Approximation of surfaces by fairness bicubic splines - Semantic Scholar
Advances in Computational Mathematics 20: 87–103, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Approximation of surfaces by fairness bicubic splines A. Kouibia and M. Pasadas Department of Applied Mathematics, Faculty of Sciences, University of Granada, Granada, Spain E-mail: {kouibia;mpasadas}@ugr.es
Received 7 February 2001; accepted 25 October 2002 Communicated by M. Gasca
In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method. Keywords: smoothing, variational surface, fairness spline, bicubic spline AMS subject classification: 65D07, 65D10, 65D17
1.
Introduction
In geology and structural geology the reconstruction of a curve or surface from a scattered data set is a commonly encountered problem. The theory of D m -splines over an open bounded set has been introduced at the first time by Attéia [1]. We have enriched this theory and extended it to the variational spline functions [6] where the early works are therein. Several works have used the variational approach specifically minimizing some fairness functional (see, for example, [4], likewise these functional also can represent the flexed energy of a thin plate [3]) on a finite element space taking advantage the suitable properties of this space (see [5,7]) in order to simplify both characterization and computation of the solution. So we have planned to resolve in this work a variational approach problem on a finite-dimensional space that is not a finite element one. This is why we focus in this paper our interest to minimize a similar fairness functional on a space of bicubic spline functions of class C 2 , meanwhile in [7] we discretize in a finite element space where in case that its functions are bicubics they turns out to be of class C 1 . The resulting function is called a fairness bicubic spline. We study some characterization of this function and we shall express it as a linear combination of the basis functions of a parametric space of bicubic splines. Moreover, under adequate hypotheses
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we prove that such fairness bicubic spline converges to a given function from which are proceeding the data. We present some numerical and graphical examples in order to show the effectiveness and the validity of the method of this paper. We remark that this work can be considered as an analogous case of the problem presented in [7], but the advantage presented here is to reach a great smoothness order. The method is presented for any open parametric surface, i.e. without boundary periodic conditions, but we think that it can be extended to any closed parametric surfaces, by restricting the space of discretization to a linear subspace formed by periodic bicubic splines. In the last example we approximate the torus by a periodic fairness bicubic spline. Some fields of applications of this problem can appear in Earth sciences, specially in geology and geophysics, as long as CAD and CAGD, etc. The remainder of this paper is organized as follows. In section 2, we briefly recall some preliminary notations and results. Section 3 is devoted to state the approximation problem and to present a method to solve it. In section 4, we compute the resulting function, while a convergence’s theorem is proved in section 5. In section 6 some numerical and graphical examples are given.
2.
Notations and preliminaries
We denote by · and ·, ·, respectively, the Euclidean norm and inner product in R3 , while we denote by ·R2 and ·, ·R2 respectively the Euclidean norm and inner product in R2 . For any real interval (a, b) ((c, d)) with a < b (c < d) we consider the rectangle R = (a, b) × (c, d) and let H 3 (R; R3 ) be the usual Sobolev space of (classes of) functions u belong to L2 (R; R3 ), together with all their partial derivative D β u with β = (β1 , β2 )T , in the distribution sense, of order |β| = β1 + β2 3. This space is equipped with the norm u =
|β|3
β
1/2
2
D u(p) dp
,
R
the semi-norms |u| =
|β|=
β
2
D u(p) dp
1/2 ,
0 3,
R
and the corresponding inner semi-products 2 D β u(p), D β v(p) dp, (u, v) = |β|=
R
0 3.
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Let Rk,3 be the space of real matrices of k rows and 3 columns equipped with the inner semi-product A, Bk,3 =
3 k
ai,j bi,j ,
i=1 j =1
with A = (aij ) 1ik , B = (bij ) 1ik and the corresponding norm 1j3
1j3
Ak,3 =
k 3
1/2 (ai,j )2
.
i=1 j =1
Moreover, for any n, m ∈ N∗ let Tn = {x0 , . . . , xn }, Tm = {y0 , . . . , ym } be some subsets of distinct points of [a, b] and [c, d], with a = x0 x1 < · · · < xn−1 xn = b and c = y0 y1 < · · · < ym−1 ym = d. We denote by S3 (Tn ) and S3 (Tm ) the spaces of cubic spline functions given by
S3 (Tn ) = s ∈ C 2 [a, b] | s|[xi−1 ,xi ] ∈ P3 [xi−1 , xi ], i = 1, . . . , n ,
S3 (Tm ) = s ∈ C 2 [c, d] | s|[yi−1 ,yi ] ∈ P3 [yi−1 , yi ], i = 1, . . . , m , where P3 [xi−1 , xi ] (P3 [yi−1 , yi ]) is the restriction on [xi−1 , xi ] ([yi−1 , yi ]) of the linear space of real polynomials with total degree less than or equal to 3. It is known that dim S3 (Tn ) = n + 3 (dim S3 (Tm ) = m + 3) and let {ϕ1 , . . . , ϕn+3 } and {ψ1 , . . . , ψm+3 } be respectively bases of functions with local support of S3 (Tn ) and S3 (Tm ). We consider the space S3 (Tn , Tm ) of bicubic spline functions given by S3 (Tn , Tm ) = span{ϕ1 , . . . , ϕn+3 } ⊗ span{ψ1 , . . . , ψm+3 }. Finally, for any n, m ∈ N∗ we define the space of parametric bicubic spline functions VN = (S3 (Tn , Tm ))3 with N = 3(n + 3)(m + 3) = dim VN constructed from S3 (Tn , Tm ) which is a Hilbert subspace of H 3 (R; R3 ) equipped with the same norm, semi-norm and inner semi-product of such space and, moreover, verifies (1) VN ⊂ H 3 R; R3 ∩ C 2 R; R3 . 3.
Fairness bicubic spline
Let ϒ0 ⊂ R3 be a surface defined by a parameterization f belonging to C 2 (R; R3 ). For each r ∈ N∗ let Ar be a finite subset of k = k(r) distinct points of R such that 1 , r → +∞. (2) sup minr p − aR2 = O a∈A r p∈R Remark 3.1. From (2) we can obtain that for r sufficiently large k(r) > r. In section 5 we will add others conditions for k(r) in order to warranty a convergence result.
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Let Lr be a Lagrangian operator defined from H 3 (R, R3) into Rk,3 by Lr v = v(a) a∈Ar and suppose that
Ker Lr ∩ P2 R; R3 = {0},
(3)
being P2 (R; R3 ) the linear space of the real polynomials with total degree less than or equal to 2 defined from R into R3 . Now, we consider the following problem: Find an approximating surface ϒ of ϒ0 parameterized by a function σ of VN that fits the data points of Ar and minimizes all the semi-norms of order less than 3 in VN . For any τ = (τ1 , τ2 , τ3 ) ∈ R3 with τ1 , τ2 belonging to R+ and τ3 > 0, let Jτr be the functional defined on H 3 (R; R3 ) by 3 2 τj |v|2j . Jτr (v) = Lr (v − f ) k,3 + j =1
Remark 3.2. The first term of Jτr (v) indicates how well v approaches f in a least discrete squares sense. The second term can represent some different conditions as, for example: fairness conditions (see [4,5]), a classical smoothness measure, etc., while the parameter vector τ weights the importance given to each conditions. Then, for any r 3 we consider the following minimization problem: Find στN,r such that
N,r στ ∈ VN , (4) ∀v ∈ VN , Jτr στN,r Jτr (v). Theorem 3.1. The problem (4) has a unique solution, called the fairness bicubic spline in VN relative to Ar , Lr and τ , which is also the unique solution of the following variational problem: Find στN,r such that N,r ∈ VN , σ τ ∀v ∈ VN , Proof.
L
r
στN,r , Lr v k,3
+
3
τj στN,r , v j = Lr f, Lr v k,3 .
j =1
Taking into account (1), (2) and that the following norm
3 2 τj |v|2j v→ [[v]] = Lr v k,3 + j =1
1/2
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is equivalent in VN to the norm · , one easily checks that the symmetric bilinear form a˜ : VN × VN → R given by
a(u, ˜ v) = L u, L v r
r
k,3
+
3
τj (u, v)j
j =1
is a continuous and VN -elliptic. Likewise, the linear form ϕ : v ∈ VN → ϕ(v) = Lr f, Lr v k,3 is continuous. The result is then a consequence of the Lax–Milgram lemma (see [2]). Remark 3.3. Obviously, the unidimensional case, i.e. the approximation problem of curves by fairness cubic splines is analogous to the bidimensional case, but not for this are simply the extension of the notations and proofs. This is why directly we deal with the bidimensional case for its interest and application.
4.
Computation
Now well, we are going to show how to obtain in practice any fairness bicubic spline but we assume that we know the parameter values associated to given data points. For any n, m ∈ N∗ we consider {w1 , . . . , w(n+3)(m+3)} a basis of elements with local support of the space S3 (Tn , Tm ) and {e1 , e2 , e3 } the canonical one of R3 . Then, the family {v1 , . . . , vN } is a basis of VN with ∀i = 1, . . . , (n + 3)(m + 3), for = 1, 2, 3, j = 3(i − 1) + ,
vj = wi e .
Thus, στN,r can be written as στN,r = N i=1 αi vi , with αi ∈ R unknown, for i = 1, . . . , N . Applying theorem 3.1 we obtain a linear system of order N as follows N 3 r αi L vi , Lr vj k,3 + τs (vi , vj )s = Lr f, Lr vj k,3 , ∀j = 1, . . . , N, i=1
s=1
that is equivalent to Cα = b where C = (cij )1i,j N , α = (α1 , . . . , αN )T , b = (b1 , . . . , bN )T ,
(5)
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with for i, j = 1, . . . , N we have 3 τs (vi , vj )s , cij = Lr vi , Lr vj k,3 + s=1 b = Lr f, Lr v j j k,3 . Finally, we point out the following result. Proposition 4.1. The matrix C is symmetric, positive definite and of band type. Proof. Obviously the matrix C is symmetric. Let now p = (p1 , . . . , pN ) ∈ RN we have pCp = T
=
N i,j =1 N
pi cij pj pi
r
r
L vi , L vj
k,3
+
i,j =1
= Lr = Lr
N
pi vi , Lr
N
2
Let w =
i=1
N
τs (vi , vj )s pj
k,3
+
pj vj
j =1
+
pi vi
i=1
N
s=1
i=1
3
3 s=1
k,3 N
τs
i=1
3
τs
N
s=1 2
pi vi ,
i=1
N j =1
pj vj s
pi vi 0. s
pi vi and we suppose that pCp T = 0 then one has
3 2 τi |w|2i = 0. Lr w k,3 + i=1
Hence [[w]]2 = 0 that implies w = 0 (where [[·]] designs the norm defined in the proof of theorem 3.1). Moreover, for the independent linearity of the family {vi }1iN we obtain that p = 0 and C is positive definite. Finally, the matrix C is of band type because for each i = 1, . . . , N the function vi has local support. In practice we use the following notations: A = Lr vi 1iN , B = (Bs )1s3 = (vi , vj )s 1i,j N 1s3 , hence system (5) is equivalent to T AA + τ T B α = ALr f.
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Remark 4.4. Taking into account that each function v defined from R into R3 can be written as v = (v 1 , v 2 , v 3 ), then the system (5) can be composed into three ones as follows l l T A A + τ T B l α l = Al Lr f l for l = 1, 2, 3, where
Al = Lr vil 1iN , B l = Bsl 1s3 = vil , vjl s 1i,j N 1s3 .
Hence, we can write the solution as στN,r = ((στN,r )l )1l3 , where
l στN,r
=
N
αil vil
for l = 1, 2, 3.
i=1
5.
Convergence
Under adequate conditions, we are going to prove that the fairness bicubic spline in VN relative to Ar , Lr f and τ converges to f when N and r tend to +∞. Before to do this we need the following results. Proposition 5.1. Let B0 = {b01 , . . . , b06 } be a P2 (R; R3 )-unisolvent subset of points of R. Then, there exists η > 0 such that if Bη stands 6-uplets B = {b1 , . . . , b6 } of points of R satisfying the condition ∀j = 1, . . . , 6,
bj − b0j R2 < η,
(6)
the application [[·]]B defined for all B ∈ Bη by 6 1/2 2 B 2 , v(bj ) + |v|3 [[v]] = j =1
is a norm on H 3 (R; R3 ) uniformly equivalent on Bη to the usual norm of Sobolev · . Proof.
It is analogous to [8, proposition 2.1].
Corollary 5.2. Suppose that the hypotheses (2) and (3) hold. Then, there exists η > 0 and, for all r ∈ N, a subset Ar0 of Ar and a constant C > 0 such that, for all r C/η, the application [[·]]r0 defined by 1/2 2 , v(a) + |v|23 [[v]]r0 = a∈Ar0
is a norm in H 3 (R; R3 ) uniformly equivalent, with respect to r, to the norm · .
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Proof. Let B0 = {b01 , . . . , b06 } be anyone P2 (R; R3 )-unisolvent subset of R. From (2) there exists C > 0 such that for all r ∈ N and all j = 1, . . . , 6, there exists ajr ∈ Ar verifying
C . r Let Ar0 = {a1r , . . . , a6r }. It suffices then to apply proposition 5.1, taking into account r that, for all r C/η, Ar0 ∈ Bη and written [[·]]r0 instead of [[·]]A0 . b0j − ajr
R2
We remember that the Tchebycheff norm in C 0 (R; R3 ) is defined by u T = max u(p) . p∈R
Let us give the easy relation between the semi-norms of order l and the Tchebycheff norm in H 3 (R; R3 ). Lemma 5.3. For all u ∈ C 2 (R; R3 ) one has |u| meas(R) D uT ,
= 0, 1, 2,
being meas(R) the area of rectangle R. Proof.
Let u ∈ C 2 (R; R3 ) then we have 2 2 2 max D j u(p) , D j u(p) dp meas(R) |u| = |j |=
R
|j |=
p∈R
which implies that 2 |u|2 meas(R)D uT . Let f ∈ C 4 (R; R3 ) and let SN be the which solves the interpolation problem SN (xi , yj ) = f (xi , yj ), ∂f ∂SN (xi , yj ) = (xi , yj ), ∂x ∂x ∂f ∂SN (x (xi , yj ), , y ) = i j ∂y ∂y ∂ 2f ∂ 2 SN (xi , yj ) = (xi , yj ), ∂x∂y ∂x∂y
parametric bicubic spline function of VN for i = 1, . . . , n; j = 1, . . . , m, for i = 1, . . . , n; j = 1, m, for i = 1, n; j = 1, . . . , m, for i = 1, n; j = 1, m.
So, from [9, section 5.2] and applying lemma 5.3 it follows that |f − SN | K h4− ,
= 0, 1, 2, 3,
(7)
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where K , = 0, 1, 2, 3, depends only on the Tchebycheff norms of mixed partial derivatives of f up through th, = 0, 1, 2, 3, and
b−a d −c , . h = max n m For the following convergence result we suppose that ∃C > 0, ∃r0 ∈ N: ∀r r0 ,
k(r) Cr 2 ,
(8)
which is an asymptotic regularity property of the distribution of the points of Ar in R. Now, let στN,r be the fairness bicubic spline in VN relative to Ar , Lr f and τ . Theorem 5.4. Let f ∈ C 4 (R; R3 ). Suppose that the hypotheses (1) and (2) hold and that τ3 = o r 2 , r → +∞, (9) r → +∞, (10) ∀i = 1, 2, τi = o(τ3 ), and rh4 1/2
τ3 Then, one has
= o(1),
r → +∞.
lim f − στN,r = 0.
N,r→+∞
Proof. The proof will be split into four steps. Step 1. First, we have Jτr (στN,r ) Jτr (SN ) which implies that N,r 2 σ 1 Lr (SN − f ) 2 + τ1 |SN |2 + τ2 |SN |2 + |SN |2 . τ 1 2 3 3 k,3 τ3 τ3 τ3 From (7) it follows that |SN |1 |SN − f |1 + |f |1 K1 h3 + |f |1 , |SN |2 K2 h2 + |f |2 and |SN |3 K3 h + |f |3 . Likewise, from (7) there exists C > 0 such that for each i = 1, . . . , r we have 2 SN (ai ) − f (ai ) Ch8 , and from (8) one has
2 r L (SN − f ) k,3 Cr 2 h8 .
(11)
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Then, finally, we obtain that there exists C > 0 such that 2 8 N,r 2 σ C r h + τ1 K1 h3 +|f |1 2 + τ2 K2 h2 +|f |2 2 + K3 h+|f |3 2 , τ 3 τ3 τ3 τ3
N → +∞.
(12) Hence, from (10)–(12) we deduced that there exist a constant C > 0 and M0 , M1 ∈ N such that N,r σ C, ∀r M0 , ∀N M1 , τ 3 and
2 Lr στN,r − f k,3 = O(τ3 ),
r → +∞, N → +∞.
(13)
Let B0 = {b01 , . . . , b06 } be a P2 (R; R3 )-unisolvent subset of points of R and let η be the constant of proposition 5.1. Obviously, there exists η ∈ (0, η] such that ∀j = 1, . . . , 6, B b0j , η ⊂ R. From (2), we have that there exists C > 0 such that C C ⊂ ∀j = 1, . . . , 6, B b0j , η − ∀r ∈ N, r > , η r
a∈Ar ∩B(b0j ,η )
C . B a, r
If Nj = card(Ar ∩ B(0j , η )) then there exists C1 > 0 verifying ∀r ∈ N,
C r > , η
∀j = 1, . . . , 6,
C 2 C1 N j η − r r2
and consequently for any r0 > C/η , there exists C2 > 0 such that C 2 2 r . ∀r r0 , ∀j = 1, . . . , 6, Nj C2 η − r0 Meanwhile, from (9) and (13) we deduce that 2 N,r στ − f (a) = o r 2 , ∀j = 1, . . . , 6,
(14)
r → +∞, N → +∞.
a∈Ar ∩B(b0j ,η )
If ajr ∈ Ar ∩ B(b0j , η ) such that N,r στ − f ajr =
(15)
min
a∈Ar ∩B(b0j ,η )
N,r στ − f (a) ,
we deduce from (14) and (15) that N,r ∀j = 1, . . . , 6, στ − f ajr = o(1),
r → +∞.
(16)
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Now, we denote by B r the set {a1r , . . . , a6r }, then by applying proposition 5.1 to B = B r , for r sufficiently close to infinity, it turns out from (12), (16) and corollary 5.2 that ∃C > 0, ∃λ > 0, ∀r λ, σ N,r C, τ
(στN,r )r,N∈N∗
which means that the family a subsequence (στNl l ,rl )l∈N , with τl = τ (rl ),
3
is bounded in VN . It follows that there exists
lim rl = +∞,
lim Nl = +∞
l→+∞
l→+∞
and an element f ∗ ∈ H 3 (R; R3 ) such that στNl l ,rl converges weakly to f ∗
in H 3 R; R3 .
(17)
Step 2. Let us now prove that f ∗ = f . We suppose that f ∗ = f . From the continuous injection of H 3 (R; R2 ) into C 1 (R; R3 ) it follows that there exist θ > 0 and an open rectangle R0 of R such that ∀p ∈ R0 , f ∗ (p) − f (p) > θ. As such injection is also compact then from (17) we obtain θ ∃l0 ∈ N, ∀l l0 , ∀p ∈ R0 , στNl l ,rl (p) − f ∗ (p) . 2 Hence, for all l l0 and all p ∈ R0 we have θ N ,r (18) στl l l (p) − f ∗ (p) f ∗ (p) − f (p) − στNl l ,rl (p) − f ∗ (p) > . 2 Now well, for l sufficiently great and using (2) we deduce that there exists a point a rl ∈ Ar ∩ R0 such that N ,r r στl l l a l − f a rl = o(1), l → +∞, which is a contradiction with (18). Consequently f ∗ = f . Step 3. As H 3 (R; R3 ) is compactly injected in H 2 (R; R3 ), using (17) and taking into account that f ∗ = f we have f = lim στNl l ,rl in H 2 R; R3 . l→+∞
Then lim
l→+∞
στNl l ,rl , f
2
= f 22 .
Using again (17) and that f ∗ = f we obtain lim στNl l ,rl , f 3 = lim στNl l ,rl , f 3 − στNl l ,rl , f 2 = f 23 . l→+∞
l→+∞
Moreover, for all f ∈ N we have N ,r σ l l − f 2 = σ Nl ,rl − f 2 + f 2 − 2 σ Nl ,rl , f 3 τl τl τl 3 3 3
(19)
(20)
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we deduce from (20) that
lim στNl l ,rl − f 3 = 0
l→+∞
which implies with (19) that
lim στNl l ,rl − f 3 = 0.
l→+∞
Step 4. Finally, by reasoning with reduction to absurd we prove that the result is true. To do this, we suppose that it is false, so there exists a real number γ > 0 and the following sequences (Nl )l ∈N , (rl )l ∈N and (τl )l ∈N with liml →+∞ rl = +∞ such that Nl ,rl στ − f 3 γ , ∀l ∈ N. (21) l N ,r
Now well, the sequence (στll l )l ∈N is bounded in VN . Hence, by following the same way of the steps 1–3 we deduce that from such sequence we can extract a subsequence that converges towards f , which produces a contradiction with (21). 6.
Numerical and graphical examples
Consider the surface ϒ0 parameterized by the function f ∈ C 4 (R; R3 ), with R = (0, 1) × (0, 1) and 3π 3π y, sin π x − y, x . f (x, y) = cos π x − 2 2 By using our smoothness method we have computed an approximating surface ϒ of ϒ0 parameterized by a FAIRNESS BICUBIC SPLINE στN,r from 500 scattered points. The parametric space VN of bicubic spline functions of class C 2 has been constructed from a partition of 7 × 7 equal rectangles which means that we have taken n = m = 7 so the dim S3 (Tn , Tm ) = 100 and dim VN = N = 3(n + 3)(m + 3) = 300 that is the order of the linear system given in (5). Notice that the basis that we have taken is constituted by a B-spline basis. Likewise, for any τ ∈ R3+ we have computed the following estimation of the relative error Er given by 1/2 10000 N,r i=1 στ (ai ) − f (ai ) Er = f (ai ) where a1 , . . . , a10000 are random points in R. From tables 1 and 2 we can conclude the effectiveness of our algorithm as an approximation method. Graphically, from k = 500 scattered points and τ = (10−7 , 10−7 , 10−9 ), figure 1 shows the graph of the original surface ϒ0 and an approximating surface ϒ of it parameterized by στN,r . It is clear that the original surface ϒ0 and its approximating ϒ are similar. In this case, we have obtained that Er = 6.22804 × 10−5 .
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Table 1 Surface ϒ0 . Some estimations of the error for various values of the parameter vector τ and k = 1000. τ3
τ2
τ1
Er
10−16
0 0 0 0 0 1 10−4 10−7 0 0 0 10−4 10−7
0 0 0 0 0 0 0 0 1 10−4 10−7 10−4 10−7
1.83708 ×10−5 1.78372 ×10−4 1.23556 ×10−3 4.00495 ×10−3 4.97581 ×10−3 2.39928 ×10−2 4.70315 ×10−5 2.72895 ×10−5 1.110185 ×10−2 5.22723 ×10−4 2.87335 ×10−5 5.40927 ×10−4 2.6681 ×10−5
10−9 10−6 10−3 1 10−9 10−9 10−9 10−9 10−9 10−9 10−9 10−9
Table 2 Surface ϒ0 . Influence of the number of points k on the estimation of error Er for τ = (10−7 , 10−7 , 10−9 ). k
Er
1000 750 500 250 100 50 25
2.6681 ×10−5 3.73732 ×10−5 6.22804 ×10−5 2.48865 ×10−4 1.2724 ×10−3 1.61421 ×10−2 4.886 ×10−2
From table 3 we conclude that the value of estimation of the air A of the approximating surface decreases whenever τ1 increases. Now, we are going to show graphically the importance of the parameter vector τ . To do this, let us consider the explicit surface ϒ1 defined in the rectangle R = (0, 1) × (0, 1) by the Franke test function f (x, y) = 0.75e−((9y−2)
2 +(9x−2)2 )/4
+ 0.5e−((9y−7)
+ 0.75e−((9y+1)
2 +(9x−3)2 )/4
2 /49+(9x+1)2/10)
− 0.2e−((9y−4)
2 +(9x−7)2 )
.
The graph of this function appears in figure 2(a). Hence, for a partition of 9 × 9 equal rectangles of R, so dim S3 (T9 , T9 ) = 144, figure 2(b) shows an approximating surface from k = 2500 scattered points and τ = (10−3 , 10−3 , 10−6 ). In this case, we have obtained that Er = 2.152 × 10−3 .
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Figure 1. Surface ϒ0 . Original surface and an approximating one of it parameterized by a BICUBIC SPLINE .
FAIRNESS
Table 3 Surface ϒ0 . Influence of τ1 on the estimation of A for fixed values of τ2 = 10−7 and τ3 = 10−9 . τ1
A
Er
10−7
1.96758 1.9248 1.9056 1.86138
4.42719×10−5 6.9043×10−5 2.02873×10−4 1.1639×10−2
10−5 10−3 10−1
(a)
(b)
Figure 2. Surface ϒ1 . Original surface and an approximating one of it parameterized by a BICUBIC SPLINE .
FAIRNESS
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Figure 3. Surface ϒ1 . From left to right, two approximating surfaces parameterized by two FAIRNESS BICUBIC SPLINES for respectively τ = {1, 1, 10−2 } and τ = {10−1 , 10−1 , 10−3 }. The approximate errors are respectively Er = 6.3562 × 10−2 and Er = 3.1032 × 10−2 .
Figure 4. Surface ϒ1 . From left to right, two approximating surfaces parameterized by two FAIRNESS BICUBIC SPLINES for respectively τ = {10−2 , 10−2 , 10−4 } and τ = {10−3 , 10−3 , 10−6 }. The approximate errors are respectively Er = 1.0536 × 10−2 and Er = 5.0917 × 10−3 .
Figures 3 and 4 show the weight of the parameter vector τ in the approximation method in agreement with the interpretation given in remark 3.2. For this, we have considered a partition of 7 × 7 equal rectangles of R and we have taken k = 1500 scattered points in all the examples. Now, we consider the torus ϒ2 , which is a closed surface, parametrized by the function f defined in R = (0, 1) × (0, 1) by f (x, y) = cos(2π x) + 2 cos(2πy), cos(2π x) + 2 sin(2πy), sin(2π x) . We compute an approximating surface of ϒ2 parametrized by a FAIRNESS BICUBIC SPLINE for k = 500 scattered points and τ = (10−7 , 10−7 , 10−9 ) (cf. figure 5). In
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(a)
(b)
Figure 5. Surface u2 . Original surface (a) and an approximating one (b) of it parameterized by a PERIODIC FAIRNESS BICUBIC SPLINE .
this case the discretization space is constituted by periodic bicubic spline functions of class C 2 and it has been constructed from a partition of 7 × 7 equal rectangles. We have taken a basis of periodic bicubic B-splines, which means that in the example we consider the problem with periodic boundary conditions. In this case, Er = 1.7585 × 10−4 . This example shows how we could effectively extend our method for the closed parametric surfaces case.
Acknowledgements The authors thank the organizing committee of MAIA 2001 for their invitation and the referees for their comments and suggestions which improved the presentation and readability of the paper. This work has been supported by the Junta de Andalucía (Research group FQM/191), while the work of the second author has been supported partially by the General Direction of Research of the Ministry of Sciences and Technology (BFM 2000-1058).
References [1] M. Attéia, Fonctions splines definies sur un ensemble convexe, Numer. Math. 12 (1968) 192–210. [2] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978). [3] J. Duchon, Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO 10(12) (1976) 5–12. [4] G. Greiner, Surface construction based on variational principles, in: Wavelets Images and Surfaces Fitting, eds. P.J. Laurent, A. Le Méhauté and L.L. Shumaker (1994) pp. 277–286. [5] A. Kouibia, M. Pasadas and J.J. Torrens, Fairness approximation by modified discrete smoothing D m splines, in: Mathematical Methods for Curves and Surfaces, Vol. II, eds. M. Daehlen, T. Lyche and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, 1998) pp. 295–302. [6] A. Kouibia and M. Pasadas, Smoothing variational splines, Appl. Math. Lett. 13 (2000) 71–75.
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[7] A. Kouibia and M. Pasadas, Approximation by discrete variational splines, J. Comput. Appl. Math. 116 (2000) 145–156. [8] M.C. López de Silanes and R. Arcangéli, Sur la convergence des D m -splines d’ajustement pour des données exactes ou bruitées, Revista Matemática Universidad Complutense Madrid 4(2/3) (1991) 279– 284. [9] P.M. Prenter, Splines and Variational Methods (Wiley-Interscience, New York, 1989).
Approximation of surfaces by fairness bicubic splines A. Kouibia and M. Pasadas Department of Applied Mathematics, Faculty of Sciences, University of Granada, Granada, Spain E-mail: {kouibia;mpasadas}@ugr.es
Received 7 February 2001; accepted 25 October 2002 Communicated by M. Gasca
In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method. Keywords: smoothing, variational surface, fairness spline, bicubic spline AMS subject classification: 65D07, 65D10, 65D17
1.
Introduction
In geology and structural geology the reconstruction of a curve or surface from a scattered data set is a commonly encountered problem. The theory of D m -splines over an open bounded set has been introduced at the first time by Attéia [1]. We have enriched this theory and extended it to the variational spline functions [6] where the early works are therein. Several works have used the variational approach specifically minimizing some fairness functional (see, for example, [4], likewise these functional also can represent the flexed energy of a thin plate [3]) on a finite element space taking advantage the suitable properties of this space (see [5,7]) in order to simplify both characterization and computation of the solution. So we have planned to resolve in this work a variational approach problem on a finite-dimensional space that is not a finite element one. This is why we focus in this paper our interest to minimize a similar fairness functional on a space of bicubic spline functions of class C 2 , meanwhile in [7] we discretize in a finite element space where in case that its functions are bicubics they turns out to be of class C 1 . The resulting function is called a fairness bicubic spline. We study some characterization of this function and we shall express it as a linear combination of the basis functions of a parametric space of bicubic splines. Moreover, under adequate hypotheses
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we prove that such fairness bicubic spline converges to a given function from which are proceeding the data. We present some numerical and graphical examples in order to show the effectiveness and the validity of the method of this paper. We remark that this work can be considered as an analogous case of the problem presented in [7], but the advantage presented here is to reach a great smoothness order. The method is presented for any open parametric surface, i.e. without boundary periodic conditions, but we think that it can be extended to any closed parametric surfaces, by restricting the space of discretization to a linear subspace formed by periodic bicubic splines. In the last example we approximate the torus by a periodic fairness bicubic spline. Some fields of applications of this problem can appear in Earth sciences, specially in geology and geophysics, as long as CAD and CAGD, etc. The remainder of this paper is organized as follows. In section 2, we briefly recall some preliminary notations and results. Section 3 is devoted to state the approximation problem and to present a method to solve it. In section 4, we compute the resulting function, while a convergence’s theorem is proved in section 5. In section 6 some numerical and graphical examples are given.
2.
Notations and preliminaries
We denote by · and ·, ·, respectively, the Euclidean norm and inner product in R3 , while we denote by ·R2 and ·, ·R2 respectively the Euclidean norm and inner product in R2 . For any real interval (a, b) ((c, d)) with a < b (c < d) we consider the rectangle R = (a, b) × (c, d) and let H 3 (R; R3 ) be the usual Sobolev space of (classes of) functions u belong to L2 (R; R3 ), together with all their partial derivative D β u with β = (β1 , β2 )T , in the distribution sense, of order |β| = β1 + β2 3. This space is equipped with the norm u =
|β|3
β
1/2
2
D u(p) dp
,
R
the semi-norms |u| =
|β|=
β
2
D u(p) dp
1/2 ,
0 3,
R
and the corresponding inner semi-products 2 D β u(p), D β v(p) dp, (u, v) = |β|=
R
0 3.
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Let Rk,3 be the space of real matrices of k rows and 3 columns equipped with the inner semi-product A, Bk,3 =
3 k
ai,j bi,j ,
i=1 j =1
with A = (aij ) 1ik , B = (bij ) 1ik and the corresponding norm 1j3
1j3
Ak,3 =
k 3
1/2 (ai,j )2
.
i=1 j =1
Moreover, for any n, m ∈ N∗ let Tn = {x0 , . . . , xn }, Tm = {y0 , . . . , ym } be some subsets of distinct points of [a, b] and [c, d], with a = x0 x1 < · · · < xn−1 xn = b and c = y0 y1 < · · · < ym−1 ym = d. We denote by S3 (Tn ) and S3 (Tm ) the spaces of cubic spline functions given by
S3 (Tn ) = s ∈ C 2 [a, b] | s|[xi−1 ,xi ] ∈ P3 [xi−1 , xi ], i = 1, . . . , n ,
S3 (Tm ) = s ∈ C 2 [c, d] | s|[yi−1 ,yi ] ∈ P3 [yi−1 , yi ], i = 1, . . . , m , where P3 [xi−1 , xi ] (P3 [yi−1 , yi ]) is the restriction on [xi−1 , xi ] ([yi−1 , yi ]) of the linear space of real polynomials with total degree less than or equal to 3. It is known that dim S3 (Tn ) = n + 3 (dim S3 (Tm ) = m + 3) and let {ϕ1 , . . . , ϕn+3 } and {ψ1 , . . . , ψm+3 } be respectively bases of functions with local support of S3 (Tn ) and S3 (Tm ). We consider the space S3 (Tn , Tm ) of bicubic spline functions given by S3 (Tn , Tm ) = span{ϕ1 , . . . , ϕn+3 } ⊗ span{ψ1 , . . . , ψm+3 }. Finally, for any n, m ∈ N∗ we define the space of parametric bicubic spline functions VN = (S3 (Tn , Tm ))3 with N = 3(n + 3)(m + 3) = dim VN constructed from S3 (Tn , Tm ) which is a Hilbert subspace of H 3 (R; R3 ) equipped with the same norm, semi-norm and inner semi-product of such space and, moreover, verifies (1) VN ⊂ H 3 R; R3 ∩ C 2 R; R3 . 3.
Fairness bicubic spline
Let ϒ0 ⊂ R3 be a surface defined by a parameterization f belonging to C 2 (R; R3 ). For each r ∈ N∗ let Ar be a finite subset of k = k(r) distinct points of R such that 1 , r → +∞. (2) sup minr p − aR2 = O a∈A r p∈R Remark 3.1. From (2) we can obtain that for r sufficiently large k(r) > r. In section 5 we will add others conditions for k(r) in order to warranty a convergence result.
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Let Lr be a Lagrangian operator defined from H 3 (R, R3) into Rk,3 by Lr v = v(a) a∈Ar and suppose that
Ker Lr ∩ P2 R; R3 = {0},
(3)
being P2 (R; R3 ) the linear space of the real polynomials with total degree less than or equal to 2 defined from R into R3 . Now, we consider the following problem: Find an approximating surface ϒ of ϒ0 parameterized by a function σ of VN that fits the data points of Ar and minimizes all the semi-norms of order less than 3 in VN . For any τ = (τ1 , τ2 , τ3 ) ∈ R3 with τ1 , τ2 belonging to R+ and τ3 > 0, let Jτr be the functional defined on H 3 (R; R3 ) by 3 2 τj |v|2j . Jτr (v) = Lr (v − f ) k,3 + j =1
Remark 3.2. The first term of Jτr (v) indicates how well v approaches f in a least discrete squares sense. The second term can represent some different conditions as, for example: fairness conditions (see [4,5]), a classical smoothness measure, etc., while the parameter vector τ weights the importance given to each conditions. Then, for any r 3 we consider the following minimization problem: Find στN,r such that
N,r στ ∈ VN , (4) ∀v ∈ VN , Jτr στN,r Jτr (v). Theorem 3.1. The problem (4) has a unique solution, called the fairness bicubic spline in VN relative to Ar , Lr and τ , which is also the unique solution of the following variational problem: Find στN,r such that N,r ∈ VN , σ τ ∀v ∈ VN , Proof.
L
r
στN,r , Lr v k,3
+
3
τj στN,r , v j = Lr f, Lr v k,3 .
j =1
Taking into account (1), (2) and that the following norm
3 2 τj |v|2j v→ [[v]] = Lr v k,3 + j =1
1/2
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91
is equivalent in VN to the norm · , one easily checks that the symmetric bilinear form a˜ : VN × VN → R given by
a(u, ˜ v) = L u, L v r
r
k,3
+
3
τj (u, v)j
j =1
is a continuous and VN -elliptic. Likewise, the linear form ϕ : v ∈ VN → ϕ(v) = Lr f, Lr v k,3 is continuous. The result is then a consequence of the Lax–Milgram lemma (see [2]). Remark 3.3. Obviously, the unidimensional case, i.e. the approximation problem of curves by fairness cubic splines is analogous to the bidimensional case, but not for this are simply the extension of the notations and proofs. This is why directly we deal with the bidimensional case for its interest and application.
4.
Computation
Now well, we are going to show how to obtain in practice any fairness bicubic spline but we assume that we know the parameter values associated to given data points. For any n, m ∈ N∗ we consider {w1 , . . . , w(n+3)(m+3)} a basis of elements with local support of the space S3 (Tn , Tm ) and {e1 , e2 , e3 } the canonical one of R3 . Then, the family {v1 , . . . , vN } is a basis of VN with ∀i = 1, . . . , (n + 3)(m + 3), for = 1, 2, 3, j = 3(i − 1) + ,
vj = wi e .
Thus, στN,r can be written as στN,r = N i=1 αi vi , with αi ∈ R unknown, for i = 1, . . . , N . Applying theorem 3.1 we obtain a linear system of order N as follows N 3 r αi L vi , Lr vj k,3 + τs (vi , vj )s = Lr f, Lr vj k,3 , ∀j = 1, . . . , N, i=1
s=1
that is equivalent to Cα = b where C = (cij )1i,j N , α = (α1 , . . . , αN )T , b = (b1 , . . . , bN )T ,
(5)
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with for i, j = 1, . . . , N we have 3 τs (vi , vj )s , cij = Lr vi , Lr vj k,3 + s=1 b = Lr f, Lr v j j k,3 . Finally, we point out the following result. Proposition 4.1. The matrix C is symmetric, positive definite and of band type. Proof. Obviously the matrix C is symmetric. Let now p = (p1 , . . . , pN ) ∈ RN we have pCp = T
=
N i,j =1 N
pi cij pj pi
r
r
L vi , L vj
k,3
+
i,j =1
= Lr = Lr
N
pi vi , Lr
N
2
Let w =
i=1
N
τs (vi , vj )s pj
k,3
+
pj vj
j =1
+
pi vi
i=1
N
s=1
i=1
3
3 s=1
k,3 N
τs
i=1
3
τs
N
s=1 2
pi vi ,
i=1
N j =1
pj vj s
pi vi 0. s
pi vi and we suppose that pCp T = 0 then one has
3 2 τi |w|2i = 0. Lr w k,3 + i=1
Hence [[w]]2 = 0 that implies w = 0 (where [[·]] designs the norm defined in the proof of theorem 3.1). Moreover, for the independent linearity of the family {vi }1iN we obtain that p = 0 and C is positive definite. Finally, the matrix C is of band type because for each i = 1, . . . , N the function vi has local support. In practice we use the following notations: A = Lr vi 1iN , B = (Bs )1s3 = (vi , vj )s 1i,j N 1s3 , hence system (5) is equivalent to T AA + τ T B α = ALr f.
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Remark 4.4. Taking into account that each function v defined from R into R3 can be written as v = (v 1 , v 2 , v 3 ), then the system (5) can be composed into three ones as follows l l T A A + τ T B l α l = Al Lr f l for l = 1, 2, 3, where
Al = Lr vil 1iN , B l = Bsl 1s3 = vil , vjl s 1i,j N 1s3 .
Hence, we can write the solution as στN,r = ((στN,r )l )1l3 , where
l στN,r
=
N
αil vil
for l = 1, 2, 3.
i=1
5.
Convergence
Under adequate conditions, we are going to prove that the fairness bicubic spline in VN relative to Ar , Lr f and τ converges to f when N and r tend to +∞. Before to do this we need the following results. Proposition 5.1. Let B0 = {b01 , . . . , b06 } be a P2 (R; R3 )-unisolvent subset of points of R. Then, there exists η > 0 such that if Bη stands 6-uplets B = {b1 , . . . , b6 } of points of R satisfying the condition ∀j = 1, . . . , 6,
bj − b0j R2 < η,
(6)
the application [[·]]B defined for all B ∈ Bη by 6 1/2 2 B 2 , v(bj ) + |v|3 [[v]] = j =1
is a norm on H 3 (R; R3 ) uniformly equivalent on Bη to the usual norm of Sobolev · . Proof.
It is analogous to [8, proposition 2.1].
Corollary 5.2. Suppose that the hypotheses (2) and (3) hold. Then, there exists η > 0 and, for all r ∈ N, a subset Ar0 of Ar and a constant C > 0 such that, for all r C/η, the application [[·]]r0 defined by 1/2 2 , v(a) + |v|23 [[v]]r0 = a∈Ar0
is a norm in H 3 (R; R3 ) uniformly equivalent, with respect to r, to the norm · .
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Proof. Let B0 = {b01 , . . . , b06 } be anyone P2 (R; R3 )-unisolvent subset of R. From (2) there exists C > 0 such that for all r ∈ N and all j = 1, . . . , 6, there exists ajr ∈ Ar verifying
C . r Let Ar0 = {a1r , . . . , a6r }. It suffices then to apply proposition 5.1, taking into account r that, for all r C/η, Ar0 ∈ Bη and written [[·]]r0 instead of [[·]]A0 . b0j − ajr
R2
We remember that the Tchebycheff norm in C 0 (R; R3 ) is defined by u T = max u(p) . p∈R
Let us give the easy relation between the semi-norms of order l and the Tchebycheff norm in H 3 (R; R3 ). Lemma 5.3. For all u ∈ C 2 (R; R3 ) one has |u| meas(R) D uT ,
= 0, 1, 2,
being meas(R) the area of rectangle R. Proof.
Let u ∈ C 2 (R; R3 ) then we have 2 2 2 max D j u(p) , D j u(p) dp meas(R) |u| = |j |=
R
|j |=
p∈R
which implies that 2 |u|2 meas(R)D uT . Let f ∈ C 4 (R; R3 ) and let SN be the which solves the interpolation problem SN (xi , yj ) = f (xi , yj ), ∂f ∂SN (xi , yj ) = (xi , yj ), ∂x ∂x ∂f ∂SN (x (xi , yj ), , y ) = i j ∂y ∂y ∂ 2f ∂ 2 SN (xi , yj ) = (xi , yj ), ∂x∂y ∂x∂y
parametric bicubic spline function of VN for i = 1, . . . , n; j = 1, . . . , m, for i = 1, . . . , n; j = 1, m, for i = 1, n; j = 1, . . . , m, for i = 1, n; j = 1, m.
So, from [9, section 5.2] and applying lemma 5.3 it follows that |f − SN | K h4− ,
= 0, 1, 2, 3,
(7)
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95
where K , = 0, 1, 2, 3, depends only on the Tchebycheff norms of mixed partial derivatives of f up through th, = 0, 1, 2, 3, and
b−a d −c , . h = max n m For the following convergence result we suppose that ∃C > 0, ∃r0 ∈ N: ∀r r0 ,
k(r) Cr 2 ,
(8)
which is an asymptotic regularity property of the distribution of the points of Ar in R. Now, let στN,r be the fairness bicubic spline in VN relative to Ar , Lr f and τ . Theorem 5.4. Let f ∈ C 4 (R; R3 ). Suppose that the hypotheses (1) and (2) hold and that τ3 = o r 2 , r → +∞, (9) r → +∞, (10) ∀i = 1, 2, τi = o(τ3 ), and rh4 1/2
τ3 Then, one has
= o(1),
r → +∞.
lim f − στN,r = 0.
N,r→+∞
Proof. The proof will be split into four steps. Step 1. First, we have Jτr (στN,r ) Jτr (SN ) which implies that N,r 2 σ 1 Lr (SN − f ) 2 + τ1 |SN |2 + τ2 |SN |2 + |SN |2 . τ 1 2 3 3 k,3 τ3 τ3 τ3 From (7) it follows that |SN |1 |SN − f |1 + |f |1 K1 h3 + |f |1 , |SN |2 K2 h2 + |f |2 and |SN |3 K3 h + |f |3 . Likewise, from (7) there exists C > 0 such that for each i = 1, . . . , r we have 2 SN (ai ) − f (ai ) Ch8 , and from (8) one has
2 r L (SN − f ) k,3 Cr 2 h8 .
(11)
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Then, finally, we obtain that there exists C > 0 such that 2 8 N,r 2 σ C r h + τ1 K1 h3 +|f |1 2 + τ2 K2 h2 +|f |2 2 + K3 h+|f |3 2 , τ 3 τ3 τ3 τ3
N → +∞.
(12) Hence, from (10)–(12) we deduced that there exist a constant C > 0 and M0 , M1 ∈ N such that N,r σ C, ∀r M0 , ∀N M1 , τ 3 and
2 Lr στN,r − f k,3 = O(τ3 ),
r → +∞, N → +∞.
(13)
Let B0 = {b01 , . . . , b06 } be a P2 (R; R3 )-unisolvent subset of points of R and let η be the constant of proposition 5.1. Obviously, there exists η ∈ (0, η] such that ∀j = 1, . . . , 6, B b0j , η ⊂ R. From (2), we have that there exists C > 0 such that C C ⊂ ∀j = 1, . . . , 6, B b0j , η − ∀r ∈ N, r > , η r
a∈Ar ∩B(b0j ,η )
C . B a, r
If Nj = card(Ar ∩ B(0j , η )) then there exists C1 > 0 verifying ∀r ∈ N,
C r > , η
∀j = 1, . . . , 6,
C 2 C1 N j η − r r2
and consequently for any r0 > C/η , there exists C2 > 0 such that C 2 2 r . ∀r r0 , ∀j = 1, . . . , 6, Nj C2 η − r0 Meanwhile, from (9) and (13) we deduce that 2 N,r στ − f (a) = o r 2 , ∀j = 1, . . . , 6,
(14)
r → +∞, N → +∞.
a∈Ar ∩B(b0j ,η )
If ajr ∈ Ar ∩ B(b0j , η ) such that N,r στ − f ajr =
(15)
min
a∈Ar ∩B(b0j ,η )
N,r στ − f (a) ,
we deduce from (14) and (15) that N,r ∀j = 1, . . . , 6, στ − f ajr = o(1),
r → +∞.
(16)
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97
Now, we denote by B r the set {a1r , . . . , a6r }, then by applying proposition 5.1 to B = B r , for r sufficiently close to infinity, it turns out from (12), (16) and corollary 5.2 that ∃C > 0, ∃λ > 0, ∀r λ, σ N,r C, τ
(στN,r )r,N∈N∗
which means that the family a subsequence (στNl l ,rl )l∈N , with τl = τ (rl ),
3
is bounded in VN . It follows that there exists
lim rl = +∞,
lim Nl = +∞
l→+∞
l→+∞
and an element f ∗ ∈ H 3 (R; R3 ) such that στNl l ,rl converges weakly to f ∗
in H 3 R; R3 .
(17)
Step 2. Let us now prove that f ∗ = f . We suppose that f ∗ = f . From the continuous injection of H 3 (R; R2 ) into C 1 (R; R3 ) it follows that there exist θ > 0 and an open rectangle R0 of R such that ∀p ∈ R0 , f ∗ (p) − f (p) > θ. As such injection is also compact then from (17) we obtain θ ∃l0 ∈ N, ∀l l0 , ∀p ∈ R0 , στNl l ,rl (p) − f ∗ (p) . 2 Hence, for all l l0 and all p ∈ R0 we have θ N ,r (18) στl l l (p) − f ∗ (p) f ∗ (p) − f (p) − στNl l ,rl (p) − f ∗ (p) > . 2 Now well, for l sufficiently great and using (2) we deduce that there exists a point a rl ∈ Ar ∩ R0 such that N ,r r στl l l a l − f a rl = o(1), l → +∞, which is a contradiction with (18). Consequently f ∗ = f . Step 3. As H 3 (R; R3 ) is compactly injected in H 2 (R; R3 ), using (17) and taking into account that f ∗ = f we have f = lim στNl l ,rl in H 2 R; R3 . l→+∞
Then lim
l→+∞
στNl l ,rl , f
2
= f 22 .
Using again (17) and that f ∗ = f we obtain lim στNl l ,rl , f 3 = lim στNl l ,rl , f 3 − στNl l ,rl , f 2 = f 23 . l→+∞
l→+∞
Moreover, for all f ∈ N we have N ,r σ l l − f 2 = σ Nl ,rl − f 2 + f 2 − 2 σ Nl ,rl , f 3 τl τl τl 3 3 3
(19)
(20)
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we deduce from (20) that
lim στNl l ,rl − f 3 = 0
l→+∞
which implies with (19) that
lim στNl l ,rl − f 3 = 0.
l→+∞
Step 4. Finally, by reasoning with reduction to absurd we prove that the result is true. To do this, we suppose that it is false, so there exists a real number γ > 0 and the following sequences (Nl )l ∈N , (rl )l ∈N and (τl )l ∈N with liml →+∞ rl = +∞ such that Nl ,rl στ − f 3 γ , ∀l ∈ N. (21) l N ,r
Now well, the sequence (στll l )l ∈N is bounded in VN . Hence, by following the same way of the steps 1–3 we deduce that from such sequence we can extract a subsequence that converges towards f , which produces a contradiction with (21). 6.
Numerical and graphical examples
Consider the surface ϒ0 parameterized by the function f ∈ C 4 (R; R3 ), with R = (0, 1) × (0, 1) and 3π 3π y, sin π x − y, x . f (x, y) = cos π x − 2 2 By using our smoothness method we have computed an approximating surface ϒ of ϒ0 parameterized by a FAIRNESS BICUBIC SPLINE στN,r from 500 scattered points. The parametric space VN of bicubic spline functions of class C 2 has been constructed from a partition of 7 × 7 equal rectangles which means that we have taken n = m = 7 so the dim S3 (Tn , Tm ) = 100 and dim VN = N = 3(n + 3)(m + 3) = 300 that is the order of the linear system given in (5). Notice that the basis that we have taken is constituted by a B-spline basis. Likewise, for any τ ∈ R3+ we have computed the following estimation of the relative error Er given by 1/2 10000 N,r i=1 στ (ai ) − f (ai ) Er = f (ai ) where a1 , . . . , a10000 are random points in R. From tables 1 and 2 we can conclude the effectiveness of our algorithm as an approximation method. Graphically, from k = 500 scattered points and τ = (10−7 , 10−7 , 10−9 ), figure 1 shows the graph of the original surface ϒ0 and an approximating surface ϒ of it parameterized by στN,r . It is clear that the original surface ϒ0 and its approximating ϒ are similar. In this case, we have obtained that Er = 6.22804 × 10−5 .
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Table 1 Surface ϒ0 . Some estimations of the error for various values of the parameter vector τ and k = 1000. τ3
τ2
τ1
Er
10−16
0 0 0 0 0 1 10−4 10−7 0 0 0 10−4 10−7
0 0 0 0 0 0 0 0 1 10−4 10−7 10−4 10−7
1.83708 ×10−5 1.78372 ×10−4 1.23556 ×10−3 4.00495 ×10−3 4.97581 ×10−3 2.39928 ×10−2 4.70315 ×10−5 2.72895 ×10−5 1.110185 ×10−2 5.22723 ×10−4 2.87335 ×10−5 5.40927 ×10−4 2.6681 ×10−5
10−9 10−6 10−3 1 10−9 10−9 10−9 10−9 10−9 10−9 10−9 10−9
Table 2 Surface ϒ0 . Influence of the number of points k on the estimation of error Er for τ = (10−7 , 10−7 , 10−9 ). k
Er
1000 750 500 250 100 50 25
2.6681 ×10−5 3.73732 ×10−5 6.22804 ×10−5 2.48865 ×10−4 1.2724 ×10−3 1.61421 ×10−2 4.886 ×10−2
From table 3 we conclude that the value of estimation of the air A of the approximating surface decreases whenever τ1 increases. Now, we are going to show graphically the importance of the parameter vector τ . To do this, let us consider the explicit surface ϒ1 defined in the rectangle R = (0, 1) × (0, 1) by the Franke test function f (x, y) = 0.75e−((9y−2)
2 +(9x−2)2 )/4
+ 0.5e−((9y−7)
+ 0.75e−((9y+1)
2 +(9x−3)2 )/4
2 /49+(9x+1)2/10)
− 0.2e−((9y−4)
2 +(9x−7)2 )
.
The graph of this function appears in figure 2(a). Hence, for a partition of 9 × 9 equal rectangles of R, so dim S3 (T9 , T9 ) = 144, figure 2(b) shows an approximating surface from k = 2500 scattered points and τ = (10−3 , 10−3 , 10−6 ). In this case, we have obtained that Er = 2.152 × 10−3 .
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Figure 1. Surface ϒ0 . Original surface and an approximating one of it parameterized by a BICUBIC SPLINE .
FAIRNESS
Table 3 Surface ϒ0 . Influence of τ1 on the estimation of A for fixed values of τ2 = 10−7 and τ3 = 10−9 . τ1
A
Er
10−7
1.96758 1.9248 1.9056 1.86138
4.42719×10−5 6.9043×10−5 2.02873×10−4 1.1639×10−2
10−5 10−3 10−1
(a)
(b)
Figure 2. Surface ϒ1 . Original surface and an approximating one of it parameterized by a BICUBIC SPLINE .
FAIRNESS
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Figure 3. Surface ϒ1 . From left to right, two approximating surfaces parameterized by two FAIRNESS BICUBIC SPLINES for respectively τ = {1, 1, 10−2 } and τ = {10−1 , 10−1 , 10−3 }. The approximate errors are respectively Er = 6.3562 × 10−2 and Er = 3.1032 × 10−2 .
Figure 4. Surface ϒ1 . From left to right, two approximating surfaces parameterized by two FAIRNESS BICUBIC SPLINES for respectively τ = {10−2 , 10−2 , 10−4 } and τ = {10−3 , 10−3 , 10−6 }. The approximate errors are respectively Er = 1.0536 × 10−2 and Er = 5.0917 × 10−3 .
Figures 3 and 4 show the weight of the parameter vector τ in the approximation method in agreement with the interpretation given in remark 3.2. For this, we have considered a partition of 7 × 7 equal rectangles of R and we have taken k = 1500 scattered points in all the examples. Now, we consider the torus ϒ2 , which is a closed surface, parametrized by the function f defined in R = (0, 1) × (0, 1) by f (x, y) = cos(2π x) + 2 cos(2πy), cos(2π x) + 2 sin(2πy), sin(2π x) . We compute an approximating surface of ϒ2 parametrized by a FAIRNESS BICUBIC SPLINE for k = 500 scattered points and τ = (10−7 , 10−7 , 10−9 ) (cf. figure 5). In
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(a)
(b)
Figure 5. Surface u2 . Original surface (a) and an approximating one (b) of it parameterized by a PERIODIC FAIRNESS BICUBIC SPLINE .
this case the discretization space is constituted by periodic bicubic spline functions of class C 2 and it has been constructed from a partition of 7 × 7 equal rectangles. We have taken a basis of periodic bicubic B-splines, which means that in the example we consider the problem with periodic boundary conditions. In this case, Er = 1.7585 × 10−4 . This example shows how we could effectively extend our method for the closed parametric surfaces case.
Acknowledgements The authors thank the organizing committee of MAIA 2001 for their invitation and the referees for their comments and suggestions which improved the presentation and readability of the paper. This work has been supported by the Junta de Andalucía (Research group FQM/191), while the work of the second author has been supported partially by the General Direction of Research of the Ministry of Sciences and Technology (BFM 2000-1058).
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[7] A. Kouibia and M. Pasadas, Approximation by discrete variational splines, J. Comput. Appl. Math. 116 (2000) 145–156. [8] M.C. López de Silanes and R. Arcangéli, Sur la convergence des D m -splines d’ajustement pour des données exactes ou bruitées, Revista Matemática Universidad Complutense Madrid 4(2/3) (1991) 279– 284. [9] P.M. Prenter, Splines and Variational Methods (Wiley-Interscience, New York, 1989).
