International Journal of Innovative Computing, Information and Control Volume x, Number 0x, x 2005
c ICIC International °2005 ISSN 1349-4198 pp. 0–0
SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA Hiroyuki Fujioka and Hiroyuki Kano Department of Information Sciences Tokyo Denki University Hatoyama, Hiki-gun, Saitama 350-0394, Japan
[email protected];
[email protected]
Magnus Egerstedt School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 30332, USA
[email protected]
Clyde F. Martin Department of Mathematics and Statistics Texas Tech University Lubbock, TX 79409, USA
[email protected]
Abstract. We consider the problem of designing optimal smoothing spline curves and surfaces for a given set of discrete data. For constructing curves and surfaces, we employ normalized uniform B-splines as the basis functions. First we derive concise expressions for the optimal solutions in the form which can be used easily for numerical computations as well as mathematical analyses. Then, assuming that a set of data in a plane is obtained by sampling some curve with or without noises, we prove that, under certain condition, optimal smoothing splines converge to some limiting curve as the number of data increases. Such a limiting curve is obtained as a functional of given curve to be sampled. The case of surfaces is treated in parallel, and it is shown that the results for the case of curves can be extended to the case of surfaces in a straightforward manner. Keywords: B-splines, optimal smoothing splines, asymptotic analysis, statistical analysis
1. Introduction. The problem of optimal design of approximating or interpolating curves and surfaces for a given set of data arises in various fields of engineering and sciences. In particular, spline functions have been used frequently in such fields as computer aided design [1], numerical analysis [2], image processing [3], trajectory planning problems [4, 5], and data analysis in general [6]. Recently, the theory of smoothing splines is used to generate cursive characters based on an idea that the underlying writing motions become smooth [7, 8]. Thus splines have been studied extensively (e.g. [6]), and in particular, an approach based on optimal control theory has been employed for generating piecewise polynomial splines as the solutions to a number of different optimal control problems [9, 10]. Moreover, 1
2
H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN
the theory of ‘dynamic splines’ provides a unified framework for generating various types of spline curves as outputs of linear dynamical systems (e.g. [11, 12, 13]. Also, the authors studied B-spline functions from the viewpoint of optimal control theory [14]. In the problem of designing surfaces by splines [15, 16], the surfaces, in most cases, are constructed by resorting to some numerical optimization techniques. Alternative approaches to this problem include the development of appropriate basis functions, e.g. wavelets, and active boundary control of partial differential equations. When we are given a set of data corrupted by noises, smoothing splines are expected to yield more feasible solutions than interpolating splines. A theoretical issue in this regard is asymptotic and statistical analyses of designed splines when the number of data increases. Such a problem is studied in [17] in dynamical systems settings, namely for spline curves generated as an output of linear dynamical systems. On the other hand, using B-splines [18] as basis functions is expected to yield simple algorithms for designing curves and surfaces. In particular, using normalized uniform Bsplines as the basis functions, we have been studying the problems of constructing optimal interpolating and approximating curves [19]. This approach enabled us to investigate how the introduction of approximation, i.e. of least-square terms in the cost function, affects the shape of the curve. Moreover, we could derive numerically tractable algorithms, and this approach makes it easier to extend the results for one-dimensional case (i.e. the case of curves) to two-dimensional case (i.e. the case of surfaces) and to even higher dimensions. The purpose of this paper is to design optimal smoothing spline curves and surfaces, and analyze their properties using B-splines as the basis functions. Assuming that a number of data is given by sampling some curve f (t) with noises, we analyze statistical properties of optimal smoothing splines and derive an expression of the splines as a functional of f (t) when the number tends to infinity. Such a design and analysis method is extended to the case of surfaces. We will see that the expressions for optimal curves and surfaces are concise, enabling us to analyze their properties easier. Moreover, extensions of results for curves to the two-dimensional case, namely the surfaces are straightforward. For designing curves x(t), we employ normalized, uniform B-spline function Bk (t) of degree k as the basis functions, m−1 X
x(t) =
τi Bk (α(t − ti )).
(1)
i=−k
Here, m is an integer, τi ∈ R is a weighting coefficient called control point, and α(> 0) is a constant for scaling the interval between equally-spaced knot points ti with 1 (2) ti+1 − ti = . α Then x(t) formed in (1) is a spline of degree k with the knot points ti . In particular, by an appropriate choice of τi ’s, arbitrary spline of degree k can be designed in the interval [t0 , tm ]. On the other hand, surfaces are generated as x(s, t) =
m 1 −1 m 2 −1 X X i=−k j=−k
τi,j Bk (α(s − si ))Bk (β(t − tj )).
(3)
SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA
3
where α, β(> 0) are constants, m1 , m2 (> 2) are integers, and si ’s, tj ’s are equally spaced knot points with 1 1 si+1 − si = , ti+1 − ti = . (4) α β This paper is organized as follows: In Section 2, we present normalized uniform Bspline functions that are used throughout the paper. In Section 3, the problem of designing optimal smoothing curves and surfaces are formulated and optimal solutions are presented. We analyze asymptotic and statistical properties of optimal curves and surfaces in Section 4, and the results are established as theorems. The results of numerical simulation studies are presented in Section 5, and concluding remarks are given in Section 6. Here we summarize some of the symbols that will be used throughout the paper: ∇2 denotes the Laplacian operator, and ⊗ the Kronecker product. Moreover, ’vec’ denotes the vec-function, i.e. for a matrix A = [a1 a2 · · · an ] ∈ Rm×n with ai ∈ Rm , vec A = £ T T ¤T a1 a2 · · · aTn ∈ Rm×n (see e.g. [20]). 2. Normalized Uniform B-Splines. The normalized, uniform B-spline function of degree k, denoted as Bk (t), is defined by Nk,k (t) 0≤t 3. Using Table 1, we can compute the integral in (89) for |p − q| ≤ 3, and the results are shown in Table 2. Note that, from this table, we h iml −1 (22) (22) obtain the matrix Rl = rp,q , which is of the form (24). p,q=−3
(ii)
(ii)
Table 2. The elements rpq of matrix Rl (00)
(11)
rpq
|p − q| = 3
151 315 397 1680 1 42 1 5040
otherwise
0
p−q =0 |p − q| = 1 |p − q| = 2
(l = 1, 2 and i = 0, 1, 2) (22)
rpq
rpq
2 3
8 3
− 18
− 32
− 15
0
1 − 12
1 6
0
0 (ij)
B.2. The case where I1 = (s0 , sm1 ) and I2 = (t0 , tm2 ). We first rewrite Rl µZ +∞ Z 0 Z +∞ ¶ ³ ´T (ij) ˆb(i) (t) ˆb(j) (t) dt. − − Rl = l l −∞
−∞
ml
Noting that the first integral was computed in Section B.2 and denoting it by have (ij) (ij) (ij) (ij) Rl = ∞ Rl − Ul − Vl where the matrices
(ij) Ul
as
∞
(ij)
Rl , we (90)
(ij)
∈ RMl ×Ml and Vl ∈ RMl ×Ml are defined by Z 0 ³ ´T (ij) (i) (j) ˆ ˆ Ul = bl (t) bl (t) dt −∞ Z +∞ ³ ´T (ij) ˆb(i) (t) ˆb(j) (t) dt. Vl = l l
(91) (92)
ml
(02)
Here, similarly as (77), we obtain Ul (02)
Ul
(11)
= Ulc − Ul
where Ulc and Vlc are given by Ulc
−1 1 −4 = 12 −1
Vlc = (20)
Obviously Ul
(02) T
= (Ul
and Vl
1 12
) and Vl
, Vl
(02)
0 0 0
1 4 1
0Ml −3,3 0Ml −3,Ml −3
(20)
03,Ml −3 = (Vl
(02)
(02) T
as
= Vlc − Vl
(11)
,
(93)
,
(94)
0Ml −3,3 1 0 −1 . 4 0 −4 1 0 −1
(95)
) .
03,Ml −3 0Ml −3,Ml −3
20
H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN (ii)
(ii)
Therefore, it remains to derive expressions for Ul and Vl for i = 0, 1, 2. Letting £ (ii) ¤ml −1 £ (ii) ¤ml −1 (ii) (ii) Ul = l up,q , Vl = l vp,q , p,q=−3 p,q=−3 these elements are given by l (ii) up,q
Z
0
Z
−∞ −p
= and
Z
(i)
B3 (t − p)B3 (t − q)dt (i)
(i)
B3 (t)B3 (t − (q − p))dt,
0
l (ii) vp,q
(i)
=
4
(i)
= ml −p
(i)
B3 (t)B3 (t − (q − p))ds.
(96)
(97)
The values of these elements are computed as shown in Table 3 and Table 4. We see that (22) (22) we obtain the matrices Ul and Vl in the form of (26) and (27) respectively. (ii)
(ii)
Table 3. The elements l upq of matrix Ul (l = 1, 2 and i = 0, 1, 2) (p, q)
l (00) upq
l (11) upq
l (22) upq
(−3, −3) (−2, −2) (−1, −1) (−3, −2) (−2, −1) (−3, −1) otherwise
599 1260 151 630 1 252 59 280 43 1680 1 84
37 60 1 3 1 20 − 11 60 7 120 1 − 10
7 3 4 3 1 3
0
0
(ii)
Table 4. The elements l vpq of matrix Vl
−1 − 21 0 0 (ii)
(l = 1, 2 and i = 0, 1, 2)
(p, q)
l (00) vpq
l (11) vpq
l (22) vpq
(ml − 3, ml − 3) (ml − 2, ml − 2) (ml − 1, ml − 1) (ml − 3, ml − 2) (ml − 2, ml − 1) (ml − 3, ml − 1) otherwise
1 252 151 630 599 1260 43 1680 59 280 1 84
1 20 1 3 37 60 7 120 − 11 60 1 − 10
1 3 4 3 7 3
0
0
− 12 −1 0 0
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SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA
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