Discrete Schemes for Gaussian Curvature and Their Convergence
|
|
- Kevin Young
- 6 years ago
- Views:
Transcription
1 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, Bejng, Chna Abstract In ths paper, several dscrete schemes for Gaussan curvature are surveyed. The convergence property of a modfed dscrete scheme for the Gaussan curvature s proved. Furthermore, a new dscrete scheme for Gaussan curvature s proposed. We show that ths new scheme converges at the regular vertex wth valence not less than 5. By constructng a counterexample, we also show that t s mpossble for buldng a dscrete scheme for Gaussan curvature whch converges over the regular vertex wth valence 4. Fnally, asymptotc errors of several dscrete schemes for Gaussan curvature are compared. AMS Subject Classfcatons: Prmary 68U07, 68U05, 65S05, 53A40. Keywords: Dscrete Gaussan curvature, dscrete mean curvature, geometrc modelng. 1 Introducton Some applcatons from computer vson, computer graphcs, geometrc modelng and computer aded desgn requre estmatng ntrnsc geometrc nvarants. It s well known that Gaussan curvature s one of the most essental geometrc nvarants for surfaces. However, n the classcal dfferental geometry, ths nvarant s well defned only for C 2 smooth surfaces. In modern computer-related The frst author s Supported by the NSFC grant The second author s supported by NSFC grant and Natonal Key Basc Research Project of Chna (2004CB318000). Emal:xuzq@lsec.cc.ac.cn Emal: xuguo@lsec.cc.ac.cn 1
2 geometry felds, one usually uses dscrete trangle meshes, whch are only C 0 contnues, representng smooth surfaces approxmately. Hence, the problem of estmatng accurately Gaussan curvature for trangle meshes s rased naturally. In the past years, a wealth of dfferent estmators have been proposed n the vast lterature of appled geometry. These methods for estmatng Gaussan curvature can be dvded nto two classes. The frst class s based on the local fttng or nterpolaton technque[2, 4, 5, 6, 17], whle the second class s to dscretze mathematcal formulatons whch present the nformaton about the Gaussan curvature[1, 3, 6, 10, 13, 15]. In ths paper, our focus s on the methods n the second class. Let M be a trangulaton of smooth surface S n R 3. For a vertex p of M, suppose {p } n =1 s the set of the one-rng neghbor vertces of p and the set {p pp +1 } ( = 1,, n) of n Eucldean trangles forms a pecewse lnear approxmaton of S around p. We let γ denote the angle p pp +1 and let the angular defect at p be 2π γ. Moreover, we use the followng conventons throughout the paper p n+1 = p 1 and p 0 = p n. A popular dscrete scheme for computng Gaussan curvature s n the form of 2π P γ,where G s a geometry quantty. In general, G one selects G as A(p)/3 and obtan the followng approxmaton K (1) := 3(2π γ ), (1) A(p) where A(p) s the sum of the areas of trangles p pp +1. In [1], another scheme s gven, where S p := K (2) := 2π γ S p (2) 1 [ 4 sn γ η η +1 cos γ 2 ] (η 2 + η+1) 2 2
3 s called the module of the mesh at p. In [13], the dscrete approxmaton K (1) s modfed as K (3) 2π := γ 1 2 area(p pp +1 ) 1 8 cot(γ, (3) )d 2 where d s the length of edges p p +1. There are several dfferent ponts of vew for explanng the reason why the angular defect s closely related to the Gaussan curvature wth ncludng the vewponts of Gaussan-Bonnet theorem, Gaussan map and Legendre s formula(see the next secton for detals). Asymptotc analyses for the dscrete schemes have been gven n [1, 6, 15]. In [6], the authors show that for the non-unform data, the dscrete scheme K (1) does not convergent to true Gaussan curvature always. In [1], Borrell et al. prove that the angular defect s asymptotcally equvalent to a homogeneous polynomal of degree two n the prncple curvatures and show that f p s a regular vertex wth valence sx, then the scheme K (2) converges to the exact Gaussan curvature n a lnear rate. Moreover, Borrell et al. show that 4 s the only value of the valence such that the angular defect depends upon the prncpal drectons. In [15], G. Xu proves that the dscrete scheme K (1) has quadratc convergence rate f the mesh satsfes the so-called parallelogram crteron. The convergence condtons presented n all these papers requre the valence of the vertex beng 6. Therefore, one hopes to construct a dscrete scheme whch converges over any dscrete mesh. But n [18], Z. Xu et al. show that t s mpossble to construct a dscrete scheme whch s convergent for any dscrete mesh. Hence, we have to content wth the dscrete schemes whch converge under some condtons. Accordng to the past experence[1, 7, 18], we regard a dscrete scheme desrable f t has the followng propertes 1. It converges at regular vertex, at least for suffcently large va- 3
4 lence; 2. It converges at the umblcal pont,.e. the ponts satsfyng k m = k M where k m and k M are two prncpal curvatures. As stated before, the prevous dscrete scheme only converges at the regular vertex wth valence 6. In [1], a method for computng the Gaussan curvature at the regular vertex wth valence unequal to 4 s descrbed. But the method requres two meshes wth valences n 1 and n 2 (n 1 4, n 2 4, n 1 n 2 ). In ths paper, we wll construct a dscrete scheme whch converges at the regular vertex wth valence not less than 5, and also at the umblcal ponts wth any valence. Moreover, the dscrete scheme requres only a sngle mesh. Hence, the new dscrete scheme s more desrable. Furthermore, we show that t s mpossble to construct a dscrete scheme whch s convergent at the regular vertex wth valence 4. Therefore, for the regular vertex, the convergent problem remanng open s the case of vertex wth valence 3. The rest of the paper s organzed as follows. Secton 2 descrbes some notatons and defntons and Secton 3 shows three dfferent vewponts for expressng the relaton between the angular defect and Gaussan curvature. In Secton 4, we study the convergence property of the modfed dscrete Gaussan curvature scheme. We present n Secton 5 a new dscrete scheme and prove that the scheme has good convergence property. In Secton 6, for the regular vertex wth valence 4, we show that t s mpossble to buld a dscrete scheme whch s convergent towards the real Gaussan curvature. Some numercal results are gven n Secton 7. 4
5 2 Prelmnares Ô Õ Æ «Õ ½ Ô ½ Ô Fg.1. Notatons. In ths secton, we ntroduce some notatons and defntons used throughout the paper(see Fg. 1). Let S be a gven smooth surface and p be a pont over S. Suppose the set {p pp +1 }, = 1,, n of n Eucldean trangles form a pecewse lnear approxmaton of S around p. The vector from p to p s denoted as pp. The normal vector and tangent plane of S at the pont p s denoted as n and Π, respectvely. We denote the projecton of p onto Π as q, and defne the plane contanng n, p and p as Π. Then we let κ denote the curvature of the plane curve S Π. The dstances from p to p and q are denoted as η and l, respectvely. The angles p pp +1 and q pq +1 are denoted as γ and β, respectvely. The two prncpal curvatures are denoted as k m and k M. Let η = max η. The followng results are presented n [1, 15, 7]: where w R. l η = 1 + O(η), β = γ + O(η 2 ), (4) w pp = w κ η O(η 3 ), (5) 5
6 Now we gve the defnton of the regular vertex usng the notatons above. Defnton 1 Let p be a pont of a smooth surface S and let p, = 1,, n be ts one rng neghbors. The pont p s called a regular vertex f t satsfes the followng condtons (1) the β = 2π, n (2) the η s all take the same value η. 3 Angular Defect and Gaussan Curvature In ths secton, we summarze three dfferent vewponts for expressng the relaton between angular defect and Gaussan curvature. These vewponts have been descrbed n dfferent lterature([6, 13, 15]). We collect them there. Throughout the secton, we use K (1) (p) to denote the dscrete Gaussan curvature, whch s obtaned usng K (1). 3.1 Gaussan-Bonnet theorem vewpont Let D be a regon of surface S, whose boundary conssts of pecewse smooth curves Γ j s. Then the local Gaussan-Bonnet theorem s as follows K(p)dA + k g (Γ j )ds + α j = 2π, D j Γ j j where K(p) s the Gaussan curvature at p, k g (Γ j ) s the geodesc curvature of the boundary curve Γ j and α j s the exteror angle at the jth corner pont p j of the boundary. If all the Γ j s are the geodesc curves, the above formula reduces to K(p)dA = 2π α j. (6) D j 6
7 Let M be a trangulaton of surface S. For vertex p of valence n, each trangle p pp +1 can be parttoned nto three equal parts, one correspondng to each of ts vertces. We let D be the unon of the part correspondng to p of trangles p pp +1. Note that γ = j α j. Assumng K(p) s a constant on D, and usng (6), we have K(p) can be approxmated by K (1) (p). 3.2 Sphercal mage vewpont We Now ntroduce another defnton of Gaussan curvature. Let D be a small patch of area A ncludng pont p on the surface S. There wll be a correspondng patch of area I on the Gaussan map. Gaussan curvature at p s the lmt of rato lm A 0 I A. Let us consder a dscrete verson of the defnton. The Gaussan map mage,.e. the sphercal mage, of the trangle p pp +1 s (p p the pont ) (p p +1 ) (p p ) (p p +1. Jon these pont by great crcle formng ) a sphercal polygon on the unt sphere. The area of ths sphercal polygon s 2π γ. Same as the above, each trangle s parttoned nto three parts, one correspondng to each vertex. Then the Gaussan curvature can be approxmated by K (1) (p). 3.3 Geodesc trangles vewpont Let T = ABC be a geodesc trangle on the surface S wth angles α, β, γ and geodesc edge lengths a, b, c. Let A B C be a correspondng Eucldean trangle wth edge lengths a, b, c and angles α, β, γ. Legendre presents the followng formulaton α α = area(t ) K(A) 3 + o(a 2 + b 2 + c 2 ), where area(t ) s the area of the geodesc trangle ABC, K(A) s the Gaussan curvature at A. 7
8 Usng Legendre s formulaton for each trangles wth p as a vertex, we arrve at the estmatng formula K (1) (p) agan. 4 Convergence of Angular Defect Schemes In [15], G. Xu gves an analyss about the scheme K (1) and proves that the scheme converges at the vertex satsfyng so-called parallelogram crteron. A numercal test shows that the scheme does not converge over the regular vertex wth valence unequal to 6 and umblcal ponts. In [1], V. Borrell et. al. gve an elegant analyss about the angular defect. Borrell et al. show that f the vertex p s regular, then the angular defct s asymptotcally equvalent to a homogeneous polynomal of degree two n the prncpal curvatures wth closed forms coeffcents. Moreover, they present another angular scheme K (2). In the followng, we show that K (2) and K (3) are equvalent, whch means these two schemes obtan the same value for the same trangle mesh. Theorem 1 For any vertex p of trangle meshes, K (2) K (3). Proof: To prove K (2) K (3), we only need to show S p 1 2 Note that area(p pp +1 ) 1 cot(γ )d 2. 8 d 2 = η 2 + η η η +1 cos γ and area(p pp +1 ) = 1 2 η η +1 sn γ. 8
9 We have 1 area(p pp 2 +1 ) 1 cot(γ )d 2 8 = [ 1 4 η η +1 sn γ 1 ] cos γ (η 2 + η+1 2 2η η +1 cos γ ) 8 sn γ = 1 [ η η +1 cos γ ] (η 2 + η 2 4 sn γ 2 +1) = S p. Hence, the dscrete schemes K (2) and K (3) are equvalent. Moreover, we have Theorem 2 For umblcal vertces, the modfed scheme K (3), and hence K (2), converges to the real Gaussan curvature as the edge lengths η tend to zero f all γ and sup η nf η are bounded (meanng that there exst two postve constants γ mn, γ max such that γ mn γ γ max ; there exst two postve constants η 1, η 2 such that η 1 sup η nf η η 2.). Proof: Snce K (2) K (3), we only need to prove K (2) converges to the real Gaussan curvature over the umblcal ponts. Accordng to Theorem 4 n [1], there exsts a postve constant C such that lm sup 2π γ η 0 K S p nc [ (km k m ) 2 + km 2 k 2 sn γ m ] 2, mn where K s the Gaussan curvature at p. For umblcal ponts, we have k M = k m. Hence, lm 2π γ η 0 K S p = 0, and the theorem holds. In [15], the author proves that the dscrete scheme K (1) quadratc convergence rate under the parallelogram crteron. the followng theorem, we shall show that the dscrete scheme K (3) has also quadratc convergence rate under the same crteron, has In 9
10 Theorem 3 Let p be a vertex of M wth valence sx, and let p j, j = 1,, 6 be ts neghbor vertces. Suppose p and p j, j = 1,, 6 are on a suffcently smooth parametrc surface F(ξ 1, ξ 2 ) R 3, and there exst u, u j R 2 such that Then p = F(u), p j = F(u j ) and u j = u j 1 + u j+1 u, j = 1,, 6. 2π γ 1 A(p, r) cot(γ (r))d 2 (r) = K(p) + O(r2 ), where, K(p) s the real Gaussan curvature at p, A(p, r) := area[p (r)pp +1 (r)], p (r) := F(u (r)), and u (r) = u + r(u u), = 1,, 6. Proof: Let A(p, r) = a 0 r 2 + a 1 r 3 + O(r 4 ) (7) and A(p, r) 2 1 cot(γ (r))d 2 (r) = b 0 r 2 + b 1 r 3 + O(r 4 ) 8 be the Taylor expansons wth respect to r. Accordng to Theorem 4.1 n [15], 3(2π γ ) A(p, r) = K(p) + O(r 2 ). Hence, to prove the theorem, we only need to show b 0 = a 0 /3, b 1 = 0. Wthout loss of generalty, we may assume u = [0, 0] T, u 1 = [1, 0] T. Then there exsts a constant a 0 and an angle θ such that u 2 = [a cos θ, a sn θ] T. Hence,u 3 = [a cos θ 1, a sn θ] T, u j+3 = u j, j = 1, 2, 3. Let u j = s j d j = s j [g j, l j ] T, j = 1,, 6, 10
11 where s j = u j, d j = 1. Then s 1 = 1, s 2 = a, s 3 = a 2 2ac + 1, s 4 = s 1, s 5 = s 2, s 6 = s 3, g 1 = 1, g 2 = c, g 3 = (ac 1)/s 3, g 4 = g 1, g 5 = g 2, g 6 = g 3, l 1 = 0, l 2 = t, l 3 = at/s 3, l 4 = l 1, l 5 = l 2, l 6 = l 3, where (c, t) := (cos θ, sn θ). Note that A(p, r) = j=1 p j (r) p 2 p j+1 (r) p 2 p j (r) p, p j+1 (r) p 2, (8) p j (r) p, p j+1 (r) p cot γ j (r) = pj (r) p 2 p j+1 (r) p 2 p j (r) p, p j+1 (r) p, 2 d 2 j(r) = p j (r) p 2 + p j+1 (r) p 2 2 p j (r) p, p j+1 (r) p. (9) (10) Let F k d j denote the kth order drectonal dervatve of F n the drecton d j. Then usng Taylor expanson wth respect to r, we have p j (r) p j 2 = s 2 jr 2 F dj, F dj + s 3 jr 3 F dj, F 2 d j s4 jr 4 F 2 d j, F 2 d j s4 jr 4 F dj, F 3 d j s5 jr 5 F 2 d j, F 3 d j s5 jr 5 F dj, F 4 d j + O(r 6 ),(11) p j (r) p, p j+1 (r) p = s j s j+1 r 2 F dj, F dj s js 2 j+1r 3 F dj, F 2 d j s2 js j+1 r 3 F dj+1, F 2 d j s2 js 2 j+r 4 F 2 d j+1, F 2 d j s js 3 j+r 4 F dj, F 3 d j s3 js j+1 r 4 F dj+1, F dj s2 js 3 j+1r 5 F 2 d j, F 3 d j s2 j+1s 3 jr 5 F 2 d j, F 3 d j s4 j+1s j r 5 F dj, F 4 d j s j+1s 4 jr 5 F 4 d j, F dj+1 + O(r 6 ). (12) 11
12 To compute all the nner products n the two equatons above, we let t = F(ξ 1, ξ 2 ) ξ, t j = 2 F(ξ 1, ξ 2 ) ξ ξ j, t jk = for, j, k, l = 1, 2. Let 3 F ξ ξ j ξ k, t jkl = 4 F ξ ξ j ξ k ξ l g j = t T t j, g jk = t T t jk, e jkl = t T t jkl, e jklm = t T t jklm, f jklm = t T jt klm. Snce F k d j can be wrtten as the lnear combnatorcs of t, t j, t jk and t jkl, all the nner products n (11) and (12) can be expressed as lnear combnatons of g j, g jk, g jkl, e jkl, e jklm and f jklm. Substtutng (11) and (12) nto (8), (9) and (10), and then substtutng (8), (9) and (10) nto the expresson 1A(p, r) cot(γ (r))d 2 (r), and usng Maple to conduct all the symbolc calculaton, we have b 0 = a 0 /3 = a 2 t 2 (g 11 g 22 g12), 2 b 1 = 0. The theorem s proved. Remark 1 The calculaton of b 0, b 1 nvolves a huge number of terms. It s almost mpossble to fnsh the dervaton by hand. Maple completes all the computaton n 26 seconds on a PC equpped wth a 3.0GHZ Intel(R) CPU. The Maple code that conducts all dervaton of the theorem s avalable n xuzq/ maple.html. The nterested readers are encouraged to perform the computaton. Remark 2 It should be ponted out that there s another dscrete scheme K (4) := 2π γ, A M (p) where A M (p) s the area of Vorono regon. Snce area(p pp +1 ) could be approxmated by 3A M (p) under some condtons, for example the condtons of Theorem 3, K (4) s easly derved from K (1). 12
13 5 A New Dscrete Scheme of the Gaussan Curvature and Its Convergence In ths secton, we ntroduce a new dscrete Gaussan curvature scheme whch converges for the umblcal ponts and regular vertces wth valence greater than 4. We frstly dscuss some propertes about the dscrete mean curvature. Usng the notatons n Fg. 1, we let H (1) := 2 (cot α + cot δ ) pp (cot α + cot δ )η 2, (13) whch denotes a dscrete scheme for the mean curvature at vertex p. Moreover, the real mean curvature and the real Gaussan curvature at p are denoted as H and K respectvely. Then, we have Lemma 1 At the regular vertex p, or the umblcal ponts, the dscrete scheme H (1) converges lnearly to the mean curvature H as η = η 0. Proof: Frstly, we consder the convergence property at the regular vertex. Snce p s a regular vertex, cot α +cot δ cot α j +cot δ j = 1 + O(η 2 ), for any dfferent and j. It follows from equaton (5), we have (cot α + cot δ ) pp = (cot α + cot δ )η 2 k + O(η 3 ). 2 Hence, H (1) = (cot α + cot δ )η 2 (cot α κ + cot δ )η 2 +O(η) = 1 κ +O(η) = H+O(η). n Secondly, we study the convergence propertes at the umblcal ponts. Over the umblcal ponts, k = k j = H for any and j. Hence, H (1) := 2 (cot α + cot δ ) pp (cot α + cot δ )η 2 = (cot α + cot δ )η 2 k + O(η 3 ) (cot α = H + O(η). + cot δ )η 2 13
14 Combnng the two results above, the theorem holds. Now, we turn to the dscrete scheme for Gaussan curvature. Let ϕ := j=1 γ j and where K (5) := 2π γ 2(S p A)(H (1) ) 2 2A S p, A := 1 ( η η +1 (1 cos 2ϕ cos 2ϕ +1 ) 4 sn γ 2 cos γ (η 2 sn 2 ϕ + η+1 2 sn 2 ϕ +1 )), 4 S p := [ 1 η η +1 cos(γ ] ) (η 2 + η 2 4 sn(γ ) 2 +1). Then, we have Theorem 4 For the regular vertces wth valence not less than 5, or the umblcal ponts, K (5) converges towards the Gaussan curvature K as η 0. Proof: We frstly consder the regular vertex case. We set θ(n) := 2π n. Snce p s a regular vertex, γ = θ(n) + O(η 2 ) for any accordng to (4). After a bref calculaton, we have A = A + O(η 4 ), S p = S p + O(η 4 ), where Hence, A = 1 16 sn θ(n) [2n n cos 2θ(n) n cos θ(n)]η2, S p = 2π γ 2(S p A)(H (1) ) 2 2A S p Note that ηmax η mn have 2π n 4 sn θ(n) [1 cos θ(n)]η2. = 2π γ 2(S p A )(H (1) ) 2 +O(η 2 ). 2A S p = 1 + O(η). Accordng to Theorem 3 n [1], we γ = A K + B (k 2 M + k 2 m) + o(η 2 ), 14
15 where, B = 1 16 sn θ(n) [n + n 2 cos 2θ(n) 3n 2 cos θ(n)]η2. Note that S p = A + 2B and A K + B (k 2 M + k 2 m) = A K + B [(k M + k m ) 2 2k M k m ] = A K + 4B H 2 2B K = (A 2B )K + 4B H 2. Hence, 2π γ = (A 2B )K+4B H 2 +o(η 2 ). S A = O(η 2 ), B = O(η 2 ) and when n 3, A 2B 0. Accordng to Lemma 1, H (1) converges to the real mean curvature. Hence, we have, when n 5, K = 2π n =1 γ 4B H 2 + o(1) A 2B = 2π γ 2(S p A )(H (1) ) 2 2A S P + o(1) = K (5) + o(1). Therefor, K (5) converges to the Gaussan curvature. Now, let us consder the umblcal pont case. ponts, each drectonal s the prncpal drecton. Lemma 4 n [1], For umblcal Accordng to 2π γ = (AK + (S p A)k 2 m) + o(η 2 ) holds over the umblcal ponts. Snce k 2 m = H 2 = K, we have 2π γ = (AK + (S p A)k 2 m) + o(η 2 ) = (AK + 2(S p A)H 2 (S p A)K) + o(η 2 ). Hence, K = 2π γ 2(S p A)(H (1) ) 2 2A S p The theorem holds. + o(1) = K (5) + o(1). Remark 3 Theorem 4 shows the new scheme K (5) converge over the regular vertex wth valence greater than 4. As shown before, the 15
16 prevous schemes only converge over the regular vertex wth valence 6, and hence the new scheme has better convergence propertes over the avalable scheme. Remark 4 In [7], the authors prove also that the dscrete scheme H (1) converges to the real mean curvature at the regular vertex. But the defnton of the regular vertex n [7] s dfferent wth our. Remark 5 Accordng to the conclusons above, the Gaussan curvature and mean curvature can be approxmated over the regular vertex wth valence greater than 4. Usng the formulaton k m = H H 2 G, k M = H + H 2 G, one can approxmate the prncpal curvature over the regular vertex wth valence greater than 4. 6 A Counterexample for the Regular Vertex wth Valence 4 In [18], we have constructed a trangle mesh and shown that t s mpossble for constructng a dscrete Gaussan curvature scheme whch converges for that mesh. But the vertex n the mesh s not regular. In ths secton, we shall show that t s also mpossble for buldng a dscrete Gaussan curvature scheme whch converges over the regular vertex wth valence 4. Suppose the xy plane s trangulated around (0, 0) by choosng 4 ponts q 1 = (r 1, 0), q 2 = (0, r 2 ), q 3 = ( r 3, 0) and q 4 = (0, r 4 ), where r 1 s gven and (r 1 2 r = + r4 1 ), = 2, 3, 4. 2 For a bvarate functon f(x, y), the graph of f(x, y),.e. F(x, y) = (x, y, f(x, y)) T, can be regarded as a parametrc surface. Let p 0 = F(0, 0) and p = F(q ), = 1, 2, 3, 4. The set of trangles p p 0 p +1 16
17 forms a trangular mesh approxmaton of F at p 0. The trangle mesh s denoted as M f. When f(x, y) s n the form of x 2 +cxy +y 2 where c R, t s easy to prove that p 0 := (0, 0, 0, ) T s a regular vertex wth valence 4. Moreover, p 1 = (r 1, 0, r1) 2 T, p 2 = (0, r 2, r2) 2 T, p 3 = ( r 3, 0, r3) 2 T, p 4 = (0, r 4, r4) 2 T. Now we show that t s mpossble to construct a dscrete scheme for Gaussan curvature whch converges over the vertex p 0 (See Fg. 2). We assume that the dscrete scheme for Gaussan curvature nvolvng one-rng neghbor vertces of p 0, whch s denoted as H(M f, p 0 ; p 1, p 2, p 3, p 4 ), s convergent for the regular vertex wth valence 4 over trangle mesh surface M f, where f(x, y) s n the form of x 2 + cxy + y 2. It s easy to calculate that the Gaussan curvature of F(x, y, z) at p 0 s 4 c 2. By the convergence property of H(M f, p 0 ; p 1, p 2, p 3, p 4 ) we have lm r1 0 H(M f, p 0 ; p 1, p 2, p 3, p 4 ) = 4 c 2. Note that the trangle mesh M f s ndependent of c,.e. for all the functon f(x, y) whch s n the form of x 2 + cxy + y 2, the trangle mesh M f s the same wth each other. Hence,lm r1 0 H(M f, p 0 ; p 1, p 2, p 3, p 4 ) s ndependent of c. A contradcton occurs. Hence, the assumpton of H(M f, p 0 ; p 1, p 2, p 3, p 4 ) beng convergent for the trangle mesh M f does not hold. Fg.2. A sequence of regular vertex wth valence n = 4 for the functon f(x, y) = x 2 + xy + y 2. At the regular vertex, t s mpossble to construct a dscrete Gaussan curvature scheme whch converges to the correct value. 17
18 Remark 6 The counterexample n ths secton justfes the concluson n [1], whch says that 4 s the only value of valence such that 2π γ depends upon the prncpal drectons. Remark 7 An open problem s to fnd a dscrete scheme for Gaussan curvature whch converges at the regular vertex wth valence 3. 7 Numercal Experments The am of ths secton s to exhbt the numercal behavors of the dscrete schemes mentoned above. For a real vector a = (a 20, a 11, a 02 ), we defne a bvarate functon f a (x, y) := a 20 x 2 + a 11 xy + a 02 y 2, and regard the graph of the functon f a (x, y) as a parametrc surface F a (x, y) = [x, y, f a (x, y)] T R 3. The Gaussan curvature of F a (x, y) at the orgn s 4a 20 a 02 a The doman around (0, 0) s trangulated locally by choosng n ponts: q k = l k (cos θ k, sn θ k ), θ k = 2(k 1)π/n, k = 1,, n. Let p k = F a (q k ) and p 0 = (0, 0, f a (0, 0)) T. Hence, the set of trangles {p k p 0 p k+1 } forms a pecewse lnear approxmaton of F a around p 0. We select (lk l4 k 1 f a(cos 2 θ k, sn θ k )) l k =, k 2 (14) 2fa(cos 2 θ k, sn θ k ) so that p 0 s a regular vertex, In Fg.3, four regular vertces wth valence 6,10,15 and 35 for x 2 xy 1.5y 2 are dsplayed. 18
19 Fg. 3. The regular vertex wth valence n = 6, 10, 15, 35 for the functon x 2 xy 1.5y 2. The dscrete scheme K (5) converges at the regular vertces. We let K () (p 0 : F a ) denote the approxmated Gaussan curvatures of F a at p 0, whch s obtaned by usng the dscrete scheme K (). Suppose A s a set consstng of M randomly chosen vectors a. Then, we let ε () (n) = K () (p 0 : F a ) (4a 20 a 02 a 2 11) /M. a A In fact, ε () (n) measures the error of the dscrete scheme K () at the regular vertex wth valence n. The convergence property and the convergence rate are checked by takng l 1 = 1/8, 1/16, 1/32,. (l k, k 2 can be obtaned by (14).) Snce p s regular, each edge has the same length η. Table 1 shows the asymptotc maxmal error ε () (n) for M = Here, the vertex valences n are taken to be 4, 5,, 8. Table 1. The asymptotc maxmal error ε () (n). n ε (1) (n) ε (2) (n) ε (4) (n) ε (5) (n) e e e e e e e e + 01η e + 01η e + 01η e + 01η e + 01η e e e e 01η e e e e + 01η Moreover, we test also the senstvty of the dscrete schemes to the nose by addng ±1% unform perturbaton along the normal drecton at the regular vertex. Table 2 shows the testng results. 19
20 Table 2. The asymptotc maxmal error ε () (n) after addng ±1% unform nose along the normal drecton. n ε (1) (η; n) ε (2) (η; n) ε (4) (η; n) ε (5) (η; n) e e e e e e e e + 02η e + 01η e + 01η η e + 01η e e e e + 01η e e e e + 01η We compute the Gaussan curvature over a randomly trangulated unt sphere by the dscrete schemes to test ther convergent property at the umblcal ponts. Fg. 4 shows the random trangulaton for the unt sphere. Denote the vertces n the random trangulaton as p, = 1,, N where N s the number of the vertces n the random trangulaton. We let K (j) (p ) denote the approxmate Gaussan curvature at the vertex p whch s calculated by K (j). Smlarly to the above, we use ε (j) = N =1 K(j) (p :) 1) /N to measure the error of dscrete scheme K (j) and use η to denote the average length of the edges. Table 3 lsts ε (j) for dfferent N. Table 3. The asymptotc error ε () over a sphere wth very rregular connectvty. N η ε (1) (η; n) ε (2) (η; n) ε (4) (η; n) ε (5) (η; n) e e e e e e e e e e e e e e e e e e e e 03 Fg. 4. Our test random trangulatons. From left to rght, the number of vertces s 60,300,1000,2000,5000 respectvely. 20
21 From these numercal results, we can draw the followng conclusons: 1. For the regular vertces wth the valence greater than 4, or the umblcal ponts, the dscrete scheme K (5) converges to the real Gaussan curvature. Ths agrees wth the theoretcal result. 2. All the dscrete schemes K (1), K (2), K (4) and K (5) are not senstve to the nose. 3. At the regular vertces and the umblcal ponts, the dfference between K (2) and K (4) s very small. References [1] V.Borrell, F.Cazals and J.-M. Morvan: On the angular defect of trangulatons and the pontwse approxmaton of curvatures, Computer Aded Geometrc Desgn 20(2003), [2] F.Cazals, M.Pouget, Estmatng dfferental quanttes usng polynomal fttng of osculattng jets. Computer Aded Geometrc Desgn 22, (2005). [3] Calladne, C.R.: Gaussan curvature and shell structures, n: J.A. Gregory, ed., The mathematcs of surfaces, Clarendon Press, Oxford, (1986). [4] Douros, I., Buxton, B.F.: Three-dmensonal surface curvature estmaton usng quadrc surface patches, In Scannng 2002 Proceedngs, Pars, 2002 [5] Martn, R.: Estmaton of prncpal curvatures from range data. Internat. J. Shape Modelng 4,99-111(1998). [6] D.S. Meek, D.J. Walton: On surface normal and Gaussan curvature approxmatons gven data sampled from a smooth surface, Computer Aded Geometrc Desgn, 17, (2000). 21
22 [7] T. Langer, A. G. Belyaev and H.P. Sedel: Analyss and desgn of dscrete normals and curvatures,techncal Report,Max- Planck-Insttut für Informatk,2005. [8] M. Desbrun, M.Meyer, P.Schröder and A.H.Barr: Implct farng of rregular meshes usng dffuson and curvature flow, SIGGRAPH99, (1999). [9] G H. Lu, Y.S. Wong, Y.F.Zhang, H. T Loh: Adaptve farng of dgtzed pont data wth dscrete curvature, Computer Aded Desgn, 34, (2002). [10] Meyer. M, Desbrun, M.,Schroder, P.,Barr,A.: Dscrete dfferental-geometry operator for trangulated 2-manfolds, n:proc. VsMath 02, Berln, Germany,2002. [11] G. Taubn, A sgnal processng approach to far surface desgn, n SIGGRAPH 95 Proceedngs (1995). [12] C. Wollmann, Estmaton of prncpal curvatures of approxmated surfaces, Computer Aded Geometrc Desgn,17, (2000). [13] Jean-Lous Maltret, Marc Danel, Dscrete curvatures and applcatons : a survey, preprnt, [14] U. F. Mayer, Numercal solutons for the surface dffuson flow n three space dmensons. Computatonal and Appled Mathematcs(to appear),2001. [15] Guolang Xu, Convergence analyss of a dscretzaton scheme for Gaussan curvature over trangular surfaces, Computer Aded Geometryc Desgn, 23, (2006). 22
23 [16] Guolang Xu, Convergenc of dscrete Laplace-Beltram operator over surfaces, Computers and Mathematcs wth Applcatons, 48, (2004). [17] Guolang Xu, Descrete Laplace-Betram operators and ther convergence, Computer Aded Geometrc Desgn 21, (2004). [18] Zhqang Xu, Guolang Xu and Jaguang Sun,Convergence analyss of dscrete dfferental geometry operators over surfaces, IMA Conference on the Mathematcs of Surfaces, (2005). 23
Computers 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 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 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 informationFair 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 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 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 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 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 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 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 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 informationMath Homotopy Theory Additional notes
Math 527 - Homotopy Theory Addtonal notes Martn Frankland February 4, 2013 The category Top s not Cartesan closed. problem. In these notes, we explan how to remedy that 1 Compactly generated spaces Ths
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 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 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 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 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 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 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 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 informationFeature Reduction and Selection
Feature Reducton and Selecton Dr. Shuang LIANG School of Software Engneerng TongJ Unversty Fall, 2012 Today s Topcs Introducton Problems of Dmensonalty Feature Reducton Statstc methods Prncpal Components
More informationLine geometry, according to the principles of Grassmann s theory of extensions. By E. Müller in Vienna.
De Lnengeometre nach den Prnzpen der Grassmanschen Ausdehnungslehre, Monastshefte f. Mathematk u. Physk, II (89), 67-90. Lne geometry, accordng to the prncples of Grassmann s theory of extensons. By E.
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 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 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 informationG 2 Surface Modeling Using Minimal Mean-Curvature-Variation Flow
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
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 informationRadial Basis Functions
Radal Bass Functons Mesh Reconstructon Input: pont cloud Output: water-tght manfold mesh Explct Connectvty estmaton Implct Sgned dstance functon estmaton Image from: Reconstructon and Representaton of
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 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 information1. Introduction. 2. Related Work
ASIAGRAPH 008 PROCEEDINGS Curvature Estmaton of 3D Pont Cloud Surfaces Through the Fttng of Normal Secton Curvatures Xaopeng Zhang* /LIAMA-NLPR, Insttute of Automaton, CAS; Hongjun L /LIAMA-NLPR, Insttute
More informationCordial and 3-Equitable Labeling for Some Star Related Graphs
Internatonal Mathematcal Forum, 4, 009, no. 31, 1543-1553 Cordal and 3-Equtable Labelng for Some Star Related Graphs S. K. Vadya Department of Mathematcs, Saurashtra Unversty Rajkot - 360005, Gujarat,
More informationUNIT 2 : INEQUALITIES AND CONVEX SETS
UNT 2 : NEQUALTES AND CONVEX SETS ' Structure 2. ntroducton Objectves, nequaltes and ther Graphs Convex Sets and ther Geometry Noton of Convex Sets Extreme Ponts of Convex Set Hyper Planes and Half Spaces
More informationthe nber of vertces n the graph. spannng tree T beng part of a par of maxmally dstant trees s called extremal. Extremal trees are useful n the mxed an
On Central Spannng Trees of a Graph S. Bezrukov Unverstat-GH Paderborn FB Mathematk/Informatk Furstenallee 11 D{33102 Paderborn F. Kaderal, W. Poguntke FernUnverstat Hagen LG Kommunkatonssysteme Bergscher
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 informationi v v 6 i 2 i 3 v + (1) (2) (3) (4) (5) Substituting (4) and (5) into (3) (6) = 2 (7) (5) and (6) (8) (4) and (6) ˆ
5V 6 v 6 î v v Ω î Ω v v 8Ω V î v 5 6Ω 5 Mesh : 6ˆ ˆ = Mesh : ˆ 8ˆ = Mesh : ˆ ˆ ˆ 8 0 = 5 Solvng ˆ ˆ ˆ from () = Solvng ˆ ˆ ˆ from () = 7 7 Substtutng () and (5) nto () (5) and (6) 9 ˆ = A 8 ˆ = A 0 ()
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 informationClassifier Selection Based on Data Complexity Measures *
Classfer Selecton Based on Data Complexty Measures * Edth Hernández-Reyes, J.A. Carrasco-Ochoa, and J.Fco. Martínez-Trndad Natonal Insttute for Astrophyscs, Optcs and Electroncs, Lus Enrque Erro No.1 Sta.
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 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 informationComputer Animation and Visualisation. Lecture 4. Rigging / Skinning
Computer Anmaton and Vsualsaton Lecture 4. Rggng / Sknnng Taku Komura Overvew Sknnng / Rggng Background knowledge Lnear Blendng How to decde weghts? Example-based Method Anatomcal models Sknnng Assume
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 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 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 informationThe Research of Ellipse Parameter Fitting Algorithm of Ultrasonic Imaging Logging in the Casing Hole
Appled Mathematcs, 04, 5, 37-3 Publshed Onlne May 04 n ScRes. http://www.scrp.org/journal/am http://dx.do.org/0.436/am.04.584 The Research of Ellpse Parameter Fttng Algorthm of Ultrasonc Imagng Loggng
More informationProblem Set 3 Solutions
Introducton to Algorthms October 4, 2002 Massachusetts Insttute of Technology 6046J/18410J Professors Erk Demane and Shaf Goldwasser Handout 14 Problem Set 3 Solutons (Exercses were not to be turned n,
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 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 informationSome Tutorial about the Project. Computer Graphics
Some Tutoral about the Project Lecture 6 Rastersaton, Antalasng, Texture Mappng, I have already covered all the topcs needed to fnsh the 1 st practcal Today, I wll brefly explan how to start workng on
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 informationElectrical analysis of light-weight, triangular weave reflector antennas
Electrcal analyss of lght-weght, trangular weave reflector antennas Knud Pontoppdan TICRA Laederstraede 34 DK-121 Copenhagen K Denmark Emal: kp@tcra.com INTRODUCTION The new lght-weght reflector antenna
More informationInverse-Polar Ray Projection for Recovering Projective Transformations
nverse-polar Ray Projecton for Recoverng Projectve Transformatons Yun Zhang The Center for Advanced Computer Studes Unversty of Lousana at Lafayette yxz646@lousana.edu Henry Chu The Center for Advanced
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 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 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 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 informationFeature-Preserving Mesh Denoising via Bilateral Normal Filtering
Feature-Preservng Mesh Denosng va Blateral Normal Flterng Ka-Wah Lee, Wen-Png Wang Computer Graphcs Group Department of Computer Scence, The Unversty of Hong Kong kwlee@cs.hku.hk, wenpng@cs.hku.hk Abstract
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 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 informationOptimization Methods: Integer Programming Integer Linear Programming 1. Module 7 Lecture Notes 1. Integer Linear Programming
Optzaton Methods: Integer Prograng Integer Lnear Prograng Module Lecture Notes Integer Lnear Prograng Introducton In all the prevous lectures n lnear prograng dscussed so far, the desgn varables consdered
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 informationThe Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique
//00 :0 AM Outlne and Readng The Greedy Method The Greedy Method Technque (secton.) Fractonal Knapsack Problem (secton..) Task Schedulng (secton..) Mnmum Spannng Trees (secton.) Change Money Problem Greedy
More informationImprovement of Spatial Resolution Using BlockMatching Based Motion Estimation and Frame. Integration
Improvement of Spatal Resoluton Usng BlockMatchng Based Moton Estmaton and Frame Integraton Danya Suga and Takayuk Hamamoto Graduate School of Engneerng, Tokyo Unversty of Scence, 6-3-1, Nuku, Katsuska-ku,
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 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 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 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 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 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 informationMOTION BLUR ESTIMATION AT CORNERS
Gacomo Boracch and Vncenzo Caglot Dpartmento d Elettronca e Informazone, Poltecnco d Mlano, Va Ponzo, 34/5-20133 MILANO boracch@elet.polm.t, caglot@elet.polm.t Keywords: Abstract: Pont Spread Functon Parameter
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 informationREFRACTION. a. To study the refraction of light from plane surfaces. b. To determine the index of refraction for Acrylic and Water.
Purpose Theory REFRACTION a. To study the refracton of lght from plane surfaces. b. To determne the ndex of refracton for Acrylc and Water. When a ray of lght passes from one medum nto another one of dfferent
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 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 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 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 informationA NOTE ON FUZZY CLOSURE OF A FUZZY SET
(JPMNT) Journal of Process Management New Technologes, Internatonal A NOTE ON FUZZY CLOSURE OF A FUZZY SET Bhmraj Basumatary Department of Mathematcal Scences, Bodoland Unversty, Kokrajhar, Assam, Inda,
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 informationAn Approach in Coloring Semi-Regular Tilings on the Hyperbolic Plane
An Approach n Colorng Sem-Regular Tlngs on the Hyperbolc Plane Ma Louse Antonette N De Las Peñas, mlp@mathscmathadmueduph Glenn R Lago, glago@yahoocom Math Department, Ateneo de Manla Unversty, Loyola
More informationOptimal Quadrilateral Finite Elements on Polygonal Domains
J Sc Comput (2017) 70:60 84 DOI 10.1007/s10915-016-0242-5 Optmal Quadrlateral Fnte Elements on Polygonal Domans Hengguang L 1 Qnghu Zhang 2 Receved: 30 January 2015 / Revsed: 21 January 2016 / Accepted:
More informationMULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION
MULTISPECTRAL IMAGES CLASSIFICATION BASED ON KLT AND ATR AUTOMATIC TARGET RECOGNITION Paulo Quntlano 1 & Antono Santa-Rosa 1 Federal Polce Department, Brasla, Brazl. E-mals: quntlano.pqs@dpf.gov.br and
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 informationVisual Curvature. 1. Introduction. y C. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), June 2007
IEEE onf. on omputer Vson and Pattern Recognton (VPR June 7 Vsual urvature HaRong Lu, Longn Jan Lateck, WenYu Lu, Xang Ba HuaZhong Unversty of Scence and Technology, P.R. hna Temple Unversty, US lhrbss@gmal.com,
More informationAn Image Fusion Approach Based on Segmentation Region
Rong Wang, L-Qun Gao, Shu Yang, Yu-Hua Cha, and Yan-Chun Lu An Image Fuson Approach Based On Segmentaton Regon An Image Fuson Approach Based on Segmentaton Regon Rong Wang, L-Qun Gao, Shu Yang 3, Yu-Hua
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 informationKinematics of pantograph masts
Abstract Spacecraft Mechansms Group, ISRO Satellte Centre, Arport Road, Bangalore 560 07, Emal:bpn@sac.ernet.n Flght Dynamcs Dvson, ISRO Satellte Centre, Arport Road, Bangalore 560 07 Emal:pandyan@sac.ernet.n
More informationAPPLICATION OF AN AUGMENTED REALITY SYSTEM FOR DISASTER RELIEF
APPLICATION OF AN AUGMENTED REALITY SYSTEM FOR DISASTER RELIEF Johannes Leebmann Insttute of Photogrammetry and Remote Sensng, Unversty of Karlsruhe (TH, Englerstrasse 7, 7618 Karlsruhe, Germany - leebmann@pf.un-karlsruhe.de
More informationGSLM Operations Research II Fall 13/14
GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are
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 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 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 informationUsing Radial Basis Functions to Solve Geodesics Equations for Body Measurements *
7659, England, UK Journal of Informaton and Computng Scence Vol. 2, No. 2, 2007, pp. 119-126 Usng Radal Bass Functons to Solve Geodescs Equatons for Body Measurements * R. Ng 1+, G.T.Y. Pong 2 and M. Wong
More informationAP PHYSICS B 2008 SCORING GUIDELINES
AP PHYSICS B 2008 SCORING GUIDELINES General Notes About 2008 AP Physcs Scorng Gudelnes 1. The solutons contan the most common method of solvng the free-response questons and the allocaton of ponts for
More informationLecture Note 08 EECS 4101/5101 Instructor: Andy Mirzaian. All Nearest Neighbors: The Lifting Method
Lecture Note 08 EECS 4101/5101 Instructor: Andy Mrzaan Introducton All Nearest Neghbors: The Lftng Method Suose we are gven aset P ={ 1, 2,..., n }of n onts n the lane. The gven coordnates of the -th ont
More informationCMPS 10 Introduction to Computer Science Lecture Notes
CPS 0 Introducton to Computer Scence Lecture Notes Chapter : Algorthm Desgn How should we present algorthms? Natural languages lke Englsh, Spansh, or French whch are rch n nterpretaton and meanng are not
More informationEdge Detection in Noisy Images Using the Support Vector Machines
Edge Detecton n Nosy Images Usng the Support Vector Machnes Hlaro Gómez-Moreno, Saturnno Maldonado-Bascón, Francsco López-Ferreras Sgnal Theory and Communcatons Department. Unversty of Alcalá Crta. Madrd-Barcelona
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 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 informationX- Chart Using ANOM Approach
ISSN 1684-8403 Journal of Statstcs Volume 17, 010, pp. 3-3 Abstract X- Chart Usng ANOM Approach Gullapall Chakravarth 1 and Chaluvad Venkateswara Rao Control lmts for ndvdual measurements (X) chart are
More information