Tuned Ternary Quad Subdivision

Transcription

1 Tued Terary Quad Subdivisio Tiayu Ni ad Ahmad H. Nasri 2 Dept. CISE, Uiversity of Florida 2 Dept. of Computer Sciece America Uiversity of Beirut Abstract. A well-documeted problem of Catmull ad Clark subdivisio is that, i the eighborhood of extraordiary poit, the curvature is ubouded ad fluctuates. I fact, sice oe of the eigevalues that determies elliptic shape is too small, the limit surface ca have a saddle poit whe the desiger s iput mesh suggests a covex shape. Here, we replace, ear the extraordiary poit, Catmull- Clark subdivisio by aother set of rules derived by refiig each bi-cubic B- splie ito ie. This provides may localized degrees of freedom for special rules so that we eed ot reach out to, possibly irregular, eighbor vertices whe tryig to improve, or tue the behavior. We illustrate a strategy how to sesibly set such degrees of freedom ad exhibit tued terary quad subdivisio that yields surfaces with bouded curvature, oegative weights ad full cotributio of elliptic ad hyperbolic shape compoets. Keywords: Subdivisio, terary, bouded curvature, covex hull Itroductio Tuig a subdivisio scheme meas adjustig the subdivisio rules or stecils to obtai a refied mesh ad surface with prescribed properties. To date, there exists o subdivisio algorithm o quadrilateral meshes that yields bouded curvature while guarateeig the covex hull property at the extraordiary odes. This paper proposes terary refiemet to obtai such a scheme. Terary subdivisio offers more parameters i a close viciity of each extraordiary ode to tue the subdivisio tha biary subdivisio does ad thereby localizes the tuig. Terary quad subdivisio geeralizes the splittig of each quad ito ie. If all odes of a quad are of valece 4, the rules for bi-cubic B-splies show i Figure 3 are applied ad we have C 2 cotiuity. The challege is with odes of valece other tha 4, called extraordiary odes. I particular, we ca devise rules at the extraordiary odes ad their ewly created direct ad diagoal eighbors to achieve eigevalues i order of magitude as, λ, λ, λ 2, λ 2, λ 2, λ i,... ad 0 λ i < 9 for i = 7,.., 2 + ; ad oegative weights. It is possible to get, i additio, λ = /3. Yet, Figure 2 shows that the macroscopic shape of the ew scheme ad Catmull-Clark are very similar (Figure 2), but, of course, the mesh is deser.

2 2. Fig.. (left) Cotrol et: x, y the characteristic map of Catmull-Clark subdivisio ad z = 50( x 2 5y 2 ); subdivisio steps 3,4,5 ad surface for Catmull-Clark subdivisio (top) ad terary subdivisio (bottom). Fig. 2. Three steps of Catmull-Clark subdivisio (left) ad terary subdivisio (right).

3 3. Backgroud Subdivisio is a widely adopted tool i computer graphics ad is also makig iroads ito geometric modelig, if oly for coceptual modelig. However, [5] poited out that Catmull-Clark subdivisio i particular, is lackig the full set of elliptic ad hyperbolic subsubdomiat eigefuctios ad therefore typically geerates saddle shapes i the limit at vertices with valece greater tha four (see Figure top). This implies that ay high-quality (stadard, symmetric) scheme eeds to have a spectrum, λ, λ, λ 2, λ 2, λ 2 followed by smaller terms. With the strategy explaied i the followig, it is possible to achieve such a spectrum by makig subdivisio stecils ear the extraordiary poit deped o the valece (Figure bottom). While such localized improvemet caot be expected to produce high-quality surfaces i all cases, it is worth seeig whether such improvemets ca lead to improved rules that are easy to substitute for the kow usatisfactory rules. A major challege is techical: there are either too few or far too may free parameters occurig i oliear iequality costraits. This makes ay selectio by geeral optimizatio [] impractical. I [3, 4], Loop proposed improvemets to his Fig. 3. The regular (bicubic B-splie refiemet) stecils of Catmull-Clark (left) ad terary quad subdivisio (right). well-kow triagular scheme o triagular meshes to achieve both bouded curvature for a biary ad terary schemes. To derive weights, he used fixed-degree iterpolatio of kow weights to set may ukow weights ad reduce the umber of free parameters. But quad mesh schemes are far more difficult tha triagular mesh schemes sice the leadig eigevalues come from a 2 by 2 diagoal block matrix rather tha from a by block. They are therefore the roots of quadratic polyomials while, i the case of triagular meshes, the eigevalue is simple. For -sided facets, the problem is eve more complex due to 4 4 blocks but ca be avoided by applyig oe iitial Catmull-Clark subdivisio step. 2 Problem Statemet We wat to derive a terary, quad subdivisio scheme that is statioary, (rotatioal ad p-mirror) symmetric ad affie ivariat. The the leadig eigevalues are, i order of

4 4 magitude,, λ, λ, µ, µ 2, µ 3. To achieve bouded curvature, the covex hull property, the eigevalues ad the weights α i, β i, γ i, δ i ad v i of the terary quad scheme (see Figure 4) have to satisfy the followig costraits: (i) C scheme > λ > µ i, ad the characteristic map is regular ad ijective. (ii) Bouded Curvature µ = µ 2 = µ 3 = λ 2. (iii) Covex Hull α i 0, β i 0, γ i 0, δ i 0, i = 0.., v, v 2 0, e 0 := P i=0 (αi + βi) 0, f0 := P i=0 (γi + δi) 0, v 0 := P 2 i= (vi) 0. (iv) Symmetry α l = α l, β l = β l, γ l = γ (+ l) mod, ad δ l = δ l We focus o the leadig eigevalues ad hece the -rig of cotrol mesh aroud extraordiary poit ad its eigestructure. As show i Figure 4, each refiemet step geerates three types of poits correspodig to vertices, edges ad faces respectively. Let be the valece of a geeric extraordiary poit. We have, offhad, 4+2 weights to determie. Symmetry reduces this to free parameters. v 2 β δ v v v v v v v 2 v v 2 α 2 e 0 α 2 v 2 β 2 δ 2 Vertex Mask Edge Mask Face Mask 0 0 α α β 0 α 0 γ 2 γ 2 f 0 γ δ 0 γ β γ δ Fig. 4. Refiemet stecils used at a extraordiary poit. 3 Spectral Aalysis The -rig subdivisio matrix S with the cotrol vertices show i Figure 4 is: v 0 v v 2 v v 2 v v 2 e 0 α 0 β 0 α β α β f 0 γ 0 δ 0 γ δ γ δ e 0 α β α 0 β 0 α 2 β 2 S := f 0 γ δ γ 0 δ 0 γ 2 δ 2 R (2+) (2+) e 0 α β α 2 β 2 α 0 β 0 f 0 γ δ γ 2 δ 2 γ 0 δ 0

5 5 Now deote the discrete Fourier trasform of a cyclic sequece {φ i } by ˆφ i := φ j ω i,j, w i,j := e 2π ij. Lemma The spectrum of S is Λ = diag(, λ + 0, λ 0, λ+ i, λ i,, λ+, λ ) where λ ± i := (ˆα i + ˆδ i ) ± (ˆα i ˆδ i ) 2 + 4ˆβ iˆγ i, i =.... () 2 Proof. Sice the 2 2 lower block matrix of S is cyclic, we ca apply the discrete Fourier trasform ( ) 0 F := 0 I 2 (ω i,j ) to obtai v 0 v v e 0 ˆα 0 ˆβ f 0 ˆγ 0 ˆδ e ˆα ˆβ 0 0 Q := F SF = f ˆγ ˆδ e ˆα ˆβ f ˆγ ˆδ The eigevalues, λ + 0, ad λ 0 come from the first 3 3 block. Each 2 2 diagoal block ( ) ˆαi ˆβi, i =..., ˆγ i ˆδi cotributes the pair of eigevalues (). By (iv) ad sice λ + i ad λ + i are complex cojugate, we aticipate the followig lemma. Lemma 2 λ + i = λ + i, λ i = λ i, ad λ+ i, λ i R. Proof. By symmetry (iv), ˆα i = α 0 + {2 2 j= α j cos 2πij 2 2 j= α j cos 2πij if is odd, + α cosπi if is eve. (2) 2 Sice cos 2πij 2π( i)j = cos ad cos(πi) = whe i is eve, ˆα i = ˆα i R. The same reasoig applies to ˆδ i, ˆβ i e π i ad ˆγ i e π i sice, e.g. ˆβ i e π i = {2 2 β j cos (2πj+π)i 2 2 β j cos (2πj+π)i Therefore, λ ± i = λ ±+ i R as claimed. + β 2 cosπi if is odd, if is eve. (3)

6 6 For = 4 the subdivisio stecils are show i Figure 3. Based o this subdivisio matrix, we ca compute the matrix Q i Sectio 3 ad therefore compute explicitly the expasios collected i Lemma 3. Lemma 3 I the regular settig ( = 4), [ˆα 4 0, ˆα 4, ˆα 4 2, ˆα 4 3] = [ , 9 8, 9, 9 8 ], [ˆβ 0 4 e π i, ˆβ 4 e π i, ˆβ 2 4 e π i, ˆβ 3 4 e π i ] = [ , 4 2 2, 0, ], [ˆγ 0 4 e π i, ˆγ 4 e π i, ˆγ 2 4 e π i, ˆγ 3 4 e π i ] = [ , 6 2 2, 0, ], [ˆδ 0, 4 ˆδ, 4 ˆδ 2, 4 ˆδ 3] 4 = [ 2 729, 8, 9, ]. (4) 8 4 Derivig Weights by Iterpolatig the Regular Case The idea of derivig extraordiary rules is ispired by Loop s approach [4]. First, we iterpolate the regular stecil by a polyomial. Such a polyomial, say β, will be evaluated to defie the coefficiets βi. For = 4, due to symmetry, the costraits o the polyomial β are β (cos π ) = ad β (cos 3π ) = The liear iterpolat to these values is egative o some subiterval of [, ] ad therefore caot give suitable coefficiets βi 0 for > 4. Addig the costrait β ( ) = 0 yields the polyomial β 4 (t) := ((44 8 2)( t) 2 + (44 9 2)2t( t) + (36 2)t 2) 729 which is oegative o [, ]. We cojecture the geeral formulae for arbitrary based o the polyomials of the same low degree ad choose α (t) := a, ( (t + b, ) 2 + c, ), β := 4 β4, γ := 6 β4, δ (t) := a,4 ( (t + b,4 ) 2 + c,4 ). (5) Now, we predict the geeral formulae for ay > 4 by evaluatig the polyomials at the x-compoet of poits equally distributed over the uit circle. This yields two of the three types of stecils at the extraordiary poit ad leaves, as degrees of freedom, the coefficiets depedig o i (5). Lemma 4 The subdivisio rules, for valece ad j = 0... ca be chose as α j := α (u), δ j := δ (u), u := cos 2πj, (6) β j := β (v + ), v + := cos 2πj + π, γj := γ (v ), v = cos 2πj π

7 7 where a, := 4 ˆα 2, b, := ˆα a,, a,4 := 4 ˆδ 2, b,4 := ˆδ a,4. (7) Note that, together with β ad γ, equatios (7) yield explicit expressios for λ ± ad λ± 2 i terms of the the coefficiets a,, b,, c,, a,4, b,4, c,4. Proof. By symmetry, the weights are a discrete cosie trasformatio of their Fourier coefficiets: α i = γ i = ˆα j cos 2πij, ˆγ j cos β i = ˆβ j 2πij πj, δi = 2πij + πj cos, (8) ˆδ j cos 2πij. For = 4, we verify the cosistecy of the choice (6) by substitutig (4) ito (8). α 4 i = ( 9 (u )2 + 5, δ 4 324) i = ( (u )2 4 ). 324 Next, we decompose the polyomial ito a vector of basis fuctios ad a vector of coefficiets: a, ( (u+b, ) 2 +c, ) = [, u, 2u 2 ][a, b 2, + 2 a,+a, c,, 2a, b,, 2 a,] T. We mimic this decompostio for > 4, by choosig ˆα i = 0 for 2 < i < 2. The α i = ˆα j cos 2πij 2πi = [, cos, cos 4πi ][ ˆα 0, 2 ˆα, 2 ˆα 2 ]T. Sice cos 2πi 4πi = u, we have cos = 2u2 ad compariso of terms yields 2a, b, = 2 ˆα, ad 2 a, = 2 ˆα 2. Solvig a,, b, i terms of ˆα, ˆα 2,, ad similarly for δi, yields (7). The rules of Lemma 4 preserve rotatioal ad p-mirror symmetry. For > 4, they reduce the free parameters from to six, amely a,i, b,i, c,i, i =, 4. (For = 3 we use the origial = 0 free parameters to eforce λ + = λ, λ = λ 2, λ + 0 = λ2, λ 0 < λ2.) It remais eed to determie the six free parameters so that for > 4 the weigths α j, δ j, v, v 2, e 0, f 0, v 0 are oegative (β j ad γ j are oegative sice β 4 0 o [, ]) ad Λ = diag(, λ +, λ+, λ+ 2, λ+ 2, λ+ 0, ν, ν 2,...,ν 2 5 ) (9) = diag(, λ, λ, λ 2, λ 2, λ 2,...,λ i,...), where 0 < λ i < λ 2.

8 8 By (), ad sice ˆγ k ad ˆβ k are fixed, each of the costraits λ + = λ ad λ+ 2 = λ2 is a equatio i the parameters, a,, b,, a,4, b,4, explicitly so due to (7). The additioal ecessary costrait, λ + 0 = λ2, λ 0 < λ2 is eforced by choice of v, v 2 i the first 3 by 3 block of Q. The result is a udercostraied system with liear iequality costraits. We fid, for example, the closed-form solutio for = stated i Lemma 5 ad yieldig the elliptic ad saddle shapes show i Figure 7. Lemma 5 For = 5..0, the followig choice satisfies (i)-(iv) ad λ = /3: a, =.43, b, =.624, c, =.229 a,4 =.260, b,4 =.286, c,4 =.058 (0) Proof. We eed oly verify (i), (ii) ad (iii) listed i Sectio 2 sice the scheme is symmetric by costructio. Table shows all eigevalues other tha, 3, 3, 9, 9, 9 for = I particular, λ = λ = 3 is the subdomiat eigevalue. Table 2 verifies oegative weights summig to. The u-differeces, scaled to uit size, fall strictly ito the lower right quadrat ad, by symmetry, the v-differeces fall strictly ito the upper right quadrat (Figure 6). The partial derivatives are a covex combiatio of the differeces ad hece all pairwise crossproducts are strictly positive. By [8], the characteristic map is regular. I additio, as show i Figure 5, the first half-segmet of the cotrol et does ot itersect the egative x-axis. By the covex hull property ad [7] Theorem 2, ijectivity of the characteristic map follows. ν i =3 =4 =5 =6 =7 =8 =9 =0 ν ν ν ν ν ν 6 ǫ ǫ ǫ ǫ ǫ ν 7 ǫ ǫ ǫ ǫ ǫ ν 8 ǫ ǫ ǫ ǫ ν 9 ǫ ǫ ǫ ǫ ν 0 ǫ ǫ ǫ ν ǫ ǫ ǫ ν 2 ǫ ǫ ν 3 ǫ ǫ ν 4 ǫ ν 5 ǫ Table. Eigevalues (see 9) other tha,,,,, for = < ǫ <.00 The mai cotributio of the paper is techical: to provide a maageable set of parameters that make it easy to satisfy the formal costraits (i) (iv). This set ca be further prued by miimizig regios of meshes where the scheme exhibits hybrid behavior [5], i.e. the Gauss curvature is ot uiquely of oe sig i the limit. It should be oted that this is typically ot eough to guaratee surface fairess, for example to avoid udue flatess whe covexity is idicated by the cotrol mesh.

9 9 Fig. 5. Cotrol polyhedro of the characteristic map for = Ijectivity test: each red fat segmet does ot itersect the egative x-axis. Fig. 6. The ormalized differeces i the u directio (gree) ad v directio (blue).

10 0 weights =3 =4 =5 =6 =7 =8 =9 =0 v v v e α β α β α β α β α β α 5.05 f δ γ = γ δ γ δ γ δ γ δ γ δ 5.05 Table 2. The subdivisio rule accordig to Lemma 5 has o-egative weights. 0 < ǫ <.00 Ackowledgmet: This work was supported by NSF Grat CCF ad America Uiversity of Beirut URB Jörg Peters gave valuable techical guidace ad assistace. Refereces. L. Barthe, ad L. Kobbelt, Subdivisio Scheme tuig aroud extraordiary vertices, Computer Aided Geometric Desig, 2: E. Catmull, ad J. Clark, Recursively Geerated B-splie Surfaces o Arbitrary Topological Meshes. Computer Aided Desig, 0(6): (978) 3. C. Loop, Smooth Terary Subdivisio of Triagle Meshes Curve ad Surface Fittig (Sait- Malo, 2002). 4. C. Loop, Bouded curvature triagle Mesh subdivisio with the covex hull property. The Visual Computer, 8 (5-6), K. Karciauskas, J. Peters ad U. Reif, Shape Characterizatio of Subdivisio Surfaces Case Studies, Comp. Aided Geom. Desig, 2(6), 2004,

11 Fig. 7. Surfaces geerated by the rules of Lemma A. Nasri, I. Hasbii, J. Zheg, T. Sederberg, Quad-based Terary Subdivisio, presetatio Dagstuhl Semiar o Geometric Modelig, 29 May - 03 Jue, U. Reif ad J. Peters. Structural Aalysis of Subdivisio Surfaces A Summary Topics i Multivariate Approximatio ad Iterpolatio, Elsevier Sciece Ltd, K. Jetter et al., 2005, G. Umlauf, Aalysis ad Tuig of Subdivisio Algorithms. Proceedigs of the 2st sprig coferece o Computer Graphics, (2005)