aa r X i v : . [ c s . G R ] M a y On Discrete Conformal Seamless Similarity Maps
MARCEL CAMPEN and DENIS ZORIN,
New York University
An algorithm for the computation of global discrete conformal parametriza-tions with prescribed global holonomy signatures for triangle meshes wasrecently described in [Campen and Zorin 2017]. In this paper we provide adetailed analysis of convergence and correctness of this algorithm. We gen-eralize and extend ideas of [Springborn et al. 2008] to show a connectionof the algorithm to Newton’s algorithm applied to solving the system ofconstraints on angles in the parametric domain, and demonstrate that thissystem can be obtained as a gradient of a convex energy.CCS Concepts: •
Computing methodologies → Mesh geometry mod-els
In [Campen and Zorin 2017, Sec. 7.3] we introduce an iterative al-gorithm that computes a discrete conformal seamless similarity map with a prescribed holonomy signature , i.e., rotation angles aroundsingularities and along homology loops; a precise definition is givenbelow.In this paper we examine the properties of this algorithm in moredetail. We establish the following properties: • If the algorithm converges, it yields a discrete conformal seam-less similarity map, and this map has the prescribed signature. • A slightly modified version of the algorithm converges unlessan infinite sequence of edge flips occurs.The question whether such an infinite sequence can actually oc-cur remains to be answered, similar to the closely related questionposed by [Luo 2004] in the context of discrete Ricci flow.
Discrete surface.
Let M = ( V , E , F ) be a closed orientable mani-fold triangle mesh of genus д , i.e. with | V | + | F | − | E | = − д . Let M c = ( V c , E c , F c = F ) be a triangle mesh obtained by cutting M todisk topology using a cut graph c (consisting of edges of M ). Discrete metric.
Let G be a discrete metric on M , i.e. an assign-ment E → R > of a positive length l Gi to each edge i . This metriccarries over to M c in a trivial manner. Discrete conformal metric.
Let G ′ be a discrete metric on M c thatis discrete conformally equivalent to G on M c , in the sense thatthere exists a discrete 0-form ϕ on M c such that l G ′ i = l Gi e ( ϕ v + ϕ w )/ (1)for all edges i ∈ E c , where i = ( v , w ) is the edge between vertices v and w . In the following we simply write l i instead of l G ′ i . Discrete conformal map.
Under the assumption that G ′ respectsthe triangle inequality for every face of F , let Θ r denote the sum ofinner triangle angles incident at vertex r of M under G ′ . A vertex r where Θ r = π is called regular under G ′ , otherwise irregular (or extraordinary ). If all inner vertices of M c (i.e. those not on the This work is supported by the National Science Foundation, under grant IIS-1320635. − α s − α s + α s − α s T s T s T s T s α a α b α c α d α e Fig. 1. Illustration of angles summed along a dual cycle, i.e. triangle strip T s (left), and around a vertex r (right), where N ( r ) = { a , b , c , d , e } . boundary due to the cut) are regular, the discrete metric G ′ is flat ;it thus implies a (continuous, locally injective) discrete conformalmap f : M c → R , unique up to a rigid transformation. Discrete seamless similarity map.
A discrete map f : M c → R iscalled a discrete seamless similarity map [Campen and Zorin 2017]if there exists a similarity transform σ i ( x ) = s i R i x + t i , where R i is a rotation by some integer multiple of π , per cut edge i of M that identifies the two images of i under f , and these transformsfulfill a cycle condition : around each regular vertex of M the compo-sition of these transforms across incident cut edges (in consistentorientation, e.g. clockwise) is the identity. Dual cycle geodesic curvature.
Let γ s be a dual cycle of M , i.e. adirected cyclic triangle strip [Crane et al. 2010]. Let the n s trianglesforming this cycle be denoted, in sequence, T sm , m = . . . n s . Thetotal geodesic curvature κ s of the cycle γ s under G ′ can then beexpressed as κ s = Õ m = ... n s d sm α sm , (2)where α sm is the angle (under G ′ ) of the triangle T sm at the vertexthat is incident to both, the preceding and succeeding triangle intriangle strip T s , and the sign d m = ± γ r enclosing a single vertex r we can write more succinctly κ r = Θ r = Õ i ∈ N ( r ) α i , (3)where N ( r ) is the set of indices i of inner triangle angles α i incidentat r in M (cf. Figure 1 right), and Θ r as above. Dual cycle basis.
Let Γ = { γ , . . . , γ | V |− + д } be a set of dualcycles of M that forms a basis of all dual cycles; concretely, we canchoose | V | − д non-contractible dual cycles [Crane et al. 2010]. Holonomy signature.
The (holonomy) signature of a discrete map f is a collection of | V | − + д values k s , defined relative to a basis Γ as k s = κ s / π . Note that this signature captures the value κ for any dual cycle (via linear combination of κ values of basis cyclesthe cycle decomposes to). :2 • Marcel Campen and Denis Zorin An important observation is that, if the map f is a discrete seam-less similarity map, κ s is a multiple of π for any cycle γ s (as wellas Θ r for any vertex r ), i.e. the signature values k s are integer num-bers. Conceptually, the algorithm [Campen and Zorin 2017, Sec. 7.3] pro-ceeds by constructing a discrete closed 1-form ξ on M , such that theimplied parametrization satisfies holonomy constraints. For a suit-ably chosen cutgraph c , on the simply-connected domain M c thisclosed 1-form is exact and can be integrated, yielding a discrete 0-form ϕ on M c . This 0-form ϕ then induces G ′ via (1).We show the following:(1) if the angles α i under G ′ satisfy the set of signature depen-dent constraints (4), then G ′ defines a discrete conformalmap (up to rigid transformation) and this map is a seamlesssimilarity map with this given signature,(2) these angles α i under G ′ can be computed directly from ξ ;they are independent of the choice of the cutgraph and theconstant of integration,(3) the iterative algorithm [Campen and Zorin 2017, Sec. 7.3]is, in its core, in fact Newton’s algorithm applied to solvethe non-linear system of equations (4) in the variables ξ .Hence, if the algorithm converges, it yields a discrete 1-form ξ ,which gives rise to a discrete 0-form ϕ , that defines a discrete con-formally equivalent metric G ′ . This metric implies a discrete, piece-wise linear, seamless similarity map for M that satisfies the holo-nomy signature constraints. Next, we consider convergence:(4) there is an energy E associated with the mesh M , such thatthe constraints (4) are the components of this energy’s gra-dient, for a particular choice of basis for closed 1-formson M . This energy is convex.The use of a globally convergent variant of Newton’s method (withline search, trust region, etc.) [Dennis and Schnabel 1996, Ch. 6]would thus allow to guarantee convergence—unless edge flips cancause an issue (cf. Section 8). A discrete metric G ′ respects a given signature { ˆ κ s = k s π , ˆ Θ r = k r π } , if and only if κ s = ˆ κ s and Θ r = ˆ Θ r , i.e. the angles α i solvethis homogeneous system of equations: Õ i ∈ N ( r ) α i − ˆ Θ r = , for all vertices r Õ m = ... n s d sm α sm − ˆ κ s = , for all cycles s (4)If the cutgraph c is chosen such that all vertices r with ˆ Θ r , π ( irregular vertices) lie on the boundary of M c (which we can andwill always assume in the remainder), this metric G ′ is flat on M c and thus implies a map f to the plane. We can now show that if the discrete metric G ′ , in addition to sat-isfying (4), is conformally equivalent to G , then this map f is aconformal seamless similarity map. Proposition 1.
Suppose for a choice of discrete closed 1-form ξ on M , the 0-form ϕ on M c obtained by integration yields a discrete metric G ′ via (1) that respects the triangle inequality for each triangle. If thisdiscrete metric G ′ satisfies (4) , then it defines a discrete conformalseamless similarity map f of M c with the designated signature. Proof.
Let σ i ( x ) = s i R i x + t i be a similarity transform per cutedge i , uniquely determined by the images of the two copies of thisedge: it maps the one image onto the other. We show: around eachregular vertex, the composition of these transforms across incidentcut edges (in consistent order, e.g. counterclockwise) is the identity.We use the example vertex depicted in Figure 2 with three incidentcut edges a , b , c , thus transforms σ a , σ b , σ c . The common case oftwo incident cut edges follows directly as a special case, and gener-alization to even more incident cut edges is trivial.We note that ϕ a − ϕ = ϕ ′ a − ϕ ′ ( = ξ ( , a ) , as ξ is defined on M ), thus ϕ ′ a − ϕ a = ϕ ′ − ϕ . Therefore, by the above definition of edge lengthsunder f , for the scale factor s a we have ln s a = ϕ ′ − ϕ = ϕ ′ a − ϕ a .Analogously, we get ln s b = ϕ ′′ − ϕ ′ and ln s c = ϕ − ϕ ′′ . Hence,ln s a + ln s b + ln s c =
0, thus s a s b s c = D f be the differential of f . Then for the rotations we have R a D f (® a ) = D f (® a ′ ) , R b D f (® b ) = D f (® b ′ ) , R c D f (® c ) = D f (® c ′ ) . Fur-thermore, let rotations R a ′ b , R b ′ c , R c ′ a be defined via R a ′ b D f ( ® a ′ ) = D f (® b ) and so forth. Then R c ′ a R c R b ′ c R b R a ′ b R a D f (® a ) = D f (® a ) , thusit follows that ( R c ′ a R b ′ c R a ′ b )( R c R b R a ) = I . Due to first set of con-ditions in (4), we know that at regular vertices angles under G ′ sumto 2 π , thus R c ′ a R b ′ c R a ′ b = I . It follows that R c R b R a = I .Finally, as σ c ◦ σ b ◦ σ a f ( p ) = f ( p ) , and σ c ◦ σ b ◦ σ a f ( p ) = s a s b s c R a R b R c f ( p ) + R a R b t c + R a t b + t a = f ( p ) + R a R b t c + R a t b + t a ,we conclude that R a R b t c + R a t b + t a =
0, thus σ c ◦ σ b ◦ σ a = I .Notice that in the special case of a vertex with two incident cutedges, the transforms on both edges are the same (or, in cyclic ori-entation, inverses of each other). Thus along a branch of the cutthe transition is constant. That the rotational part of these constanttransforms is a multiple of π / γ (possibly composed ϕ c ϕ ′ c ϕ ′′ ϕ ϕ ′ cb ® a ® a ′ ϕ a ϕ ′ a ϕ ′ b ϕ b p Fig. 2. Example cut configuration around a regular vertex. Cuts are bold. n Discrete Conformal Seamless Similarity Maps • 1:3 from basis cycles) that crosses the cut: the rotation between the twoimages of the cut edge is ˆ κ γ .We conclude that, as the transforms σ are similarities that iden-tify the cut images by definition and satisfy the cycle condition, themap f obtained from ξ is a discrete seamless similarity map.As G ′ satisfies (4), this resulting map has the desired signature. (cid:3) α FROM 1-FORM ξ The angles α i non-linearly depend on the edge lengths l i of G ′ ,which in turn depend on ϕ , which in turn depends, due to an arbi-trary choice of cut and arbitrary constant of integration not evenuniquely, on ξ . However, we observe that the angles α i themselvesare uniquely defined by ξ (independent of the cut and the constantof integration). This allows us to express (4) directly in terms of the1-form ξ . Notation.
We consider closed discrete 1-forms ξ on M . We num-ber directed halfedges on the mesh, and associate a variable ξ i witheach halfedge. We use the following notation: for a halfedge i , i ′ isthe sibling halfedge, corresponding to the same edge; h ( i ) and t ( i ) are the halfedge head and tail vertices, respectively. Each halfedge iii jj kk belongs to a unique triangle, it is oriented coun-terclockwise, and the opposite angle in this trian-gle is denoted α i . For a value assignment ξ to de-fine a 1-form, we require ξ i ′ = − ξ i . The 1-formis closed if for a triangle with halfedges ( i , j , k ) itholds ξ i + ξ j + ξ k = Proposition 2.
Suppose a discrete 0-form ϕ is computed by inte-gration of a closed 1-form ξ on M c , so that ξ i = ϕ k − ϕ j on eachtriangle ( i , j , k ) , and the edge lengths l i = l Gi e ( ϕ j + ϕ k )/ satisfy thetriangle inequality for every triangle.. Then the angles α i implied bythese edge lengths l depend on the 1-form ξ only, not on the choice ofthe cut or the constant of integration. Proof.
For M c , the (double) logarithmic edge lengths λ i are de-fined by λ i = l i = λ Gi + ϕ j + ϕ k , with λ Gi = l Gi .Furthermore, define the value µ i = λ Gi + ( ξ k − ξ j ) . We have µ i = λ Gi + ϕ j + ϕ k − s = λ i − s , where s = ( ϕ i + ϕ j + ϕ k ) . Inother words, µ i (computed from ξ ) differs from λ i (computed from ϕ ) by − s , a common scaling factor for all three edges of a triangle.As uniformly scaling a triangle does not change its angles, α i canbe computed from the edge lengths e µ i , e µ j , e µ k (which form atriangle that is similar to the triangle formed by l i , l j , l k ). (cid:3) Let L ( ξ ) = L is a nonlinear function of ξ .Newton’s method for finding a solution to L ( ξ ) = J L and, starting from the initialization ξ =
0, proceedsby iterating: ξ n + ← ξ n − J L ( ξ n ) − L ( ξ n ) . (5)We compare this to the central line of the iterative algorithm givenin [Campen and Zorin 2017, Sec. 7.3] ξ ← A ( G ′ ) − b ( G ′ ) , (6) (where ξ n vanishes because it is ’baked’ into the metric G ′ aftereach step by method rescale , as a merely technical difference fornotational simplicity), and show in the following that the matrix A used in the algorithm equals J L , and the vector b equals − L . Proposition 3.
The iterative algorithm of [Campen and Zorin 2017,Sec. 7.3] is (except for the degeneracy handling) Newton’s method ap-plied to the nonlinear system (4) in the variables ξ . Thus, as estab-lished by the previous propositions, if it converges, the resulting ξ satisfies (4) and, for any choice of cut, defines a discrete conformalseamless similarity map with the given signature. Proof.
Observe, following the derivation of [Springborn et al.2008, Eq. 14], that for a triangle ( i , j , k ) the first-order change in α i relative to ξ is dα i =
12 cot α k dξ k −
12 cot α j dξ j Thus, the linearized form of the first type of constraint in (4) canbe written as Õ i ∈ N ( r )
12 cot α k ∆ ξ k −
12 cot α j ∆ ξ j = ˆ Θ r − Θ r , for every r , (7)where j , k are the other two halfedge indices in the triangle con-taining halfedge i . Note that ∆ ξ here denotes the Newton step, thechange in the 1-form ξ that is solved for in each iteration of (5)(called ξ in (6), because ξ n = O ( r ) be theset of outgoing halfedges at vertex r , i.e. halfedges i with t ( i ) = r .With this and ξ i = − ξ i ′ , (7) can be rearranged to Õ i ∈ O ( r ) ( cot α i + cot α i ′ ) ∆ ξ i = ˆ Θ r − Θ r , for every r . (7 ′ )Similarly, the second type of constraints in (4) is linearized as Õ m = ... n s d sm ( cot α k ∆ ξ k − cot α j ∆ ξ j ) = ˆ κ s − κ s , for every s , (8)where j , k are the two halfedge indices in triangle T sm not oppositeto angle α sm . Let E ( s ) denote the set of all halfedges between succes-sive triangles T sm , T sm + n s pointing consistently from the leftboundary of the strip to the right boundary (in accordance withthe definition of the sign d sm ). With this, and grouping factors perhalfedge in E ( s ) , we can rearrange (8) to Õ i ∈ E ( s ) ( cot α i + cot α i ′ ) ∆ ξ i = ˆ κ s − κ s , for every v . (8 ′ )Realizing that, in terms of [Campen and Zorin 2017], κ s = κ tot [ γ ds ] and Θ r = π − K G [ r ] , one observes that (7 ′ ) and (8 ′ ) equal equationsystem 6 in [Campen and Zorin 2017], which defines A and b . Thus(6) is indeed a Newton step for solving (4). (cid:3) Analogously to the simply-connected case [Springborn et al. 2008](where only elementary dual cycles, i.e. constraints of the first typein (4) play a role), L , the vector of constraint left-hand sides, canbe shown to be a gradient of a convex function E . This suggeststhat global convergence of the algorithm can be ensured if one :4 • Marcel Campen and Denis Zorin augments it with a line search (or other global convergence tech-niques, such as a trust region approach [Dennis and Schnabel 1996,Ch. 6])—assuming triangle inequalities do not get violated (cf. Sec-tion 8).To show this, we introduce an auxiliary variable ψ i per halfedge,satisfying ψ i = ( ξ k − ξ j ) in the triangle ( i , j , k ) . Note that then µ i = λ Gi + ψ i . We also observe that, due to closedness, ξ i = ( ψ l − ψ k ) .We define a triangle function д ( ψ i , ψ j , ψ k ) , similar to the func-tion f defined in [Springborn et al. 2008, Eq. 8] involving Milnor’sLobachevsky function Λ , but depending on the values of ψ (andthus, by a linear change of variable, ξ ) only. д ( ψ i , ψ j , ψ k ) = Õ ℓ ∈ i , j , k ( λ G ℓ + ψ ℓ ) α ℓ + Λ ( α ℓ ) , (9)It differs from the function f by a linear change of variables andaddition of a linear function, thus, it is convex as well. This functionhas the following property, similar to the corresponding propertyof f : ∂ ψ i д ( ψ i , ψ j , ψ k ) = α i (10)with similar equations for derivatives with respect to ψ j and ψ k obtained by cyclic permutation of ( i , j , k ) . This property makes itpossible to formulate conditions on angles (such as in (4)) as com-ponents of the gradient of an energy constructed from д ( ξ i , ξ j , ξ k ) for individual triangles. Towards this goal we make the followinggeneral observation. General observation.
Suppose for an energy E = E ( x ) , with thevariables x satisfying constraints Cx =
0, we make a change ofvariables x = Py , where the variables y are independent, i.e., thecolumns of P are a basis of the null space of C . Then ∇ y E ( Py ) = P T ∇ x E ( x ) . Let P + = ( P T P ) − P T be the pseudoinverse of P . Then y = P + x .Consider an augmented energy E ′ ( x ) = E ( x ) − b T P + x = E ( Py ) − b T y . Then ∇ y E ′ = P T ∇ x E − b . Let p i be the columns of P (rowsof P T ). We conclude that for any choice of basis y in the null-spaceof the constraint matrix C , the value of x ∗ corresponding to an ex-tremum of E ( x ) , satisfies the constraints p i ∇ x E ( x ∗ ) − b i = Basis change.
We define a basis for closed 1-forms w vr , w ℓ s , r = . . . | V | − s = . . . д , jointly denoted by w t , t = . . . | V | + д − w vr is obtained by applying the discrete exterior derivative to thehat function centered at a vertex r , i.e. each halfedge i with h ( i ) = r is assigned value 1, and t ( i ) = r is assigned − γ ds , w ℓ s is defined by setting each interior halfedgeof the cycle to 1 if it crosses the midline from right to left (rela-tive to the direction of the cycle), and − w ℓ s . Together, the vertex and cycle basis forms are denoted w t , t = . . . | V | − + д .For a closed 1-form ξ , we denote the coefficients of the form rep-resented in this basis by y t , forming a vector y , ξ = W y , where W is the matrix with columns w t .We also consider the basis z i , i = . . . | F | , with basis formsassociated with vertices, corresponding to the variables ψ i , which z i w vr w ℓ s Fig. 3. Illustration of employed basis elements for closed 1-forms, w ℓ s and w vr , as well as basis elements z i for ψ . have − j , 1 on the halfedge k and 0 elsewhere.Then ξ = Zψ , where Z is the matrix with columns z i .We observe that w vr = Í i ∈ N ( r ) z i , and w ℓ s = Í m = m ℓ d ℓ m z ℓ m , with d ℓ m defined as above, and z ℓ m are the basis functions correspondingto vertices on the boundary of the cycle triangle strip T ℓ m . Noticethat with these basis choices, the coefficients for expressing w t interms of z i are exactly the coefficients in (4), expressing total anglesin terms of angles α i .In a general form, we write these relations as w t = Zp t , i.e., W = ZP , where the columns of P are p i . From this we infer that ξ = W y = ZPy ; comparing to ξ = Zψ , we conclude that ψ = Py .These definitions allow us to formulate the following proposi-tion: Proposition 4.
Define the energy E ( ψ ) = Õ T ijk д ( ψ i , ψ j , ψ k ) , and let E ′ ( ψ ) = E ( ψ ) − b T P + ψ , where the components of b t of b are,for t = . . . | V |− , the target angle sums ˆ Θ t at vertices, excluding one(implied by the theorem of Gauss-Bonnet), and for t = | V | . . . | V | + д − , the target geodesic curvatures ˆ κ t on dual cycles γ t −| V | + . Thematrix P + is the pseudoinverse of P formed by the coefficient vectors p t of the closed 1-form basis y t , t = . . . | V | − + д , expressed interms of the basis z i defined above. Then the left-hand-sides of (4) arethe components of ∇ ψ E . Proof.
Using the relation ψ = Py , we rewrite our energy as E ′ ( ψ ) = E ( Py )− b T y , and computing the gradient yields P T ∇ ψ E − b . As the columns p t of P coincide with the coefficients of (4), and ∂ ψ i E = ∂ ψ i д T ( i ) = α i , where д T = д ( ψ i , ψ j , ψ k ) for the triangle T ( i ) containing the halfedge i , each row has the form p t · α − b t , t = . . . V − + д , which is exactly the left-hand sides of (4). (cid:3) In our analysis we have generally assumed that the discrete metric G ′ (at intermediate steps as well as in the end) respects the triangleinequality for every triangle of M c ; otherwise angles α are not well-defined.[Springborn et al. 2008] extend definitions of gradient and Hes-sian to violating states, thereby allowing their algorithm to proceedeven if the triangle inequality is violated at an intermediate step. If,however, the triangle inequality is violated in the end, the metric n Discrete Conformal Seamless Similarity Maps • 1:5 G ′ does not well-define a map. Note that, depending on the meshconnectivity and the desired signature, such violations can be in-herent rather than be artifacts of the algorithm; no discrete mapwith the desired properties might exist on a given mesh. Preventing violations
In [Campen and Zorin 2017] a strategy is used that prevents vio-lations, instead of extending the energy definition to cover them,and that modifies the mesh connectivity where necessary. In a waysimilar to a line search strategy, the Newton step is truncated if itwould lead to a violation, such that instead a degenerate configura-tion is obtained (that necessarily occurs before a violation occurs),which is immediately resolved by an edge flip. This closely followsa continuous technique proposed by [Luo 2004] to deal with singu-larities of the discrete Yamabe flow.
Effect on convergence
These edge flips are performed intrinsically, i.e. values Θ r and κ s ,and thus the energy E , are preserved. If finitely many edge flipsoccur in the course of the algorithm, it obviously proceeds withnormal Newton steps after the last one, converging if a globallyconvergent variant is employed.Questions left to be answered are: • can an infinite sequence of edge flips occur? (note, e.g., thatone cannot guarantee meeting Wolfe’s curvature condition ifoccurring degeneracies can truncate the steps in an arbitrarymanner). The same question applies to discrete Yamabe flowwith edge flips, as discussed in [Luo 2004]. • can two adjacent triangles degenerate simultaneously in sucha manner that the effected intrinsic edge flip leads to an edgeof zero length? REFERENCES
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