SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA. Hiroyuki Fujioka and Hiroyuki Kano. Magnus Egerstedt. Clyde F. Martin

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1 Internationa Journa of Innovative Computing, Information and Contro ICIC Internationa c 5 ISSN Voume x, Number x, x 5 pp. SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA Hiroyuki Fujioka and Hiroyuki Kano Department of Information Sciences Tokyo Denki University Hatoyama, Hiki-gun, Saitama 5-94, Japan fujioka@j.dendai.ac.jp; kano@j.dendai.ac.jp Magnus Egerstedt Schoo of Eectrica and Computer Engineering Georgia Institute of Technoogy Atanta, GA, USA magnus@ece.gatech.edu Cyde F. Martin Department of Mathematics and Statistics Texas Tech University Lubbock, TX 7949, USA martin@math.ttu.edu Abstract. We consider the probem of designing optima smoothing spine curves and surfaces for a given set of discrete data. For constructing curves and surfaces, we empoy normaized uniform B-spines as the basis functions. First we derive concise expressions for the optima soutions in the form which can be used easiy for numerica computations as we as mathematica anayses. Then, assuming that a set of data in a pane is obtained by samping some curve with or without noises, we prove that, under certain condition, optima smoothing spines converge to some imiting curve as the number of data increases. Such a imiting curve is obtained as a functiona of given curve to be samped. The case of surfaces is treated in parae, and it is shown that the resuts for the case of curves can be extended to the case of surfaces in a straightforward manner. Keywords: B-spines, optima smoothing spines, asymptotic anaysis, statistica anaysis. Introduction. The probem of optima design of approximating or interpoating curves and surfaces for a given set of data arises in various fieds of engineering and sciences. In particuar, spine functions have been used frequenty in such fieds as computer aided design [], numerica anaysis [], image processing [], trajectory panning probems [4, 5], and data anaysis in genera [6]. Recenty, the theory of smoothing spines is used to generate cursive characters based on an idea that the underying writing motions become smooth [7, 8]. Thus spines have been studied extensivey e.g. [6]), and in particuar, an approach based on optima contro theory has been empoyed for generating piecewise poynomia spines as the soutions to a number of different optima contro probems [9, ]. Moreover,

2 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN the theory of dynamic spines provides a unified framework for generating various types of spine curves as outputs of inear dynamica systems e.g. [,, ]. Aso, the authors studied B-spine functions from the viewpoint of optima contro theory [4]. In the probem of designing surfaces by spines [5, 6], the surfaces, in most cases, are constructed by resorting to some numerica optimization techniques. Aternative approaches to this probem incude the deveopment of appropriate basis functions, e.g. waveets, and active boundary contro of partia differentia equations. When we are given a set of data corrupted by noises, smoothing spines are expected to yied more feasibe soutions than interpoating spines. A theoretica issue in this regard is asymptotic and statistica anayses of designed spines when the number of data increases. Such a probem is studied in [7] in dynamica systems settings, namey for spine curves generated as an output of inear dynamica systems. On the other hand, using B-spines [8] as basis functions is expected to yied simpe agorithms for designing curves and surfaces. In particuar, using normaized uniform B- spines as the basis functions, we have been studying the probems of constructing optima interpoating and approximating curves [9]. This approach enabed us to investigate how the introduction of approximation, i.e. of east-square terms in the cost function, affects the shape of the curve. Moreover, we coud derive numericay tractabe agorithms, and this approach makes it easier to extend the resuts for one-dimensiona case i.e. the case of curves) to two-dimensiona case i.e. the case of surfaces) and to even higher dimensions. The purpose of this paper is to design optima smoothing spine curves and surfaces, and anayze their properties using B-spines as the basis functions. Assuming that a number of data is given by samping some curve ft) with noises, we anayze statistica properties of optima smoothing spines and derive an expression of the spines as a functiona of ft) when the number tends to infinity. Such a design and anaysis method is extended to the case of surfaces. We wi see that the expressions for optima curves and surfaces are concise, enabing us to anayze their properties easier. Moreover, extensions of resuts for curves to the two-dimensiona case, namey the surfaces are straightforward. For designing curves xt), we empoy normaized, uniform B-spine function B k t) of degree k as the basis functions, xt) = m i= k τ i B k αt t i )). ) Here, m is an integer, τ i R is a weighting coefficient caed contro point, and α> ) is a constant for scaing the interva between equay-spaced knot points t i with t i+ t i = α. ) Then xt) formed in ) is a spine of degree k with the knot points t i. In particuar, by an appropriate choice of τ i s, arbitrary spine of degree k can be designed in the interva [t, t m ]. On the other hand, surfaces are generated as xs, t) = m m i= k j= k τ i,j B k αs s i ))B k βt t j )). )

3 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA where α, β> ) are constants, m, m > ) are integers, and s i s, t j s are equay spaced knot points with s i+ s i = α, t i+ t i = β. 4) This paper is organized as foows: In Section, we present normaized uniform B- spine functions that are used throughout the paper. In Section, the probem of designing optima smoothing curves and surfaces are formuated and optima soutions are presented. We anayze asymptotic and statistica properties of optima curves and surfaces in Section 4, and the resuts are estabished as theorems. The resuts of numerica simuation studies are presented in Section 5, and concuding remarks are given in Section 6. Here we summarize some of the symbos that wi be used throughout the paper: denotes the Lapacian operator, and the Kronecker product. Moreover, vec denotes the vec-function, i.e. for a matrix A = [a a a n ] R m n with a i R m, vec A = [ ] a T a T a T T n R m n see e.g. []).. Normaized Uniform B-Spines. The normaized, uniform B-spine function of degree k, denoted as B k t), is defined by N k,k t) t < N k,k t ) t < B k t) = N,k t k) k t < k + t < or t k +. Here the basis eements N j,k t) j =,,, k) are obtained recursivey by the foowing agorithm []. Let N, t) and, for i =,,, k, compute N,i t) = t N i,i t) N j,i t) = i j+t N k j,i t) + +j t N i j,i t), j =,, i N i,i t) = t i N i,i t). Thus, B k t) is a piece-wise poynomia of degree k with integer knot points and is k times continuousy differentiabe. It is noted that B k t) for k =,,, is normaized in the foowing sense k N j,k t) =, t, 7) and this yieds j= B k t)dt = k+ 5) 6) B k t)dt =. 8) For the sake of ater reference, we show the function B t) and its derivatives in the foowing tabe.

4 4 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN Tabe. Cubic B-spine B t) and its derivatives. B t) B ) t) B ) t) B ) t) t < 6 t t t t < 6 t + t t + 4) t + 8t 4) t + 4 t < 6 t 4t + 6t 44) t 6t + ) t 8 t < 4 6 t) 4 t) t + 4 t <, t 4. Resuts on Standard Smoothing Spine Probems. In this section, we present basic resuts on smoothing spine probems. Since cubic spines are most frequenty used for practica purposes, we restrict ourseves to the case of k = in the seque... Smoothing Spine Curves. Equation ) in the case of k = is written as xt) = Suppose that we are given a set of data m i= τ i B αt t i )). 9) D = {u i ; d i ) : u i [t, t m ], d i R, i =,, N}, ) and et τ R M M = m + ) be the weight vector defined by τ = [ τ τ τ m ] T. ) Then, a basic probem of optima smoothing spine curves is to find a curve xt), or equivaenty a vector τ R M, minimizing a cost function, Jτ) = λ x ) t) ) N dt + w i xu i ) d i ), ) I where λ>) is a smoothing parameter, w i w i ) are weights for approximation errors, and the integration interva I is taken as either I =, + ) or I = t, t m ). This probem can be soved as foows. In order to express the right hand side of ) in terms of τ, we introduce the foowing notations: Let a vector bt) R M be i= bt) = [ B αt t )) B αt t )) B αt t m )) ] T, ) and a matrix B R M N be B = [ bu ) bu ) bu N ) ]. 4) Then, noting that xt) is expressed as xt) = τ T bt), we can show that the cost function is written as Jτ) = λτ T Qτ + B T τ d) T W B T τ d). 5)

5 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA 5 Here, Q R M M is a Gramian defined by d bt) d b T t) Q = dt, 6) dt dt and I W = diag{w, w., w N }, 7) d = [ d d d N ] T. 8) We then see that optima weight τ minimizing Jτ) in 5) is obtained as a soution of λq + BW B T )τ = BW d. 9) Note that this equation has at east one soution, since in genera the reation ranks + UU T ) = rank[s + UU T, Uv] ) hods for any matrices S = S T, U and vector v of compatibe dimensions. Obviousy, the soution of 9) is unique if and ony if λq + BW B T >. By changing the integration variabe in 6), the Gramian Q can be transformed as Q = α R. ) Here R R M M is a Gramian defined by ˆb) T R = ˆb) t) t)) dt, ) Î where Î =, + ) if I =, + ) and Î =, m) if I = t, t m ), and ˆbt) = [ B t )) B t )) B t m )) ] T. ) Then we can show that the foowing properties hod see Appendix A. for the proof). Lemma.. The Gramian Q in 6) is nonsinguar if I =, + ), and singuar if I = t, t m ). Moreover, using the function B ) t) in Tabe, we can compute the integra in ). Denoting the matrix R for Î =, + ) by R, we obtain R = 6 and R F for Î =, m), is given by R F = m ) , 4) ˆb) t) ˆb) t)) T dt = R R + R + ), 5)

6 6 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN where and R = R + = + m b ) t) b ) t) ) T dt = 6 b ) t) b ) t) ) T dt = ,M M, M,M M,M M,,M , 6). 7) Remark.. If I =, + ), then λq + BW B T is nonsinguar by Lemma., and thus 9) has a unique soution. On the other hand, when I = t, t m ), the matrix λq + BW B T may be singuar. In such a case, we empoy the so-caed minimum norm soution for 9), which is guaranteed to be unique... Smoothing Spine Surfaces. Equation ) in the case of k = is written as xs, t) = m m i= j= τ i,j B αs s i ))B βt t j )). 8) Then, choosing appropriate weighting coefficient τ i,j caed contro point, xs, t) can represent arbitrary spine surface on the rectanguar domain Now suppose that a set of spatia data S = [s, s m ] [t, t m ] R. 9) D = {u i, v j ; d ij ) : u i, v j ) S, d ij R, i =,,, N, j =,,, N } is given. Letting τ = [τ i,j ] R M M be the weight matrix with M = m + and M = m +, we consider the foowing cost function for a design of smoothing surface: Jτ) = λ xs, t) ) N N dsdt + w ij xu i, v j ) d ij ), ) I I where I, I are some intervas in, + ), λ> ) is a smoothing parameter, and w ij w ij ) denotes weights for approximation errors. Minimization of this cost function proceeds simiary as in the previous case. In order to express the right hand side of ) in terms of τ, we introduce the foowing notations. Let b t) R M and b t) R M be i= b s) = [B αs s )) B αs s )) B αs s m ))] T ) b t) = [B βt t )) B βt t )) B βt t m ))] T. ) Then with ˆτ = vec τ R M M ), xs, t) in 8) can be written as j= and the cost function in ) is obtained in terms of ˆτ as xs, t) = b t) b s)) T ˆτ, ) Jˆτ) = λˆτ T Qˆτ + B B ) T ˆτ d ) T W B B ) T ˆτ d ). 4)

7 Here, Q R M M M M SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA 7 Q = Q ) Q ) + Q ) where Q ij) R M M are given by Q ij) = is a Gramian defined by ) T + d i b t) I dt i Note that Q = Q T since ˆτ T Qˆτ = I Q ) Q ) ) T Q ) + Q ) Q ), 5) d j b T t) dt j dt, =,, ; i, j =,,. 6) I xs, t) ) dsdt ˆτ. Moreover, matrices B R M N and B R M N are defined by and W R N N N N and d R N N by B = [ b u ) b u ) b u N ) ], B = [ b v ) b v ) b v N ) ], 7) W = diag{w, w,, w N,, w N, w N,, w N N } d = [ d, d,, d N,, d N, d N,, d N N ] T. 8) Then the optima weight ˆτ minimizing the cost function in 4) is obtained as a soution of λq + B B )W B B ) ) T ˆτ = B B )W d. 9) Note that, by ), this equation is consistent and aways has at east one soution. For the sake of anayses and numerica computations of Q matrix, it is convenient to introduce the matrices R ij) R M M defined by R ij) = ˆbi) Î t) ˆbj) t)) T dt, =,, ; i, j =,,, 4) where Î =, + ) if I =, + ) for =,, and Î =, m ), Î =, m ) if I = s, s m ), I = t, t m ). More over the vector ˆb t) R M is defined by ˆb t) = [ B t )) B t )) B t m )) ] T. 4) Note that the matrices R ij) 6) and are reated to Q ij) Thus the matrix Q is rewritten in terms of R ij) Q = α β R) R ) + αβr ) in 4) are obtained by changing the integration variabe in as Q ij) = α i+j R ij), Q ij) = β i+j R ij). 4) R ) as ) T + αβ R ) ) T ) R + β α R) R ). 4) Regarding the Gramian Q R M M M M in 5), the foowing properties hod see Appendix A. for the proof). Lemma.. The matrix Q in 5) is nonsinguar if I = I =, + ), and singuar if I = s, s m ) and I = t, t m ).

8 8 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN The matrix Q can be formed easiy as in the case of Section., and is shown in Appendix B. Moreover, by Lemma., there aways exists a unique soution of 9) when I = I =, + ), whereas such a uniqueness is not guaranteed in the case of I = s, s m ) and I = t, t m ). 4. Smoothing Spine Curves for Samped Data. In this section, we anayze various properties of smoothing spine curves derived in Section.. In particuar, we anayze asymptotic and statistica properties assuming that the data is obtained by samping some curve with or without noises. Let the data d i in ) be obtained by samping a curve ft) which is assumed to be continuous in the interva [t, t m ]. In order to anayze asymptotic properties of spine curves as the number of data points N increases, we consider the foowing cost function instead of ), J N τ) = λ x ) t) ) dt + I N N xu i ) fu i )). 44) When the data d i is obtained by samping the function ft) with additive noises we consider the foowing cost function JNτ) ɛ = λ x ) t) ) dt + N I i= d i = fu i ) + ɛ i, i =,,, N, 45) N xu i ) fu i ) ɛ i ). 46) We assume that the noises are zero-mean and white, namey E{ɛ i } = and E{ɛ i ɛ j } = σ δ ij for a i, j. Moreover, in order to anayze the asymptotic properties, we introduce the cost function: J c τ) = λ x ) t) ) tm dt + xt) ft)) dt. 47) I t The soutions that minimize the cost functions J N τ), JN ɛ τ) and J cτ) are obtained as foows. The first two cases foow directy from the resut in the previous section: The soution τ N minimizing J N τ) is obtained as a soution of λq + N ) BBT τ N = Bf, 48) N where Q and B are given in 6) and 4) respectivey, and f = [fu ) fu ) fu N )] T. Obviousy, τn ɛ minimizing J N ɛ τ) is a soution of λq + N ) BBT τn ɛ = Bf + ɛ), 49) N where ɛ = [ ɛ ɛ ɛ N ] T. On the other hand, Jc τ) can be written as i= where tm J c τ) = τ T λq + Q ) )τ τ T bt)ft)dt + t Q ) = tm tm t f t)dt, 5) t bt)b T t)dt. 5)

9 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA 9 Thus optima τ denoted by τ c is obtained as a soution of λq + Q ) )τ c = tm t bt)ft)dt. 5) By Appendix A., we see that Q ) = Q ) ) T >, hence optima τ c exists uniquey. Moreover, Q ) can be obtained expicity as in the case of R F in 5). Convergence properties are estabished under the foowing assumption. A) The sampe points u i, i =,,, N, are such that im N N N gu i ) = i= for every continuous function gt) in [t, t m ]. tm We now have the foowing resuts for the case I =, + ). t gt)dt 5) Theorem 4.. Assume that the integration interva is I =, + ) and that the condition A) hods. Then, i) The optima soutions τ N, τ ɛ N and τ c exist uniquey. ii) τ N converges to τ c as N. iii) E{τ ɛ N } = τ N and τ ɛ N converges to τ c as N in the mean squares sense. Proof: i) As noted in the previous section, the Gramian Q in the case of I =, + ), namey Q, is positive-definite, and hence equations 48), 49) and 5) have unique soutions. ii) In 48) and 5), we show that im N im N Regarding the first assertion, 4) and ) yied N BBT = Q ) 54) tm N Bf = BB T = t bt)ft)dt. 55) N bu i )b T u i ), and denoting BB T = [c jk ] m j,k=, we get N c jk = B αu i t j ))B αu i t k )). i= i= Then noting that the function g jk t) given by g jk t) = B αt t j ))B αt t k )) is continuous in [t, t m ], and using the assumption A), it hods that im N N c jk = tm t g jk t)dt.

10 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN Thus im N [ ] m tm tm N BBT = im N N c jk = [g jk t)] m j,k= dt = bt)b T t)dt = Q ). j,k= t t The second assertion 55) foows simiary by noting that ft) is assumed to be continuous in [t, t m ]. iii) Taking expectations of both sides of 49) and noting E{ɛ} =, we get λq + N ) BBT E{τN} ɛ = Bf, 56) N and we see that E{τ ɛ N } = τ N hods. On the other hand, 49) and 56) yied λq + N BBT ) τ ɛ N E{τ ɛ N}) = N Bɛ. Letting P N = E{τN ɛ E{τN}) ɛ τn ɛ E{τN}) ɛ T }, and using E{ɛɛ T } = σ I N, we have λq + N ) BBT P N λq + N ) T ) BBT = σ N N BBT. Noting 54), this equation in the imit of N reduces to λq + Q ) )P λq + Q ) ) =, and P =. Since E{τ ɛ N } = τ N, we see that τ ɛ N converges to τ c in mean squares sense. Q.E.D.) On the other hand, when I = t, t m ), the matrix Q is singuar. Thus it is possibe that the coefficient matrix λq + N BBT in 48) and 49) becomes singuar depending on the matrix B, i.e. on the data points u i, i =,, N. In such a case, we empoy the minimum norm soution as before, and the foowing coroary hods. Coroary 4.. Assume that integration interva I is I = t, t m ) and that the condition A) hods. Then, the same assertions hod as in Theorem 4. with the understanding that we empoy the minimum norm soutions for equation 48). 5. Smoothing Spine Surfaces for Samped Data. We assume that the data d ij in ) for constructing smoothing surfaces is obtained by samping a function fs, t) which is assumed to be continuous in both variabes in [s, s m ] [t, t m ]. Corresponding to the cost function Jτ) in ), we consider the foowing cost functions, J N,N τ) = λ and JN ɛ,n τ) = λ I I I xs, t) ) dsdt + N N I xs, t) ) dsdt + N N N N i= N N i= xu i, v j ) fu i, v j )), 57) j= xu i, v j ) fu i, v j ) ɛ ij ). 58) In 58), the noises ɛ ij are assumed to be zero-mean and white, i.e. E{ɛ ij } = i, j, E{ɛ ij ɛ k } = σ for i = j = k =, and E{ɛ ij ɛ k } = otherwise. Moreover, in order to j=

11 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA anayze convergence properties of soutions as N, N, we introduce the foowing cost function. J c τ) = λ xs, t) ) sm tn dsdt + xs, t) fs, t)) dsdt. 59) I t I Using the resuts estabished in Section., we easiy see that the optima soutions τ N,N and τn ɛ,n minimizing the cost functions in 57) and 58) are obtained respectivey as soutions of A N,N ˆτ N,N = B B )f, 6) N N and A N,N ˆτ N ɛ,n = B B )f + ɛ), 6) N N where A N,N R M M M M is given by s A N,N = λq + N N B B )B B ) T, 6) and the vectors f R N N and ɛ R N N are defined by f = [ fu, v ), fu, v ),, fu N, v ),, fu, v N ), fu, v N ),, fu N, v N )] T, ɛ = [ ɛ, ɛ,, ɛ N,, ɛ N, ɛ N,, ɛ N N ] T. On the other hand, we can show that the optima soution for the cost function in 59) is obtained as the soution of ) λq + R ) R ) ˆτ c = φ, 6) where φ R M M is defined by φ = Now we introduce the foowing assumption: I I fs, t) b t) b s)) dsdt. 64) A) The sampe points u i, v j ), i =,, N, j =,, N, are such that N N sm tm im gu i, v j ) = gs, t)dsdt N,N N N i= for every continuous function gs, t) in [s, s m ] [t, t m ]. j= Then we obtain the foowing resut. Theorem 5.. Assume that integration intervas I, I are I = I =, + ) and that the condition A) hods. Then, i) The optima soutions τ N,N, τn ɛ,n and τ c exist uniquey. ii) τ N,N converges to τ c as N, N. iii) E{τN ɛ,n } = τ N,N and τn ɛ,n converges to τ c as N, N in the mean squares sense. s t

12 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN Proof: i) As noted in the previous section, the matrix Q in the case of I = I =, + ) is positive-definite, and hence equations 6), 6) and 6) have unique soutions. ii) In view of 6) and 6), it suffices to show that In 65), it hods that Then, using 54), we get which proves 65). Now etting we get im N,N im N,N B B )B B ) T = R ) R ) N N 65) B B )f = φ. N N 66) N N B B )B B ) T = ) ) B B T B B T. N N im B i Bi T = R ) i, i =,, N i N i f j = [ fu, v j ) fu, v j ) fu N, v j ) ] T, j =,,, N, B B )f = = = N j= N j= i= N b v j ) B )f j = b v j ) B f j ) j= N ) b v j ) fu i, v j )b u i ) N N Using Assumption A), it hods that i= fu i, v j ) b v j ) b u i )) j= N N im [B B )f] N,N N N k = im fu i, v j ) [b v j ) b u i )] N,N N N k i= j= = fs, t) [b t) b s)] k dsdt = [φ] k, I I where [ ] k k =,,, M M ) denotes the k-th eements of respective vectors. Thus the reation 66) hods. iii) Taking expectations of both sides of 6) and by noting E{f + ɛ} = f, we get A N,N E {ˆτ ɛ N,N } = N N B B )f, 67) and the first assertion foows by the uniqueness of soutions of 6) and 67).

13 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA Regarding the second assertion, we first get the foowing reation from 6) and 66): where and In 68), it hods that A N,N P N,N A T N,N = σ GG T, 68) { P N,N = E ˆτ ɛ N,N E {ˆτ }) N ɛ ɛ,n ˆτ N,N E {ˆτ }) } N ɛ T,N, 69) im N,N GGT = by 65), and moreover im N,N G = N N B B ). 7) N N N N B B )B B ) T =, im A N,N = λq + R ) R ), N,N which is positive-definite. Thus we concude that im N,N P N,N =. Q.E.D.) In the case where the integration intervas are I = s, s m ) and I = t, t m ), the coefficient matrix A N,N in 6) and 6) may be singuar since the matrix Q is singuar, in which case we use the minimum norm soution. Note that 6) aways has a unique soution since the matrix R ) R ) is positive-definite by Appendix A.. Thus we have the foowing coroary. Coroary 5.. Assume that integration intervas I, I are I = s, s m ) and I = t, t m ) and that the condition A) hods. Then, the same assertions hod as in Theorem 5. with the understanding that we empoy the minimum norm soutions for equations 6) and 6). 6. Simuation Studies. We construct optima smoothing curves and surfaces for the case k =. The discrete data are generated by samping the foowing function. fs, t) = + 5exp{ s 5)t 5) /} + sins) cost). First we consider the case of curves where we set α = and m =, t =, t m =. We generated noisy discrete data i.e. N = ) using the function f, t) i.e. we fix s = in fs, t)) at equay-spaced samping points with σ =.. The resuts are shown in Figure. We see that, as the smoothing parameter λ decreases, the imiting curve x c t) as we as the optima smoothing curve x ɛ N t) approximate more cosey the origina function f, t). Next we construct optima smoothing surfaces for the case α = β =, s = t =, m = n = and thus s m = t n =. The surface data is generated by samping the function fs, t) with the noise magnitude σ =.5. As we see from Figure, the surface x ɛ N,N s, t) approximates the origina surface fs, t) more accuratey as the number of data N, N increases, and eventuay converges to the surface x c s, t).

14 4 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN "!#%$& ' "!*) +#,!*-./ ε,!#-.,!*- "!#%$& ' "!*) +#,!*-./ ε,!#-.,!*- ε ε a) λ =.5 b) λ =.5 Figure. The curves ft) red ine), x c t) green ine) and x ɛ N t) back ine) with N = and σ =.. 7. Concuding Remarks. We considered the probem of designing optima smoothing spine curves and surfaces for a given set of discrete data. By empoying normaized uniform B-spines as the basis functions, we derived concise expressions for the optima curves and surfaces, and their numerica computation procedures are straightforward. Then, assuming that a set of data in a pane is obtained by samping some continuous curve with or without noises, we proved that, under a very natura condition, optima smoothing spines converge to some imiting curve as the number of data increases. The case of surfaces is treated in parae, and we see that the extensions of the resuts for one-dimensiona case i.e. curves) to the two-dimensiona case i.e. surfaces) are straightforward. This indicates that the same ine of approach can be used for extensions to sti higher dimensions. The resuts estabished in this paper provide theoretica basis for further studies on spine probems incuding observationa data processing, contour modeing of objects such as iving bodies, image processings, reinforcement earning and so forth. Appendix A. Proofs of Lemmas. and.. A.. Proof of Lemma.. By ), it suffices to prove that the matrix R R M M M = m + ) in ), i.e. ˆb) T R = ˆb) t) t)) dt, 7) Î is nonsinguar when Î =, + ), and singuar when Î =, m). Here ˆbt) R M is given by ˆbt) = [ B t )) B t )) B t m )) ] T. 7) When Î =, + ), we see from Tabe that the eements in the vector ˆb ) t) are ineary independent in the interva, + ). Thus the matrix R is positive-definite or nonsinguar.

15 xs,t) xs,t) SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA 5 contro points s s t t a) f s, t) b) xc s, t) c) x²n,n s, t) with N = N = d) x²n,n s, t) with N = N = 4 Figure. The surfaces f s, t), xc s, t) and x²n,n s, t) with λ =.5. Next we consider the case where Iˆ =, m). Let us introduce an M -dimensiona vector M with a its entries unity, i.e. T M = RM, 7) and consider Z m RM = ³ T b t) b t) M dt. ) ) On the other hand, we have T b t)m = m X B t i) i= and, since B t i) vanishes for i < j and i > j outside of each interva [j, j + ] j =,,, m ), it hods that T b t)m = j X i=j B t i) = X = N, t j), t [j, j + ].

16 6 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN Note that the ast equaity foows from 5). But since 7) shows that N, t j) = t [j, j + ], = we get ˆb T t) M = t [, m], and hence it hods that ˆb) t) ) T M = d dtˆb T t) M =, ˆb) t) ) T M =, t [, m]. 74) Thus R M = and R is singuar. A.. Proof of Lemma.. Reca that the matrix Q R M M M M is given by 4), where R ij) R M M are defined in 4), i.e. R ij) ˆbj) T = t) t)) dt, =,, ; i, j =,,. 75) ˆbi) Î Here, the integration interva Î is either Î =, + ) or Î =, m ), and ˆb t) R M is given by 4), namey ˆb t) = [ B t )) B t )) B t m )) ] T. 76) We first consider the case where Î =, + ). Regarding the second and third terms in 4), we get + R ) ) T ) T + + ) T = ˆb t) t) dt = ˆb) t) t) ) t) t) dt = R 77) Moreover, it is easiy seen that the symmetric matrices R ii) are a nonsinguar and hence positive-definite for =, and i =,,, since the eements B ii) t j) in the vector ˆbii) t) are ineary independent in the interva, + ). Thus a the four terms in 4), consisting of Kronecker products of symmetric positive-definite matrices, are symmetric and positive-definite, and so is Q. Next we consider the case where the integration interva is finite, i.e. I =, m ), =,. From the resut in Appendix A. 74), in particuar), we see that R ) M = and R ) M = hod for =,. On the other hand, it can be easiy verified that T j+k A B) j+k = ) ) T j A j T k B k hods for any matrices A R j j and B R k k. Then, for the matrix Q in 4), we have ) ) T M +M Q M +M = α T M β R ) M T M R ) M ) ) +αβ T M R ) M T M R ) M ) ) + β T M α R ) M T M R ) M =, and hence Q is singuar.

17 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA 7 A.. Positive-Definiteness of Matrix Q ) in 5). Here we show that the Gramian Q ) R M M defined in 5) is positive-definite for both I =, + ) and I = t, t m ). Equivaenty we show that the Gramian R ) = ˆbt)ˆbT t)dt, 78) Î is positive-definite for both Î =, + ) and Î =, m), since Q) = α R) hods. Here ˆbt) R M is given by ˆbt) = [ B t )) B t )) B t m )) ] T. 79) In the case of Î =, + ), the fact that the matrix R) is positive-definite foows from the same arguments as for the matrices R ii) in Appendix A.. Thus, in the seque, we show that the same hods for the case of Î =, m), namey For a vector z = [z z Then we get R ) = m ˆbt)ˆbT t)dt >. 8) z m ] T R M, et m = z T R ) z = z Tˆbt) ) dt. 8) z Tˆbt) m = i= z i B t i) = t [, m]. 8) Here note that the boundaries t =, m can be incuded since z Tˆbt) is a continuous function in, + ). From Tabe, we have B ) = 6, B ) = 4 6, B ) = 6, B j) = for other integers, and writing 8) for t =,,, m yieds where the matrix C R M ) M is defined by C = C z =, 8) ) Differentiating 8) with respect to t, and noting the continuity of z T ˆbt) in, + ), we get m z T ˆbt) = z i Ḃ t i) = t [, m]. i=

18 8 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN Since Ḃ) =, Ḃ ) = and Ḃj) = for any other integers j, writing this equation for t =,,, m yieds C z =, 85) where C R M ) M is given by C = ) Combining 8) and 85), we have [ C C ] z =. 87) Here, the matrix 4 4 is nonsinguar, since this is the resutant matrix e.g. []) for poynomias fx) = x + 4x +, gx) = x + [ ] C which have no common roots. Thus we see that rank = M in 87) and hence z =. C By 8), we concude that the matrix R ) is nonsinguar and positive-definite. Appendix B. Computation of Matrix Q in Section.. Here we show how to obtain the matrix Q in 9) for constructing smoothing surfaces. Since Q is given as in 4), our main concern is to obtain the matrices R ij). As we wi see, these matrices are of very specia structure and can be readiy set up using precomputed vaues. B.. The case where I =, + ). In this case, 77) hods and the matrix Q in 4) becomes Q = α β R) R ) + αβr ) R ) + β α R) R ). 88) Thus we ony need to compute R ii) R M M for i =,,. Noting that the eements in [ ] matrices R ii) and R ii) are simiary defined, we can denote R ii) as R ii) = r p,q ii) m p,q= with r ii) p,q = + B i) t p) B i) t q) dt. Then, since B i) t) vanishes outside of the interva [, 4], we get r ii) p,q = 4 p q B i) t) B i) t p q )) dt, 89)

19 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA 9 and we see that r p,q ii) = for p q >. Using Tabe, we can compute the integra in 89) for p q, and the resuts are shown in Tabe. Note that, from this tabe, we obtain the matrix R ) = [ r ) p,q ] m p,q=, which is of the form 4). Tabe. The eements r ii) pq of matrix R ii) =, and i =,, ) p q = 5 5 r pq ) r pq ) r pq ) 97 p q = 68 8 p q = 4 5 p q = 54 6 otherwise 8 B.. The case where I = s, s m ) and I = t, t m ). We first rewrite R ij) as + + ) R ij) ˆbj) T = ˆbi) t) t)) dt. Noting that the first integra was computed in Section B. and denoting it by R ij), we have where the matrices U ij) R ij) = R ij) R M M and V ij) U ij) = V ij) = + m m U ij) V ij) 9) R M M are defined by ˆbi) t) Here, simiary as 77), we obtain U ) and V ) as where U c and V c ˆbj) t)) T dt 9) ˆbj) T ˆbi) t) t)) dt. 9) U ) = U c U ), V ) = V c V ), 9) are given by U c = V c = Obviousy U ) = U ) ) T and V ) = V ) ) T. 4 4,M M, M,M M,M M,,M 4 4, 94). 95)

20 H. FUJIOKA, H. KANO, M. EGERSTEDT AND C.F. MARTIN Therefore, it remains to derive expressions for U ii) U ii) these eements are given by = [ ] u p,q ii) m, p,q= u ii) p,q = = p V ii) = [ v ii) p,q and V ii) ] m, p,q= B i) t p)b i) t q)dt for i =,,. Letting B i) t)b i) t q p))dt, 96) and 4 v p,q ii) = B i) t)b i) t q p))ds. 97) m p The vaues of these eements are computed as shown in Tabe and Tabe 4. We see that we obtain the matrices U ) and V ) in the form of 6) and 7) respectivey. Tabe. The eements u ii) pq of matrix U ii) =, and i =,, ) p, q) u ) pq, ) 599 6, ) 5 6, ) 5 u ) pq, ) 59 8, ) 4 68 u ) pq , ) 84 otherwise Tabe 4. The eements v ii) pq p, q) v ) pq m, m ) 5 m, m ) 5 6 m, m ) m, m ) 4 68 of matrix V ii) =, and i =,, ) v ) pq v ) pq m, m ) m, m ) 84 otherwise REFERENCES [] Hosaka, M., Modeing of Curves and Surfaces in CAD/CAM, Springer-Verag, 99. [] Kress, R., Numerica Anaysis, Springer-Verag, 998.

21 SMOOTHING SPLINE CURVES AND SURFACES FOR SAMPLED DATA [] Wang, L.-J. et a., A Fast Efficient Computation of Cubic-Spine Interpoation in Image Codec, IEEE Trans. Signa Processing, Vo.49, No.6, pp.89-97,. [4] Khai, W. and E. Dombre, Modeing, Identification and Contro of Robots, Hermes Penton Ltd.,. [5] Crouch, P. and J. Jackson, Dynamic Interpoation and Appication to Fight Contro, J. of Guidance, Contro and Dynamics, Vo. 4, pp. 84-8, 99. [6] Wahba, G., Spine modes for observationa data, CBMS-NSF Regiona Conference Series in Appied Mathematics, 59, Society for Industria and Appied Mathematics SIAM), Phiadephia, PA, 99. [7] Nakata, H. and H. Kano, Generation of Japanese Cursive Sentences Using Optima Smoothing Spines, J. of the Information Processing Society of Japan, 44, no., pp.4 4,. [8] Fujioka, H., H. Kano, H. Nakata and H. Shinoda, Constructing and Reconstructing Characters, Words and Sentences by Synthesizing Writing Motions, IEEE Trans. SMC Part A, to appear. [9] Mangasarian, O.L. and L.L. Schumaker, Spines via Optima Contro, in Approximation with Specia Emphasis on Spine Functions, Academic Press, New York, 969. [] Schumaker, L.L., Spine Functions: Basic Theory, John Wiey & Sons, New York, 98. [] Zhang, Z., J. Tominson and C. F. Martin, Spines and Linear Contro Theory, Acta Appicandae Mathematicae, 49, pp.-4, 997. [] Sun, S., M. Egerstedt, and C. F. Martin, Contro theoretic smoothing spines, IEEE Trans. Automat. Contro, 45, no., pp.7 79,. [] Egerstedt, M. and C. F. Martin, Optima trajectory panning and smoothing spines, Automatica, 7, pp.57 64,. [4] Kano, H., M. Egerstedt, H. Nakata and C. F. Martin, B-spines and Contro Theory, Appied Mathematics and Computation, 45, issues -, pp.6-88,. [5] Sinha, S. and B. Schunck, A Two Stage Agorithm for Discontinuity-Preserving Surface Reconstruction, IEEE Trans. on Pattern Anaysis and Machine Inteigence, Vo.4, No., pp.6 55, 99. [6] Greiner, G. and K. Hormann, Interpoating and approximating scattered D data with hierarchica tensor product B-Spines, Surface Fitting and Mutiresoution Methods, pp.6 7, Vanderbit University Press, 997. [7] Egerstedt, M. and C. F. Martin, Statistica Estimates for Generaized Spines, Contro, Optimisation and Cacuus of Variations, Vo.9, pp.55-56,. [8] de Boor, C., A practica guide to spines, Springer-Verag, New York, 978. [9] Kano, H., H. Nakata and C. F. Martin, Optima Curve Fitting and Smoothing Using Normaized Uniform B-Spines: A too for studying compex systems, Appied Mathematics and Computation, to appear. [] Lancaster, P. and M. Tismenetsky, The Theory of Matrices, Second Edition, Academic Press, 985. [] Takayama, K. and H. Kano, A New Approach to Synthesizing Free Motions of Robotic Manipuators Based on a Concept of Unit Motions, IEEE Trans. SMC., Vo. 5, No., pp , March, 995. [] Bocher, M., Introduction to Higher Agebra, Dover Pub. Inc., 964.