Octahedral Frames for Feature-Aligned Cross-Fields
Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, Justin Solomon
OOctahedral Frames for Feature-Aligned Cross-Fields
PAUL ZHANG,
Massachusetts Institute of Technology, USA
JOSH VEKHTER,
University of Texas at Austin, USA
EDWARD CHIEN,
Massachusetts Institute of Technology, USA
DAVID BOMMES,
University of Bern, Switzerland
ETIENNE VOUGA,
University of Texas at Austin, USA
JUSTIN SOLOMON,
Massachusetts Institute of Technology, USA
Fig. 1. A variety of feature-aligned cross fields computed using our novel cross field formulation.
We present a method for designing smooth cross fields on surfaces that auto-matically align to sharp features of an underlying geometry. Our approachintroduces a novel class of energies based on a representation of cross fieldsin the spherical harmonic basis. We provide theoretical analysis of theseenergies in the smooth setting, showing that they penalize deviations fromsurface creases while otherwise promoting intrinsically smooth fields. Wedemonstrate the applicability of our method to quad-meshing and includean extensive benchmark comparing our fields to other automatic approachesfor generating feature-aligned cross fields on triangle meshes.CCS Concepts: •
Computing methodologies → Mesh models ; Shapeanalysis .Additional Key Words and Phrases: discrete differential geometry, geometryprocessing, total variation, singularities, feature alignment
Authors’ addresses: Paul Zhang, Massachusetts Institute of Technology, 77 Mas-sachusetts Avenue, Cambridge, MA, 02139, USA, [email protected]; Josh Vekhter, Uni-versity of Texas at Austin, Austin, TX, 78712, USA, [email protected]; Edward Chien,Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA,02139, USA, [email protected]; David Bommes, University of Bern, Hochschul-strasse 6, Bern, 3012, Switzerland, [email protected]; Etienne Vouga, Univer-sity of Texas at Austin, Austin, TX, 78712, USA, [email protected]; Justin Solomon,Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA,02139, USA, [email protected] to make digital or hard copies of part or all of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).© 2020 Copyright held by the owner/author(s).0730-0301/2020/1-ART1https://doi.org/10.1145/3374209
ACM Reference Format:
Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga,and Justin Solomon. 2020. Octahedral Frames for Feature-Aligned Cross-Fields.
ACM Trans. Graph.
1, 1, Article 1 (January 2020), 13 pages. https://doi.org/10.1145/3374209 N -rotationally symmetric (RoSy) tangential vector fields over sur-faces are ubiquitous in computer graphics. 2-RoSy fields can be usedto generate stripe patterns due to their ambivalence to rotation by π about the normal. 4-RoSy fields (cross fields) are heavily used inboth surface parameterization and quadrilateral (quad) meshing,thanks to their symmetry with respect to rotations by π about thesurface normal.Depending on the application, n -RoSy field design algorithmsmust trade off between several desirable properties of the field.In almost all cases, n -RoSy fields are expected to be as smooth aspossible. For surfaces with boundary, constraints on how the fieldaligns to the boundary are common, and for artistic applications,users may wish to prescribe a sparse set of streamlines that the fieldmust follow. For meshing applications, alignment of n -RoSy fieldsto salient geometric features is also desirable as a means to identifyor preserve mesh detail. Our focus will be on improving this latteraspect for the important case of 4-RoSy fields.There are two broad strategies for achieving feature alignment.The first is to optimize only for smoothness, under the assumptionthat a well-chosen functional for measuring cross-field smoothnesswill automatically penalize fields that fail to align to geometric fea-tures. The most commonly-used smoothness functionals (including ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. a r X i v : . [ c s . G R ] J u l :2 • Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon Fig. 2. Two surfaces (the three-cylinder-intersection and wavy-box )whose maximal curvature directions (blue lines) contradict its feature curves(red lines). the Dirichlet energy and its variants) are intrinsic , and recover so-lutions that are unique only up to rotation [Knöppel et al. 2013].These are ambivalent to isometric deformations of the surface andignore extrinsic features such as creased folds.An alternative strategy is to include energy terms that explicitlyenforce alignment to an input guiding field of principal curvaturedirections during cross-field design [Brandt et al. 2018; Knöppelet al. 2013]. Drawbacks include the difficulty of robustly computingprincipal curvature directions on noisy meshes, the fact that forcingalignment to a guiding field based on local geometry may excludecross-field designs that are globally more optimal, and more criticallythe fact that principal curvature directions are often different fromfeatures e.g. Figure 2.Our main observation is that neither of the above strategiesadequately identify those features most important to generatinghigh-quality quad meshes. Often the surface being modelled is con-structed from smooth patches that are joined along sharp extrin-sic feature curves where the normal direction is discontinuous orchanges rapidly. On the one hand, such features are invisible tointrinsic smoothness functionals; on the other, the orientation ofthe feature curves often contradict that of nearby curvature lines.Consider the surfaces shown in Figure 2: Neither existing strategywill promote alignment to the features curves shown in red. Bothof these shapes are developable away from a sparse set of conesingularities at the corners; the gaussian curvature is nearly zero atcreased edges and curved facets, and so purely intrinsic approacheshave no hope of aligning to the creases. Augmenting with a guid-ing field based on extrinsic curvature is counter-productive, as thecurvature lines (blue) are not compatible with the surface’s moreimportant crease features curves (red).We approach feature alignment in a new way, which detects andaligns cross fields to sharp features in a stable fashion. Our methodis based on an extrinsic representation of cross fields using spher-ical harmonic (SH) basis functions. SH functions have been usedsuccessfully in volumetric octahedral field problems for hexahedralmeshing [Huang et al. 2011; Ray et al. 2016; Solomon et al. 2017],and we argue that this representation is well-suited not only forcomputing octahedral fields in volumes but also for field compu-tation on surfaces. In particular, we apply an SH representationof octahedral frames, or frames of three orthogonal directions in R , proposed by Huang et al. [2011]; when one of its directions isconstrained to the surface normal, it exhibits the same symmetry as a two-dimensional cross. We use this fact to devise a class of crossfield energies that promote intrinsic smoothness in smooth regionsof the surface. Over sharp creases, however, our energy aligns thefield to the crease direction, achieving automatic feature alignmentwithout the need for explicit computation of extrinsic curvaturedirections or feature curves. Contributions.
In this work, we • introduce spherical harmonic functions for the computation ofsurface cross fields; • propose a family of field smoothness energies whose optima arefeature-aligned cross fields; • provide a theoretical analysis of the behavior of a few importantmembers of this family; and • introduce cross fields with soft normal alignment for increasedversatility/robustness.Our approach is able to extract feature-aligned fields with compara-ble levels of efficiency to those of purely intrinsic algorithms. Wetest our algorithm extensively on over 200 different meshes withresults in both §6 and in supplementary materials. We leverage ouralgorithm to produce feature-aligned cross fields and demonstratetheir usefulness for quad meshing. The generation of tangential n -RoSy fields over surfaces has manyapplications in computer graphics ranging from surface BRDF mod-ification [Brandt et al. 2018] to meshing [Bommes et al. 2009; Jakobet al. 2015] to texture synthesis [Knöppel et al. 2015] and sketch-based modeling [Bessmeltsev and Solomon 2019; Iarussi et al. 2015].Surveys of n -RoSy field design methods are provided in [Vaxmanet al. 2016] and [de Goes et al. 2015]. Cross fields ( n =
4) have been especially well-studied since their π -symmetry allows them to behave like local coordinate systems,resulting in intuitive seamless surface parameterizations.Methods to compute intrinsically-smooth cross fields with align-ment and singularity constraints were studied by Ray et al. [2008],Crane et al. [2010] and Knöppel et al. [2013]. More similar to ourwork, Jakob et al. [2015] instead formulate an extrinsic smoothnessfunctional on cross fields, in an attempt to automatically align tosurface features. Their method penalizes an extrinsic distance be-tween neighboring crosses that does not use a shared tangent spaceor connection. The resulting energy is non-convex but is minimizedto local optimality, often resulting in more singularities than neces-sary. Huang and Ju [2016] analyze this extrinsic energy, and findthat it can be decomposed into an energy expressed in terms ofintrinsic twisting and alignment to extrinsic curvature directions.We perform a similar analysis of the energy we introduce in thesupplemental documents.When using cross fields for quad mesh parameterization or pro-cessing, methods [Bommes et al. 2009; Brandt et al. 2018; Campenet al. 2016; Knöppel et al. 2013] often promote feature alignmentby including a loss term penalizing disagreement with curvaturedirections. However, as we argue in the introduction and illustratein Figure 2, alignment to curvature directions is often less important ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. ctahedral Frames for Feature-Aligned Cross-Fields • 1:3 than alignment to sharp creases. Other parameterization methodssuch as [Bommes et al. 2013, 2009; Campen et al. 2015] allow featurealignment but just assume that such feature curves are provided asinput.
The three-dimensional generalization of a cross field is an octahedralfield . Octahedral fields are often used in volumetric problems likehexahedral meshing [Nieser et al. 2011]. A single octahedral frameconsists of three mutually-orthogonal vectors and their negations.Huang et al. [2011] introduced a particularly convenient represen-tation of an octahedral frame as a rotation of the spherical function д ( x , y , z ) : ( x , y , z ) ∈ S (cid:55)→ x + y + z , encoded by coefficients inthe spherical harmonic (SH) basis. Ray et al. [2016] used this repre-sentation to generate volumetric normal-aligned octahedral fields,and Solomon et al. [2017] combined the SH representation withthe boundary element method to remove the need for a volumetricmesh. Both of these methods use normal-alignment constraints atthe surface to enforce alignment of the octahedral frame with thevolume’s boundary.While there is no canonical three-dimensional generalization ofarbitrary n -RoSy fields, the spherical harmonic representation al-lows for frames that mimic the symmetries of all platonic solids [Shenet al. 2016], including octahedral fields [Corman and Crane 2019;Liu et al. 2018; Solomon et al. 2017]. Algebraic characterization ofthe orbit of д ( x , y , z ) under the space of rotations as a subset of allpossible SH coefficients was presented by Palmer et al. [2019] andChemin et al. [2018]. Since our formulation relies heavily on both the spherical harmonicrepresentation of octahedral frames and vectorial total variation,we present a preliminary introduction to these topics. e [ v ] −→−→ e v · L rotation of octahedral frameAs introduced by Huang et al.[2011], the canonical axis-alignedoctahedral frame can be repre-sented by spherical harmonics asa function д : S → R writtenas д = (cid:113) Y + (cid:113) Y , where Y lm denotes the basis for realspherical harmonics. The function д can be understood as the scaled projection of x + y + z ontothe fourth band ( l =
4) of spherical harmonics. Written differently,we can encode д as a vector of coefficients in the full basis offourth-band spherical harmonics Y (− ) , . . . , Y : f = (cid:34) , , , , (cid:114) , , , , (cid:114) (cid:35) T . The space of octahedral frames can be described as all rotationsof the canonical octahedral frame, that is, the orbit of f under thegroup of 3D rotations SO ( ) [Palmer et al. 2019, Definition 3.5]. Wewrite this via exponentiation of the Lie algebra elements: the set of octahedral frames is V = (cid:110) f (cid:12)(cid:12)(cid:12) there exists v ∈ R with f = e v · L f (cid:111) , where v · L = v x L x + v y L y + v z L z and L x , L y , L z are the angularmomentum operators expressed in the basis of band-four sphericalharmonics. In this basis, L x , L y , L z are each 9 × v can be interpreted as an axis-angle representation of rotation, with corresponding rotation matrix e [ v ] . [ v ] denotes the skew-symmetric matrix that acts as [ v ] u = v × u .Accordingly, e v · L д encodes the octahedral frame whose directionsare ˆ x , ˆ y , ˆ z rotated by e [ v ] , whereˆdenotes normalization (see insetabove). ++= SH frame as sum of three lobes
Using such SH ro-tations, we can presentan alternative inter-pretation of the octa-hedral frame f as thesum of three orthog-onal SH lobe-shapedfunctions. The z -alignedlobe is l = [ , , , , (cid:113) , , , , ] and is depicted in the inset. Lobescan be rotated in the same way that frames can i.e. by applying e v · L .The canonical octahedral frame can therefore be equivalently ex-pressed as f = l + e π L x l + e π L y l .The space of octahedral frames that are aligned to a unit vector ˆ n can be described by the set (cid:110) e v n · L e θ L z f (cid:12)(cid:12)(cid:12) θ ∈ S (cid:111) , where v n is any axis-angle rotation taking ˆ z to ˆ n , (e.g., the vectorparallel to ˆ z × ˆ n and has magnitude equal to the angle from ˆ z to ˆ n )and θ encodes an additional twist of the frame about ˆ n . The firstrotation about ˆ z can be written in explicit form [Huang et al. 2011]as e θ L z f = (cid:34)(cid:114)
512 cos 4 θ , , , , (cid:114) , , , , (cid:114)
512 sin 4 θ (cid:35) T . The above allows us to formulate the set of all octahedral frames д aligned to a given direction ˆ n in terms of two constraints: ∥ f ∥ = , W n f = u = (cid:34) , , , (cid:114) , , , (cid:35) T , (1)where W n is the second through eighth rows of e − v n · L . The linearconstraint rotates the frame from normal alignment to ˆ z alignment,and the norm constraint ensures that the first and last componentsare of the appropriate form.Lastly, we will make use of the projection operator π V : R → V onto the space of octahedral frames V , as defined in [Palmer et al.2019, §5.5]. We will later make use of a total variation energy (amongst others) toanalyze the behavior of our cross fields on creased surfaces. Here, weintroduce total variation and vectorial total variation definitions in
ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. :4 • Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon R n and provide intuition about their use. The extension to functionson a Riemannian manifold is straightforward, using the standardintrinsic gradient and divergence operators.The total variation of a differentiable scalar function h : Ω → R is TV [ h ] = ∫ Ω ∥∇ h ∥ dA where Ω ⊂ R n [Ambrosio et al. 2000]. Fornon-differentiable h , the relevant definition is: TV [ h ] = sup ϕ ∈ C c , ∀ x ∥ ϕ ( x )∥ ≤ (cid:18)∫ Ω h ∇ · ϕ dA (cid:19) , where C c denotes differentiable, compactly-supported vector fields.For smooth h , equivalence to ∫ Ω ∥∇ h ∥ follows from integrationby parts and Stokes’s theorem. In this case, the maximizing ϕ is −∇ h / ∥∇ h ∥ . If h is the indicator function of a suitably regular (e.g.,non-fractal) subset A ⊂ Ω , then TV [ h ] is the perimeter of A .When h : Ω → R m is vector-valued rather than scalar-valued,there are many different definitions for the vectorial total variation VTV [ h ] [Sapiro 1996]. We use one proposed by Di Zenzo [1986],which for differentiable h is given by VTV [ h ] = ∫ Ω ∥∇ h ∥ F dA , where ∥ · ∥ F is the Frobenius norm. More generally, we can take VTV [ h ] = sup ϕ ∈ C c , ∀ x ∥ ϕ ( x )∥ F ≤ (cid:32) m (cid:213) i = ∫ Ω h i ∇ · ϕ i (cid:33) , (2)where h = ( h , h , . . . , h m ) , and ϕ = ( ϕ , ϕ , . . . ϕ m ) is a differen-tiable, compactly-supported m -tuple of vector fields. This definitionis not equivalent to a sum of m independent scalar total variations:The constraint on ϕ introduces nontrivial coupling between thedimensions. This definition of total vectorial variation is consideredin the case where Ω is a surface in R by Bresson and Chan [2008],but without specific analysis for discontinuous h . We use normal-aligned octahedral fields to encode tangent crossfields on surfaces, with the goal of computing a smooth cross fieldon a surface aligned to sharp features. The SH representation willenable us to capture features even when they are purely extrinsic. Tothis end, our next task is to define a means of measuring smoothnessby examining the gradient of a SH field along the surface.
To calculate ∥∇ f ∥ , we first express it in an appropriate local coor-dinate system that simplifies the formulas in coordinates and betterreveals the structure. Following the notation in §3.1, an octahedralfield f ( r ) : Ω → V ⊂ R can be parameterized relative to a point r ∗ by v ( r ) : Ω → R , where v ( r ) is the axis-angle rotation from f ( r ∗ ) to f ( r ) . This implies that v ( r ∗ ) = [ , , ] . Without loss ofgenerality, we rotate the surface so that the normal of Ω at r ∗ isˆ z . We can then compute the gradient ∇ f at the point r ∗ from theformula f ( r ) = e v ( r )· L f ( r ∗ ) : ∇ f ( r )| r ∗ = | | | L x f ( r ∗ ) L y f ( r ∗ ) L z f ( r ∗ )| | | | | |∇ x v ∇ y v ∇ z v | | | r ∗ . (3) As the field f ( r ) encodes an extrinsically embedded frame at eachpoint, we take the gradient ∇ to be the component-wise derivativeof the field’s nine scalar functions rather than a covariant or Liederivative along the surface in order to capture the extrinsic geome-try of the surface. We use [Rossmann 2002, §1.2.5] and the fact that v ( r ∗ ) = [ , , ] to derive Equation (3).By combining facts about the SH representation and standardresults in differential geometry we show that the squared norm ∥∇ f ( r )∥ at r ∗ can then be expressed in the following more intuitiveway:Proposition 4.1. Let f ( r ) : Ω → V ⊂ R be a normal-alignedoctahedral field over a smooth surface Ω . Then at every point r ∗ ∈ Ω , ∥∇ f ∥ F = k + k + w , where k and k are the principal curvaturesand w measures the intrinsic tangential twist of the octahedral field.Using mean and Gauss curvatures H and K , we can write ∥∇ f ∥ F = H − K + w . We leave the full proof of this formula to supplementary materials.Proposition 4.1 gives a more intuitive form for Equation (3) andrelates the spherical harmonic representation of an octahedral frameto properties of the frame it represents. Most notably, the Dirichletenergy of the SH representation can be effectively decoupled intoextrinsic dependence of ∥∇ f ∥ F on the surface Ω and the intrinsictangential twisting of the normal-aligned octahedral field f ( r ) . Thevalues of H and K simply contribute a fixed quantity dependingon Ω rather than the field. Therefore, the influence of f on ∥∇ f ∥ F is just in w , the intrinsic twist of the cross field it represents. Westress that this behavior is quite different from the behavior ofthe component-wise derivative evaluated on vectors, as studiedin [Huang and Ju 2016], where their smoothness energy promotesalignment to extrinsic curvature directions. L p Smoothness Energy of SH Cross Fields
Suppose we wish to measure smoothness of a normal-aligned octa-hedral field in the SH representation. We define the following classof convex smoothness energies using the L p -norm of ∥∇ f ∥ F overthe surface Ω for p ≥ E p ( Ω , f ) = (cid:18)∫ Ω ∥∇ f ∥ pF dA (cid:19) p . (4)We now analyze the behavior of the E p energy for cross fields inseveral select cases. p = : Dirichlet Energy. We begin with a common choicein geometry processing when smoothness is desirable: the Dirichletenergy E . Given Proposition 4.1, we can write the Dirichlet energyas ∫ Ω H − K + w . Since H and K are independent of the octahe-dral field f , they have no influence over the f that minimizes E .Therefore on smooth Ω , we recover intrinsically smooth cross fields.Since the Dirichlet energy may diverge at singularities [Knöp-pel et al. 2013], this choice of energy has the theoretical drawbackof diverging for all f in the neighborhood of creases which breakoctahedral symmetry. In the discretized setting, however, the behav-ior of E is dependent on mesh resolution and empirically leads tostrong feature alignment as demonstrated in §6. It also leads to aneasily-solved optimization problem described in §5.2. ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. ctahedral Frames for Feature-Aligned Cross-Fields • 1:5 p = : Vectorial Total Variation. As noted in the previoussection, the conventional means of measuring field smoothness failsto be well-defined for our field representation on creased surfaces.We show here that the E energy is not only finite across sharpedges and around singular points but also provides an intuitivemeasure of field quality that captures both smoothness and featurealignment. It is also known as the vectorial total variation .Consider a function f : Ω → R that is piecewise smooth on n closed patches Ω j intersecting in a finite-length curve network Γ = (cid:208) sk = γ k , where each γ k is a C curve. Equivalently Γ = (cid:208) nj = ∂ Ω j ,and the vectorial total variation can be decomposed into integralsover each patch and Γ .Proposition 4.2. For compact Ω and f as above, VTV [ f ] is finiteand given by the following equation: VTV [ f ] = n (cid:213) j = ∫ ˚ Ω j ∥∇ f ∥ F dA + s (cid:213) k = ∫ γ k ∥ f + − f − ∥ dL , (5) where f + and f − refer to the limiting values of f on either side of γ k ,and ˚ Ω j denotes the interior of Ω j . The basic argument starts from Equation (2), splits it into integralsover the patches, applies integration by parts, and utilizes partitionsof unity to construct a maximizing sequence of ϕ ’s. The full argu-ment is contained in supplementary materials §2. An analogousresult, which applies to arbitrary functions on R n with boundedvariation, is contained in [Ambrosio et al. 2000], with the additionof a third term representing the Cantor part of f . Since our f ispiecewise smooth, however, we can safely ignore the Cantor part.The second term is often referred to as the jump part in the totalvariation literature.The formula (5) provides an intuitive description of the totalvariation of an octahedral field in the SH basis as a measure ofintrinsic smoothness with extra jump terms. Letting f represent anormal-aligned octahedral field we obtain: VTV [ f ] = n (cid:213) j = ∫ ˚ Ω j (cid:112) H − K + w dA + s (cid:213) k = ∫ γ k (cid:13)(cid:13) f + − f − (cid:13)(cid:13) dL (6) Generalizing to Creased Surfaces.
While the above result is de-rived for smooth surfaces Ω and discontinuous f , we can furthergeneralize the result to a surface Ω constructed from smooth openpatches Ω j joined along a network of sharp creases Γ = (cid:208) sk = γ k .As there is neither a consistent metric nor a consistent tangentspace on Ω across Γ , there is no well-defined choice of gradient. Wetherefore use equation (6) as the definition for E on such a creasedsurface. Since f is a normal-aligned octahedral field, it is necessarilydiscontinuous across creases, resulting in contributions to the jumpterm.The jump ∥ f + − f − ∥ , where f + and f − represent octahedralframes aligned to different normal directions, is minimized if f + and f − are both aligned to the axis of rotation from one normal tothe other. We formalize this property by Proposition 4.3Proposition 4.3. Let Ω + and Ω − be smooth patches of a surfacewith normal directions ˆ n + and ˆ n − that meet at a crease. Let ˆ d denote the intersection of their tangent spaces at the crease. Let f + θ and f − ϕ be the octahedral frames on either side of the crease aligned to ˆ n + and ˆ n − respectively. θ and ϕ denote their deviation from alignment to ˆ d .The cost ∥ f + θ − f − ϕ ∥ is minimized by θ = ϕ = . f − ϕ ˆ n − ˆ n + ˆ d f + ϕ Ω − Ω + Octahedral frames near crease
Proof of this proposition isleft to supplementary materi-als §3.The setup is depicted on theright, showing discontinuousnormal directions ˆ n + and ˆ n − as the left and right red ar-rows respectively. The creasedirection ˆ d is shown by themiddle red arrow. We emphasize that this proposition implies (lo-cally) crease alignment always minimizes the VTV. We extensivelytest and show in supplemental materials that this crease alignmenttends to globally hold on surfaces with complicated geometry andtopology as well. p ≥ . By Equation 4, and Proposition 4.1, E p incen-tivizes intrinsic smoothness for all p on smooth domains. On creaseddomains, we have demonstrated (local) crease-alignment for the p = p ≥
2, the value of E p diverges for a creased surface.However, we find empirically that minimizing E p (by recoveringsolutions to Equation 7) on a discretized surface leads to strongerfeature alignment as p increases. This behavior may be explained byProposition 4.3, which affects all edges regardless of p . The p simplyexponentiates the energy across each edge before accumulating itinto the total E p . Local to a single edge, the energy-minimizing con-figuration is unaffected by p . Based on our experiments, we furtherconjecture that the sequence of fields obtained by minimizing E on an increasingly dense discrete approximation of Ω convergesto a feature-aligned cross field. This intrinsically smooth feature-alignment is empirically shown in Figure 8. We leave proof of thisconjecture to future work. We achieve an additional prop-erty for all values of p through our use of SH octahedral frames.Consider the case of Ω being a cube: minimizers of E p will havezero energy, despite the cube’s sharp corners, since the field’s oc-tahedral symmetry allows it to simultaneously align to all threecreases at each corner. Effectively a surface with many angle- π turns and cube-corners can have just as low of an energy as onewith no creases at all. More generally, if Ω is a polycube surface, E p ( Ω , f ) = f to be a facet aligned uniform framefield. Our discussion above provides a new class of energies based onthe SH representation of cross fields, which naturally promote bothintrinsic smoothness and extrinsic crease alignment without theneed for feature curve detection or reliance on potentially noisylocal curvature estimates. For this reason, we propose solving the
ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. :6 • Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon following variational problem to find a cross field f ∗ over a surface: f ∗ = arg min f E p ( Ω , f ) subject to W n ( x ) f ( x ) = u . (7)Recall that the constraint encodes normal alignment of the framefield (see equation (1)). Some past algorithms have an extra || f ( x )|| = Ω . This constraint makes the problem nonconvex, andcauses the functional to diverge in the neighborhood of field singu-larities, which are unavoidable on generic surfaces by the Poincaré-Hopf Theorem. Accordingly, a relaxation is naturally required; wedrop the constraint ∥ f ( x )∥ =
1, yielding a convex problem withglobally-optimal solution. Dropping the ∥ f ( x )∥ = It is sometimes beneficial to relax the normal alignment constraint,e.g., in cases where the mesh contains sliver triangles with unstablenormal directions. In these cases, a smoother cross field can beobtained by deviating slightly from exact normal alignment. Thisrelaxation changes the optimization problem from Equation (7) intothe following: f ∗ = argmin f E p ( Ω , f ) subject to ∥ W n ( x ) f ( x ) − u ∥ ≤ ϵ . (8)This problem imposes a point-wise normal alignment constraintwith tolerance ϵ . When ϵ =
0, we recover the hard normal align-ment formulation (7). On the opposite side of the spectrum, as ϵ → ∥ u ∥ = (cid:113) ≈ .
76, the solution to (8) approaches a constantoctahedral field. This is the case where normal alignment has re-laxed so far that the octahedral frames are effectively unconstrained.Soft normal alignment: ϵ M a x i m u m d e v i a t i o n d e g r ee For values inbetween we per-form the follow-ing experimentto obtain a roughcorrespondence be-tween soft nor-mal alignment pa-rameter ϵ andmaximum angledeviation from nor-mal alignment: For each value of ϵ between 0 and . . ϵ -perturbations of a ˆ z − aligned frame, extract the frame they represent, and compute its maximum angle deviationfrom the ˆ z -axis. Results are shown in the inset.We highlight that this parameter encodes a point-wise constraintuniformly applied over the mesh. As such its interpretation does notchange with different meshes. Please see the supplemental materialsfor results on over 200 different meshes using a variety of values of ϵ . The benefit of soft normal alignment is demonstrated in Figure 5.Due to the influence of a sliver triangle in the buste mesh withunstable normal direction, the hard-normal-aligned cross field isforced to create a localized artifact. By using soft normal alignment,the sliver triangle’s unstable normal direction has less influenceover the resulting cross field, therefore increasing the quality of theresult. A similar benefit is demonstrated on the duck and armchair meshes shown in the supplementary materials.Additionally we test soft normal alignment on a cube-mesh withartificial noise added in Figure 7. With hard normal alignment thecross fields exhibit undesirable alignment to noise which increaseswith p . With soft-normal alignment, the cross fields show signifi-cantly decreased sensitivity to noise. Now we describe how to construct smooth cross-fields by numer-ically optimizing a discretization of E p . We assume the surface Ω has been triangulated into a manifold mesh M = ( V , E , F ) . Let n t be the normal direction of triangle t ∈ F . We represent a cross on M as a normally-aligned octahedral frame f t ∈ V ⊂ R per triangle.We use the shorthand f to denote the concatenation of all f t into asingle 9 | F | × V is a n v × n v is the number of vertices. E denotes the n e × n e is the number of edges. The energy E p can be discretizedas E p = (cid:32)(cid:213) e ∈ E w e ∥ f t − f t ∥ p (cid:33) p (9)where t and t are triangles adjacent to edge e , and w e are weightscorresponding to the dual Laplacian. We use w e = ∥ e ∥∥ e ∗ ∥ , where ∥ e ∥ is the length of edge e and ∥ e ∗ ∥ is the distance between barycentersof t and t .For ϵ =
0, the normal alignment constraint is discretized by thelinear constraint
W f = u , where W is a sparse block-diagonal matrixwith a block W n t for each triangle. It has dimensions 7 | F | × | F | . Thevector u is a repetition of u for each triangle, resulting in a 7 | F | × ϵ >
0, the normal alignment constraint is discretized bya second-order cone constraint: ∥ W n t f t − u ∥ ≤ ϵ per triangle.For the case p = Ω , our discretiza-tion agrees with the standard discretization of total variation inimage processing [Chambolle et al. 2010; Rudin et al. 1992]. When the mesh contains boundaries, we do not enforce any bound-ary conditions; in PDE parlance, this choice corresponds to “naturalboundary conditions.” While it is tempting to expect the naturalboundary conditions for p = ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. ctahedral Frames for Feature-Aligned Cross-Fields • 1:7 (a) hard normal-alignedstreamlines (b) hard normal-alignedcrosses (c) mesh normal direc-tions near sliver triangle(zoom × × buste mesh is shown with p = ∞ and varying normal alignment: ϵ = for the top figure and ϵ = . on the bottom. (a) Hard normal aligned streamlines; (b) magnified crossesshows a small patch of diagonal crosses in an otherwise regular region; (c)magnified triangle normals visualized with sliver triangle 4611 shaded inblue; (d) soft normal aligned streamlines; (e) magnified crosses no longershows diagonal artifacts; (f) extra-magnified triangle normals visualizedwith sliver triangle 4611 shaded in blue. While the normal direction of theregion points diagonally up and right, the sliver triangle’s normal directionpoints almost completely to the right. conditions [Stein et al. 2018], the SH representation vector is com-plicated by being constrained to a spatially varying linear subspace.We simply allow the cross on the boundary to be that which mini-mizes total energy. If desired, one can enforce a constraint that thecross field on the boundary be aligned to the boundary through themethod described in §5.4. (a) before (b) afterFig. 6. Octahedral fields obtained by minimizing E ∞ on the hand meshbefore and after adding manual direction. The manually-added streamlineis shown by the inset black arrow. This constraint removes a singularityfrom the original octahedral field. To support manual guidance of the octahedral frame field, we canprescribe alignment of the frame field to streamlines. Streamline con-straints combined with normal alignment result in a fully-determinedframe. Therefore, prescribing streamlines is equivalent to prescrib-ing the value of f t on a subset of triangles T p . Denote the prescribedoctahedral frame on triangle t as F t . We then add a new linearconstraint that ∀ t ∈ T p , f t = F t . (10)This technique is demonstrated in Figure 6. As a result of dropping the unit-norm constraint from Equation 7,we have no explicit guarantee that the tangential components ofoctahedral frames do not degenerate to zero. On a surface with acrease, however, the normal alignment constraint on one side of thecrease imposes that the magnitude of the tangential component onthe other side of the crease is close to one. As a result, we observeempirically that the vast majority of our octahedral frames do notdegenerate.In the case that octahedral frames do degenerate significantly,their norms can be too small to project robustly. We locate theseby using the octahedral projection from [Palmer et al. 2019] tomeasure the distance from f t to the octahedral variety V : d ( f t ) = ∥ π V ( f t ) − f t ∥ , and thresholding by d ( f t ) > . In its most general form, our problem formulation consists of mini-mizing a mixed-norm objective, with both linear and second-ordercone constraints. This results in a convex problem that we solvewith Mosek 9 [ApS 2017]. The normal alignment constraint becomes (cid:2) ϵ , ( W n t f t − u ) T (cid:3) ∈ L , where L is the 8-dimensional Lorentzcone. Likewise, the energy is formulated using a single p -norm cone.Our code is written in Matlab with a mex interface to Mosek; it builds ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. :8 • Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon p = , ϵ = p = , ϵ = p = , ϵ = p = , ϵ = p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . p = , ϵ = . Fig. 7. As ϵ increases or as p decreases, the cross fields become less sensitiveto noise added to the cube-mesh . cross-platform. Since our problem is convex, any dependence oninitialization would entirely be due to non-unique solutions, whichwe do not observe in practice. Furthermore, we use the interiorpoint method, which does not accept manual initialization. In thespecific case of ϵ = , p =
2, solving this optimization is equivalentto solving a linear system.
We begin with a comparison of the behavior of our energy fordifferent values of p . This experiment is depicted in Figure 9. We ob-serve that our cross fields naturally align to features with increasingstrength for higher p . In the case p =
1, our cross field is discontinu-ous over all creases, but while it is provably incentivized to align,it sometimes deviates due to the influence of neighboring creases,e.g. on the top surface of the fandisk. For p =
2, our cross fieldsachieve close alignment to the upper half of the shallow crease, aswell as alignment on the top face where the p = p = ∞ our fields align down the entirety of the shallow crease.While in theory, E p for p ≥ p = p = M e s h e s
100 Triangles 400 Triangles 2K Triangles 4K TrianglesFig. 8. On this developable surface, our cross fields are intrinsically smoothin the limit of refinement, but exhibit some mesh sensitivity on coarsemeshes, particularly for higher p values. They are crease-aligned for allresolutions. Note that the extrinsic curvature of the cylindrical bend has noeffect on the cross fields at higher resolutions. geometry in Figure 10 and observe that in all cases, we achievecrease alignment. Supplementary Materials.
In our supplementary document weperform an empirical study to evaluate the performance of ourmethod. We evaluate our method on a number of models drawnfrom the Thingi10k [Zhou and Jacobson 2016] dataset, as well as anumber of other commonly used benchmark models to demonstrateeffective crease alignment on real-world models. We also compareour approach to several baseline methods ([Brandt et al. 2018; Jakobet al. 2015; Knöppel et al. 2013]), by generating fields on the modelsin the “Robust Field-Aligned Global Parametrization” dataset [Myleset al. 2014], taking care to sample the relevant parameter space foreach formulation. While it is difficult to precisely quantify the qualityof a vector field, we highlight a number of cases where our methodrecovers fields which more faithfully conform to mesh features thanbaseline methods on real-world models.Our runtimes are shown for a set of meshes with 240 to 76K ver-tices and 480 to 152K faces in Figure 13. Runtimes naturally increasewith mesh size and appear to grow linearly with number of trianglesin our mesh test set. Memory costs are incurred to store a W n t pertriangle, a single u , w e per edge, and f t per triangle. Hence, storageis linear in size of the mesh. More detailed information regardingparameter choices and runtimes is provided in the supplementarymaterials. Table 1 shows a summary of our runtimes in comparisonto that of other methods. Our runtimes are on the same scale as[Knöppel et al. 2013] and to the bases setup step in [Brandt et al.2018]. ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. ctahedral Frames for Feature-Aligned Cross-Fields • 1:9 (a) Fandisk mesh (b) p = (c) p = (d) p = ∞ Fig. 9. Cross fields generated by minimizing E p for p = , , ∞ on the fandisk mesh. The shallow crease of the fandisk mesh is marked in red. Our crossfields naturally align to the shallow crease with increasing strength for higher p . Number of Bases Biharmonic Instant Globally Ourstriangles setup solve meshes optimal3K 5 .005 .026 .85 2.812K 24 .005 .053 20.46 15.05820K 44 .005 .080 21.913 25.89569K 170 .006 .141 62.733 135.0980K 181 .006 .222 71.15 112.3
Table 1. Runtimes in seconds for computing cross-fields using differentmethods on meshes with a varying number of triangles. Methods listed arethose of Brandt et al. [2018], Jakob et al. [2015], Knöppel et al. [2013], andour own. Runtimes for fields from [Brandt et al. 2018] are split into timeneeded for the setup of 500 bases eigenfields and the field computationseparately because of drastically differing timescales.
Comparison to Explicit Feature Curves.
Next, we compare ourfeature-aligned cross fields to those produced with the help ofexplicitly-computed feature curves. We obtain feature curves onthe 1904-triangle
Moai mesh from [Gehre et al. 2016, Fig. 9]. Wecompute a cross field with additional hard constraints as describedin §5.4 to enforce alignment to the precomputed feature curves. Wecompare the resulting field with and without explicit feature-curvealignment in Figure 11. While the feature curves help guide thecross field, just a few artifacts in the computed features drasticallyinfluence the resulting cross field to have more singularities andbe less smooth without clear benefit. The
Moai is shown from anangle where these differences are most pronounced.
Effect of Mesh Resolution on Crease-Alignment vs Extrinsic Curva-ture.
In this experiment we test on a geometry where a sharp creaseis mis-aligned to extrinsic curvature directions. We generate meshesof this geometry at varying resolution to see how crease-alignmentinteracts with extrinsic curvature. Results of this experiment aredepicted in Figure 8. As mesh resolution increases our cross fieldsbecome crease-aligned and intrinsically smooth, agreeing with thetheory. For very low mesh resolution, the cross fields are moresensitive to the underlying meshing pattern. (a) 1.3k (b) 5.4k (c) 21.8k (d) 11.6k (e) MultiresFig. 10. Cross fields generated by minimizing E on different meshings ofthe three-cylinder-intersection with number of faces indicated. Crossfield (d) is computed on the multi-resolution mesh (e). Notice that we obtainthe same feature-aligned cross field each time. Comparison to 3D Octahedral Fields.
Due to similarity of framerepresentation, we compare our method to surface cross fields ob-tained by optimizing a volumetric octahedral field. Algorithms likethose of [Huang et al. 2011; Ray et al. 2016] can generate surfacecross fields by approximating the surface with the limiting behaviorof a thin layer of tetrahedra or prism elements. However, prismelements are non-standard and both element types will be poorlyconditioned without introducing further restrictions such as zeronormal gradient to mimic a triangle mesh. We instead opt to com-pare with the Boundary Element Method (BEM) [Solomon et al.2017] which acts directly on surface triangle meshes. We use the2500 triangle fandisk mesh for this comparison. As observed earlier,our method has increasing feature alignment with increased valuesof p . In comparison, Figure 12 shows that the BEM field fully ignoresthe shallow crease of the fandisk , running through it at a 45 ◦ offset.Moreover, despite the fact that the BEM only needs boundary dataas input, its runtime is close to 50 times slower than ours. Challenging Test Cases.
We compare feature alignment of ourcross fields with that of existing methods on several meshes illustra-tive examples in Figure 14. As pointed out in the introduction, a keyadvantage of our technique is that it recovers crease aligned fieldson models whose maximal curvature directions disagree with theircreases. This occurs naturally when models are specified by theintersections of developable patches, a very common primitive inCAD tools. We introduce two benchmark models for testing creasealignment when creases disagree with intrinsic notions of curvature.
ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. :10 • Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon (a)
Moai and explicitfeature curves(rededges) (b) Explicit FeatureCurve Alignment: p = , ϵ = (c) Our method: p = , ϵ = Fig. 11. Comparison of our feature-aligned cross fields to those generatedwhen adding additional explicit feature curve alignment constraints. Explicitfeature curves were obtained from [Gehre et al. 2016, Fig. 9] Despite theextra cost of precomputing explicit feature curves, slight artifacts in thefeature curves (most pronounced on the side) force the explicitly guidedcross field to have lower quality.(a) Ours p = , ϵ = :3.9s (b) Ours p = ∞ , ϵ = :3.5s (c) BEM: 161sFig. 12. Cross field and runtime comparison of our method to a methodoptimizing volumetric octahedral frames [Solomon et al. 2017]. The fandisk used contains 2.5k triangles. The three-cylinder-intersection mesh is composed of 12 quadri-lateral patches where each patch is a subset of a cylinder and hasmaximal curvature directions making π angles with its boundarycreases. The wavey-box example has the same creases as a standardcube, with the modification that each of its faces has a sine waveripple running diagonally through it. These two cases are shownin Figure 2. The fandisk mesh is another example of a challengingcase for feature-alignment due to its shallow crease with strongnon-aligning neighboring creases which is representative of oneway that such features arise in real-world models.Our cross fields on these test cases are shown in Figure 14. Weobserve proper feature alignment in our fields and while othermethods can sometimes be tuned per model to achieve the samefeature alignment, there is no choice of parameters that worked onall test cases. In particular: • fields from [Jakob et al. 2015] are distracted by extrinsic curvatureon the three-cylinder-intersection and entirely pave over theshallow crease of the fandisk . Their results on wavey-box , and wedge are successfully aligned to the creases.; • fields from [Brandt et al. 2018] are challenging to tune with λ representing alignment to a guiding extrinsic curvature field.We show their method for the biharmonic energy ( m =
2) as apoint of contrast to Dirichlet energy. We choose two values of λ , λ = − . λ = − . T i m e i n s e c o n d s Fig. 13. Runtimes to compute cross fields over various mesh sizes. for stronger extrinsic curvature alignment. Their fields are unableto align to features of the three-cylinder-intersection in bothcases, and specifically for λ = − . fandisk mesh. Their fields aresuccessfully crease-aligned for the wedge mesh; • we compare against both the anti-holomorphic and Dirichletenergies of [Knöppel et al. 2013] with the curvature alignmentparameter λ set to − .
1. This results in good alignment on the three-cylinder-intersection , but noisy or unaligned fields forthe remaining test cases.In contrast, our method for p = p and ϵ in the supplementary materials. The fields are crease-alignedfor all creased meshes and are otherwise intrinsically smooth. Forcomparison, we include fields from [Brandt et al. 2018] and [Knöppelet al. 2013] on a larger range of λ . We also include fields from[Brandt et al. 2018] for m = Quad Meshing.
Feature alignment is especially important whenusing cross fields to guide high-fidelity quad meshing. We generatequad meshes using [Campen et al. 2015] to parameterize our crossfields. We compare against a standard quad meshing pipeline usingcross fields from [Bommes et al. 2009] and [Campen et al. 2015] forparameterization. We also test against parameterization by [Campenet al. 2016], which introduces extra guidance to encourage extrinsiccurvature alignment.For the fandisk mesh prior methods generate quad meshes thatare influenced by the shallow crease, but do not manage to captureit sharply (see Figure 15). We observe that by placing singularitiesnear the shallow crease of the fandisk , our quad meshes manage toalign much more sharply. The quad mesh generated by minimizing E ∞ aligns even better than for E .We also compare quad meshes generated from our cross fieldsagainst the prior art on the anchor , spot , moomoo , and three-cylinder-intersection meshes. These results are shown in Fig-ure 16. We observe generally better alignment in the quad meshesgenerated from our method. By placing singularities on the cylindri-cal region of the anchor , our quad meshing manages to align better ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. ctahedral Frames for Feature-Aligned Cross-Fields • 1:11 (a) Ours p = (b) [Jakob et al. 2015] (c) [Brandt et al. 2018] λ = − . Biharmonic (d) [Brandt et al. 2018] λ = − . Biharmonic (e) [Knöppel et al. 2013] λ = − . A-holomorphic (f) [Knöppel et al. 2013] λ = − . DirichletFig. 14. Various cross field methods compared on several meshes with complex features and geometry. We test on the three-cylinder-intersection, wavey-box,wedge, and fandisk meshes and compare against the following works: [Brandt et al. 2018; Jakob et al. 2015; Knöppel et al. 2013] with various parameters. Weuse normal aligned octahedral fields generated by minimizing E . We achieve crease-alignment on all test cases where other methods succeed sporadically.(a) E (b) E ∞ (c) QGP (d) Curvature filterFig. 15. Quad meshes of the fandisk mesh generated using cross fields from E , E ∞ , MIQ + QGP [Campen et al. 2015], and MIQ + Curvature filter [Campenet al. 2016] respectively. Our methods achieve sharp alignment to the shallow crease with increased depth for higher p . Alternative methods are influenced bythe crease only to a shallower extent. to its creases. On the spot mesh we see a straighter connection be-tween the ear and the head. For the three-cylinder-intersection ,the quad mesh generated from our fields clearly align better. Since the moomoo is a relatively smooth mesh, we do not see particularlydefining differences in quality. ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. :12 • Paul Zhang, Josh Vekhter, Edward Chien, David Bommes, Etienne Vouga, and Justin Solomon (a) Anchor Mesh with E (b) Anchor Mesh with [Bommes et al.2009; Campen et al. 2015] (c) Spot Mesh with E (d) Spot Mesh with [Bommes et al.2009; Campen et al. 2015](e) Moomoo Mesh with E (f) Moomoo Mesh with [Bommes et al.2009; Campen et al. 2015] (g) 3 Cylinder Intersection with E (h) 3 Cylinder Intersection with[Bommes et al. 2009; Campen et al.2015]Fig. 16. Quad meshes of the anchor , spot , moomoo , and three-cylinder-intersection meshes. We compare quad meshes generated using cross fields fromour E energy with quad meshes generated through [Campen et al. 2015] and [Bommes et al. 2009]. Our methods achieve sharper feature alignment on the anchor , spot (on the ear), and three-cylinder-intersection meshes. Feature alignment is a desirable property in many geometry pro-cessing applications. In the context of cross fields and remeshing,we consider features to be creases where the surface changes non-smoothly. Quality of feature detection and alignment can signifi-cantly impact quality of the remeshing and the usefulness of theresulting cross fields. While significant effort has been put into ex-trinsic alignment of cross fields to curvature directions, they arenot always appropriate substitutes for crease alignment. By specifi-cally targeting discontinuities of the surface we have created a newclass of octahedral frame field energies parameterized by p ≥ E p for p ≥ ϵ as a single parameterper mesh, it could also be defined as a scalar field representing“trust” in the quality of a mesh. It would be interesting to explore aspatially-varying ϵ dependent on triangle quality or other metrics in ACM Trans. Graph., Vol. 1, No. 1, Article 1. Publication date: January 2020. ctahedral Frames for Feature-Aligned Cross-Fields • 1:13 the future. If we treat the mesh itself as variables, soft normal align-ment enables a surface flow towards meshes with lower cross-fieldenergy. Our analysis can be further extended to SH representationsof n -RoSy fields or even platonic solid symmetries [Shen et al. 2016].We also conjecture that with mild assumptions the solution to ourproblem is unique, but a proof is out of scope for this work; we leaveexploration of these ideas to future work.Even without these extensions, our method provides a practicalsolution to a challenging problem. By using a new representationof cross fields we achieve crease-aligned cross fields on surfaces. ACKNOWLEDGMENTS
The authors would like to thank Christopher Brandt for help obtain-ing comparison fields, Amir Vaxman for support with Directionaland rendering, Michal Adamaszek for help debugging Mosek, andDavid Palmer for discussions and help with his code base. We thankWenzel Jacob, Keenan Crane, and Qingnan Zhou for their opensource code implementations. Finally, we thank Ryan Viertel formany valuable discussions and Panini Pals for the panini press.Paul Zhang acknowledges the generous support of the Depart-ment of Energy Computer Science Graduate Fellowship. EtienneVouga acknowledges the generous support of Adobe, SideFX, andNSF IIS-1910274. David Bommes acknowledges the generous sup-port of the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program (AlgoHex,grant agreement no. 853343). Justin Solomon acknowledges the gen-erous support of Army Research Office grant W911NF-12-R-0011,National Science Foundation grant IIS-1838071, Air Force Office ofScientific Research award FA9550-19-1-0319, and a gift from AdobeSystems. Any opinions, findings, and conclusions or recommenda-tions expressed in this material are those of the authors and do notnecessarily reflect the views of these organizations.
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