Curvature normal vector driven interpolatory ... - Semantic Scholar

IEEE INTERNATIONAL CONFERENCE ON SHAPE MODELING AND APPLICATIONS (SMI) 2009

Curvature normal vector driven interpolatory subdivision Huanxi Zhao ∗ , Xia Qiu, Luming Liang ∗,† , Chuan Sun, Beiji Zou School of Information Science and Engineering, Central South University Changsha 410083, People’s Republic of China

Abstract—We present an intrinsically nonlinear interpolatory subdivision scheme with geometric information and some free parameters via discrete curvatures normal vector. Our scheme can produce fair G1 -continuous curves, which can avoid the potential pitfalls and unacceptable cases appeared in the four-point subdivision scheme. Furthermore, with the proper parameter choice, the proposed scheme is convexity-preserving, and reproduces the conic curve. Finally, the experimental results show our scheme is effective. Keywords—Nonlinear subdivision; Curvature normal vector; Convexity preserving 1. I NTRODUCTION In an interpolative curve subdivision, new vertices will be inserted into old polyline, and limit curves will pass through all the initial data points. In the 4-point subdivision scheme [4], locations of newly interpolated vertices are 1 k k pk+1 2i+1 = (pi + pi+1 ) 2 (1) + ω[(pki − pki−1 ) + (pki+1 − pki+2 )]. This scheme is linear because positions of new vertices are obtained by linear combination of old vertices, uniform and stationary since the same mask are used everywhere along the curve for each iteration of the subdivision process. The 4-point scheme was popular in the history of linear subdivision, but the potential pitfalls and unacceptable cases may appear in the subdivision process with four-point subdivision scheme (1). For example,(b) and (c) in Fig.8 demonstrates that the potential pitfalls are produced when generating an interpolatory 1 curve via the subdivision scheme (1) with ω = 16 . In another hand, the four-point scheme can not reproduce the basic curves, such as conic curves, triangular function and exponent function curves. Therefore a lot of researches have focused on improving this scheme as well as extending it into various non-stationary or nonlinear cases. More complex, nonuniform, non-stationary and non-linear, four-point subdivision scheme can be found [3], [6]–[8], [13], [14], [16]. Micchelli [1] introduced a revision of the 4-point scheme, which could reproduce Project supported in part by NSFC grant 10871208, 60673093 and Major Research Plan of NSFC(90715043). ∗ Correspounding Author:[email protected]; [email protected] †Now, he is a Ph.D. student in the Department of Mathematical & Computer Science, Colorado School of Mines, US.

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any function in the form of p(x)el(x) , where p(x) was a polynomial and l(x) was a linear function. Moreover, Floater and Micchelli [15] designed a family of interpolatory nonlinear subdivision schemes by replacing the algebraic average in classical four-point scheme with the harmonic one. This effort preserved the convexity of the original data. Schaefer et al. [18] proposed a new family of nonlinear schemes through nonlinear averages, and therefore acquired a nonlinear four-point scheme, which can reproduce the triangular function curve. Recently, geometrical information such as tangent/normal vectors, chord length, chord tangent angles are used in several nonlinear four-point schemes [13], [14], [16]. Marinov et al. [13] changed the tension parameter ω from a fixed value into a local chord length function, which enhances the geometric feature of the scheme, and this improvement eliminated potential pitfalls produced by the classical four-point scheme when one of the adjoining edges in the control polygon is short and the other is too long. Aspert [14] gave another nonlinear fourpoint scheme by constructing local spherical coordinates at every old vertices. The univariate scheme proposed by Aspert can be easily adapted to subdivide discrete surfaces. If the Marinov’s method can be regarded as the edge length averaging, while the Aspert’s improvement can be considered as the angles averaging, a normal vector averaging is introduced by Yang. In his method, the positions of newly interpolated vertices are pk+1 2i−1 =

1 k + pki ) + ω(λki nki−1 + μki nki ), (p 2 i−1

where nki is the normal vector at pki , λki and μki are projections of pki−1 pm and pm pki onto nki−1 and nki , respectively. Further details is described in [16]. The Yang’s scheme can reproduce the circular arc and line with the proper parameter choice. Inspired by the mentioned above works and the following fact: the force is related to the acceleration. The formula p¨(s) = kn(s is arc length parameter) implies us that the curvature vector is related to the force, so we think the displacement of each new point is related to curvature vector on that point, and we propose a more intrinsical nonlinear fourpoint scheme as follows, 1 k + pki ) (p 2 i−1 + ωik ((1 − λki )κki−1 nki−1 + λki κki nki ),

pk+1 2i−1 =

where κki nki is the curvature normal vector at the point pki and the tension parameter ωik varies adaptively with edge length (see Section 2 for more details). Because the

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curvature is intrinsical geometric, the proposed scheme can eliminate potential pitfalls produced by the classical four-point scheme when one of the adjoining edges in the control polygon is short and the other is too long, and reproduces the conic curve(see Section 5 for details). The rest of this paper is organized as follows: in Section 2, we supply more details for our proposed subdivision scheme. Section 3 is dedicated to the smoothness analysis of the scheme. In Section 4, we illustrate how to develop a convexity-preserving scheme by explicit choosing parameters. Based on it, we then propose a revised scheme for reproducing conic curve in Section 5. Several examples will be shown in the Section 6, and the last section is for conclusions. 2. D EFINITION OF THE SCHEME Let {p0i }, i = 0, 1, ..n be a sequence of initial control points, the curvature normal vector driven subdivision scheme is defined as = pki pk+1 2i (2) k k k k k pk+1 2i−1 = (1 − si )pi−1 + si pi + di . Here dki = ((1 − λki )κki−1 nki−1 + λki κki nki )ωik , ski and λki are free parameters which can be set by users in the range of (0,1/2). According to the convergence analysis of our scheme, we find that ωik in (2) is determined by ωik = clik /( lk 1+lk + lk +l1k ), where c is a constant and i−1

i

i

i+1

c 12 ski (1 − ski ). By Taylor expansions we can give a discrete approximation of the curvature vector κki nki κki nki =

pki−1 − pki pki+1 − pki 2 ( + ), k k lik + li+1 lik li+1

nki = (−Tik .y, Tik .x).

Here, Tik .x and Tik .y denote the x and y components of Tik . As shown in Fig. 1, we let chord tangent angles αik and βik be the unsigned angles between the chord pki−1 pki k and the tangents Ti−1 and Tik . Then, we define θk as the largest chord tangent angle in the kth level of the control polygon. k+1 k+1 k k Let β 2i−1 = ∠pk+1 = ∠pk+1 2i−1 pi−1 pm , α2i 2i−1 pi pm pk

Lemma 2: Assume that 0 < φ1 , φ2 < r sin φ2 with r 1, then we have

k i

k+1 2i 1

pk+1 2i 1

k+1 2i

2

k i

k+1 2i 1

pm

k i

(6) π 2,

if sin φ1
0,

|dki | |dki | k k (1 − si )li (1 − ski )lik a non-convex edge, when κki−1 · κki 0. −− → − → k k k k k k k ω max{1 − λi , λi }(|ki−1 ni−1 | + |ki ni |) Remark. The discrete curvature can be calculated by i (1 − ski )lik κki = κki nki · nki , where κki nki is defined in equation (9) 4 max{1 − λki , λki }tki 1 and nki is defined in equation (5). = · k k 1 − si li Definition 3. The polygon P k is strictly convex if every edge in P k is convex. sin αik sin βik ·( k + k ). Theorem 4. If we choose subdivision parameters k li−1 + lik li + li+1 λki = sin αik /(sin αik + sin βik ), ski = 1 − λki and n 0 < c < 0.25 at each subdivision level, the scheme If we denote max{αik , βik } by θk , we have i=1 (2) becomes a strictly convexity-preserving scheme. Proof: To prove this theorem, we only need to verify 4 max{1 − λki , λki }ωik k+1 that P k+1 is strictly convex when P k is strictly convex, sin β 2i−1 (1 − ski )lik that is to say, no interior angle in newly generated 1 1 polygon is larger than π. Here, we investigate interior ·( k + k ) sin θk . k li−1 + lik li + li+1 angles of P k+1 in two categories: (1)Interior angles at newly added vertices. k p , using sine law, we can get Similarly, in Δpki pk+1 Recalling the basic scheme (2) and the geometric 2i−1 m the bound of the other chord tangent angle correspond- meaning of curvature normal vectors, for a convex edge ing to pki−1 pki pki−1 pki , the newly added vertex pk+1 2i−1 and two edges k k k k , p p will be located in different half-plane, p p i−1 i i−1 i 4 max{1 − λki , λki }ωik k+1 as shown in Fig. 1. The inner angle at pk+1 sin α2i 2i−1 is k k si li k+1 k 0 < ∠pki−1 pk+1 1 1 2i−1 pi = π − 2α2i−1 < π ·( k + ) sin θ . k k li−1 + lik lik + li+1 (2)Interior angles at old vertices in the kth level. k+1

sin β 2i−1 = sin γ k+1 2i−1

As ωik 12 ski (1 − ski )lik /( lk

1

k i−1 +li

+

1 ), k lik +li+1

we have

pki

2 max{1 − λki , λki }(1 − ski ) sin θk sin αk+1 2i

k i+1

k+1

sin β 2i−1 2 max{1 − λki , λki }ski sin θk Paying attention to Lemma 2, we have k+1 α2i 2 max{1 − λki , λki }(1 − ski )θk k+1

β 2i−1 2 max{1 − λki , λki }ski θk

k+1 2i+1

pk+1 2i 1

i

i

i+1

then chord tangent angles αik and βik approach zero, namely lim θk = 0. Furthermore, the scheme is conk→∞

k+1 2i

(10)

Now, applying Lemma 1, Lemma 2 and the relationship k+1 (10) of αk+1 2i , β 2i−1 and θk , it is not difficult to fulfill the convergence and smoothness analysis of the presented scheme, namely, we have the following the Theorem 1. Theorem 1. In the subdivision scheme defined in (2), if ωik = clik /( lk 1+lk + lk +l1k ), and c 12 ski (1 − ski ), i−1

pk+1 2i+1

vergence andlimit curve generated by this scheme is G1 smooth. The proof of Theorem 1 is similar to the one of corresponding results in [16]. Here, we left it to readers.

Fig. 3.

Computing inner angles at old vertices.

According to the sine law, we have, k+1

sin β 2i+1

|dki+1 | k (1 − ski+1 )li+1

4 =4

( lk +l1k i

i+1

+

1 k )ωi+1 k k li+1 +li+2

k (1 − ski+1 )li+1

·

k k sin βi+1 sin αi+1 k k sin αi+1 + sin βi+1

k k sin βi+1 sin αi+1 c · k k k λi sin αi+1 + sin βi+1

k =4c sin βi+1 ,


where 0 < c < 0.25, then we get k+1

k sin β 2i+1 < sin αi+1 ,

and sin αk+1 < sin βik . 2i k+1

k According to (7), we have β 2i+1 < αi+1 and αk+1 < 2i k+1 k k+1 k βi . In this case, the new inner angle ∠p2i−1 pi p2i+1 < k βik + ∠pki−1 pki pki+1 + αi+1 = π. The Theorem 4 is proven. Remark. The explicit choosing of parameters in the Theorem 4 will not affect the convergence and smoothness.

5. T HE SCHEME FOR REPRODUCING CONIC CURVES 5.1 Reproducing circles It’s not difficult to develop a revised scheme for generating circles from regular control polygons(see Fig.4). Based on the convexity-preserving scheme, we change the constant coefficient c into a variable cki . To determine the magnitude of cki , we need 2 steps. (Step 1)Calculate dki by just using constant coefficient c. (Step 2)Take dki as a feed back for recomputing cki . We set − → λki κki nki lik /( lk 1+lk + lk +l1k ) 1 i−1 i i i+1 · c. (12) cki = 2 (1 + cos βik )dki Suppose the new displace vector is ddki , we have ck ddki = ci dki according to (12), then the new point corresponding to pki−1 pki can be calculated by pk+1 2i−1 = pm + ddki . Figures below are circles generated by this revised scheme. Because the original control polygon is regular, it’s 0 0 simple to see that li−1 = li0 = li+1 , i = (1, 2, ...) and for each edge p0i−1 p0i , αi0 = βi0 , thus λ0i = 0.5. In this case, it is not difficult to see that dd0i is perpendicular to p0i−1 p0i . Let γ = ∠p12i−1 p0i−1 p0i , we have tan γ =

|dd0i | = s0i li0

1 2

0.5|ki0 n0i |li0 /(

1 + 0 10 l0 +l0 l +l i−1 i i i+1 0 (1+cos βi )|d0i |

)

· |d0i |

1 0 2 li

1 sin αi0 1 = = tan βi0 . 2 1 + cos βi0 2

(13) Note that 0 < γ, 12 βi0 < π2 , γ = 21 βi0 = 12 αi0 1 1 1 1 and α2i−1 = β2i−1 = α2i = β2i = 12 αi0 , also 1 1 1 li−1 = li = li+1 , i = (1, 2, ...). When new vertexes have been computed and added repeatedly, the limit curve is a circle. 5.2 Reproducing conics In order to generate conics, let us consider a scheme ⎧ k+1 k ⎪ ⎨ p2i = pi k k k k (14) pk+1 2i+1 = (1 − si )pi + si pi+1 ⎪ ⎩ k k k k k k k + ((1 − λi )κi−1 ni−1 + λi κi ni )ω ,

Fig. 4. Circles modeling: the upper row is the subdivision process of modeling circle from the square; the lower one is the corresponding process when the original polygon is the hexagon.

If parameters λki is determined by λki = 1/(1 +

k k + lik )li−1 li−1 ) k k k li + li+1 )li+1

and parameter ski is defined by ski = 1/2 + (

k k k − li−1 )(1 − λki )ω k li+1 2(li+1 k k lk ) (li−1 + lik )(li−1 i

Then the newly inserted point pk+1 2i+1 can be rewritten as 1 k k k k k k pk+1 2i−1 = ( + ω )(pi + pi−1 ) − ω (pi−2 + pi+1 ), 2 (15) here, ω k is no longer a constant value, and it is defined as 1 ωk = k k , k 0, (16) 8v (v + 1) 1+v k where v 0 > −1 and v k+1 = 2 . According to Beccari’s paper [2], if initial control points are all uniformly sampled on a conic section, we can easily generate conics by scheme (14) with an appropriate


tension parameter(see Fig.5). More details can be found in reference [2]. (a) original polygon

(b) λi = 0.2

(b) v 0 = cosh(1)

(a)

(c) λi = 0.5

(d) λi = 0.8 (d) v 0 = 0

(c)

(f) v 0 = 0

(e)

Fig. 5. The process of reproducing conics. (a),(c),and (e) are original control polygon which are sampled uniformly from ellipse,parabola and hyperbola,respectively. (b),(d) and (f) are the subdivision result of (a),(c),(e), respectively.

Fig. 6. The result of a wave-like shape by using our scheme with different λi s. Other two parameters si = 0.5 and c = 0.125 are same for (b)-(d).

(a) original polygon

(b) ski = 0.3

6. E XPERIMENTS In this section, we give some experiments to show the presented subdivision is effective. (c) ski = 0.5

6.1 Modeling power The goal of our first two experiments is to reveal the modeling power of our scheme by choosing different parameters. From Fig. 6, we can see that parameter λi influences the slope of the corresponding vertex on the final limit curve, that is, λki influence the movement direction of the interpolated point. The following Fig. 7 show us that ski decides the movement position of the interpolated point. As mentioned in the introduction, the classical fourpoint scheme may generate artifacts such as self intersections [13], [18](also see (c) in Fig. 8). To eliminate them, Marinov have already given us a solution in [13]. However, this solution is not a perfect one, since the converge speed will be slowed down. Our scheme eliminates the artifacts and converges relatively faster. From the unit tangent plotting in Fig. 8, we can see that after 8 iterations, the angles between adjacent tangents in (d)(produced by Marinov’s scheme) are larger than (e)(shape produced by ours). Actually, the

(d) ski = 0.7 Fig. 7. The result of a wave-like shape by using our scheme with different ski s. Other two parameters λki = 0.5 and c = 0.0625 are same for (b)-(d).

largest chord tangent angle in (d) is 0.2379, while the largest chord tangent angle in (e) is 0.0561. The(b) and (c) of Fig. 9 show that the tension parameter in Marinov’s scheme has limited influence on the shape of the limit curve. However, there are three free parameters in our scheme, which can largely influence the shape of the final limit curve. The follow Fig. 9 shows us the difference between the Marinov’s scheme and ours.


lating subdivision scheme because of numercial stability. Here, we will test our convexity preserving scheme in this case. We find out that our scheme works well from the Fig. 10. (a)

(b)

(c) (a)

(d)

(e)

Fig. 8. Subdivision of the shape shown in (a). (b) is the result by 1 using the classical four-point scheme, where ω = 16 . (c) is the right buttom part of (b). (d) and (e) are the results with unit tangent plotting (red lines) by using Marinov’s scheme and our scheme, respectively. (b)

(a) original polygon

(b) Marinov’s scheme with ωik = 0.05

(c) Fig. 10. Subdivision under the situation where two almost-straight triples meet. (a)original polygon (b)subdivision result with curvature plotting (c)subdivision result with unit tangent plotting. (c) Marinov’s scheme with ωik = 0.10

(d) Our scheme with c = 0.125

7. C ONCLUSIONS

(e) Our scheme with c = 0.0625 Fig. 9. The comparison of results of Mariniv’ scheme and the proposed scheme with different parameter c and Other two parameters λki = 0.5 and ski = 0.5 are same for (d)-(e).

6.2 Performances of the convexity-preserving scheme The situation where two almost-straight triples meet is inherently hard for any convexity preserving interpo-

We have presented a novel curve interpolating scheme via curvature normal vector in this paper. Three parameters with quite wide ranges can be adjusted to control the shape of the subdivision curves. The limit curves produced by this curvature normal vector driven scheme are G1 smooth and fair. Furthermore, corresponding convexity-preserving scheme is also developed. In addition, with the proper parameter choice, the proposed scheme reproduces the conic curve. ACKNOWLEDGMENTS The authors would like to thank all the reviewers for their valuable comments and suggestions.


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