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論 文

Discrete-Time Control Systems Approach for Optimal Smoothing Splines

*

Hiroyuki KANO•E Hiroyuki FUJIOKA

Discrete-Time Control Systems Approach for Optimal Smoothing Splines *

Hiroyuki KANO and Hiroyuki FUJIOKA

We consider the problem of designing optimal smoothing spline curves by employing an approach based on linear control systems. First, the problem is formulated using continuous-time, time-invariant systems with piecewise constant inputs. Then by introducing discrete time-varying systems, the solutions for optimal splines including periodic splines are derived. The existence conditions for unique optimal solutions are established and linked to the concepts of controllability and observability. The computational procedures for the optimal splines are straightforward. The design method for periodic splines is applied to a shape synthesizing problem using jellyfish as the example.

1. Introduction

Spline functions have been used in various fields including computer graphics, computer-aided design, numerical analysis, image processing, trajectory plan- ning of robot and aircraft, and data analysis in general (see e.g. [4,9,13,14,21,22]). Recently, using B-splines [2], the authors studied smoothing splines and applied to generating cursive characters [6,16] and modeling of Dow-Jones industrial data [11].

The basic problems of spline interpolation or smoothing are to design curves or surfaces that pass through or are close to given data points. A tra- ditional way of solving such problems is to design polynomial splines with the knot points taken at the data points using some fundamental splines such as N- and B-splines, etc.(e.g. [20,21])

Another approach based on control systems is well-suited and highly desirable for such problems, and in fact there have been relatively new de- velopments, called dynamic splines (e.g. [4,23]).

Namely, it has been shown that, by considering linear continuous-time control systems, various types of spline functions can be generated and that spline

interpolation and smoothing problems can be treated

in a unified framework [5,18]. The admissible set of controls is taken as L2(0,T), and the knot points coincide with the data points.

On the other hand, by introducing a class of discrete-time systems, the problem of 'multilevel' interpolation is considered in [17]. Namely, not only the function value but also its derivatives are interpolated. This idea of discrete-time systems and multilevel problem is interesting, but the problem is limited to interpolation. The smoothing counterparts of this method is desirable especially when the data contains noise.

The smoothing splines have been considered in various papers, for example, in [18] by dynamic splines approach and in [11,7] by B-splines approach.

Moreover, these two approaches are used to develop the theory of periodic smoothing splines (see, e.g. [19]

and [12]).

Inspired by the idea of discrete-time systems in [17] and using dynamic splines approach, we here

develop a method of designing optimal smoothing splines and periodic splines but with piecewise con- stant control inputs. This restriction on controls is introduced so that the conventional polynomial splines can be incorporated into the present formu- lation. As the result, the control problem is of finite- dimensional and the computational load of optimal splines becomes very small when the underlying linear continuous-time system is taken to yield polynomial

* 原 稿 受 付2007年6月18日

Division of Science, School of Science and Engineering, Tokyo Denki University; Hatoyama, Hiki-gun, Saitama 350-0394, JAPAN

Key Words: smoothing splines, periodic splines, linear control systems, observability, controllability.

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KANO and FuJIOKA:Discrete-Time Control Systems Approach for Optimal Smoothing Splines 61

splines and the knot points are taken to be equally

spaced. This is in contrast to the usual dynamic

splines approach. On the other hand, the present

method can generate splines other than polynomial splines, e.g. exponential and trigonometric functions,

and is more flexible as the spline generator than

other methods in the sence that the knot points can arbitrarily be placed.

The continuous-time system is sampled to yield

a discrete-time system, which is time-varying when

the specified knot points are not equally spaced.

Necessary and sufficient conditions are established for the existence of unique optimal solutions to the

smoothing and periodic smoothing spline problems.

We see that the conditions are closely related to, and in fact in some special cases are exactly, the con- trollability and observability of the sampled system.

Moreover the computational procedure of optimal

splines is straightforward. The periodic splines can

be used, for example, to model contours or shapes [1]

of various objects.

The rest of this paper is organized as follows. In

Section 2., we formulate the smoothing and peri-

odic smoothing spline problems. In Section 3., we

establish necessary and sufficient conditions for the existence of unique optimal solutions. Some special cases including the case of the standard splines are treated in Section 4.. In Section 5., we apply the results on periodic splines to synthesizing the shape of jellyfish from its image data. Concluding remarks are given in Section 6..

2. Problem Statement

Let y(t), t•¸[0,T], be a polynomial spline of degree n with the knot points tk,

(1) Since y (t) is a piecewise polynomial of degree n and is (n-1)-times continuously differentiable, it can be

written as

(2)

with the continuity conditions

(3)

for k= 1,2,•cm-1. We rewrite (2) as

(4)

for t•¸ [tk,tk+i), k= 0,1,•c,m-1. Then, letting x•¸Rn

be

(5) y (t) is expressed as the output of the following

continuous-time linear system:

(6)

Here u•¸U is a piecewise constant control input with

(7)

and A •¸Rn•~n, b•¸ Rn, c•¸Rl•~n are defined by

(8)

Thus the design of spline functions is regarded as that of the control input u E U and the initial state xo E Rn .

Now suppose that we are given a set of data

(9)

where we assume si•‚si for i•‚ j. Then we consider the following two problems.

[Problem 1] (smoothing spline) Find an optimal

control u* and an optimal initial state 4, such that

where

(10)

λ>0,and ω ん>0∀k with ω1十 ω2十 … 十 ωN=1.

It is noted that, in (10), the first term governs the smoothness of spline y(t) , while the second term describes the approximation error from the given data. The so-called smoothing parameter A is used to control the tradeoff between the two terms.

[Problem 2] (periodic smoothing spline) Find an optimal control u* and an optimal initial state x'(;' such that

subject to the constraint

(11) for J(u,xo) in (10).

In the next section, we derive the optimal solu- tions, where we do not restrict the matrices A,b,c to be of the forms in (8). Moreover note that, for given data set 7) in (9), we can arbitrarily set the knot points, i.e. the number m as well as the location tk in (1). Of course, one way is to choose the knot points tk coinciding with the data points sk. Then, in Section 4., the three cases where the knot points tk are equally-spaced, the data points sk are equally-spaced, and A,b,c are in the forms in (8) are treated as the special cases.

In order to solve these problems, it is conve- nient to introduce a discrete-time system obtained by sampling (6) at the knot points. Namely, with

(3)

xk = x(tk), yk = y(tk) and the sampling interval (12)

we consider the following system,

(13)

where

(14)

(15)

Once the state Xk and the input uk are determined for k = 0,1,•c,m- 1, the spline is obtained in each interval [tk,tk+i) by

(16) where g (t) is defined by

(17)

In actual numerical computations, we evaluate x(t)

and y(t) for t E (tk,tk+i) at discrete points, say tk ,i = tk +i•¸iE, i = 1,2,•c for some small•¸ > 0, by the following relation

(18)

where

(19) In this way the computations of exponential and its integral values in (16) for each t may be avoided.

3. Optimal Solutions

We study optimal solutions for Problems 1 and 2 in Sections 3.1 and 3.2 respectively.

3.1 Smoothing Splines

First we solve Problem 1. In the cost function Au,x0) in (10), the integral term is expressed as

where u•¸ Rm and H E Rm'm are defined by

(20) (21) The second term in (10) is written as

where

(22)

Here y(sk) is obtained from (6) as

By introducing ft (T):

(23) y(sk) is expressed as

(24)

where dk E Rrn is defined by

(25)

Thus 9 is obtained as

(26)

where the matrices C E RN" and D E RNxm are defined by

(27)

(28) Thus the cost function J(u,x0) is expressed in terms of ft and xo as

(29)

Taking the derivatives with respect to ft and x0 and setting zero yield the following set of algebraic equations.

(30)

The optimal solution of Problem 1, if it exists, is obtained as the solution of (30).

[Theorem 1] Problem 1 has a unique optimal

solution u* and x0* if and only if rank C = n.

(Proof) Denoting the coefficient matrix in (30) by A •¸ R(m+n)•~(m+n) , it may be expressed as

(31)

(4)

KANO and FuJIOKA:Discrete-Time Control Systems Approach for Optimal Smoothing Splines 63

andhence.A≧0.Moreover, suppose that. Aa=O fOr

avectora∈Rm+π,and let a=[aT1aT2]T with a1∈

Rm,a2∈Rn.Then we get

(λH1十DTWD)a1+DTWCa2=0 oTWDa1+CTWCa2=0.

Manipulating these equations, we obtain

(32) Thus a1 = 0 since H > 0, and hence Ca2 = 0 since W > 0. It is then obvious that rank C= n is necessary and sufficient for a2 = 0 or regularity of A. In other words, J(u,x0) is strictly convex if and only if rank

C = n holds. (QED)

[Remark 1] For any matrices S = ST > 0, U and vector v of compatible dimensions, it holds that rank (S -HUTU, UT v)= rank(S UTU). Hence, in view of (30) and (31), we easily see that (30) is always consistent. Thus if rank C = n does not hold, there are infinitely many optimal solutions for Problem 1. Then the Moore-Penrose matrix inverse may be employed yielding the minimum-norm solution of (30).

3.2 Periodic Smoothing Splines

Next we consider Problem 2 with the constraint (11), i.e.

(33) From (13), xm is obtained as

(34)

where the matrix G •¸Rn•~m is given by

(35) Now, using the cost function in (29), we form the following Lagrangian

(36)

where p, E Rn is the Lagrangian multiplier. Taking the derivatives with respect to u, x0 and f/ yields

(37)

Thus the optimal solution is obtained as the solution of this algebraic equation.

We now have the following theorem.

[Theorem 2] Problem 2 has a unique optimal solution u* and x0* if and only if rank C = n and rank G = n .

(Proof) Denoting the coefficient matrix of (37) by

A, we first examine its regularity. Letting

Aa=0,a∈Rm+2n,

and partitioning a into three vectors al E Rrn , a2,a3 e Rn compatibly with A yield

(38) (39) (40) Premultiplying arT to (38), premultiplying arl to (39), and using (40), we obtain

λ││a1││2H+││Da1+Ca2││2W=0,

the same equation as (32). Thus we get al = 0, Ca2 = 0, and further GTa3 = 0 by (38): We then see that (37) has a unique solution, i.e. Aa= 0 implies a = 0, if and only if rank C = n and rank G = n. Moreover, under these rank conditions, the cost function J(u,x0) in (36) is strictly convex in ii, and xo as shown in the proof of Theorem 1. Thus the solution of (37) provides the optimal solution to Problem 2. (QED) [Remark 2] The condition rank G = n is the con- trollability condition

(41)

Finally, in this subsection, we examine in details the structure of the matrix G in (35). Denoting its i-th column by Gi, i = 1,2,•c,m, we obtain

(42) Moreover, by changing the integration variable and using (17), we get

Thus Gi is written as

(43)

where ƒÂi is defined by

(44) In particular, 6m = 0 and hence Gm = g(6,1)

3.3 Computational Procedures

For convenience, we summarize the procedure for computing the optimal splines.

(5)

[Procedure 1] (Optimal smoothing splines) (SO) Input data: knot points tk,k = 0,1,•c,m, data

D, and weights wk,k = 1,2,••cN.

(Si) Set up H ER' in (21), W CRNxN and a e

RN in (22), and compute C E RN" in (27) and

D ERN' in (28).

(S2) Set up and solve the algebraic equation (30) for is and x0.

(S3) Compute xk E Rn and Yk E R for k = 0,1,••c On by (13).

(S4) For each interval (tk,tk+1) (k = 0,1,••c,m-1), compute x(t) E Rn by (18) as x(tk

,i) (= x(tk + i)) for small 6 > 0 and i =1,2,•c, and then the optimal spline y(t) E R as y(tk

,i) = ex (t k ,i) The case of periodic splines follows similarly.

[Remark 3] In the foregoing development, we as-

sumed that the smoothing parameter A is pre-

assigned. For an optimal choice of A, we may employ the so-called cross validation method (e.g. [21]): Let Di be the data set obtained from V in (9) by deleting the l-th data, i.e.

(45) and let y).,i(t) be the splines constituted from the above optimal solutions for the data Di. Then an optimal A is obtained as the minimizer of the following cross validation function,

(46)

4. Some Special Cases

The results established in the previous section are examined in details for the following three special cases.

4.1 Equally-Spaced Knot Points

We consider the case where the knot points tk are equally spaced, namely

(47) Then, the sampled system in (13) is time-invariant, and is given by

(48)

where ƒÓ and g are given from (14) and (15) as

(49) In this case, the matrix G in (35) is written as

(50) Thus the condition rank G=n is the same as the controllability of the pair (4P,g), and Theorem 2 can be restated as follows.

[Corollary 1] Assume that the knot points tk are equally spaced. Then, Problem 2 has a unique

optimal solution u* and x*o(; if and only if rank C = n and the pair (0,g) is controllable.

It is desirable if we could re-state the controllabil- ity condition in terms of that for the original system, namely the continuous-time system in (6). Regarding the relation of controllability between a continuous-

time system and the sampled discrete-time system,

the following result holds (see e.g. [3], Theorem

C-2), where •¬[•E] and •¬[•E] respectively denotes real and imaginary part of complex numbers.

[Lemma 1] Assume that the pair (A,b) in (6) is

controllable. Then the pair (0,g) in (48) is con-

trollable if and only if a[Ai (A)-Aj (A)] 27k/it for

k=•}1,•}2,•c, whenever R[Ai (A)-Aj (A)] = O.

Thus, unless the matrix A has complex conjugate eigenvalues Fy + j 6 satisfying 6 = irk/h for some k=

+1,•}2,•c, the controllability condition of (0,g) in

Corollary 1 can be replaced with the controllability of (A,b).

4.2 Equally-Spaced Data Points

Let us examine the case where the data points sk in (9) are equally spaced as

(51) Then the matrix C in (27) is written as

(52)

where

(53)

and

(54)

It is obvious that, when N > n, rank C = rank CO,N-1= rank CO,n-1. Thus the condition rank C= n in Theorems 1 and 2 is the observability of the pair

(c,ψ):

[Corollary 2] Assume that the data points sk are equally spaced. Then, Problem 1 has a unique

optimal solution u* and 4, if and only if the pair (c,ĵ) is observable. Problem 2 has a unique optimal solution if and only if the pair (c, ĵ) is observable and rank G = n.

Again, using Lemma 1 and the duality, the

observability of the pair (c,W) is linked to that of (c, A) for continuous-time system (6): The observability condition of the pair (c,W) is equivalent to that of (c, A) unless the matrix A has complex conjugate

eigenvalues ƒÁ j6 satisfying 6 =ƒÎk/i3 for some k =

±1,±2,….

4.3 The Case of A,b,c given in (8) We consider the case where A, b, and c in (6) are given by (8). This case corresponds to the standard

(6)

KANO and FuJIOKA:Discrete-Time Control Systems Approach for Optimal Smoothing Splines 65

polynomial splines.

It is then possible to examine in details the existence conditions in Theorems 1 and 2, and moreover we can derive more explicit expressions for the matrices and vectors arising in the optimal solutions.

First of all, eAt becomes

(55)

and g (t) in (17) is computed as

(56) We now show that the condition rank C=n always holds when N > n. In (27), we get

(57)

Thus, denoting the matrix consisting of the first n rows of matrix C by C, we see that

(58)

where S = diag11/1!1/2!1/(

n-1)1 Noting si si for i j, the result on Vandermonde matrix shows that the matrix C is nonsingular, and hence rank C = n always holds.

Next we consider the matrix G in (35). Since its i-th column Gi is given by (43) with Gm= g(6m-i),

it holds that

(59)

Now, assuming that m > n, let us consider the square sub-matrix G E R" n defined as

(60) Then using the function g (t) in (56), G may be

written as

where S= diag{1/

n!1/(n-1)!1/1!}and

△= diag{δ0δ1… δn-1}.Sinceδi≠ δj fori≠j,we

see that the matrix G is nonsingular, and hence rank G = n always holds.

Thus, Theorems 1 and 2 may be written as follows.

[Corollary 3] Assume that A,b,c are given by (8).

The condition N > n is necessary and sufficient for Problem 1 to have a unique optimal solution u* and

x*0. On the other hand, the conditions N > n and

m•† n are necessary and sufficient for Problem 2 to possess a unique optimal solution.

Finally, in this subsection, we present the matrix D in (28) in a more explicit form. Consider the vector dk in (25), and let j (0 < j < m) be such that ti <sk<

tj•}1 . Then dk is of the following form

(61)

Here, dk,i is computed as

(62)

for i=1,2,•c,j, and dk,i+i is obtained by

(63) 5. Experimental Results

We apply the above method for designing optimal periodic smoothing splines to a contour synthesis problem. In particular, we model the shape of a jellyfish from its image data'.

The data D in (9) is obtained as follows. First the given image is converted to a binary image by using the Otsu's binarization method (see e.g.[8]) and the centroid (i.e. the center of mass) of the object (i.e.

the jellyfish) is computed. Fixing an O-pq plane with its origin 0 at the centroid, the distance a (in pixels) is computed from the origin to the boundary pixel by increasing an angle s by 6s = 1 (degree).

In the following design example, we selected the data consisting of 36 data points (i.e. N = 36) which correspond to the measurement at every 10 degrees.

Thus the data points are equally spaced. The weights wk are set as wk =1/N Vk. We used (A,b,c) of the form in (8) with n = 3 and equally spaced knot points with tk+1-tk=h= 10. Thus Corollary 3 guarantees

1 Educational Image Collections, Information- technology Promotion Agency (IPA), Japan.

http://www2.edu.ipa.go.jp/gz/

(7)

the existence of unique optimal periodic solution.

In Fig. 1, we show the converted binary image to- gether with the centroid (`+' mark) and the measured 36 data points (`4,' mark). Fig. 2 shows the results of designed optimal periodic spline y(t) (solid line) together with the data points (`*' mark). Dashed and dotted lines show y(l) (t) and y(2) (t) respectively.

Note that, in the present case of (A,b,c) and n = 3, the third derivative y(3) (t) coincides with the control input u(t), and will be shown later. In Fig. 3, the value of the cross validation function V(A) in (46) is

plotted in the interval [10-0.5,104.5]. We confirmed that, outside of this interval, V(A) increases further, and very fast beyond 104.5 in particular. The optimal value of the smoothing parameter was found to be

λ*= 398.1072. Fig. 4 shows the reconstructed contour in the 0 pq plane, and the optimal control input u*(t) is plotted in Fig. 5. The original image is overlaid with the constructed contour in Fig. 6. From these figures, we may observe that we have recovered fairly good contour by the present model.

Fig. 1 Converted binary image with the centroid and the measured 36 data points

Fig. 2 Optimal periodic spline y (t) and its derivatives

Fig. 3 Cross validation function

Fig. 4 Constructed contour

Fig. 5 Optimal control input

Fig. 6 Original image and the constructed contour

(8)

KANO and FunoKA:Discrete-Time Control Systems Approach for Optimal Smoothing Splines 67

6. Concluding Remarks

We considered the problem of designing optimal smoothing spline curves by employing an approach based on linear control systems. The problem is formulated using continuous-time, time-invariant sys- tems with piecewise constant inputs and the sampled discrete (in general, time-varying) systems. We es- tablished necessary and sufficient conditions for the existence of unique optimal smoothing and periodic smoothing splines, and the computational procedures for the optimal splines are straightforward. The de- sign method for periodic splines is applied successfully to synthesizing the shape of jellyfish from its image data.

By this approach, the existence conditions of optimal splines are shown to be linked to the concepts of controllability and observability of the sampled sys- tem. Moreover, we believe that the present approach will provide new insight and systematic treatments for various types of spline problems, e.g. splines for noisy measurement data in a stochastic systems context, multi-level interpolation and smoothing problems, two-dimensional splines for surface design as studied by distributed parameter systems [15] , etc.

Acknowledgments

The first author was supported in part by Japan Ministry of Education, Culture, Sports, Science and Technology under Grant No.19560455 and the COE HAM project, and the second author in part under Grant No.18700186.

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References

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