers, where higher winding numbers correspond to warmer colors (the actual mapping is slightly adjusted foreach image to better visualize the distribution).Optimized 9.28 9.01 Figure 3: An example representing a torso with two small objects attached to the shoulder and the side. Ourapproach results in those small attachments oriented inside-out.4For an intuitive explanation of this behavior, it is useful to consider the closure of a patch P denotedas P¯ [JKSH13], where the union of P and P¯ bounds a finite volumetric domain ΩP (i.e., ∂ΩP = P ∪ P¯).Denoting the winding number defined by a patch P as wP(p), the following holds:wP(p) + wP¯(p) ={1, p ∈ ΩP0, otherwise. (27)Since the right hand side of (27) is piecewise-constant, the following also holds:∇wP(p) = −∇wP¯(p) . (28)Let us now consider a simple case with two patches P1 and P2 as shown in Figure 4. We can decomposethe gradient and apply the above observation as:∇wP1∪P2(p) = ∇wP1(p) +∇wP2(p) (29)= −∇wP¯1(p)−∇wP¯2(p) . (30)It is clear in the figure that the current configuration of P¯1 and P¯2 yields smaller Dirichlet energy than theother configuration, and hence P2 is oriented this way in our approach.= + = − − = − Figure 4: An example showing the reason of why small attachments are oriented inside-out in our approach.Note again that the mapping between the winding number and the color is adjusted in each image to bettervisualize the function.4 Conclusion and future workWe conclude that the presented approach has a fundamental flaw and is not applicable for most handmademeshes. One direction for further exploration is to apply a similar approach to surface reconstruction froman unoriented point cloud [MDGD+10, LW10], where only the positions of the points are given and findinggood normals for the points is critical. In this case, the unknowns are the continuous normal directions,potentially making the problem easier. A key question, however, would be how to generalize the definitionof winding numbers to point clouds.AcknowledgementsWe are grateful to Peter Kaufmann, Wenzel Jakob, Nobuyuki Umetani, Melina Skouras, Ilya Baran, andRyan Schmidt for their advice and feedback. This work was supported in part by the ERC grant iModel5(StG-2012-306877), by an SNF award 200021 137879, by an NSF grant IIS-1350330 and by a gift from AdobeResearch. Kenshi Takayama’s work was funded by JSPS Postdoctoral Fellowships for Research Abroad. AlecJacobson’s work was supported by an Intel Doctoral Fellowship.References[BHT07] Endre Boros, PeterL. Hammer, and Gabriel Tavares. Local search heuristics for quadraticunconstrained binary optimization (qubo). Journal of Heuristics, 13(2):99–132, 2007.[BZK04] Pavel Borodin, Gabriel Zachmann, and Reinhard Klein. Consistent normal orientation forpolygonal meshes. In Proceedings of the Computer Graphics International, CGI ’04, pages18–25, 2004.[JKSH13] Alec Jacobson, Ladislav Kavan, and Olga Sorkine-Hornung. Robust inside-outside segmenta-tion using generalized winding numbers. ACM Trans. Graph., 32(4):33:1–33:12, 2013.[LW10] Shengjun Liu and Charlie C.L. Wang. Orienting unorganized points for surface reconstruction.Computers & Graphics, 34(3):209–218, 2010.[MDGD+10] Patrick Mullen, Fernando De Goes, Mathieu Desbrun, David Cohen-Steiner, and Pierre Alliez.Signing the unsigned: Robust surface reconstruction from raw pointsets. Computer GraphicsForum, 29(5):1733–1741, 2010.6

Consistently Orienting Facets in Polygon Meshes by Minimizing the Dirichlet Energy of Generalized Winding Numbers

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