Iterative Computation of Moment Forms for Subdivision Surfaces

Transcription

1 Iterative Computation of Moment Forms for Subivision Surfaces Jan P. Hakenberg ETH Zürich Ulrich Reif TU Darmstat Figure : Solis boune by Catmull-Clark an Loop subivision surfaces with centrois inicate by ashe lines, an inertia inicate by the principal axes of the ellipsoi with equivalent inertia. The iterative erivation of multilinear forms etermines the moments with unpreceente accuracy. Abstract The erivation of multilinear forms use to compute the moments of sets boune by subivision surfaces requires solving a number of systems of linear equations. As the support of the subivision mask or the egree of the moment grows, the corresponing linear system becomes intractable to construct, let alone to solve by Gaussian elimination. In the paper, we argue that the power iteration an the geometric series are feasible methos to approximate the multilinear forms. The tensor iterations investigate in this work are shown to converge at favorable rates, achieve arbitrary numerical accuracy, an have a small memory footprint. In particular, our approach makes it possible to compute the volume, centroi, an inertia of spatial omains boune by Catmull-Clark an Loop subivision surfaces. Keywors: Stationary subivision, volume, centroi, inertia. Concepts: Theory of computation Computational geometry; Computing methoologies Shape analysis; Introuction Volume, centroi, an inertia are important shape characteristics in 3D moeling an animation. Accurate formulas for the etermination of the moments of solis boune by stationary subivision schemes have only recently become available. The approach is restricte to the computation of moments of low egree an to subivision schemes with weight masks of small support. The iterative methos presente in this publication exten the limits of what is possible. Using the new framework to compute moment forms, solis boune by subivision surfaces such as Catmull- Clark or Loop can be esigne with exact volume, centroi, an inertia up to machine precision. The moment formula for soli shapes in the context of subivision has the following applications: By translation of vertices in the control mesh, limit surfaces can be eforme subject to preservation of volume an centroi of the interior set. If a subivision surface is the contour of an animate entity with constant mass ensity, the formulas help to make the motion physically more accurate. In the absence of applie torques, the inertia (secon moment) is use to etermine motions that preserve angular momentum.. Relate Work [Peters an Nasri 997] approximate the moment of solis boune by [Doo an Sabin 978] an [Catmull an Clark 978] surfaces by increasing the level of refinement aroun non-regular vertices of a mesh. As facets ajacent to non-regular vertices ecrease in size, their moment contribution can be approximate using a planar surface patch. The approach requires regular submeshes to have a polynomial parametrization. Moments efine by the Butterfly scheme by [Dyn et al. 990] are not covere by their framework. [Schwal 999] achieves accurate results for solis boune by the Doo-Sabin algorithm. The moment contribution of a non-regular facet is expresse as a multilinear form in the coorinates of the vertices of the control mesh. In orer to compute the form coefficients, the volume contributions of all spline rings are taken into account. This is achieve by eigen-ecomposition of the subivision matrix. The contributions of infinite sequences of scale eigen-rings form a geometric series. The exact limit of the series is etermine algebraically. [Warren an Weimer 200] pp calculate the exact area form for the 4-point curve subivision scheme by [Dubuc 986]. The erivation is the first application of the homogeneous refinement equation () in the literature. [Hakenberg et al. 204] erive the exact moment forms for facets ajacent to non-regular vertices as well as sharp creases from the general refinement equation (7). The limitation of the approach is the size of the non-sparse linear systems that are solve to etermine the coefficients of the tensors. As the complexity of the subivision scheme, patch topology, an egree of moment grows, the corresponing linear system becomes intractable to store in memory an solve by Gaussian elimination. For instance, the inertia form for a qua facet ajacent to a vertex of valence 5 in a Catmull- Clark mesh has coefficients..2 Contributions We buil upon the work of [Hakenberg et al. 204] by introucing two iterative methos that approximate the coefficients of the moment forms up to arbitrary numerical precision. The first metho is the power iteration that etermines the form as an eigenvector associate with the largest eigenvalue of a linear system. The secon

2 iteration type is a geometric series. Eigen-analysis guarantees that the iterations converge at favorable rates for stanar subivision schemes so that builing the matrix of the linear system becomes unnecessary. Catmull-Clark surfaces are the e facto stanar for 3D moeling in the entertainment inustry. Our approach allows to calculate volume, centroi, an inertia of solis boune by these surfaces with unpreceente accuracy an minimal computational effort. 2 Preliminaries The moment E p,q,r(ω) of a boune set Ω R 3 for non-negative integers p, q, r is efine as the integral E p,q,r(ω) := x p y q z r xyz. () Ω The egree is efine as := p + q + r. Moments of low egree have interpretations as volume(ω) = E 0,0,0(Ω), centroi(ω) = (E,0,0(Ω), E0,,0(Ω), E0,0,(Ω)), volume(ω) inertia(ω) = (E 2,0,0(Ω), E 0,2,0(Ω), E 0,0,2(Ω), E,,0(Ω), E 0,,(Ω), E,0,(Ω)). The mass ensity is assume to be constant across the omain Ω. The ivergence theorem with vector fiel G : R 3 R 3 as G(x, y, z) = ( p+ xp+ y q z r, 0, 0), an iv G = x p y q z r reformulates () to an integral over the bounary Ω E p,q,r(ω) = x p+ y q z r n xa, (2) p + where n x enotes the x-component of the surface normal. 2. Multilinear Forms Let M be an orientable mesh of control points. Accoring to [Schwal 999] an [Hakenberg et al. 204], the moment of the set Ω boune by the subivision limit surface Ω = S (M) is compute as a sum over all facets of the mesh E p,q,r(ω) = E p,q,r(f). (3) Ω f M The contribution of the facet f to the global moment is E p,q,r(f) = p + M τ(f). x. {{.x. y..y. z. {{ {{.z.y.z. (4) p+-times q-times r-times The control points that etermine the surface patch associate to f are arrange into the coorinates vectors x, y, z. For instance, in a Catmull-Clark control mesh, each qua is a facet. The control points in the one ring of the qua etermine the surface patch. The tensor M τ(f) epens on the topology τ(f) of the facet, an the egree of the moment. [Peters an Nasri 997] establish that the coefficients of the tensor M τ(f) have the following integral expression ) i,...,i +,j,k = b i b i+ ( sb j tb k tb j sb k )st. (5) D For qua-base subivision schemes the omain is D = [0, ] 2, an for triangle base schemes D = {(s, t) R 2 0 s, t s+t. The basis function b i : D R is associate to control point i. For surfaces that consist entirely of B-spline, or Bézier-Bernstein patches, [Gonzalez-Ochoa et al. 998] integrate (5) symbolically in orer to yiel the exact tensor coefficients. For facets in subivision meshes that are ajacent to non-regular vertices, the basis functions typically o not have a close-form expression. 2.2 Recursive Formulation We summarize the erivation given in [Hakenberg et al. 204]. One roun of subivision partitions a facet f M into a number of smaller facets f h := S h (f). For stanar surface schemes h {, 2, 3, 4, see Figures 3 an 4, but this is not a requirement. The contribution E p,q,r(f) to the global moment of Ω is ientical to the total contribution of the subivie parts E p,q,r(f) = h E p,q,r(s h (f)). (6) Substituting x S h.x, y S h.y, an z S h.z in (4) to expan the rhs of (6) yiels the general tensor equation inepenent of control points M τ(f) = h M τ(f h) [S h ]. (7) The tensor transformation M[S] is shorthan for the basis transformation along each imension of tensor M with S, i.e. (M[S]) i,...,i m := The matrix S oes not have to be square. j,...,j m M j,...,j m S j i S jm i m. In orer to solve the coefficients of M τ(f), equation (7) is written as a linear system m = c + A.m. Matrix A encoes the linear relationship between the sought coefficients of M τ(f) that make up vector m. In case of homogeneous refinement, i.e. τ(f) = τ(f h ) for all h, it follows that vector c = 0. Then, the problem is reuce to ientifying m as an eigenvector to eigenvalue of matrix A m = A.m. (8) Once establishe, the eigenvector m has to be scale to the proper length in a calibration step using a configuration of control points x, y, z for which the moment E p,q,r(f) of the surface patch is known analytically. The calibration proceure is etaile in [Hakenberg et al. 204] an [Hakenberg an Reif 206b]. In case of non-homogeneous refinement, i.e. τ(f) τ(f h ) for at least one h, vector c aggregates the contribution of the subivie facets for which the moment forms M τ(f h) are alreay known. The resulting linear system is of the form (I A).m = c. (9) For non-regular vertices, the facet topology τ(f) typically appears once in the sum of the rhs of (7), see Figure 4. Then, the system matrix A = +3 S is the ( + 3)-fol Kronecker prouct of the square subivision matrix S. S maps the coorinates of the control points associate to facet f to those of the subivie non-regular facet f h with τ(f) = τ(f h ). The eigenvalues of A follow from the eigenvalues of S. For common subivision schemes, A has

3 eigenvalue with multiplicity, an all other eigenvalues λ with λ <. That means I A is rank eficient by. The nullspace of I A correspons to a symmetric tensor. But m is skew in the last two imensions as seen from (5), that means m follows uniquely from (9). 2.3 Tensor Symmetry When arranging the coefficients of M τ(f) into a vector m, reunant entries stemming from tensor symmetries shoul be omitte in orer to reuce the complexity of the linear system. Two symmetry relations are intrinsic to the inner prouct (5): Invariance uner permutation of the first + inices ) i,...,i +,j,k = an skew symmetry in the last two imensions ) i,...,i +,j,k = ) sort(i,...,i + ),j,k, ) i,...,i +,k,j. Permutations of control points uner which the image of the surface patch is invariant also contribute to tensor symmetries. For instance, rotation ρ of control point inices of a regular facet results in the same tensor coefficient ) i,...,i +3 = ) ρ(i ),...,ρ(i +3 ). Mirroring σ of control point inices of a facet along an axis of mirror symmetry flips the sign ) i,...,i +3 = ) σ(i ),...,σ(i +3 ). For egree = 0, one may aitionally assume that the trilinear form is alternating, i.e. ( ˆM τ(f) 0 ) i,j,k = 6 π S 3 sign(π) 0 ) π(i,j,k), (0) where S 3 is the permutation group of orer 3! = 6. [Hakenberg an Reif 206b] treat alternating volume forms for refinable basis functions b i of which subivision surfaces are a special case. Figure 2: Rotation (pink arrow) an mirror (purple axis) symmetries of facets f. Examples The surface patch associate to a regular Catmull-Clark qua facet f is etermine by 6 control points. The left iagram in Figure 2 illustrates the symmetries. The moment contribution of the facet is invariant up to sign uner rotation an mirroring of the control points. The number of unique coefficients in the centroi form M 4 is boune by The surface patch associate to a Loop triangular facet f with a non-regular vertex of valence τ(f) = 7 is etermine by 3 control points. The moment contribution of the facet is invariant up to sign uner mirroring of the control points across the axis through the non-regular vertex, see the right iagram in Figure 2. The number of unique coefficients in the inertia form M 7 2 is boune by Iteration of Forms Despite taking coefficient symmetries in M τ(f) into account, the linear systems (8) an (9) may still be too large to buil an etermine m. Moreover, the system matrix is typically non-sparse an has a large conition number. Memory constraints an loss of numerical accuracy prevent the application of Gaussian elimination. In this section, we argue that the power iteration an the geometric series are feasible methos to approximate the moment forms. In the subsequent treatment, the egree an facet topology τ(f) are fixe. We rop these two inices to simplify the notation. 3. Homogeneous Refinement Figure 3: Subivision of a regular Loop facet f into f h for h {, 2, 3, 4. S h are matrices of imension 2 2 that map the coorinates of the control points in the -ring of f to the control points of f h. In case of homogeneous refinement, all moment forms in (7) are ientical. The equation simplifies to M = h M[S h ]. () The system of linear equations is of the form m = A.m. An eigenvector m is approximate by the power iteration, if is the largest eigenvalue of matrix A, see [von Mises an Pollaczek-Geiringer 929]. The smaller the subominant eigenvalue of A, the faster the convergence of the iteration. We are not aware of a metho to euce the eigenvalues of A irectly from the collection of subivision matrices S h. For the subivision schemes that we investigate, the largest eigenvalue of matrix A is. For egree = 0, an assuming that M is an alternating trilinear form (0), the multiplicity of eigenvalue is. For {, 2, the multiplicity is 2. For Doo-Sabin, Loop, an Catmull-Clark subivision an {0,, 2 the eigenvalues of A were experimentally etermine to be of the form 2 2n for n {0,, 2,... For the Butterfly scheme with tension parameter w = /6 an = 0, the set of eigenvalues {, /4, 3/32, /6,... contains values outsie of this pattern. The construction of matrix A is not require to perform the power iteration. In tensor notation, the sequence is equivalent to M 0 := π(ranom) M k := M k / M k ( ) M k+ := π M k [S h ], h (2) which is more memory efficient. Coefficient symmetries as erive in Section 2.3 are enforce by a linear projection π in between iteration steps. After the iteration has converge, scaling of the form M = µ lim k M k with an appropriate factor µ R is performe analogous to the calibration of eigenvector m in (8).

4 Figure 4: Subivision of a Catmull-Clark facet f with a vertex of valence τ(f) := 5 into f h for h {, 2, 3, 4. S, S 2, S 3 are 6 8 matrices, S 4 is of imension Non-Homogeneous Refinement For a facet ajacent to a non-regular vertex, (7) simplifies to M = C + M[S]. (3) The tensor M = M τ(f) is unknown. Square matrix S maps the coorinates of the control points that support f to the control points of f h with τ(f) = τ(f h ). C is the constant tensor aggregate from the known terms. The eigenanalysis of the linear system (9) carrie out in Section 2.2 permits to write m as the geometric series m = (I A).c = lim k i=0 k A i.c. In tensor notation, the geometric series is compute as M 0 := C M k+ := C + M k [S] (4) with M = lim k M k. The subominant eigenvalue of A is the subominant eigenvalue λ of S. The smaller λ, the faster the convergence, see Figure 6. 4 Summary The moment forms M τ(f) for a specific subivision scheme have to be erive only once. Subsequently, the tensors apply universally to any close, orientable control mesh M using (3) an (4). The methos to erive the forms are selecte base on the type of basis functions, number of coefficients of M τ(f), an requirements on accuracy. In the following overview, methos 3 solve the homogeneous refinement equation (), whereas methos 4 5 solve the non-homogeneous refinement equation (3). Symbolic Integration The basis functions that parametrize regular surface patches of Doo-Sabin an Catmull-Clark surfaces are polynomials. [Peters an Nasri 997] compute the moment form for arbitrary egree by explicit integration of (5). 2 Nullspace an Calibration For the Loop, an Butterfly subivision schemes, the basis functions of the regular facet are not reaily available. (8) is rewritten to (I A).m = 0. The nullspace of matrix I A can be compute algebraically. For Loop s algorithm, the size of matrix A is 43, 874, 4032 square for = 0,, 2 respectively, see [Hakenberg et al. 204]. For the volume form of the Butterfly scheme, the size of matrix A is 508 square, see [Hakenberg an Reif 206b]. Subsequent calibration of the eigenvector m yiels the exact, rational tensor coefficients. 3 Power Iteration The regular patch of the Butterfly scheme epens on 29 control points. The centroi form is etermine by 2294 coefficients. We estimate that the system matrix A has subominant eigenvalue of λ = /4. We use the power iteration (2) to approximate the tensor. Machine precision in the coefficients is surpasse after few iterations, see Figure Figure 5: Coefficients sorte by magnitue of the centroi form M k for the regular facet of the Butterfly scheme in the power iteration for k =, 2,..., 7. 4 Gaussian Elimination [Hakenberg et al. 204] solve the linear system (9) by Gaussian elimination to obtain the exact moment forms for moerate non-regular vertex valences. If the subivision weights are rational numbers, the tensor coefficients are also rational. For the Doo-Sabin scheme, the egree is limite to {0,, 2; for Loop, {0, ; an for Catmull-Clark, = 0. The construction of matrix A for moments beyon the liste egrees is prohibitive ue to the large number of form coefficients. 5 Geometric Series We use the tensor iteration (4) to approximate the inertia forms for the Loop scheme, as well as the centroi an inertia forms for Catmull-Clark surfaces an moerate vertex valences τ(f). The collection of tensors is compute within a ay using an implementation in Mathematica. The memory footprint of the iteration is low, since only the tensor M k nees to be store. The numerical accuracy of the coefficients increases with each iteration step. Figure 6 shows the rate of convergence Figure 6: Convergence of Catmull-Clark inertia forms for nonregular valences up to 8. The slope is etermine by the subominant eigenvalue λ of the subivision matrix: For vertex valence 3, λ 0.4; for vertex valence 8, λ 0.6. Machine precision in the form coefficients is surpasse after no more than 50 iteration steps. 5 Future Work We plan to investigate the use of the iterative methos to calculate inner proucts ifferent from (5). Such inner proucts arise for instance when moeling a solution space to elliptic partial ifferential equations. Acknowlegements The authors thank Brian Paen for his suggestions that le to the improvement of the article

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