Numerical Integration of Geometric Flows for Space Curves

Transcription

1 Numercal Integraton of Geometrc Flows for Space Curves by Guolang Xu Report No. ICMSEC January 2013 Research Report Insttute of Computatonal Mathematcs and Scentfc/Engneerng Computng Chnese Academy of Scences

2 Numercal Integraton of Geometrc Flows for Space Curves Guolang Xu LSEC, Insttute of Computatonal Mathematcs and Scentfc/Engneerng,Academy of Mathematcs and Systems Scence, Chnese Academy of Scences, Bejng, Chna January 23, 2013 Abstract Several second-order and fourth-order geometrc flows have been constructed for space curves n an earler paper (see 12]). Some propertes on the changng rates of the arc-length and the area swept by the curve have been dscussed. In ths paper, dscretzaton schemes of these flows, usng both dvded dfference and mxed fnte element methods, are consdered. The convergence results on the dscretzed dfferental operators are establshed. Two methods on space curve regularzaton are presented. Numercal tests are conducted, showng that these flows can be used for curve smoothng and each of them has dstnct feature. 1 Introducton Smooth space curves often need to be generated from nosy nput data n many desgn and constructon problems, such as reverse engneerng, mult-component mesh optmzaton and ndustral geometrc desgn from wreframe. An effcent way to acheve ths goal to evolve the curve by utlzng some geometrc partal dfferental equatons that have smoothng effect. In paper 12], several second-order and fourth-order geometrc flows for evolvng space curves are constructed. Some propertes on the changng rates of the arclength of the evolved curves and the areas swept by the curves are dscussed. Short term and long term behavors of the evolved curves are llustrated. These flows can be used for space curve farng or denosng. However, the numercal ntegratons of these flows are not presented n 12]. The am of ths paper s to present effcent and robust numercal methods for solvng these geometrc flows for space curve embedded n R 3. Supported n part by NSFC under the grants and , NSFC key project under the grant and Funds for Creatve Research Groups of Chna (grant No ). 1

3 2 Guolang Xu ICM-RR Prevous works on Space Curves Smmothng. Space curve smoothng or denosng s often requred n reverse engneerng, CAD modelng, anmaton, computer vson, mage regstraton, and mesh optmzaton 9]. Ths problem has been well studed n the area of computer aded geometrc desgn 4]. Wallner et al. 10] propose an approach to compute far webs, whch amounts to smooth a set of connected curves to mnmze the total squared-norm of frst- or second-order dervatves of the curve wthn the gven underlyng surface. Theoretcal results of optmal curve set are presented n 10], as an extenson of the work on energy-mnmzng splne curves on surfaces 7]. These results are manly developed for graphcs applcatons such as remeshng, parameterzaton and surface restoraton. To smooth a dscrete space curve (a sequence of ponts) wth nose, B-splne were used n 5] to approxmate the dscrete data. In reference 3], the nosy scattered data were smoothed by mnmzng a functonal combnng the L 2 norm and the smoothng norm. An effcent farng technque of wreframe defned by two sets of space curves was presented n 13]. In the farng process, they fx the parameters and then release them by an teratve gradent descent optmzaton method. In reference 8], a 3D curve smoothng algorthm usng the movng least square (MLS) projecton s proposed wth the prmary focus on smoothng boundares of pont clouds obtaned durng reverse engneerng laser scanned models. An algorthm for farng of space curves wth respect to spatal constrants based on a vector-valued curvature functon s presented n 6]. It descrbes a curve smoothng flow that satsfes strct spatal constrants and allows smultaneous control of both curvature functons. In reference 1], the authors show how to use the curvature flow on space to defne a natural contnuaton of the planar soluton for all the tme. Man Content of Ths Paper. Our attenton s focused on the numercal computaton of geometrc flows for space curves. Dscretzaton schemes of these flows are consdered. The convergence results on the dscretzed dfferental operators are establshed. Dscrete curve evoluton by the geometrc flows may generate sngulartes (edge length vanshng). To solve ths problem, two methods on space curve regularzaton are also presented. Numercal tests are conducted, showng that these flows can be used for curve smoothng and each of them has dstnct features. Organzaton of the Paper. The remander of ths paper s organzed as follows. In Secton 2, we ntroduce the used notatons. In Secton 3, we revew several second-order geometrc flows constructed n 12], followng wth the computatonal detals for ntegratng these flows usng explct dvded dfference method. Convergence results on the dscretzed dfferental operators are also presented. In Secton 4, we revew several fourth-order geometrc flows from 12], Integraton methods on these fourth-order flows usng mxed sem-

4 ICM-RR Numercal Integraton of Geometrc Flows 3 mplct fnte element method are dscussed n ths secton. Two regularzaton schemes of the space curve n the polygon form are descrbed n Secton 5. Fnally, we conclude the paper wth a summary n Secton 6. 2 Notatons In ths secton, we ntroduce some used notatons and several dfferental operators on space curve, ncludng the defntons of curvature, normal, gradent, dvergence and Laplace operator. Curvature and Normal. For a gven smooth space curve C defned by C = { x(u) = x(u), y(u), z(u)] T R 3 : u Ω = a, b] }, let κ be the curvature, whch s gven as (see 2], page 25) κ(u) = x (u) x (u) x (u) 3 = x (u) 2 x (u) 2 (x (u) T x (u)) 2 x (u) 3, (1) where x (u) and x (u) denote the frst- and second-order dervatves of x(u) wth respect to u. Note that κ 0. Ths s dfferent from the mean curvature of surface whch can be ether postve or negatve. The prncpal normal s gven as n(u) = x (u) x (u) 2 x (u)x (u) T x (u) x (u) x (u) 2 x (u) 2 (x (u) T x (u)) 2. (2) It should be noted that n(u) s well defned for all the u such that x (u) and x (u) are lnearly ndependent. If x (u) and x (u) are lnearly dependent, n(u) s arbtrarly defned. We artfcally defne t as a zero vector. The curvature κ = 0 s well defned. From (2), we can derve that n(u) does not depend on the re-parametrzaton of the curve. Hence, n(u) s unquely defned. In addton to the curvature and normal, we defne curvature vector as It s easy to see that κ(u) = x (u) x (u) 2 x (u)x (u) T x (u) x (u) 4. (3) κ(u) = κ(u), n(u) = κ(u) κ(u) f κ(u) > 0. (4) Hence, n(u) s not contnuous at the pont κ(u) = 0. functon, we defne n ϵ (u) = κ(u) κ(u) + ϵ, To obtan a contnuous normal

5 4 Guolang Xu ICM-RR where ϵ s a small postve number. In the followng, all the ϵ stands for the same number. Laplace Operator. Suppose f s a C 2 smooth functon defned on the curve x(u), then we defne Laplace operator c (here the subscrpt c stands for that the operator s defned on curve) applyng to f as c f = dv c ( c f), (5) where the gradent and dvergence operators on a space curve are defned as c f = x f x 2 = x (x ) T f x 2, (v ) T x + v T x ] x 2 (x ) T x v T x dv c v = x 4. If we take f as a vector value functon f = f 1, f 2, f 3 ] T, then we defne c f = c f 1, c f 2, c f 3 ]. Then the Laplace operator can be wrtten as c f(x) = f (x) x 2 f (x)(x ) T x x 4, (6) where f and f denote the dervatves of f(x(u)) wth respect to u, 3 Second-order Geometrc Flows for Space Curves and Ther Dscretzatons In ths secton, we frst recall several second-order geometrc flows from 12]. Some propertes of these flows are brefly revewed. Then we consder the dscretzatons of these flows. 3.1 Second Order Curve Flows Curvature Flow. Let x(u) be a gven space curve wth u a, b]. The curvature flow s defned as x t = κn or x = κ. (1) t Curvature flow s arc-length shortened. In general, the curvature flow may not be area shrnkng. However, f the curve s convex, then the curvature flow s area shrnkng. Averaged Curvature Flow. Let κ 0 = C κ 2 / κ + ϵ C ds κ κ + ϵ ds,

6 ICM-RR Numercal Integraton of Geometrc Flows 5 where C = {x(u) R 3 : u a, b]}, L s the arc-length of the curve. Then the averaged curvature flow s gven as x t = (κ κ 0(t))n. (2) The averaged curvature flow s arc-length shortened for a space curve. For plane curves, If the curve s convex, the flow s area preservng. Area-preserved Averaged Curvature Flow. Let κ 0 = 1 κ s ds, L where κ s s the sgned curvature of the curve C. Then the flow C x t = (κ s κ 0 )N. (3) s arc-length shortenng and area preservng. Here vector N R 3 stands for the orentated normal, such that, such that k s N = κ. 3.2 Dsctrzaton Usng Explct Dvded Dfference Method Let x 0 x 1 x n ] be a polygonal lne. To dscretzed the flows presented n the prevous subsecton, we need to dscretze the dscretzatons. Fast verson κ and n. In the followng, we present two versons of Let e = x x 1, and x (s) the quadratc nterpolaton of the ponts x 1, x and x +1 on the knots e, 0 and e +1. Regardng s as the arc-length parameter, we obtan the followng dscretzaton: κ (1) = t +1 t s, (4) where n (1) = κ(1) κ (1), (5) s = e + e +1, t = x x 1, e = x x 1. 2 e Vectors κ (1) and n (1) stand for the dscretzatons of the curvature vector κ and the normal n, respectvely. More accurate verson

7 6 Guolang Xu ICM-RR Dscretzatons (4) and (5) are obtaned by regardng parameter s n x (s) as arclength. However, s s only an approxmate arc-length. A more accurate dscretzaton s to compute the frst- and second-order dervatves of x (s), and then compute κ and n usng (3) and (4). In ths way, we obtan the followng dscretzaton: κ (2) = x (0) x (0) 2 x (0)x (0)T x (0) x (0) 4, (6) where n (2) = κ(2) κ (2), (7) x (0) = e +1t + e t +1 e + e +1, x (0) = 2(t +1 t ) e + e +1 = κ (1). Numercal tests We use numercal experments to compare the performance of the two dscretzaton schemes for curvature vector κ. We frst take three space curves as follows. x 1 (u) = x 2 (u) = x 3 (u) = cos t 2, sn t 2, e t ] T, t ( 1, 1), (8) t 1 + t 2, t t 2, t 3 ] T 1 + t 2, t ( 1, 1), (9) e t, sn π 2 (1 + t), cos π 2 (1 + t) + log tan π ] T 4 (1 + t), t ( 1, 1). (10) We then take 32 unformly dstrbuted ponts, t 1, t 2, t 32, n the nterval ( 1, 1). Around Table 1: Maxmal errors e (1) j (h) = max κ (1) κ(x j (t )) and e (2) j (h) = max κ (2) κ(x j (t )) for j = 1, 2, 3, h = k = 1, l = 5,, l h = 2 5 h = 2 6 h = 2 7 h = 2 8 h = 2 9 h = 2 10 h = 2 11 e (1) 1 (h) 6.075h h h h h h h 2 e (1) 2 (h) 2.782h h h h h h h 2 e (1) 3 (h) h h h h h h h 2 e (2) 1 (h) 3.934h h h h h h h 2 e (2) 2 (h) 1.995h h h h h h h 2 e (2) 3 (h) h h h h h h h 2 each t, we sample the gven space curves on three knots t h, t and t + k, obtanng

8 ICM-RR Numercal Integraton of Geometrc Flows 7 three curve ponts. We fnally compute the dscrete curvature vectors usng these ponts and compute the errors wth the exact curvature vectors from the gven curves. For h and k, we consder two scenaros. One s h = k = 2 l, l = 5, 6,..., 11. The other s h = 2 l, l = 5, 6,..., 11 and k = 2h. Table 1 and Table 2 lst the maxmal errors e (1) j (h) = max κ (1) κ(x j (t )) and e (2) j (h) = max κ (2) κ(x j (t )) for each of the three curves and for the frst and second scenaros, respectvely. These numercal results show the followng facts: 1. Both κ (1) and κ (2) are convergent. If h = k, the convergence rate s quadratc. Otherwse, the convergence rate s lnear. These facts justfy the theoretcal analyss n the next subsecton. 2. In all the cases, κ (2) s lttle better than κ (1). But the convergence orders are the same. Table 2: Maxmal errors e (1) j (h) = max κ (1) κ(x j (t )) and e (2) j (h) = max κ (2) κ(x j (t )) for j = 1, 2, 3, h = 1, l = 5,, 11 and k = 2h. 2 l h = 2 5 h = 2 6 h = 2 7 h = 2 8 h = 2 9 h = 2 10 h = 2 11 e (1) 1 (h) 3.745h 3.650h 3.603h 3.579h 3.567h 3.561h 3.559h (h) 2.316h 2.307h 2.304h 2.302h 2.301h 2.300h 2.300h e (1) 2 e (1) 3 e (2) 1 e (2) 2 e (2) 3 (h) 4.088h 4.036h 4.010h 3.997h 3.990h 3.987h 3.985h (h) 3.043h 3.021h 3.013h 3.011h 3.010h 3.009h 3.009h (h) 2.028h 2.016h 2.011h 2.009h 2.008h 2.008h 2.007h (h) 3.598h 3.585h 3.580h 3.578h 3.577h 3.577h 3.577h 3.3 Convergence of the Dscretzed Operators Gven a space curve x(u). We sample the curve at u 1, u and u +1, and obtan the curve ponts x 1 = x(u 1 ), x = x(u ) and x +1 = x(u +1 ). Wthout loss of generalty, we assume u = 0. Let h = u u 1, k = u +1 u. Now we prove that the dscretzed curvature vectors κ (l) and dscretzed normals n (l) converge to the exact curvature vector and normal of x(u) as h and k go to zero. Usng Taylor expanson, we have x +1 = x + kx (0) k2 x (0) k3 x (0) + O(k 4 ), (11) x 1 = x hx (0) h2 x (0) 1 6 h3 x (0) + O(h 4 ). (12)

9 8 Guolang Xu ICM-RR Then e 2 +1 = k 2 (a 2 + bk + ck 2 ) + O(k 5 ), (13) e 2 = h 2 (a 2 bh + ch 2 ) + O(h 5 ). (14) wth a = x (0), b = x (0) T x (0), c = 1 3 x (0) T x (0) x (0) 2. Usng the expanson we have Hence and c0 + c 1 h + c 2 h 2 + = c 0 + c 1 2 c 0 h + 4c 0c 2 c 2 1 8c 0 c0 h 2 +, ( e +1 = ka 1 + b 2a 2 k + 4a2 c b 2 ) 8a 4 k 2 + O(k 4 ), (15) ( e = ha 1 b 2a 2 h + 4a2 c b 2 ) 8a 4 h 2 + O(h 4 ). (16) s = (k + h)a 2 e 1 +1 = 1 ( 1 b ka e 1 = 1 ha + (k2 h 2 )b 4a 2a 2 k 4a2 c 3b 2 ) 8a 4 k 2 ( 1 + b 2a 2 h 4a2 c 3b 2 8a 4 h 2 + O(k 3 ) + O(h 3 ), (17) + O(k 2 ), (18) ) + O(h 2 ). (19) Hence t +1 t = 1 ( x (0)α + 1 ) a 2 x (0) (k + h) ( + x (0)β 1 2 x (0)α + 1 ) ] 6 x (0) (k 2 h 2 ) + O(k 3 ) + O(h 3 ) (20) where α = b 2a 2, β = c 3b 2 4a2 8a 4. From (17) and (20), we know that x (0) = κ (1) = 2 ( x a 2 (0)α + 1 ) 2 x (0) + O(k h) + O(k 2 ) + O(h 2 ) = x (0) x (0) 2 x (0)x (0) T x (0) x (0) 4 + O(k h) + O(k 2 ) + O(h 2 ) (21) = κ(u ) + +O(k h) + O(k 2 ) + O(h 2 ). (22)

10 ICM-RR Numercal Integraton of Geometrc Flows 9 Hence, Therefore, the convergence of κ (1) and n (1) κ (2) and n (2). Set n (1) = n(u ) + O(k h) + O(k 2 ) + O(h 2 ). s proved. Now we prove the convergence of Then, we have A = 1 + b 2a 2 k + 4a2 c b 2 8a 4 k 2, B = 1 b 2a 2 h + 4a2 c b 2 8a 4 h 2. x (0) = kh 1 A 2 (x x 1 ) + hk 1 B 2 (x +1 x ) + O(h 4 ) + O(k 4 ) a(ka + hb)ab + O(h 4 ) + O(k 4 ) = ka2 x (0) 1 2 hx (0) + O(h 2 )] + hb 2 x (0) kx (0) + O(k 2 )] + O(h 4 ) + O(k 4 ) a(ka + hb)ab + O(h 4 ) + O(k 4 ) x (0) 1 + b (k h) + O(hk) + O(h 2 ) + O(k 2 ) ] a = 2 x (0) 1 + b (k h) + O(hk) + O(h a 2 ) + O(k 2 ) ] 2 = x (0) 1 + O((k h) 2 x ) + O(hk) + O(h 2 ) + O(k 2 ) ]. (23) (0) From (21) and (23), we can obtan that κ (2) = κ(u ) + O(k h) + O(kh) + O(k 2 ) + O(h 2 ), n (2) = n(u ) + O(k h) + O(kh) + O(k 2 ) + O(h 2 ), Therefore, we have proved the followng results Theorem 1 Let x(u) be a gven C 2 smooth space curve. Assume that the curve s sampled at u 1, u and u +1 obtanng the curve ponts x j = x(u j ), j = 1,, + 1. Then κ (k) and n (k), k = 1, 2, converge to the exact κ(u ) and n(u ), respectvely, n the lnear rate as u 1 u and u +1 u wth fxed u. Furthermore, f u u 1 = u +1 u, then the convergent rate s quadratc. 3.4 Numercal Solutons and Performance Comparsons Let us take equaton (1) as an example for the dscretzaton. Other equatons s solved n the same way. Equaton (1) s solved usng the explct Euler scheme x (k+1) = x (k) + τκ, = 1,, n 1, (24)

11 10 Guolang Xu ICM-RR where τ s a temporal step-sze, x (0) = x, and x (k) 0 = x (k+1) 0 = x 0, x (k) n = x (k+1) n = x n. κ and n can be computed by (4)-(5) or (6)-(7), by takng x = x (k), = 1,, n 1. To compare the performance of three second-order equatons. We use followng strateges: Gven an ntal nosy space curve. We use each of the three flows to smooth the curve wth maxmal allowable temporal step-sze. The maxmal allowable temporal step-sze s determne as follows: Gven a small temporal step-sze τ = τ 0, we use ths τ to evolve the curve 50 tmes. If ths τ s good for each of the eratons, then we ncrease τ to 1.01τ, to try agan, untl the τ s not good. Then the prevous τ s the maxmal allowable temporal step-sze. The goodness of a τ s checked usng the arc-length. If the arc-length s shorten for each of the 50 teratons, then the τ s good. Otherwse, t s not good. Table 3: Maxmal allowable temporal step-szes. p = 0.0 p = 0.1 p = 0.2 p = 0.4 Flow (1) e e e e-05 Flow (2) e e e e-04 Flow (3) e e e e-04 Snce these three flows are arc-length shortenng, an allowable temporal step-sze should make the arc-length decreasng. A flow that makes the maxmal allowable temporal stepsze larger, the flow s more stable. Fg. 1 shows the ntal space curves, the perturbed curves and the evolved curves. At each vertex x of the ntal curve, a perturbaton p rand()/randmax s added to each of ts components, where rand() s the random number generator n C, RANDMAX s the maxmal of the random numbers. In Table 3, we lst the maxmal allowable temporal step-sze. These test results show that the dfference of the the maxmal allowable temporal stepszes for each p s not sgnfcant. Hence, the stabltes of these three flows are smlar. It should be noted that as the perturbaton level p ncreases, the maxmal allowable temporal step-sze decreases. Fg. 1 shows that these second-order flows have smoothng effects for nosy curves. 4 Fourth-order Geometrc Flows for Space Curves and Ther Dscretzatons In ths secton, several fourth-order flows, ncludng curve dffuson flow, quas-curve dffuson flow, projectve quas-curve dffuson flow, mnmal curvature flow and mnmal squared-curvature flow, are revewed. Snce these hgher order flows are rather dffcult

12 ICM-RR Numercal Integraton of Geometrc Flows 11 Fg 1: Each fgure shows a curve (thn lne) to be evolved and an evolved curve (thck lne). The curves to be evolved n the second, thrd and fourth columns are the perturbed curves of the curves to be evolved n the frst column wth p = 0.1, 0.2 and 0.4, respectvely. The evolved curves n the frst, second and thrd rows are the evoluton results usng curvature flow (1), averaged curvature flow (2) and area-preserved averaged curvature flow (3), respectvely, wth teraton number 50. The temporal step-sze used for each evolved curve s taken from Table 3 correspondngly. to solve, we use sem-mplct mxed fnte element method to solve them. Hence, mxed weak-forms of these flows are also gven. Then we consder the dscretzatons of these flows. 4.1 Fourth Order Curve Flows Curve Dffuson Flow. Curve dffuson flow s defned as x t = cκ s N, (1) where k s s sgned curvature, N s the orentated normal, such that k s N = κ. Ths flow s arc-length shortenng and area-preservng. Let y = κ s = κ T N, then the weak-form of

13 12 Guolang Xu ICM-RR (1) s wrtten as dx dt ϕds = N c yϕ ds, ϕ C0 2(Ω), (2) yψ ds = N c x T ψ ds, ψ C0 2(Ω), where C 2 0 (Ω) conssts of C2 smooth functons over Ω wth support n the nteror of Ω. Quas-Curve Dffuson Flow. Quas-curve dffuson flow s defned as dx dt = cκ. (3) Flow (3) s arc-length shortenng for a close space curve. Set y = κ, then the weak-form of the quas-curve dffuson flow s wrtten as dx dt ϕds = c y T c ϕds, ϕ C0 2(Ω), (4) yψ ds = c x T c ψ ds, ψ C0 2(Ω). Projectve Quas-Curve Dffuson Flow. The quas-curve dffuson flow contans the motons n the tangent and bnormal drectons. defne the projectve quas-curve dffuson flow as follows: To dsable the tangental moton, we dx dt = (I 3 tt T ) c κ, (5) where t s the unt tangent vector of the curve, I 3 s the unt matrx n R 3. Flow (5) s arc-length shortenng for evolvng close space curves. The weak-form of (5) can be wrtten as: Fnd (x, y) such that dx dt ϕds = yψ ds = (I 3 tt T ) c yϕ ds, ϕ C 2 0 (Ω), c x T c ψ ds, ψ C 2 0 (Ω). (6) Mnmal Squared-Curvature Flow. By mnmzng the followng energy E 1 (C) = κ 2 ds, C we obtan the followng flows dx dt = 2 cκ 3κ 2 κ. (7)

14 ICM-RR Numercal Integraton of Geometrc Flows 13 We name ths geometrc flow as mnmal squared-curvature flow. The weak-form of ths flow s wrtten as dx dt ϕds = yψ ds = 2 c y c ϕ + 3κ 2 c x c ϕ ] ds, ϕ C 2 0 (Ω). c x T c ψ ds, ψ C 2 0 (Ω). (8) Mnmal Curvature Flow. By mnmzng the followng energy E 2 (C) = we can derve the followng flow under the assumpton n 0. C κ ds, (9) dx dt = cn + κκ]. (10) Ths flow s named as mnmal curvature flow. For a plane convex curve, flow (10) s area preservng. The weak-form of the flow (10) can be wrtten as dx dt ϕds = yψ ds = y c ϕ κ ] κ c x T c ϕ ds, ϕ C 2 0 (Ω). c x T c ψ ds, ψ C 2 0 (Ω). (11) The rght-hand sde of (11) has sngularty when κ = 0. To overcome ths dffculty, we replace the term 1 κ wth 1 κ+ϵ, where ϵ > 0 s a small number (we take t as n our mplementaton). 4.2 The Dscretzaton of the Flows Snce the hghest order of the dervatves nvolved n the mxed weak-forms of the flows presented n the prevous subsecton s two, we therefore dscretze these flows n the space of cubc B-splne functons. Let x 0 x 1 x n ] be a gven control polygon of the cubc splne curve. The curve consdered can be opened and closed. For each vertex x of a close space control polygon, we assocate t wth a cubc B-splne bass functon ϕ (u), whch s defned on the knots { 2, 1,, +1, +2}. All the bass functons ϕ 0,, ϕ n are defned on the knots 2, 1, 0, 1,, n, n + 1, n + 2. For a open curve defned on 0, n], the bass functons are ϕ 1, ϕ 0,, ϕ n, ϕ n+1, whch are defned on the knots 0, 0, 0, 0, 1, 2,, n 1, n, n, n, n. n the followng, we consder only the case of

15 14 Guolang Xu ICM-RR close curve. Dscretzaton of (2). Let x(u) = y(u) = n x j ϕ j (u), c x(u) = j=0 n y j ϕ j (u), c y(u) = j=0 n x j c ϕ j (u). (12) j=0 n y j c ϕ j (u). (13) Substtutng these nto (2), and then takng ϕ = ψ = ϕ, we obtan the followng lnear system M (k) X (k+1) l + τ j=0 L (k) 1 Y (k+1), L (k) 2 Y (k+1), L (k) 3 Y (k+1) ] = M (k) X (k) l, l = 1, 2, 3, M (k) Y (k+1) = L (k) 1 X(k+1) 1 + L (k) 2 X(k+1) 2 + L (k) 3 X(k+1) 3, for the unknowns X (k+1) 1, X (k+1) 2, X (k+1) 3 and Y (k+1), where X (k+1) and τ s a gven temporal step-sze, X (k) = x (k) 0,, x(k) n ] T, Y (k) = y (k) 0,, y(k) n T, ] n M (k) = ϕ ϕ j ds C (k) L (k) l =,,j=0 C (k) N l ϕ c ϕ j ds N 1, N 2, N 3 ] T := N s computed from the curve C (k). C (k) = Equaton (14) can be equvalently wrtten as ] n, l = 1, 2, 3,,j=0 (14) ] X (k+1) 1, X (k+1) 2, X (k+1) 3 := { x (k) (u) = } n j=0 x(k) j ϕ j (u) : u Ω. M (k) + τl (k) ]X (k+1) = M (k) X (k), (15) where M (k) = dag M (k), M (k), M (k)], L (k) = X (k+1) = L (k) (M (k) ) 1 L (k) j ] 3,j=1, (X (k+1) 1 ) T, (X (k+1) 2 ) T, (X (k+1) 3 ) T ] T. Dscretzaton of (4). Let n y(u) = y j ϕ j (u), n c y(u) = c ϕ j (u)yj T. (16) j=0 j=0

16 ICM-RR Numercal Integraton of Geometrc Flows 15 Substtutng (12) and (16) nto (4), takng ϕ = ψ as ϕ and approxmatng dx dt scheme, we obtan the followng lnear system M (k) X (k+1) τl (k) Y (k+1) = M (k) X (k), M (k) Y (k+1) = L (k) X (k+1), by Euler (17) where L (k) = C (k) ( c ϕ ) T c ϕ j ds ] n.,j=0 Dscretzaton of (6). approxmatng dx dt Substtutng (12) and (16) nto (6), takng ϕ = ψ as ϕ and by Euler scheme, we obtan the followng lnear system M (k) X (k+1) τl (k) 1 Y(k+1) = M (k) X (k), M (k) Y (k+1) = L (k) 2 X(k+1), (18) where M (k) = X (k) = Y (k) = L (k) 1 = L (k) 2 = ] n,,j=0 I 3 ϕ ϕ j ds C (k) ] (x (k) 0 )T,, (x (k) n ) T T, ] (y (k) 0 )T,, (y n (k) ) T T, (I 3 tt T )ϕ c ϕ j ds C (k) ] n I 3 ( c ϕ ) T c ϕ j ds C (k) ] n.,j=0,,j=0 The tangent vector t n L (k) 1 s computed from x (k). Dscretzaton of (8) and (11). Smlar to the dscretzatons of (4) and (6) we obtan the followng lnear system M (k) X (k+1) τl (k) x X (k+1) + L (k) y Y (k+1) ] = M (k) X (k), where L (k) x = M (k) Y (k+1) = L (k) X (k+1), C (k) 3κ 2 ( c ϕ ) T c ϕ j ds ] n, L (k) y =,j=0 C (k) 2( c ϕ ) T c ϕ j ds ] n (19), for (8),,j=0

17 16 Guolang Xu ICM-RR ] n L x (k) = κ( ϕ ) T c ϕ j ds, L (k) y = ] cϕ ϕ n j C (k),j=0 C (k) κ + ϵ ds, for (11),,j=0 κ and c are computed from the curve C (k). Hence (19) s a lnear system. Gven a ntal set of the control ponts {x (0) } n =0 and a temporal step-sze τ, we can obtan teratvely a sequence of X (1),, X (k) by solvng system (19). 4.3 Performance Comparson To compare the performance of the fourth-order equatons. We use the same strateges as for the second-order flows. Here, the goodness of a τ s checked by the followng rules: 1. For flow (2), (4) and (6), the arc-length s shorten. 2. For flow (8), the energy E 1 (C) decreases. 3. For flow (11), the energy E 2 (C) decreases. Table 1: Maxmal allowable temporal step-szes. p = 0.0 p = 0.1 p = 0.2 p = 0.4 Flow (2) e e e e-05 Flow (4) Flow (6) e e e e-02 Flow (8) e e e e-10 Flow (11) e e e e-07 Snce (2), (4) and (6) are arc-length shortenng flow, an allowable temporal step-sze should make the arc-length decreasng. For flow (8) and flow (11), they mnmze the energy E 1 (C) and the energy E 2 (C), respectvely. Hence, we requre the energes decrease at each step. Fg. 1 shows the ntal curves, perturbed curves and the evolved curves. At each vertex x of the ntal curve, a perturbaton p(rand()/randmax 0.5) n the normal drecton of the curve s added wth p = 0.0, 0.1, 0.2 and 0.4. In Table 1, we lst the maxmal allowable temporal step-sze. These test results show that flow (4) s most stable. There s no lmtaton on the allowable temporal step-sze. After flow (4), the flows (6) and (2) follow. Flow (11) and (8) are the most unstable. Fg. 1 shows that these fourth-order flows have smoothng effects for nosy curves.

18 ICM-RR Numercal Integraton of Geometrc Flows 17 Fg 1: Each fgure shows a curve (thn lne) to be evolved and an evolved curve (thck lne). The curves to be evolved n the second, thrd and fourth columns are the perturbed curves of the curves to be evolved n the frst column wth p = 0.1, 0.2 and 0.4, respectvely. The evolved curves n the frst, second, thrd, fourth and ffth rows are the evoluton results usng flows (2), (4), (6), (8) and (11), respectvely. The temporal step-sze used for each evolved curve s taken from Table 1 correspondngly. For flow (4), the temporal step-sze s taken as one. From the frst row to the ffth row, the used teraton numbers are 5, 2, 1, 500 and 500, respectvely. 5 Geometrc Flows for Curve Regularzaton The curve evoluton may lead to sngulartes due to the edge collapse (edge length vanshng). To avod the happenng of the sngulartes, curve regularzaton technques can be

19 18 Guolang Xu ICM-RR ntroduced. In ths secton, two curve regularzaton technques are descrbed. One s to move the curve n the tangental drecton. Another s to move the curve by reparametrzaton. Two geometrc flows are then derved. Solvng methods for these flows are also descrbed. 5.1 Curve Regularzaton from Tangental Moton Gven a smooth parametrc space curve C(u) wth u a, b]. We regularze the curve by changng the speed of the curve from mnmzng the followng energy where E(x) = b a r = 1 b a x (u) 2 r 2] 2 du, (1) b a x (v) dv (2) s the average of the length of the tangent vector. Now we compute the Euler Lagrange equaton of the energy (1). Let x ε (u) = x(u) + εϕ(u)t(u) be a perturbaton of the curve x(u) n the tangental drecton, where t(u) = x (u)/ x (u) s the unt tangental vector. Then from de(x ε) dε = 0, we obtan b a ( x, x r 2 ) x, t ϕ + ( x, x r 2 ) x, t ϕ ] du = 0. Suppose ϕ s compactly supported, then from Green s formula we obtan the followng Euler-Lagrange equaton ( r 2 3 x, x ) x, t = 0. The correspondng L 2 gradent flow s defned as x t = t ( 3 x, x r 2) x, t ]. (3) Flow (3) descrbe a moton n the tangental drecton of the curve, we therefore name t as tangental moton flow. 5.2 Forward Euler Scheme and Temporal Step Sze Usng forward Euler scheme, the tangental moton flow s dscretzed as x (k+1) = x (k) + τϕ(x (k) ), (4)

20 ICM-RR Numercal Integraton of Geometrc Flows 19 where ( ϕ(x (k) ) = t(x (k) ) 3 (x (k) ), (x (k) ) r 2) ] (x (k) ), t(x (k) ). Let E(τ) = b a (x (k) ) + τϕ (x (k) ) 2 r 2] 2 du. (5) We determne a τ such that E(τ) s mnmzed. From ths obtan that τ E (0) E (0) = b a b ( a (x (k) ) 2 r 2) (x (k) ), ϕ (x (k) ) du 2 (x (k) ), ϕ (x (k) ) 2 + ( (x (k) ) 2 r 2) (x (k) ), ϕ (x (k) ) ] du. (6) 5.3 Spatal Dscretzaton of the Tangental Moton Flow Gven a polygon {x 0, x 1,, x n }. Assume x 0,, x n are the samplng ponts of a parametrc space curve at 0, 1,, n. Snce flow (3) s of the second-order, t s qute easy to solve. We smply use the followng explct Euler scheme to dscretze t. x (k+1) = x (k) + τt (k) R (k), (7) where ( R (k) = 3 x (k), x (k) r 2) x (k), t (k). x (k) and x (k) are dscretzed as x (k) = ( ) x (k) +1 x(k) 1 /2, (8) x (k) = x (k) +1 + x(k) 1 2x(k). (9) Ths dscretzaton s obtaned from the quadratc nterpolaton of three vertces x (k) 1, and x (k) +1 over the knots 1,, + 1, The ntal values of the teraton are gven as = x, = 0,, n. The number r s computed from (2). The ntegral s dscretzed by x (k) x (0) the trapezod rule. t (k) R (k) 0.01 s computed as x (k) ( x (k) 1 x(k) / x (k) + x (k) +1 x(k). The teraton stops f the condtons ), = 0,, n, hold. The temporal step-sze s computed by (6), where the ntegratons are dscretzed by the trapezod rule. ϕ (x (k) ) s computed as ϕ (x (k) ) = (t (k) +1 R(k) +1 t(k) 1 R(k) 1 )/2. dstrbuted un- Note that the am of the regularzaton s to make the vertces x (k) formly. Therefore, we ntroduce the followng defnton.

21 20 Guolang Xu ICM-RR Defnton 1 If the vertces {x k) } n the teraton scheme (7) are fxed ponts under the condton that the edge lengths x (k) s structure preserved. x (k) +1 are the same, then we say the teraton scheme Theorem 1 If x (k) and x (k) are defned by (8) and (9), respectvely, then the teraton scheme (7) s structure preserved. Proof. If s = x (k) x (k) +1 = s s ndependent of. Then t s easy to see that x (k), t (k) = x (k), x (k) /s = 0. Therefore, R (k) = 0 and all the x (k) are fxed. 5.4 Curve Regularzaton by Reparametrzaton Gven a smooth parametrc space curve x(u) wth u a, b]. We regularze the curve by reparameterze the curve from mnmzng the followng energy E(ϕ) = b a x(ϕ(u)) 2 r 2] 2 du, (10) where r s defned by (2), ϕ(u) s an ncreasng functon from a, b] to a, b]. That s, ϕ (u) > 0, ϕ(a) = a, ϕ(b) = b. Compute the frst-order varaton, we obtan the followng Euler-Lagrange equaton 3 x (ϕ) 2 x (ϕ), x (ϕ) (ϕ ) x (ϕ) 4 (ϕ ) 2 ϕ r 2 ( x (ϕ), x (ϕ) (ϕ ) 2 + x (ϕ) 2 ϕ ) ] = 0. The L 2 gradent flow s ϕ t = 3 x (ϕ) 2 x (ϕ), x (ϕ) (ϕ ) x (ϕ) 4 (ϕ ) 2 ϕ r 2 ( x (ϕ), x (ϕ) (ϕ ) 2 + x (ϕ) 2 ϕ ). (11) Flow (11) s a second-order flow, whch descrbes a moton of parameter of the curve, we therefore call t parameter moton flow. 5.5 Dscretzaton of the Parameter Moton Flow For smplcty, we use one step temporal drecton dscretzaton of (11). Then we consder the dscretzaton n spatal drecton.

22 ICM-RR Numercal Integraton of Geometrc Flows 21 One Step Temporal Dscretzaton. as Usng forward Euler scheme, (11) s dscretzed ϕ (k+1) = ϕ (k) + τ 3 x (ϕ (k) ) 2 x (ϕ (k) ), x (ϕ (k) ) ((ϕ (k) ) ) x (ϕ (k) ) 4 ((ϕ (k) ) ) 2 (ϕ (k) ) r 2 ( x (ϕ (k) ), x (ϕ (k) ) ((ϕ (k) ) ) 2 + x (ϕ (k) ) 2 (ϕ (k) ) ) ]. (12) Gven an ntal ϕ (0) (u) = u, then (ϕ (0) ) = 1, (ϕ (0) ) = 0. The one step teraton s ϕ (1) (u) = u + τ(3 x (u) 2 r 2 ) x (u), x (u). (13) Let where E(τ) = b a x(u + τψ(u)) τ ψ(u) = (3 x (u) 2 r 2 ) x (u), x (u). 2 r 2 ] 2 du, (14) We determne a τ such that E(τ) s mnmzed. From ths obtan that where E (0) = 4 E (0) = 4 b a b a ψ 2 du, τ E (0) E (0), (15) { 2 x, x 2 + ( x 2 r 2) ( x, x + x 2 ) ] ψ 2 + ( 2 x 2 r 2) 3 x, x ψψ + x 2 (ψ ) 2]} du Spatal Dscretzaton. Scheme (13) s dscretzed, n the spacal drecton, as ϕ (1) () = + τ(3 x () 2 r 2 ) x (), x (), = 1,, n 1, (16) x () and x () are computed as x (0) and x (0) by (8) and (9), respectvely. Averaged length r of tangent vector s computed from (2). After ϕ (1) () s obtaned, we compute x (1) as x(ϕ (1) ()). Suppose ϕ (1) () j, j + 1], then x(ϕ (1) ()) s computed from cubc nterpolaton of the four ponts x (0) j 1, x(0) j, x (0) j+1 and x(0) j+2 over the knots j 1, j, j +1, j +2.

23 22 Guolang Xu ICM-RR Temporal step-sze τ n (16) s computed by (15), where the ntegratons are dscretzed by the trapezod rule. ψ() s computed as ψ() = ( 3 x () 2 r 2) x (), x (). Snce we use the quadratc nterpolaton for computng dervatves of x(u), the thrd-order dervatves x (u) n (15) s taken as zero. Usng the algorthm for obtanng {x (1) from {x (k 1) } from {x (0) }. The teraton stops when 3 x () 2 x (), x () r 2 x (), x () < ϵ, for all. It s easy prove the followng result. }, we can teratvely compute {x (k) } Theorem 2 The regularzaton algorthm by reparametrzaton s structure preserved. 6 Concluson Dscretzaton schemes of several second-order and fourth-order geometrc flows have been constructed. The convergence results on the dscretzed dfferental operators are establshed. Two structure preserved algorthms for space curve regularzaton have been presented. Numercal tests are conducted, showng that these flows can be used for curve smoothng and each of them has dstnct features. References 1] S. J. Altschuler and M. A.. Grayson. Shortenng space curves and flow through sngulartes. J. Dfferental Geom., 35(2): , ] M. P. Do Carmo. Dfferental Geometry of Curves and Surfaces. Englewood Clffs, New Jersey, ] P. Derckx. Curve and Surface Fttng wth Splnes. Oxford Unversty Press, ] G. Farn. Curves and Surfaces for Computer Aded Geometrc Desgn: A Practcal Gude, Second Edton. Academc Press Inc., ] A. Guzec and N. Ayache. Smoothng and matchng of 3-d space curves. The Internatonal Journal of Computer Vson, 12(1):79 104, ] K. Hldebrandt, K. Polther, and E. Preuss. Evoluton of 3d curves under strct spatal constrants. In Proceedngs of the Nnth Internatonal Conference on Computer Aded Desgn and Computer Graphcs, CAD-CG 05, pages 40 45, Washngton, DC, USA, IEEE Computer Socety.

24 ICM-RR Numercal Integraton of Geometrc Flows 23 7] M. Hofer and H. Pottmann. Energy-mnmzng splnes n manfolds. In Computer Graphcs Proceedngs, Annual Conference seres, ACM SIGGRAPH04, pages , ] E. Cohen L. Tekumalla1. Smoothng Space Curves wth the MLS Projecton. In Geometrc Modelng, Vzualzaton and Graphcs, Salt Lake Cty, July ] J. Leng, Y. Zhang, and G. Xu. A novel geometrc flow-drven approach for qualty mprovement of segmented tetrahedral meshes. In Proceedngs of the 20th Internatonal Meshng Roundtable, pages , ] J. Wallner, H. Pottmann, and M. Hofer. Far webs. The Vsual Computer, 23(1):83 94, ] G. Xu. Geometrc Partal Dfferental Equaton Methods n Computatonal Geometry. Scence Press, Bejng, Chna, ] G. Xu. Geometrc Flows for Space Curves. Research Report No. ICM-12-13, Insttute for Computatonal Mathematcs and Scentfc/Engneerng Computng, ] Y. Lu Y. La and, Y. Zang, and S. Hu. Farng wreframes n ndustral surface desgn. In 2008 IEEE Internatonal Conference on Shape Modelng and Applcatons, pages 29 35, Stony Brook, New York, June ACM.