An algorithm for improving the quality of an initial triangulation on a fixed set of vertices is suggested. The edge flip operation is performed consecutively, aiming to minimise the discrete Willmore energy over a triangulated surface (or mesh). The Will- more energy of a surface is a function of Gaussian and mean curvature, and measures local deviation from a sphere. Virtual points are introduced in the triangulation to overcome the local invariance of Willmore energy under edge flips. Some experimental results are given

Alboul, Lyuba

Brink, Willie

Rodrigues, Marcos

Sheffield Hallam University Research Archive

Mesh optimisation based on Willmore energyALBOUL, Lyuba, BRINK, Willie and RODRIGUES, MarcosAvailable from Sheffield Hallam University Research Archive (SHURA) at:http://shura.shu.ac.uk/7891/This document is the author deposited version.  You are advised to consult the publisher's version if you wish to cite from it.Published versionALBOUL, Lyuba, BRINK, Willie and RODRIGUES, Marcos (2006). Mesh optimisation based on Willmore energy. In: 22nd European Workshop on Computational Geometry, Delphi, Greece, March 27-29 2006. Repository use policyCopyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any article(s) in SHURA to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain.Sheffield Hallam University Research Archivehttp://shura.shu.ac.ukMesh optimisation based on Willmore energyWillie Brink∗ Lyuba Alboul† Marcos Rodrigues‡AbstractAn algorithm for improving the quality of an ini-tial triangulation on a fixed set of vertices is sug-gested. The edge flip operation is performed consec-utively, aiming to minimise the discrete Willmore en-ergy over a triangulated surface (or mesh). The Will-more energy of a surface is a function of Gaussian andmean curvature, and measures local deviation from asphere. Virtual points are introduced in the triangu-lation to overcome the local invariance of Willmoreenergy under edge flips. Some experimental resultsare given.1 IntroductionComputer-based modelling and visualisation has nu-merous applications in engineering, science and the in-dustry. In particular, the digital capturing and recon-struction of a physical 3D object has applications incomputer graphics and vision, computational geome-try, reverse engineering, terrain modelling, tomogra-phy and medical imaging.A polyhedral surface, or mesh, is a piecewise planarsurface and is commonly used in computer graphicsfor approximating smooth surfaces. We will be con-cerned with triangular meshes, hereafter referred toas triangulations.Given a set of vertices sampled from the surfaceof some physical 3D object, surface reconstructionis concerned with finding an “optimal” triangulationthat in some sense best approximates the original sur-face. If the coordinates of the vertices are fixed thesetriangulations are said to be data-dependent.Some examples of methods for surface reconstruc-tion from scattered data may be found in [3, 6, 7].These methods aim at constructing a triangulationthat defines an underlying smooth surface (e.g. bymeans of subdivision) that somehow approximates thesurface of the original 3D object in an optimal man-ner. Most of the reconstruction methods thereforecontain an optimisation step that changes some ini-tial triangulation on the scattered data such that theresulting triangulation induces a smooth surface thatis optimal. This paper focusses on that step.∗Materials and Engineering Research Institute, SheffieldHallam University, UK, W.Brink@shu.ac.uk (corresponding au-thor)†idem, L.Alboul@shu.ac.uk‡idem, M.Rodrigues@shu.ac.ukThe problem that is addressed may be stated asfollows: given a fixed set of points in 3D, typically ac-quired from the surface of a physical object by meansof a 3D scanner, and some initial triangulation onthese points, improve (or optimise) the local qualityof this triangulation. This is generally referred to asmesh optimisation.Some mesh optimisation algorithms based on min-imising certain geometric properties of a polyhedralsurface have been proposed. Examples include min-imising the total area of the triangular mesh [10],minimising discrete analogues of the integral Gaus-sian curvature [1] and absolute mean curvature [2].See also [5] for algorithms based on minimising thesecurvatures. The question of which of these algorithmsproduce the “best” result remains unanswered.We propose an algorithm similar in structure tothose mentioned above. However, we incorporate ageometric property known as Willmore energy. This,to our knowledge, is a new approach to mesh optimi-sation.The rest of the paper is structured as follows: Sec-tion 2 briefly outlines the theory of discrete curva-tures, with specific focus on the Willmore energy ofa surface. Section 3 describes the mesh optimisationalgorithm. Some experimental results are given inSection 4 and Section 5 concludes.2 Discrete curvaturesFor a smooth surface with Gaussian curvature K andmean curvature H the following geometric propertiesare of importance: the area,∫dA; the total Gaussiancurvature,∫K dA; the total mean curvature,∫H dA;and the total Willmore energy,∫(H2 −K) dA.Consider a simplicial triangular surface (i.e. a tri-angulation) with vertex set V , face set F and edge setE. Discretisations for the first three of these proper-ties are well known for this type of surface (see forexample [1, 8]).The discrete area is simply the sum of areas of allthe triangular faces.The discrete analogue of the Gaussian curvature ata vertex v ∈ V is defined asG(v) = 2pi −∑iαi,where αi denotes the angle at v of every triangle shar-ing v. The total discrete Gaussian curvature is then1given by G =∑v∈V G(v).The discrete analogue of the mean curvature at anedge e ∈ E is defined asM(e) = θ|e|,where |e| denotes the length of the edge and θ the an-gle between the normals of the two adjacent faces.The total mean curvature is then given by M =∑e∈EM(e).Bobenko [4] recently proposed a discrete analogueof the Willmore energy for a triangulated surface. Ata vertex v it is defined asW (v) =∑e∋vβ(e)− 2pi,where the sum is taken over all incident edges of v.For each edge e the angle β(e) is calculated as fol-lows: let vi and vj denote the endpoints of e, and vkand vℓ the other two vertices of the adjacent faces, asshown in Figure 1. The value of β(e) is then definedto be the external angle of intersection between thecircumcircles of the two triangles vjvivk and vivjvℓ.    ✁✂✁✄✁☎✁✆✝ ✞✟✠Figure 1: The angle β(e) of an edge e = (vi, vj).The total discrete Willmore energy of the mesh isthen given by W =∑v∈V W (v).Bobenko [4] derives some properties of this energy,most notably that W (v) ≥ 0 and W (v) = 0 if andonly if v is convex and v and all its neighbours lie ona common sphere (possibly with infinite radius, i.e. aplane).It is therefore expected that a surface with mini-mum Willmore energy would be smooth and visuallypleasing. This motivates the need for developing amesh optimisation algorithm that aims at minimisingWillmore energy.3 Mesh optimisationConsider a given set of vertices V and some initialtriangulation on these points. Our mesh optimisationalgorithm attempts to minimise the discrete Willmoreenergy of this surface by changing the triangulation.Following the methodology of [2], the triangulationis changed with the edge flip operation illustrated inFigure 2. A cost function is defined for a specifictriangulation and edge flips that result in a decreasein this cost function are then performed successivelyuntil a minimum is reached.   ✁✁✁Figure 2: The edge flip operation.The algorithm assigns to each edge a value thatreflects the difference between the cost function beforeand after flipping the corresponding edge. We refer tothis value as the cost reduction value.The order in which the edges are flipped may bechosen by an optimisation method such as simulatedannealing, but at this stage our algorithm is greedy innature, flipping an edge that maximally reduces thecost function at every step. It is important to notethat this may not always lead to a global minimum inthe cost function and the algorithm may terminate ata local minimum. Techniques to escape from such alocal minimum are currently under investigation. Pos-sibile strategies include implementing multiple edgeflips at each step [9].Since the aim of our algoritm is to minimise theWillmore energy of the surface we want to define thecost function to be the total discrete Willmore energyof the current triangulation. However, as also men-tioned in [4], for an edge e and its flipped version e′,β(e) = β(e′). This may lead to the Willmore energybeing locally invariant under flips.Attempting to overcome this we introduce “virtual”points to the triangulation. To assign a cost reductionvalue to an edge e we implement a simple subdivisionscheme by adding a vertex v in the middle of e andconnect it as shown in Figure 3. The total Willmoreenergy is calculated for this new triangulation andthen compared to the total Willmore energy of thetriangulation resulting from flipping e to e′, with apoint v′ in the middle of e′. The cost reduction valueof e is then taken to be the difference between theseenergies. Since in general the positions of v and v′would differ there would also be a difference in theWillmore energy before and after the flip.We call the points v and v′ virtual since they onlyappear in calculating the cost reduction values, not inthe resulting triangulation. We refer to the Willmoreenergy of the triangulation with added points as thevirtual Willmore energy.2  ✁✂✂✄☎✆✝✝✁✞✟✟✠✡✆✞Figure 3: Adding points to the triangulation.In one of the proofs of Bobenko [4] the combina-torics of a mesh is changed by adding points to everyedge and connecting them in a similar way as depictedin Figure 3. This is done in order to render an ab-stract simplicial sphere inscribable and is thereforeessentially quite different from what we are doing.In the implementation of the algorithm there aresome types of edges that cannot be flipped withoutchanging the topological type of the triangulation. Anedge e with flipped version e′ should not be flippedif e′ is already an edge of the triangulation. For suchedges we define the cost reduction function to be −∞.Note also that β(e) is undefined for a boundaryedge e (i.e. an edge with only one adjacent face). Itis also clear that a boundary edge cannot be flipped.Hence the cost reduction function of boundary edgesare also defined as −∞.It is important to realise that the algorithm de-scribed above does not necessarily minimise the totaldiscrete Willmore energy of the triangulated surfacesince this energy might remain unchanged under edgeflips. The virtual Willmore energy (VWE) might bea new type of energy somehow related to the Will-more energy. It can be said that our algorithm at-tempts to minimise the VWE over a triangulation inthe hopes of minimising the Willmore energy of theunderlying smooth surface induced by the triangula-tion. Whether or not this would always be achievedis still under investigation although experiments dosuggest that it would.4 ResultsThis section provides some experimental results fromapplying our mesh optimisation algorithm on a fewtest models.The data of the first model comprises of the 8 cor-ners of a cube with 6 vertices added to the centres ofeach face, slightly inside the cube. Figure 4 shows onthe left an initial triangulation on these vertices. Theresult of applying our algorithm on this triangulationis shown on the right of the figure.What is interesting about this result is that the ini-tial triangulation is a so-called tight triangulation [1],i.e. a triangulation on the data with minimum totalabsolute extrinsic curvature. Our algorithm changesthis triangulation to what appears to be a triangu-lation with minimum total area. An algorithm thatminimises total absolute curvature [2] has the exactopposite effect on this data. Exactly how these al-gorithms relate to each other is a topic for furtherstudy.Figure 4: Test model I - a cube with 6 added vertices,initial (left) and optimised (right).The second test model consists of points sampledon the surface of a torus. Figure 5 shows an initialtriangulation on the left. On the right of the figurethe result of applying our algorithm is shown.The resulting triangulation is clearly more regular(the triangles are more or less equal in size). The tri-angulation is also visually smoother. This may be dueto the fact that for a region of a smooth surface thatclosely resembles a sphere the corresponding Willmoreenergy is close to zero. It would seem that our algo-rithm attempts to extract regions in a triangulationthat is spherelike.Figure 5: Test model II - points sampled on a torus,initial (left) and optimised (right).The data for the third test model was acquired witha 3D scanner and consists of points on the surfaceof a human face. An initial triangulation, shown onthe top left of Figure 6, was obtained by parameter-ising the data and applying 2D Delaunay triangula-tion. The result from our algorithm is shown on the3top right of the figure. The figure also shows shadedversions of the top parts of these two triangulations.There are many “vertical” edges visible in the initialtriangulation. This is a parameterisation artifact andresults in the underlying smooth surface to have manyvertical creases (see lower left part of Figure 6). Ouralgorithm does seem to smooth out these creases (aresult from flipping most of the vertical edges in theinitial triangulation), as can be seen on the lower rightof Figure 6.Figure 6: Test model III - points sampled on a face,initial (left) and optimised (right).Regarding the compexity of the algorithm, considera triangulation with n edges. By keeping record ofadjacency in the mesh (edges to faces) a single edgeflip can be performed in O(1) time. Searching througha priority queue for which edge to flip would requireO(logn) time. The worst case scenario, in which everyedge is flipped, would thus be O(n log n).5 Conclusion and future workWe presented a new mesh optimisation algorithm.The algorithm attempts to minimise the so-called vir-tual Willmore energy of a triangulation by performingthe edge flip operation successively.An important area for further research is to studyand develop methods for escaping from local minimain the cost function. Implementing other optimisationstrategies such as simulated annealing, rather thanour greedy method, might prove to be useful.Another important issue in the algorithm is thatsome edge flips can result in the surface intersectingitself. Detecting these edges and avoiding such inter-sections is still an open problem.Based on results obtained experimentally our algo-rithm seems to be promising. It should be stressedthat the arguments upon which the algorithm is builtare mostly heuristic in nature and a comprehensiveanalytical analysis is necessary.Topics of current ongoing research also include de-termining under what circumstances the choice ofwhere to position the virtual points on the edges af-fects the outcome, and the relationship between thisalgorithm and other algorithms such as minimisingarea or mean curvature.The discrete analogue of Willmore energy is rela-tively new. It would be interesting to learn how dif-ferent areas in shape modelling could benefit from thisconcept.References[1] L. Alboul and R. van Damme. Polyhedral metricsin surface reconstructions. In: The Mathematics ofSurfaces VI, G. Mullineux (Ed.), Clarendon Press,Oxford, 171–200, 1996.[2] L. Alboul, G. Kloosterman, C. Traas and R. vanDamme. Best data-dependent triangulations. Jour-nal of computational and applied mathematics, 119:1–12, 2000.[3] F. Bernardini and C.L. Bajaj. Sampling and recon-structing manifolds using alpha-shapes. Proceedingsof the 9th Canadian conference of computational ge-ometry, 193–198, 1997.[4] A.I. Bobenko. A conformal energy for simplicial sur-faces. Combinatorial and computational geometry,52:133–143, 2005.[5] N. Dyn, K. Hormann, S.J. Kim and D. Levin. Op-timizing 3D triangulations using discrete curvatureanlysis. Mathematical methods for curves and sur-faces, Oslo, 135–146, 2000.[6] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald andW. Stuetzle. Surface reconstruction from unorganizedpoints. ACM SIGGRAPH ’92, 71–78, 1992.[7] K. Hormann. From scattered samples to smoothsurfaces. Proceedings of the 4th Israel-Korea bi-national conference on geometric modeling and com-puter graphics, 1–5, 2003.[8] M. Meyer, M. Desbrun, P. Schro¨der and A.H. Barr.Discrete differential-operators for triangulated 2-manifolds. In: Visualization and mathematics III,Hege and Polthier (Eds.), 35–57, 2003.[9] A. Netchaev. Triangulations and their applicationin surface reconstruction. PhD thesis, University ofTwente, The Netherlands, September 2004.[10] J. O’Rurke. Triangulation of minimal area as 3D ob-ject models. Proceedings of the international jointconference on AI 81, Vancouver, 664–666, 1981.4

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