G 2 Surface Modeling Using Minimal Mean-Curvature-Variation Flow
|
|
- Lynette Hudson
- 5 years ago
- Views:
Transcription
1 G 2 Surface Modelng Usng Mnmal Mean-Curvature-Varaton Flow Guolang Xu 1 Qn Zhang 2 1,2 LSEC, Insttute of Computatonal Mathematcs, Academy of Mathematcs and System Scences, Chnese Academy of Scences, Bejng , Chna 2 Department of Basc Courses, Bejng Informaton Scence and Technology Unversty, Bejng , Chna Abstract Physcal and geometrc based varatonal technques for surface constructon have been shown to be advanced methods for desgnng hgh qualty surfaces n the felds of CAD and CAGD. In ths paper, we derve an Euler-Lagrange equaton from a geometrc nvarant curvature ntegral functonal the ntegral about the mean curvature gradent. Usng ths Euler-Lagrange equaton, we construct a sxthorder geometrc flow, whch s solved numercally by a dvded-dfference-lke method. We apply our equaton to solvng several surface modelng problems, ncludng surface blendng, N-sded hole fllng and pont nterpolatng, wth G 2 contnuty. The llustratve examples provded show that ths sxthorder flow yelds hgh qualty surfaces. Keywords: Euler-Lagrange equaton; Mnmal mean-curvature-varaton flow; Energy functonal; Dscretzaton; Surface modelng 1 Introducton Varatonal surface desgn. Surface farng (see [12] for bascs on the subject), free-from surface desgn (see [25]), surface blendng (see [27] and Fg. 5.3) and N-sded hole fllng (see [31] and Fg. 1.1) have been mportant ssues n the areas of CAD and CAGD. These problems can be effcently solved by an energy-based varatonal approach (e. g. [2, 4, 5, 10, 11, 21, 22, 24]). Roughly speakng, the varatonal approach s to pursue a Project supported n part by NSFC grant and Natonal Key Basc Research Project of Chna (2004CB318000). The second author s also supported n part by the NSFC grant and the Bejng Educatonal Commttee Foundaton KM curve or surface whch mnmzes certan type of energy smultaneously satsfyng prerequste boundary condtons. A problem one meets wthn ths approach s the choce of energy models. Energy models prevously used can be classfed nto the categores of physcal based and geometrc based. The class of physcal models encompasses membrane energy E 1 and stran energy E 2 of thn elastc plate (see [7, 24]): E 1 (f) := (fx 2 + fy 2 )dxdy, (1.1) E 2 (f) := (fxx 2 + 2fxy 2 + fyy)dxdy, 2 (1.2) where f(x, y) and are surface parametrzaton and ts doman, respectvely. These energes are generalzed as ( E 3 (M) := α11 r 2 u + 2α 12 r u r v + α 22 r 2 vrg ) dudv ( + β11 r 2 uu + 2β 12 r 2 uv + β 22 r 2 vv 2rg ) dudv by Terzopoulos et al. n [23] for a parametrc surface M := {r(u, v); (u, v) } whch can be regarded as a combnaton of E 1 (f) and E 2 (f), where α, β, g(u, v) are gven parameters and a vectorvalued functon. Recently, energy functonals based on geometrc nvarants begn to lead n ths feld. As s well-known, area functonal and total curvature functonal (see [13]) E 4 (M) := da, E 5 (M) := (k1 2 + k2)da 2 M are the most frequently used energes, where k 1 and k 2 are the prncpal curvatures. The mnmzng surfaces of E 4 (M) and E 5 (M) are mnmal surfaces and Wllmore surfaces, respectvely. The energy E 6 (M) := M [ (dk1 M de 1 ) 2 + ( dk2 de 2 ) 2 ] da 1
2 (a) (b) (c) Fg 1.1: (a) shows a head mesh wth several holes. (b) shows an ntal fller constructon. (c) s the smooth fllng surface, after 50 teratons, generated by usng equaton (3.8). proposed by Moreton et al. n [17] punshes the varaton of the prncpal curvatures, where e 1 and e 2 are prncpal drectons correspondng to the prncpal curvatures k 1 and k 2. The advantage of utlzng physcal based models s that the resultng equatons are lnear and therefore easy to solve. The dsadvantage s that the resultng equatons are parameter dependent. That means when a reparametrzaton s performed, dentcal surfaces may have dfferent energes. Energy models based on geometrc nvarants have no such a dsadvantage, whch are not affected by the choce of the parametrzaton. For certan specal cases, these two knds of functonal models are compatble when surfaces are sometrcally parameterzed. For nstance, E 4 (M) and E 5 (M) concde wth E 1 (f) and E 2 (f), respectvely. In fact, parametrzaton dependent functonals can be regarded as the lnear substtutes for geometrc nvarants. But n the general cases ths equvalency s not correct any more. Another crtcal problem of the varatonal approach s how to fnd out those surfaces whch mnmze these energy functonals. Two approaches have been employed to solve ths problem. One method s usng the optmzaton approach (see [11, 17, 20, 24]), whch starts from a gven surface, and searches teratvely a next surface that has less energy. Usng local nterpolaton or fttng, or replacng dfferental operators wth dvded dfference operators, the optmzaton problems are dscretzed to arrve at fnte dmensonal lnear or nonlnear systems. Approxmate solutons are then obtaned by solvng the constructed systems. Another wdely accepted method s based on varatonal calculus. The frst step of ths method s to calculate the Euler-Lagrange equatons for the energy functonals, then solve these equatons for the ultmate surfaces. Ths method s superor to the optmzaton technque n general because optmzaton s lack of local shape control and computatonally expensve. Gradent descent flow method. Generally speakng, Euler-Lagrange equatons of the geometrc energy functonals are hghly nonlnear. Except for a very lmted number of smple cases do these equatons gve analytc and smple soluton, drectly solvng of the equatons s dffcult. Gradent descent flow method s therefore ntroduced to crcumvent ths problem. For nstance, from the Euler-Lagrange equaton H = 0 of E 4 (M), whch s also the defnton of mnmal surface nvestgated snce 250 years ago, we can construct a flow, called mean curvature flow, r t = Hn, here n s the normal vector feld of the surface, the auxlary varable t s denoted as a tme-marchng parameter. When the steady state of the flow s acheved, we obtan H = 0. Smlarly, Wllmore surfaces (see [26]), the soluton of the Euler-Lagrange equaton H + 2H(H 2 K) = 0 of the energy E 7 (M) := H 2 da, (1.3) M can be constructed by ths gradent descent flow method. Note that functonal (1.3) s equvalent to E 5 (M) after Gauss-Bonnet-Chern formula beng taken nto account. For the purpose of volumepreservng for closed surfaces, surface dffuson flow (see [16]) r t = Hn s sometmes employed, whch can be regarded as a smplfed verson of the Wllmore flow. Contnuty. It s well-known that the second-order flows, such as mean curvature flow or averaged mean curvature flow (see [8]), yeld G 0 contnuous surfaces at the boundares of the constructed surfaces. The fourth-order flows, such as surface dffuson flow and Wllmore flow ([14, 26]), result n G 1 contnuty. However, hgher order contnuty are sometmes requred n the ndustral and engneerng applcatons. For nstance, n the shape desgn of the 2
3 streamlned surfaces of arcraft, shps and cars, G 2 contnuous surfaces are crucal. Therefore, hgher order flows need to be consdered. On ths aspect, Xu et al. have utlzed a sxth-order flow n [31] to acheve G 2 contnuty and Zhang et al. have used another sxth-order PDE n [32, 33] to obtan C 2 contnuty. A sxth-order equaton s also proposed n [3] by Botsch and Kobbelt to conduct real-tme freeform medelng. But all these sxth-order flows and PDEs are nether physcal based nor geometrc based n the sense mentoned above. Our Contrbutons. In ths paper, a sxth-order geometrc based PDE s ntroduced. It s derved from the Euler-Lagrange equaton of the energy functonal F (M) := H 2 da, (1.4) M whch punshes the total varaton of mean curvature. Ths functonal s smlar to but dfferent from the functonal E 8 (M) := M (Hn) 2 da, whch s proposed by Grener n [11] for smplfyng energy E 6 (M) and s used to far splne surfaces by an optmzaton approach. A surface whch mnmzes functonal (1.4) s called Mnmal Mean-Curvature- Varaton Surface. We expect that G 2 contnuty can be acheved usng ths sxth-order PDE n solvng the surface modelng problems, such as surface blendng, N-sded hole fllng and scattered ponts nterpolaton. The expermental results show that hgh qualty surfaces are obtaned. The rest of ths paper s organzed as follows. In Secton 2, some used notatons and prelmnares are ntroduced. One sxth-order flow s derved n Secton 3. The numercal solvng technque of the flow s dscussed n Secton 4. The applcaton and examples are provded n Secton 5. Secton 6 concludes ths paper. 2 Notatons and Prelmnares In ths secton, we ntroduce some notatons and several dfferental operators defned on surfaces. Let M be a regular parametrc surface represented as r(u, v) R 3, (u, v) R 2, whose unt normal vector s n = ru rv r u r v after sutable orentaton beng chosen, where the subscrpt of r denotes ts partal dervatve and x := x, x 1 2 := (x T x) 1 2 s the usual Eucldean norm. Superscrpt T stands for the transpose operaton. We assume at least r C 6 (, R 3 ). The coeffcents of the frst and the second fundamental forms are g 11 = r u, r u, g 12 = r u, r v, g 22 = r v, r v, b 11 = n, r uu, b 12 = n, r uv, b 22 = n, r vv. To smplfy the notaton we sometmes wrte w = (u, v) and u 1 = u, u 2 = v and [ g αβ ] = [ g αβ ] 1, g = det[ g αβ ], [ b αβ ] = [ b αβ ] 1, b = det[ b αβ ]. To ntroduce the mean curvature and Gaussan curvature, let us frst ntroduce the concept of Wengarten map. The Wengarten map or shape operator of surface M s a self-adjont lnear map on the tangent space T r M := span{r u, r v } defned by (see [9]) S : T r M T r M, S(v r ) = D v n, where v r s an arbtrary tangent vector of M at pont r and v s a tangent vector feld satsfyng v(r) = v r, and D v s the drectonal dervatve operator along drecton v. We can represent ths lnear map by a matrx as [ ] 1 b11 g S = 22 b 12 g 12 b 12 g 11 b 11 g 12 g 11 g 22 g12 2. b 12 g 22 b 22 g 12 b 22 g 11 b 12 g 12 The trace dvded by 2 and determnant of S, H = b 11g 22 + b 22 g 11 2b 12 g 12 2(g 11 g 22 g12 2 ) = k 1 + k 2, 2 K = b 11b 22 b 2 12 g 11 g 22 g12 2 = k 1 k 2 are the mean curvature and Gaussan curvature, respectvely. Now let us ntroduce some used dfferental operators defned on surface M. Tangental gradent operator. Let f be a smooth functon on M. Then the tangental gradent operator actng on f s gven by f = [r u, r v ][ g αβ ][f u, f v ] T R 3. (2.1) Second tangent operator. Let f be a smooth functon on M. Then the second tangent operator actng on f s gven by f = [r u, r v ][ K b αβ ][f u, f v ] T R 3. (2.2) These dfferental operators are all geometrc ntrnsc, though expressons (2.1) and (2.2) depend on a local surface parametrzaton. That s we have the followng lemma: 3
4 Lemma 2.1 Let r = r(u, v) and r = r(ū, v) be two dfferent parametrc representaton of surface M. Provded that the transformaton (u, v) (ū, v) preserves the orentaton of the surface, then f = f, f C 1 (M), (2.3) f = f, f C 1 (M), (2.4) where, and, are two groups of operators on M under dstnct coordnates. Proof. We only prove equaton (2.4). Smlar proof can be performed for (2.3). Assume that the change of parameters s σ : (ū, v) (u, v). then the transformaton of bass of the tangent space s [ ] [ u ] [ [ v rū = ū ū ru ru =: J r v u v v v r v ] r v ]. For the transformaton preservng the orentaton of the surface, n = n holds. Therefore the transformatons between the coeffcents of the frst and second fundamental forms are Hence [ ḡ αβ ] = J [ g αβ ] J T, [ bαβ ] = J [ b αβ ] J T. f = [ rū, r v ][ K b αβ ][ fū, f v ] T = [ r u, r v ]J T J T [ Kb αβ ]J 1 J[ f u, f v ] T = f. Dvergence operator. Let v be a C 1 smooth vector feld on M. Then the dvergence of v s defned by dv(v) = 1 [ g u, ] [ g [ g αβ ] [r u, r v ] T v ]. v Note that f v s a normal vector feld of M, dv(v) = 0. Laplace-Beltram operator. Let f C 2 (M). Then f s a smooth vector feld on M. The Laplace-Beltram operator (LBO) applyng to f s defned by f = dv( f) (see [6]). From the defnton of and dv, we can derve that f = 1 [ g u, ] [ g ] [ g αβ ] [f u, f v ] T. v It s easy to see that s a second-order dfferental operator whch relates closely to the mean curvature normal by the relaton r = 2H, where H := Hn s the mean curvature normal. It s well-known that these two dfferental operators are also geometrc ntrnsc though they are parametrcally defned. 3 A Sxth-Order Geometrc Flow In ths secton, we frst derve an Euler-Lagrange equaton for the functonal (1.4) and then construct a sxth-order geometrc flow. Theorem 3.1 Let F (M) be defned as (1.4). Then the Euler-Lagrange equaton of F (M) s 2 H+2(2H 2 K) H+2 H, H 2H H 2 = 0. (3.1) Proof. At frst, we can rewrte the functonal (1.4) as F (M) = H 2 gdu 1 du 2, (3.2) whch s parameter-nvarant. Consder now an extremal M of functonal (3.2) and a famly of normal varaton r(w, ε) of M defned by r(w, ε) = r(w) + εϕ(w)n(w), w, ε 1, where ϕ Cc () := {φ C (, R); suppφ }. Then we obtan 0 = d dε F (M(, ε)) ε=0 =: δf (M, ϕ), (3.3) where δf (M, ϕ) = δ( H 2 ) gdu 1 du 2 [ + H 2 (δ g)/ g ] gdu 1 du 2. (3.4) From δ(g αβ ) = 2ϕb αβ, δ(g) = 4gHϕ, δ( g) = 2Hϕ g, δ(h) = (2H 2 K)ϕ ϕ, we can deduce that δ( H 2 ) = δ( 1 g )(g 22H 2 u 1 + g 11H 2 u 2 2g 12H u 1H u 2) 4
5 + 1 g δ(g 22H 2 u 1 + g 11H 2 u 2 2g 12H u 1H u 2) = 4Hϕ H [ (b 22 Hu 2 g + b 11H 2 1 u 2b 12H 2 u 1H u 2)ϕ +(g 22 H u 1 g 12 H u 2) [(2H2 K)ϕ ϕ] u 1 +(g 11 H u 2 g 12 H u 1) [(2H2 K)ϕ ϕ] ] u 2 = 4Hϕ H 2 2 H, H ϕ +2 H, [(2H 2 K)ϕ + 1 ϕ]. (3.5) 2 In the above dervaton, we have used the fact that the varaton operaton δ and dervatve operaton u are commutatve. Substtutng (3.5) nto (3.4), α we arrve at [ δf (M, ϕ) = 2H H 2 ϕ 2 H, H ϕ +2 H, [(2H 2 K)ϕ + 1 ] gdu 2 ϕ] 1 du 2 [ = 2H H 2 ϕ 2 H, H ϕ 2(2H 2 K) Hϕ H ϕ] gdu 1 du 2. Usng the Green s formula, f 2 h gdu 1 du 2 = f h gdu 1 du 2, for all f Cc (), we eventually wrte (3.3) as [ 2H H 2 2 H, H ] ϕ gdu 1 du 2 [ 2(2H 2 K) H + 2 H ] ϕ gdu 1 du 2 = 0, for any ϕ Cc (). In the end, the Euler-Lagrange equaton of functonal (1.4) s (3.1) and the theorem s proved. Obvously, equaton (3.1) s of sxth-order. It s easy to see that surfaces wth constant mean curvature, such as Delaunay surfaces (see [18], pp ) (nclude undulod and nodod), sphere, cylnder, and mnmal surfaces, are the solutons of the equaton. But tor and cone are not the soluton surfaces of the equaton. It s not dffcult to derve that Theorem 3.2 Equaton (3.1) s nvarant under the transforms of rotaton, translaton and scalng. Here the nvarant means that a soluton surface of (3.1) s stll a soluton under the three transforms mentoned. Now let us ntroduce the sxth-order flow used n ths paper. Let M 0 be a compact mmersed orentable surface n R 3. A curvature drven geometrc evoluton conssts of fndng a famly {M(t) : t 0} of smooth mmersed orentable surfaces n R 3 whch evolve accordng to the flow equaton r t = nu, M(0) = M 0. (3.6) Here r(t) s a surface pont on M(t), U denotes the normal velocty of M(t), whch depends on the curvatures H and K of M(t). Let M(t) be a closed surface wth outward normal. Then t has been shown that (see [15], Theorem 4) da(t) dt = 2 UHdA, M(t) dv (t) dt = M(t) UdA, (3.7) where A(t) denotes the area of surface M(t) and V (t) denotes the volume of the regon enclosed by M(t). If da(t) dt = 0, we say that the flow s areapreservng. Smlarly, the flow s volume-preservng dv (t) f dt = 0. Let M 0 be a compact orentable surface n R 3 wth boundary Γ. Then the sxth-order flow constructed from the Euler-Lagrange equaton (3.1) s r t = [ 2 H + 2(2H 2 K) H + 2 H, H 2H H 2] n, r M(t), M(0) = M 0, M(t) = Γ. (3.8) If M(t) s a closed constant mean curvature surface, (3.7) mples that da(t) dv (t) dt = 0 and dt = 0 for the flow (3.8). In general, ths area-preservng or volume-preservng propertes are not vald. In ths paper we name ths newly ntroduced flow as mnmal mean-curvature-varaton flow (abbrevated as MMCVF). Though the problems of the exstence and the unqueness of the solutons of the geometrc flow (3.8) are currently left open, the numercal solvng of the equaton could be conducted by ether the dvded-dfference-lke (generalzed dvded dfference) method or the fnte element approach. For smplcty, we solve t n ths paper by the dvded-dfference-lke method. 5
6 4 Numercal Solvng of the GPDE 4.1 Dscretzatons of curvatures and geometrc dfferental operators To solve the geometrc PDE (3.8) over a trangular surface mesh M wth vertex set {r } usng a dvdeddfference-lke method, dscrete approxmatons of the mean curvature, Gaussan curvature and varous dfferental operators are requred. In order to use a sem-mplct scheme, we requre the approxmatons of the dfferental operators mentoned above at r to have the followng form Θf(r ) = j N 1() w Θ jf(r j ), where Θ represents one of above mentoned dfferental operators and w Θ j R or wθ j R3, N k () s the ndex set of the k-rng neghbor vertces of r. Although there are several dscretzaton schemes of Laplace-Beltram operator and Gaussan curvature (see [28, 30] for a revew), the dscretzatons of Gaussan curvature are not n the requred form and may be not consstent n the followng sense. Defnton 4.1 A set of approxmatons of dfferental geometrc operators s sad consstent f there exsts a smooth surface S, such that the approxmate operators concde wth the exact counterparts of S. Here we use a bquadratc fttng of the surface data and functon data to calculate the approxmate dfferental operators. The algorthm we adopted s from [29]. Let r be a vertex of M wth valence n, r j be ts neghbor vertces for j N 1 (). Then approxmatons of the used dfferental operators are represented as (see [29] for detal) f(r ) j N 1() w j f(r j), w j R3, f(r ) j N 1() w j f(r j), w j R3, f(r ) j N 1() w j f(r j), w j R, K(r ) j N 1() (wk j )T r j, w K j R3. Usng the relaton r = 2H = 2Hn, we have H(r ) 1 2 H(r ) 1 2 j N 1() j N 1() w j r j, w j n(r ) T r j. Remark 4.1. Now we explan why we use the dscretzed dfferental operators based on the parametrc fttng rather than other methods. The frst reason s that ths fttng scheme yelds a convergent approxmaton as the mesh sze (the maxmal edge length) h 0. The second reason s that the computaton of these operators s consstent. The thrd reason s that the fttng scheme yelds the requred form expressons, whch are ready for use n the sem-mplct dscretzaton of the PDE. The last reason s that all the dfferental operators used n ths paper nvolve the frst and second order dervatves of the surface or functons on the surface. Hence, quadratc functon s enough to provde these partal dervatve data. 4.2 Sem-mplct dscretzaton of the GPDE Let us now consder the dscretzaton of (3.8). An explct scheme for solvng the equaton (3.8) n general s unstable, therefore requres a small tme stepsze. To make the evoluton process more effcent, an mplct scheme s more desrable. However, snce the used PDE s hghly nonlnear, a complete mplct scheme s hard to solve. In the followng we present a sem-mplct scheme, whch leads to a lnear system of equatons. The basc dea for formng the lnear equatons s to decompose each of the terms of (3.1) as a product of a lnear term and a remanng term. The lnear term s dscretzed usng the dscretzed dfferental operator. The remanng term s computed from prevous approxmaton of the surface. Specfcally, the terms of the equaton (3.8) are approxmated as follows: r t r(k+1) r (k), τ 2 H ( H (k+1) ), 2(2H 2 K) H (2H (k+1) H (k) +[2(H (k) ) 2 K (k) ] H (k+1), 2 H, H H (k+1), H (k) + H (k), H (k+1), 2H H, H H (k+1) +H (k) H (k+1), H (k), K (k+1) H (k), H (k) ) H (k) where the subscrpt denotes the correspondng quantty s evaluated at the vertex r, the superscrpt (k) denotes the quantty s at the tme kτ, the superscrpt (k + 1) denotes the quantty s at the tme (k + 1)τ. The quanttes at (k + 1)τ are 6
7 unknowns. Usng these approxmatons, we can dscretze the equaton (3.8) recursvely, and derve a lnear system wth r (k+1) as unknowns. For nstance, n ( H (k+1) ) n 1 2 Smlarly, j N 1() w j H (k+1) j wj j N 1() l N 1(j) wj j N 1() l N 1(j) j N 1() w jl(n n T l )H (k+1) l w jl(n n T l ) m N 1(l) n H (k+1), H (k) wj, H (k) n H (k+1) j 1 2 j N 1() j N 1() where n := n(r (k) w j, H (k) (n n T j )H (k+1) j wj, H (k) (n n T j ) l N 1(j) w lmr (k+1) m. w jlr (k+1) l, ) s the surface normal at r (k). Note that the dscretzed equaton at the vertex r nvolves three-rng neghbor vertces. 4.3 Boundary condton Suppose we are gven a trangular surface mesh M wth certan vertces are tagged as nteror. The nteror vertces are subject to change. The remanng vertces are fxed. Usng the above mentoned approxmaton of dfferental operators, we can dscretze recursvely the GPDE for each nteror vertex r and fnally derve a lnear equaton. Ths equaton s a lnear combnaton of the three-rng neghbor vertces of r. r (k+1) + τ j N 3() w j r (k+1) j = r (k), w j R 3 3. If an nvolved vertex r (k+1) j s not an nteror one, r (k+1) j = r j s fxed and the term τw j r (k+1) j s moved to the rght hand sde of the equaton. Such a treatment of the boundary condton leads to a system of n equatons wth n unknowns. Here n s the number of nteror vertces. The dea of ths boundary treatment s adopted from [31]. 4.4 Solvng the lnear system The result system s hghly sparse. An teratve approach for solvng the system s desrable. We employ Saad s teratve method (see [19]), named GM- RES, to solve the system. The experment shows that ths teratve method works very well. 5 Illustratve Examples 5.1 Recover property to some surfaces We have mentoned that constant mean curvature surfaces are the solutons of equaton (3.1). Fg. 5.1 and 5.2 are used to llustrate that constant mean curvature surfaces can be recovered from ther perturbed counterparts by the sxth-order geometrc flow. The test s performed as follows. We frst replace certan parts of a gven constant mean curvature surface wth another surface, and then we use our geometrc flow to evolve the surface. The frst row of Fg. 5.1 shows that a cylnder s recovered, where (a) s a cylnder wth certan parts mssng. Fgure (b) shows the mnmal surface fllng of the mssng parts. Ths mnmal surface acts as an ntal surface M 0 for the geometrc flow. (c) shows the evoluton result. It can be seen that the cylnder s correctly recovered. The second row of Fg. 5.1 shows that a sphere s recovered, where (d) s a wre-frame of a sphere wth eght openngs. These openngs are flled wth mnmal surfaces as shown n (e). These mnmal surfaces act as ntal surfaces M 0 of the geometrc flow. (f) shows the evoluton result. It s easy to see that the sphere s perfectly recovered. Fg. 5.2 shows a more complex shaped surface example, where fgure (b) shows a Delaunay surface patch wth a mssng part, whch s flled wth a surface patch as shown n (c). Ths Delaunay surface s generated by rotatng the parametrc curve { x(u) = sn(4u), y(u) = u 1+3 sn(4t) dt, sn(4t) as shown n (a). (d) shows evoluton result of the fllng surface. (e) shows the nput surface wth the fllng surface. It s not dffcult to observe that the Delaunay surface s recovered. It should be ponted out that the recovery property s vald only f the provded perturbed surface s nearby the soluton surface. 7
8 (a) (b) (c) (d) (e) (f) Fg 5.1: (a) s a cylnder wth certan parts are removed. (b) shows the mnmal surface fllng of the removed parts. (c) shows the evoluton results. (d) s a wre-frame of a sphere wth eght openngs. These openngs are flled wth mnmal surfaces as shown n (e). (f) shows the evoluton result (a) (b) (c) (d) (e) Fg 5.2: (b) shows a Delaunay surface wth a mssng part, whch s flled wth a surface patch as shown n (c). The Delaunay surface s the rotaton result of the curve as shown n (a). (d) shows the evoluton result of the fllng surface. (e) shows the nput surface wth the fllng surface. 5.2 Smooth blendng of surfaces Gven a collecton of surface meshes wth boundares, we construct a far surface to blend smoothly the meshes at the boundares. Fg 5.3 shows the case, where surfaces to be blended are gven (fgure (a), (d) and (g)) wth ntal mnmal surface constructons (fgure (b), (e) and (h)) usng [1] and then mean curvature flow. The surfaces (c), (f) and () are the blendng meshes generated usng our sxthorder flow. These fgures show the results after 60 teratons wth tme step szes N-sded hole fllng Gven a surface mesh wth holes, we construct a far surface to fll smoothly the holes wth G 2 contnuty at the boundary. Fg 1.1 shows such an example, where a head mesh wth several holes n the nose, face and jaw subregons s gven as the nput (fgure (a)). An ntal G 0 fller of the holes are shown n (b) usng [1] and then evolved wth the mean curvature flow. The blendng surface (c) s generated usng flow (3.8). 5.4 Pont nterpolaton For the pont nterpolaton problem, we are gven some ponts as the nput data, and we wsh to construct a far surface mesh to nterpolate ths multdmensonal data. Fg. 5.4 shows ths surface constructon approach, where a dodecahedron s served as nput as shown n fgure (a). The constructed surface s requred to nterpolate the vertces of the nput polygon. Each face of the nput polygon s trangulated by subdvdng the 5-sded face nto three trangles. Then each trangle s subdvded nto 64 sub-trangles. The GPDE s appled to the trangulated polygon wth the nput vertces fxed. (b) s 8
9 (a) (b) (c) (d) (e) (f) (g) (h) () Fg 5.3: Fgure (a), (d) and (g) show surfaces to be blended wth ntal mnmal surface constructons (fgure (b), (e) and (h)). The surfaces (c), (f) and () are the blendng meshes generated usng the sxth-order flow. the evoluton result usng the sxth-order flow. (c) shows an ntermedate result of the evoluton. 5.5 Comparson wth fourth-order flows Now we compare the used sxth-order flow MM- CVF wth three well-known lower order flows (see [31]): mean curvature flow (MCF), surface dffuson flow (SDF) and Wllmore flow (WF). From the defnton of MMCVF, we know that the man dfference of MMCVF from the lower order flows s that the former yelds G 2 and mean curvature unformly dstrbuted surfaces. Fg. 5.5 shows the evoluton results of the sxth- and fourth-order flows for the nput (d) of Fg. 5.3, where (a), (b) and (c) show the mean curvature plots of the evoluton results usng the MMCVF, SDF and WF, respectvely. From these fgures, we can observe that the surface produced by MMCVF s mean-curvature contnuous at the blendng boundares, whle the surfaces produced by SDF and WF are not. Fg. 5.6 and 5.7 show the dfferent effects of MM- CVF from the MCF and WF. Fg. 5.6 (a), (b) 9
10 (a) (b) (c) Fg 5.4: (a) shows the nput dodecahedron. (b) s the evoluton result of the sxth-order flow. The surface s requred to nterpolate the vertces of the nput polygon. (c) shows an ntermedate result of the evoluton process. (a) (b) (c) Fg 5.5: (a), (b) and (c) are the mean curvature plots of the evoluton results of MMCVF, SDF and WF, respectvely. (a) (b) (c) Fg 5.6: (a), (b) and (c) are the evoluton results of MCF, WF and MMCVF of an nput rng wth two fxed crcles on the rng. and (c) show the evoluton results (mean curvature plots) usng the MCF, WF and MMCVF from an nput rng wth two fxed crcles on t. It s easy to see that MMCVF produces more far surface (the rng s recovered). In Fg. 5.7, the surface to be evolved s defned as a graph of a functon g: x(u, v) = [u, v, g(u, v)] T, g(u, v) = e(u, v) + e(u + 1, v) + e(u, v + 1) + e(u + 1, v + 1), wth (u, v) := [ 1, 1] 2 and e(u, v) = exp[ (u 0.5)2 + (v 0.5) 2 ]. Ths surface s unformly trangulated usng a grd over the doman. We evolve a part of the surface, where g > 1.5. Fgures (a), (b) and (c) show the results of MCF, WF and MMCVF, respectvely. Fgures (d), (e) and (f) are the mean curvature plots of (a), (b) and (c), respectvely. It s easy to see that the second and the fourth-order flows are not curvature contnuous at the boundares of the evolved surface patch. 10
11 (a) (b) (c) (d) (e) (f) Fg 5.7: (a), (b) and (c) are the evoluton results of the MCF, WF and MMCVF. (d), (e) and (f) are the mean curvature plots of (a), (b) and (c), respectvely. 5.6 Runnng Tmes We summarze n Table 5.1 the computaton tme needed by some of our examples. The algorthm was mplemented n C++ runnng on a Dell PC wth a 3.0GHz Intel CPU. All the examples presented n ths secton are the approxmate steady soluton (t ). Hence, the total tme costs depend greatly on how far we go n the tme drecton, whch n turn depend on how far the ntal surface away from the fnal soluton. In Table 5.1, we lst the costs for a sngle teraton. The second column n Table 5.1 s the number of unknowns. These numbers are counted as 3n 0 (each vertex has x, y, z varables). Here n 0 s the number of nteror vertces. The thrd column s the used tme step-sze. The fourth column n the table s the tme (n seconds) for formng the coeffcent matrx. The ffth column s the number of evoluton steps. The last column s the tme for solvng the lnear systems for one tme step. 6 Conclusons We have derved a sxth-order nonlnear geometrc flow from the functonal F = M H 2 da. Ths flow can be used to solve several surface modelng problems, such as surface denosng, surface blendng, N-sded hole fllng and free-form surface desgn, when G 2 contnuty at the boundary s requred. The expermental results show that the dvded-dfference-lke method for solvng the nonlnear equaton s effcent and the flow yelds hgh qualty and hgh order contnuty surfaces. References [1] C. Bajaj and I. Ihm. Algebrac surface desgn wth Hermte nterpolaton. ACM Transactons on Graphcs, 11(1):61 91, [2] G. P. Bonneau, H. Hagen, and St. Hahmann. Varatonal surface desgn and surface nterrogaton. Computer Graphcs Forum, 12(3): , [3] M. Botsch and L. Kobbelt. An ntutve framework for real-tme freeform modelng. ACM Transacton on Graphcs, 23(3): , Proceedngs of the 2004 SIGGRAPH Conference. [4] G. Brunnett, H. Hagen, and P. Santarell. Varatonal desgn of curves and surfaces. Surv. Math. Industry, 3:1 27, [5] G. Celnker and D. Gossard. Deformable curve and surface fnte-elements for free-form shape desgn. ACM Computer Graphcs, 25(4): , [6] M. P. do Carmo. Remannan Geometry. Boston, [7] H. Du and H. Qn. Dynamc PDE-based surface desgn usng geometrc and physcal constrant. Graphcal Models, 67(1):43 71, [8] J. Escher and G. Smonett. The volume preservng mean curvature flow. Proceedngs of the Amercan Mathematcal Socety, 126(9): ,
12 Table 5.1: Runnng Tmes Examples # unknowns Tme step-sze Form matrx # steps Solvng Tme Fg 5.1(c) Fg 5.1(f) Fg 5.3(c) Fg 5.3(f) Fg 5.3() Fg 1.1(c) Fg 5.7(c) [9] A. Gray. Modern Dfferental Geometry of Curves and Surfaces wth Mathematca. CRC Press, second edton, [10] G. Grener. Surface constructon based on varatonal prncples. In P. J. Laurent, A. LeMéhauté, and L. L. Schumaker, edtors, Wavelets, Images, and Surface Fttng, pages Wellesley MA, AKPeters. [11] G. Grener. Varatonal desgn and farng of splne surface. Computer Graphcs Forum, 13: , [12] J. Hoschek and D. Lasser. Grundlagen der geometrschen Datenverarbetung. B. G. Teubner, Stuttgart, [13] M. Kallay. Constraned optmzaton n surface desgn. In B. Falcdeno and T. L. Kun, edtors, Modelng n Computer Graphcs, pages Sprnger-Verlag, Berln, [14] E. Kuwert and R. Schätzle. The Wllmore flow wth small ntal energy. J. Dfferental Geom., 57(3): , [15] H. B. Lawson. Lectures on Mnmal Submanfolds. Publsh or Persh, Berkeley, CA, [16] U. F. Mayer. Numercal solutons for the surface dffuson flow n three space dmensons. Comput. Appl. Math., 20(3): , [17] H. P. Moreton and C. H. Séqun. Functonal optmzaton for far surface desgn. SIGGRAPH 92 Conference Proceedngs, pages , [18] J. Oprea. Dfferental Geometry and Its Applcatons. Pearson Educaton, Inc., second edton, [19] Y. Saad. Iteratve Methods for Sparse Lnear Systems. Second Edton wth correctons, [20] N. Sapds. Desgnng Far Curves and Surfaces: Shape Qualty n Geometrc Modelng and Computer-Aded Desgn. Socety for Industral and Appled Mathematcs, Phladelpha, PA, USA, [21] R. Schneder and L. Kobbelt. Geometrc far meshes wth G 1 boundary condtons. In Geometrc Modelng and Processng, pages , Hongkong, Chna. [22] R. Schneder and L. Kobbelt. Geometrc farng of rregular meshes for free-form surface desgn. Computer Aded Geometrc Desgn, 18(4): , [23] D. Terzopoulos, J. Platt, and A. Barr. Elastcally deformable models. ACM Computer Graphcs, 24(4): , [24] W. Welch and A. Wtkn. Varatonal surface modelng. Computer Graphcs, 26: , [25] W. Welch and A. Wtkn. Free-from shape desgn usng trangluated surfaces. In SIGGRAPH 94 Proceedngs, volume 28, pages , [26] T. J. Wllmore. Remannan Geometry. Clarendon Press, Oxford, England, [27] L. R. Woodwark. Blends n geometrc modellng. In R. R. Martn, edtor, The Mathematcs of Surface II, pages Oxford Unversty Press, [28] G. Xu. Dscrete Laplace-Beltram operators and ther convergence. Computer Aded Geometrc Desgn, 21(8): , [29] G. Xu. Consstent approxmaton of some geometrc dfferental operators. Research Report No. ICM-06-01, Insttute of Computatonal Mathematcs, Chnese Academy of Scences, [30] G. Xu. Convergence analyss of a dscretzaton scheme for Gaussan curvature over trangular surfaces. Computer Aded Geometrc Desgn, 23(2): , [31] G. Xu, Q. Pan, and C. L. Bajaj. Dscrete surface modellng usng partal dfferental equatons. Computer Aded Geometrc Desgn, 23(2): , [32] L. H. You, P. Comnnos, and J. J. Zhang. PDE blendng surfaces wth C 2 contnuty. Computers and Graphcs, 28(6): , [33] J. J. Zhang and L. H. You. Fast surface modellng usng a 6th order PDE. Comput. Graph. Forum, 23(3): ,
Fair Triangle Mesh Generation with Discrete Elastica
Far Trangle Mesh Generaton wth Dscrete Elastca Shn Yoshzawa, and Alexander G. Belyaev, Computer Graphcs Group, Max-Planck-Insttut für Informatk, 66123 Saarbrücken, Germany Phone: [+49](681)9325-414 Fax:
More informationAn Accurate Evaluation of Integrals in Convex and Non convex Polygonal Domain by Twelve Node Quadrilateral Finite Element Method
Internatonal Journal of Computatonal and Appled Mathematcs. ISSN 89-4966 Volume, Number (07), pp. 33-4 Research Inda Publcatons http://www.rpublcaton.com An Accurate Evaluaton of Integrals n Convex and
More informationDiscrete surface modelling using partial differential equations
Computer Aded Geometrc Desgn 23 (2006) 125 145 www.elsever.com/locate/cagd Dscrete surface modellng usng partal dfferental equatons Guolang Xu a,1,qngpan a, Chandrajt L. Bajaj b,,2 a State Key Laboratory
More informationHermite Splines in Lie Groups as Products of Geodesics
Hermte Splnes n Le Groups as Products of Geodescs Ethan Eade Updated May 28, 2017 1 Introducton 1.1 Goal Ths document defnes a curve n the Le group G parametrzed by tme and by structural parameters n the
More informationAccounting for the Use of Different Length Scale Factors in x, y and z Directions
1 Accountng for the Use of Dfferent Length Scale Factors n x, y and z Drectons Taha Soch (taha.soch@kcl.ac.uk) Imagng Scences & Bomedcal Engneerng, Kng s College London, The Rayne Insttute, St Thomas Hosptal,
More informationParallelism for Nested Loops with Non-uniform and Flow Dependences
Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr
More informationWavefront Reconstructor
A Dstrbuted Smplex B-Splne Based Wavefront Reconstructor Coen de Vsser and Mchel Verhaegen 14-12-201212 2012 Delft Unversty of Technology Contents Introducton Wavefront reconstructon usng Smplex B-Splnes
More informationVery simple computational domains can be discretized using boundary-fitted structured meshes (also called grids)
Structured meshes Very smple computatonal domans can be dscretzed usng boundary-ftted structured meshes (also called grds) The grd lnes of a Cartesan mesh are parallel to one another Structured meshes
More informationComputers and Mathematics with Applications. Discrete schemes for Gaussian curvature and their convergence
Computers and Mathematcs wth Applcatons 57 (009) 87 95 Contents lsts avalable at ScenceDrect Computers and Mathematcs wth Applcatons journal homepage: www.elsever.com/locate/camwa Dscrete schemes for Gaussan
More informationLECTURE : MANIFOLD LEARNING
LECTURE : MANIFOLD LEARNING Rta Osadchy Some sldes are due to L.Saul, V. C. Raykar, N. Verma Topcs PCA MDS IsoMap LLE EgenMaps Done! Dmensonalty Reducton Data representaton Inputs are real-valued vectors
More information2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements
Module 3: Element Propertes Lecture : Lagrange and Serendpty Elements 5 In last lecture note, the nterpolaton functons are derved on the bass of assumed polynomal from Pascal s trangle for the fled varable.
More informationHigh-Boost Mesh Filtering for 3-D Shape Enhancement
Hgh-Boost Mesh Flterng for 3-D Shape Enhancement Hrokazu Yagou Λ Alexander Belyaev y Damng We z Λ y z ; ; Shape Modelng Laboratory, Unversty of Azu, Azu-Wakamatsu 965-8580 Japan y Computer Graphcs Group,
More informationNUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS
ARPN Journal of Engneerng and Appled Scences 006-017 Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana
More informationS1 Note. Basis functions.
S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type
More informationA Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme
Mathematcal and Computatonal Applcatons Artcle A Fve-Pont Subdvson Scheme wth Two Parameters and a Four-Pont Shape-Preservng Scheme Jeqng Tan,2, Bo Wang, * and Jun Sh School of Mathematcs, Hefe Unversty
More informationAn Application of the Dulmage-Mendelsohn Decomposition to Sparse Null Space Bases of Full Row Rank Matrices
Internatonal Mathematcal Forum, Vol 7, 2012, no 52, 2549-2554 An Applcaton of the Dulmage-Mendelsohn Decomposton to Sparse Null Space Bases of Full Row Rank Matrces Mostafa Khorramzadeh Department of Mathematcal
More informationCluster Analysis of Electrical Behavior
Journal of Computer and Communcatons, 205, 3, 88-93 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/cc http://dx.do.org/0.4236/cc.205.350 Cluster Analyss of Electrcal Behavor Ln Lu Ln Lu, School
More informationAdaptive Fairing of Surface Meshes by Geometric Diffusion
Adaptve Farng of Surface Meshes by Geometrc Dffuson Chandrajt L. Bajaj Department of Computer Scences, Unversty of Texas, Austn, TX 78712 Emal: bajaj@cs.utexas.edu Guolang Xu State Key Lab. of Scentfc
More informationInterpolation of the Irregular Curve Network of Ship Hull Form Using Subdivision Surfaces
7 Interpolaton of the Irregular Curve Network of Shp Hull Form Usng Subdvson Surfaces Kyu-Yeul Lee, Doo-Yeoun Cho and Tae-Wan Km Seoul Natonal Unversty, kylee@snu.ac.kr,whendus@snu.ac.kr,taewan}@snu.ac.kr
More informationIn the planar case, one possibility to create a high quality. curve that interpolates a given set of points is to use a clothoid spline,
Dscrete Farng of Curves and Surfaces Based on Lnear Curvature Dstrbuton R. Schneder and L. Kobbelt Abstract. In the planar case, one possblty to create a hgh qualty curve that nterpolates a gven set of
More informationType-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data
Malaysan Journal of Mathematcal Scences 11(S) Aprl : 35 46 (2017) Specal Issue: The 2nd Internatonal Conference and Workshop on Mathematcal Analyss (ICWOMA 2016) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES
More information12/2/2009. Announcements. Parametric / Non-parametric. Case-Based Reasoning. Nearest-Neighbor on Images. Nearest-Neighbor Classification
Introducton to Artfcal Intellgence V22.0472-001 Fall 2009 Lecture 24: Nearest-Neghbors & Support Vector Machnes Rob Fergus Dept of Computer Scence, Courant Insttute, NYU Sldes from Danel Yeung, John DeNero
More informationSolitary and Traveling Wave Solutions to a Model. of Long Range Diffusion Involving Flux with. Stability Analysis
Internatonal Mathematcal Forum, Vol. 6,, no. 7, 8 Soltary and Travelng Wave Solutons to a Model of Long Range ffuson Involvng Flux wth Stablty Analyss Manar A. Al-Qudah Math epartment, Rabgh Faculty of
More informationAn Optimal Algorithm for Prufer Codes *
J. Software Engneerng & Applcatons, 2009, 2: 111-115 do:10.4236/jsea.2009.22016 Publshed Onlne July 2009 (www.scrp.org/journal/jsea) An Optmal Algorthm for Prufer Codes * Xaodong Wang 1, 2, Le Wang 3,
More informationA Binarization Algorithm specialized on Document Images and Photos
A Bnarzaton Algorthm specalzed on Document mages and Photos Ergna Kavalleratou Dept. of nformaton and Communcaton Systems Engneerng Unversty of the Aegean kavalleratou@aegean.gr Abstract n ths paper, a
More informationBarycentric Coordinates. From: Mean Value Coordinates for Closed Triangular Meshes by Ju et al.
Barycentrc Coordnates From: Mean Value Coordnates for Closed Trangular Meshes by Ju et al. Motvaton Data nterpolaton from the vertces of a boundary polygon to ts nteror Boundary value problems Shadng Space
More informationChapter 6 Programmng the fnte element method Inow turn to the man subject of ths book: The mplementaton of the fnte element algorthm n computer programs. In order to make my dscusson as straghtforward
More informationDiscrete Schemes for Gaussian Curvature and Their Convergence
Dscrete Schemes for Gaussan Curvature and Ther Convergence Zhqang Xu Guolang Xu Insttute of Computatonal Math. and Sc. and Eng. Computng, Academy of Mathematcs and System Scences, Chnese Academy of Scences,
More informationFor instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)
Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A
More informationLearning the Kernel Parameters in Kernel Minimum Distance Classifier
Learnng the Kernel Parameters n Kernel Mnmum Dstance Classfer Daoqang Zhang 1,, Songcan Chen and Zh-Hua Zhou 1* 1 Natonal Laboratory for Novel Software Technology Nanjng Unversty, Nanjng 193, Chna Department
More informationQuality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation
Intellgent Informaton Management, 013, 5, 191-195 Publshed Onlne November 013 (http://www.scrp.org/journal/m) http://dx.do.org/10.36/m.013.5601 Qualty Improvement Algorthm for Tetrahedral Mesh Based on
More informationA Geometric Approach for Multi-Degree Spline
L X, Huang ZJ, Lu Z. A geometrc approach for mult-degree splne. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 27(4): 84 850 July 202. DOI 0.007/s390-02-268-2 A Geometrc Approach for Mult-Degree Splne Xn L
More informationA Newton-Type Method for Constrained Least-Squares Data-Fitting with Easy-to-Control Rational Curves
A Newton-Type Method for Constraned Least-Squares Data-Fttng wth Easy-to-Control Ratonal Curves G. Cascola a, L. Roman b, a Department of Mathematcs, Unversty of Bologna, P.zza d Porta San Donato 5, 4017
More informationMathematics 256 a course in differential equations for engineering students
Mathematcs 56 a course n dfferental equatons for engneerng students Chapter 5. More effcent methods of numercal soluton Euler s method s qute neffcent. Because the error s essentally proportonal to the
More informationPositive Semi-definite Programming Localization in Wireless Sensor Networks
Postve Sem-defnte Programmng Localzaton n Wreless Sensor etworks Shengdong Xe 1,, Jn Wang, Aqun Hu 1, Yunl Gu, Jang Xu, 1 School of Informaton Scence and Engneerng, Southeast Unversty, 10096, anjng Computer
More informationReading. 14. Subdivision curves. Recommended:
eadng ecommended: Stollntz, Deose, and Salesn. Wavelets for Computer Graphcs: heory and Applcatons, 996, secton 6.-6., A.5. 4. Subdvson curves Note: there s an error n Stollntz, et al., secton A.5. Equaton
More informationProblem Definitions and Evaluation Criteria for Computational Expensive Optimization
Problem efntons and Evaluaton Crtera for Computatonal Expensve Optmzaton B. Lu 1, Q. Chen and Q. Zhang 3, J. J. Lang 4, P. N. Suganthan, B. Y. Qu 6 1 epartment of Computng, Glyndwr Unversty, UK Faclty
More informationSum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints
Australan Journal of Basc and Appled Scences, 2(4): 1204-1208, 2008 ISSN 1991-8178 Sum of Lnear and Fractonal Multobjectve Programmng Problem under Fuzzy Rules Constrants 1 2 Sanjay Jan and Kalash Lachhwan
More informationLecture 5: Multilayer Perceptrons
Lecture 5: Multlayer Perceptrons Roger Grosse 1 Introducton So far, we ve only talked about lnear models: lnear regresson and lnear bnary classfers. We noted that there are functons that can t be represented
More informationR s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges
More informationA MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS
Proceedngs of the Wnter Smulaton Conference M E Kuhl, N M Steger, F B Armstrong, and J A Jones, eds A MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS Mark W Brantley Chun-Hung
More informationA new paradigm of fuzzy control point in space curve
MATEMATIKA, 2016, Volume 32, Number 2, 153 159 c Penerbt UTM Press All rghts reserved A new paradgm of fuzzy control pont n space curve 1 Abd Fatah Wahab, 2 Mohd Sallehuddn Husan and 3 Mohammad Izat Emr
More informationModeling, Manipulating, and Visualizing Continuous Volumetric Data: A Novel Spline-based Approach
Modelng, Manpulatng, and Vsualzng Contnuous Volumetrc Data: A Novel Splne-based Approach Jng Hua Center for Vsual Computng, Department of Computer Scence SUNY at Stony Brook Talk Outlne Introducton and
More informationActive Contours/Snakes
Actve Contours/Snakes Erkut Erdem Acknowledgement: The sldes are adapted from the sldes prepared by K. Grauman of Unversty of Texas at Austn Fttng: Edges vs. boundares Edges useful sgnal to ndcate occludng
More informationModule 6: FEM for Plates and Shells Lecture 6: Finite Element Analysis of Shell
Module 6: FEM for Plates and Shells Lecture 6: Fnte Element Analyss of Shell 3 6.6. Introducton A shell s a curved surface, whch by vrtue of ther shape can wthstand both membrane and bendng forces. A shell
More informationA New Approach For the Ranking of Fuzzy Sets With Different Heights
New pproach For the ankng of Fuzzy Sets Wth Dfferent Heghts Pushpnder Sngh School of Mathematcs Computer pplcatons Thapar Unversty, Patala-7 00 Inda pushpndersnl@gmalcom STCT ankng of fuzzy sets plays
More informationSupport Vector Machines
/9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.
More informationVectorization of Image Outlines Using Rational Spline and Genetic Algorithm
01 Internatonal Conference on Image, Vson and Computng (ICIVC 01) IPCSIT vol. 50 (01) (01) IACSIT Press, Sngapore DOI: 10.776/IPCSIT.01.V50.4 Vectorzaton of Image Outlnes Usng Ratonal Splne and Genetc
More informationS.P.H. : A SOLUTION TO AVOID USING EROSION CRITERION?
S.P.H. : A SOLUTION TO AVOID USING EROSION CRITERION? Célne GALLET ENSICA 1 place Emle Bloun 31056 TOULOUSE CEDEX e-mal :cgallet@ensca.fr Jean Luc LACOME DYNALIS Immeuble AEROPOLE - Bat 1 5, Avenue Albert
More informationParameterization of Quadrilateral Meshes
Parameterzaton of Quadrlateral Meshes L Lu 1, CaMng Zhang 1,, and Frank Cheng 3 1 School of Computer Scence and Technology, Shandong Unversty, Jnan, Chna Department of Computer Scence and Technology, Shandong
More informationClassification / Regression Support Vector Machines
Classfcaton / Regresson Support Vector Machnes Jeff Howbert Introducton to Machne Learnng Wnter 04 Topcs SVM classfers for lnearly separable classes SVM classfers for non-lnearly separable classes SVM
More informationCS 534: Computer Vision Model Fitting
CS 534: Computer Vson Model Fttng Sprng 004 Ahmed Elgammal Dept of Computer Scence CS 534 Model Fttng - 1 Outlnes Model fttng s mportant Least-squares fttng Maxmum lkelhood estmaton MAP estmaton Robust
More informationAn Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed
More informationFinite Element Analysis of Rubber Sealing Ring Resilience Behavior Qu Jia 1,a, Chen Geng 1,b and Yang Yuwei 2,c
Advanced Materals Research Onlne: 03-06-3 ISSN: 66-8985, Vol. 705, pp 40-44 do:0.408/www.scentfc.net/amr.705.40 03 Trans Tech Publcatons, Swtzerland Fnte Element Analyss of Rubber Sealng Rng Reslence Behavor
More informationMesh Editing in ROI with Dual Laplacian
Mesh Edtng n ROI wth Dual Laplacan Luo Qong, Lu Bo, Ma Zhan-guo, Zhang Hong-bn College of Computer Scence, Beng Unversty of Technology, Chna lqngng@sohu.com, lubo@but.edu.cn,mzgsy@63.com,zhb@publc.bta.net.cn
More informationRobust Curvature Estimation and Geometry Analysis of 3D point Cloud Surfaces
Robust Curvature Estmaton and Geometry Analyss of 3D pont Cloud Surfaces Xaopeng ZHANG, Hongjun LI, Zhangln CHENG, Ykuan ZHANG Sno-French Laboratory LIAMA, Insttute of Automaton, CAS, Bejng 100190, Chna
More information6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour
6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the
More informationApplications of DEC:
Applcatons of DEC: Flud Mechancs and Meshng Dscrete Dfferental Geometry Overvew Puttng DEC to good use Fluds, fluds, fluds geometrc nterpretaton of classcal models dscrete geometrc nterpretaton new geometry-based
More informationRelated-Mode Attacks on CTR Encryption Mode
Internatonal Journal of Network Securty, Vol.4, No.3, PP.282 287, May 2007 282 Related-Mode Attacks on CTR Encrypton Mode Dayn Wang, Dongda Ln, and Wenlng Wu (Correspondng author: Dayn Wang) Key Laboratory
More informationMultiblock method for database generation in finite element programs
Proc. of the 9th WSEAS Int. Conf. on Mathematcal Methods and Computatonal Technques n Electrcal Engneerng, Arcachon, October 13-15, 2007 53 Multblock method for database generaton n fnte element programs
More informationF Geometric Mean Graphs
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 2 (December 2015), pp. 937-952 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) F Geometrc Mean Graphs A.
More informationAnalysis of Continuous Beams in General
Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support,
More informationA Fast Visual Tracking Algorithm Based on Circle Pixels Matching
A Fast Vsual Trackng Algorthm Based on Crcle Pxels Matchng Zhqang Hou hou_zhq@sohu.com Chongzhao Han czhan@mal.xjtu.edu.cn Ln Zheng Abstract: A fast vsual trackng algorthm based on crcle pxels matchng
More informationThe Shortest Path of Touring Lines given in the Plane
Send Orders for Reprnts to reprnts@benthamscence.ae 262 The Open Cybernetcs & Systemcs Journal, 2015, 9, 262-267 The Shortest Path of Tourng Lnes gven n the Plane Open Access Ljuan Wang 1,2, Dandan He
More informationShape Control of the Cubic Trigonometric B-Spline Polynomial Curve with A Shape Parameter
Internatonal Journal of Scences & Appled esearch www.jsar.n Shape Control of the Cubc Trgonometrc B-Splne Polynomal Curve wth A Shape Parameter Urvash Mshra* Department of Mathematcs, Mata Gujr Mahla Mahavdyalaya,
More informationHarmonic Coordinates for Character Articulation PIXAR
Harmonc Coordnates for Character Artculaton PIXAR Pushkar Josh Mark Meyer Tony DeRose Bran Green Tom Sanock We have a complex source mesh nsde of a smpler cage mesh We want vertex deformatons appled to
More informationA Unified, Integral Construction For Coordinates Over Closed Curves
A Unfed, Integral Constructon For Coordnates Over Closed Curves Schaefer S., Ju T. and Warren J. Abstract We propose a smple generalzaton of Shephard s nterpolaton to pecewse smooth, convex closed curves
More informationReview of approximation techniques
CHAPTER 2 Revew of appromaton technques 2. Introducton Optmzaton problems n engneerng desgn are characterzed by the followng assocated features: the objectve functon and constrants are mplct functons evaluated
More informationIntra-Parametric Analysis of a Fuzzy MOLP
Intra-Parametrc Analyss of a Fuzzy MOLP a MIAO-LING WANG a Department of Industral Engneerng and Management a Mnghsn Insttute of Technology and Hsnchu Tawan, ROC b HSIAO-FAN WANG b Insttute of Industral
More informationAMath 483/583 Lecture 21 May 13, Notes: Notes: Jacobi iteration. Notes: Jacobi with OpenMP coarse grain
AMath 483/583 Lecture 21 May 13, 2011 Today: OpenMP and MPI versons of Jacob teraton Gauss-Sedel and SOR teratve methods Next week: More MPI Debuggng and totalvew GPU computng Read: Class notes and references
More informationResearch Article Quasi-Bézier Curves with Shape Parameters
Hndaw Publshng Corporaton Appled Mathematcs Volume 3, Artcle ID 739, 9 pages http://dxdoorg/55/3/739 Research Artcle Quas-Bézer Curves wth Shape Parameters Jun Chen Faculty of Scence, Nngbo Unversty of
More informationImage Representation & Visualization Basic Imaging Algorithms Shape Representation and Analysis. outline
mage Vsualzaton mage Vsualzaton mage Representaton & Vsualzaton Basc magng Algorthms Shape Representaton and Analyss outlne mage Representaton & Vsualzaton Basc magng Algorthms Shape Representaton and
More informationInvestigations of Topology and Shape of Multi-material Optimum Design of Structures
Advanced Scence and Tecnology Letters Vol.141 (GST 2016), pp.241-245 ttp://dx.do.org/10.14257/astl.2016.141.52 Investgatons of Topology and Sape of Mult-materal Optmum Desgn of Structures Quoc Hoan Doan
More informationThe Codesign Challenge
ECE 4530 Codesgn Challenge Fall 2007 Hardware/Software Codesgn The Codesgn Challenge Objectves In the codesgn challenge, your task s to accelerate a gven software reference mplementaton as fast as possble.
More informationRange images. Range image registration. Examples of sampling patterns. Range images and range surfaces
Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples
More informationStructured Grid Generation Via Constraint on Displacement of Internal Nodes
Internatonal Journal of Basc & Appled Scences IJBAS-IJENS Vol: 11 No: 4 79 Structured Grd Generaton Va Constrant on Dsplacement of Internal Nodes Al Ashrafzadeh, Razeh Jalalabad Abstract Structured grd
More informationCategories and Subject Descriptors B.7.2 [Integrated Circuits]: Design Aids Verification. General Terms Algorithms
3. Fndng Determnstc Soluton from Underdetermned Equaton: Large-Scale Performance Modelng by Least Angle Regresson Xn L ECE Department, Carnege Mellon Unversty Forbs Avenue, Pttsburgh, PA 3 xnl@ece.cmu.edu
More informationGeometric Error Estimation
Geometrc Error Estmaton Houman Borouchak Project-team GAMMA 3 UTT Troyes, France Emal: houman.borouchak@utt.fr Patrck Laug Project-team GAMMA 3 INRIA Pars - Rocquencourt, France Emal: patrck.laug@nra.fr
More informationLobachevsky State University of Nizhni Novgorod. Polyhedron. Quick Start Guide
Lobachevsky State Unversty of Nzhn Novgorod Polyhedron Quck Start Gude Nzhn Novgorod 2016 Contents Specfcaton of Polyhedron software... 3 Theoretcal background... 4 1. Interface of Polyhedron... 6 1.1.
More informationSubspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;
Subspace clusterng Clusterng Fundamental to all clusterng technques s the choce of dstance measure between data ponts; D q ( ) ( ) 2 x x = x x, j k = 1 k jk Squared Eucldean dstance Assumpton: All features
More informationConstructing Minimum Connected Dominating Set: Algorithmic approach
Constructng Mnmum Connected Domnatng Set: Algorthmc approach G.N. Puroht and Usha Sharma Centre for Mathematcal Scences, Banasthal Unversty, Rajasthan 304022 usha.sharma94@yahoo.com Abstract: Connected
More informationContent Based Image Retrieval Using 2-D Discrete Wavelet with Texture Feature with Different Classifiers
IOSR Journal of Electroncs and Communcaton Engneerng (IOSR-JECE) e-issn: 78-834,p- ISSN: 78-8735.Volume 9, Issue, Ver. IV (Mar - Apr. 04), PP 0-07 Content Based Image Retreval Usng -D Dscrete Wavelet wth
More informationMulti-stable Perception. Necker Cube
Mult-stable Percepton Necker Cube Spnnng dancer lluson, Nobuuk Kaahara Fttng and Algnment Computer Vson Szelsk 6.1 James Has Acknowledgment: Man sldes from Derek Hoem, Lana Lazebnk, and Grauman&Lebe 2008
More informationAssignment # 2. Farrukh Jabeen Algorithms 510 Assignment #2 Due Date: June 15, 2009.
Farrukh Jabeen Algorthms 51 Assgnment #2 Due Date: June 15, 29. Assgnment # 2 Chapter 3 Dscrete Fourer Transforms Implement the FFT for the DFT. Descrbed n sectons 3.1 and 3.2. Delverables: 1. Concse descrpton
More informationAPPLICATION OF MULTIVARIATE LOSS FUNCTION FOR ASSESSMENT OF THE QUALITY OF TECHNOLOGICAL PROCESS MANAGEMENT
3. - 5. 5., Brno, Czech Republc, EU APPLICATION OF MULTIVARIATE LOSS FUNCTION FOR ASSESSMENT OF THE QUALITY OF TECHNOLOGICAL PROCESS MANAGEMENT Abstract Josef TOŠENOVSKÝ ) Lenka MONSPORTOVÁ ) Flp TOŠENOVSKÝ
More informationCHAPTER 2 DECOMPOSITION OF GRAPHS
CHAPTER DECOMPOSITION OF GRAPHS. INTRODUCTION A graph H s called a Supersubdvson of a graph G f H s obtaned from G by replacng every edge uv of G by a bpartte graph,m (m may vary for each edge by dentfyng
More informationSolutions to Programming Assignment Five Interpolation and Numerical Differentiation
College of Engneerng and Coputer Scence Mechancal Engneerng Departent Mechancal Engneerng 309 Nuercal Analyss of Engneerng Systes Sprng 04 Nuber: 537 Instructor: Larry Caretto Solutons to Prograng Assgnent
More informationLine Clipping by Convex and Nonconvex Polyhedra in E 3
Lne Clppng by Convex and Nonconvex Polyhedra n E 3 Václav Skala 1 Department of Informatcs and Computer Scence Unversty of West Bohema Unverztní 22, Box 314, 306 14 Plzeò Czech Republc e-mal: skala@kv.zcu.cz
More informationShape Preserving Positive and Convex Data Visualization using Rational Bi-cubic Functions
Shape Preservng Postve and Conve Data Vsualzaton usng Ratonal B-cubc unctons Muhammad Sarfraz Malk Zawwar Hussan 3 Tahra Sumbal Shakh Department of Informaton Scence Adala Campus Kuwat Unverst Kuwat E-mal:
More informationSnakes-based approach for extraction of building roof contours from digital aerial images
Snakes-based approach for extracton of buldng roof contours from dgtal aeral mages Alur P. Dal Poz and Antono J. Fazan São Paulo State Unversty Dept. of Cartography, R. Roberto Smonsen 305 19060-900 Presdente
More informationSolving two-person zero-sum game by Matlab
Appled Mechancs and Materals Onlne: 2011-02-02 ISSN: 1662-7482, Vols. 50-51, pp 262-265 do:10.4028/www.scentfc.net/amm.50-51.262 2011 Trans Tech Publcatons, Swtzerland Solvng two-person zero-sum game by
More informationFitting: Deformable contours April 26 th, 2018
4/6/08 Fttng: Deformable contours Aprl 6 th, 08 Yong Jae Lee UC Davs Recap so far: Groupng and Fttng Goal: move from array of pxel values (or flter outputs) to a collecton of regons, objects, and shapes.
More informationSLAM Summer School 2006 Practical 2: SLAM using Monocular Vision
SLAM Summer School 2006 Practcal 2: SLAM usng Monocular Vson Javer Cvera, Unversty of Zaragoza Andrew J. Davson, Imperal College London J.M.M Montel, Unversty of Zaragoza. josemar@unzar.es, jcvera@unzar.es,
More informationDetermining the Optimal Bandwidth Based on Multi-criterion Fusion
Proceedngs of 01 4th Internatonal Conference on Machne Learnng and Computng IPCSIT vol. 5 (01) (01) IACSIT Press, Sngapore Determnng the Optmal Bandwdth Based on Mult-crteron Fuson Ha-L Lang 1+, Xan-Mn
More informationProper Choice of Data Used for the Estimation of Datum Transformation Parameters
Proper Choce of Data Used for the Estmaton of Datum Transformaton Parameters Hakan S. KUTOGLU, Turkey Key words: Coordnate systems; transformaton; estmaton, relablty. SUMMARY Advances n technologes and
More informationPolyhedral Surface Smoothing with Simultaneous Mesh Regularization
olyhedral Surface Smoothng wth Smultaneous Mesh Regularzaton Yutaka Ohtake The Unversty of Azu Azu-Wakamatsu Cty Fukushma 965-8580 Japan d800@u-azu.ac.jp Alexander G. Belyaev The Unversty of Azu Azu-Wakamatsu
More informationNonlocal Mumford-Shah Model for Image Segmentation
for Image Segmentaton 1 College of Informaton Engneerng, Qngdao Unversty, Qngdao, 266000,Chna E-mal:ccluxaoq@163.com ebo e 23 College of Informaton Engneerng, Qngdao Unversty, Qngdao, 266000,Chna E-mal:
More informationOutline. Midterm Review. Declaring Variables. Main Variable Data Types. Symbolic Constants. Arithmetic Operators. Midterm Review March 24, 2014
Mdterm Revew March 4, 4 Mdterm Revew Larry Caretto Mechancal Engneerng 9 Numercal Analyss of Engneerng Systems March 4, 4 Outlne VBA and MATLAB codng Varable types Control structures (Loopng and Choce)
More informationConstructing G 2 Continuous Curve on Freeform Surface with Normal Projection
Chnese Journal of Aeronautcs 3(010 137-144 Chnese Journal of Aeronautcs www.elsever.com/locate/cja Constructng G Contnuous Curve on Freeform Surface wth Normal Projecton Wang Xaopng*, An Lulng, Zhou Lashu,
More informationNAG Fortran Library Chapter Introduction. G10 Smoothing in Statistics
Introducton G10 NAG Fortran Lbrary Chapter Introducton G10 Smoothng n Statstcs Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Smoothng Methods... 2 2.2 Smoothng Splnes and Regresson
More information