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1 Thin-Plte Splines Dvid Eberly, Geometric Tools, Redmond WA This work is licensed under the Cretive Commons Attribution 4.0 Interntionl License. To view copy of this license, visit or send letter to Cretive Commons, PO Box 1866, Mountin View, CA 94042, USA. Creted: Mrch 1, 1996 Lst Modified: Jnury 21, 2018 Contents 1 Introduction 2 2 The Clculus of Vritions Functionls of f nd f Functionls of f, f, nd f Cubic Splines nd Green s Functions Euler-Lgrnge Equtions for Multivrite f Thin-Plte Splines in n Dimensions 5 4 Smoothed Thin-Plte Splines 7 1

2 1 Introduction Recll tht nturl cubic splines re piecewise cubic polynomil nd exct interpolting functions for tbulted dt (x i, f(x i. The globlly constructed spline hs continuous second-order derivtives. The second derivtives t the endpoints re zero (no bending t endpoints. It is lso possible to clmp the endpoints by specifying zero first derivtives there. The spline curve represents thin metl rod tht is constrined not to move t the smple points x i. The concept pplies eqully s well in two dimensions. A thin-plte spline is physiclly motivted 2D interpoltion scheme for rbitrrily spced tbulted dt (x i, y i, f(x i, y i. These splines re the generliztion of the nturl cubic splines in 1D. The spline surfce represents thin metl sheet tht is constrined not to move t the smple points (x i, y i. The construction is bsed on choosing function tht minimizes n integrl tht represents the bending energy of surfce. The origins of thin-plte splines in 2D ppers to be [1, 2]. In fct, the concept pplies in ny dimension for rbitrrily spced tbulted dt (x i, f(x i. The method of construction for ll dimensions is presented in [3] nd is bsed on functionl nlysis. In n dimensions, the ide of thin-plte splines is to choose function f(x tht exctly interpoltes the dt points (x i, y i, sy, y i = f(x i, nd tht minimizes the bending energy, E[f] = D 2 f 2 dx (1 R n where D 2 f is the mtrix of second-order prtil derivtives of f nd D 2 f 2 is the sum of squres of the mtrix entries. The infinitesiml element of hypervolume is dx = dx 1 dx n, where x i re the components of x. It is lso possible to formulte the problem with smoothing prmeter for regulriztion [4]. A function f is chosen tht does not necessrily exctly interpolte ll the dt points but tht does minimize E[f] = f(x i y i 2 + λ D 2 f 2 dx (2 R n The smoothing prmeter is λ > 0 nd is chosen priori. The summtion mkes it cler tht there re m dt points. 2 The Clculus of Vritions The ides re presented in mthemticlly informl mnner. 2.1 Functionls of f nd f To motivte the minimiztion, consider functionl tht is n integrl involving function F tht depends on n independent vrible x, on function f nd on the derivtive function f, E[f] = F (x, f, f dx (3 2

3 For exmple, F (x, f, f = f, in which cse E[f] = f(x dx is just the definite integrl of f for the intervl [, b]. Another exmple is F (x, f, f = 1 + (f 2, in which cse E[f] = 1 + (f (x 2 dx is the rclength of the grph of f for the intervl. We wish to construct f for which E[f] of eqution (3 is minimum. The clculus of vritions llows us to do this, process tht is the extension of directionl derivtives for multivrite functions to directionl derivtives of functions whose independent inputs re themselves functions. Consider the function E s f vries in the direction of nother function g, φ(t = E[f + tg] = F (x, f + tg, f + tg dx (4 We ssume tht g does not chnge f t the intervl endpoints, so g( = 0 nd g(b = 0. For ech sclr t we obtin the rel number φ(t from the integrtion. If f is function tht minimizes E, then we expect φ(t = E[f + tg] E[f] = φ(0 for t ner zero. From stndrd clculus, for φ(0 to be minimum we expect tht its derivtive with respect to t is zero: φ (0 = 0. If we formlly differentite eqution (4 with respect to t, we obtin φ (t = (x, f + tg, f + tg f g + (x, f + tg, f + tg f g dx (5 The integrnd is n ppliction of the chin rule to differentite F (x, f + tg, f + tg. Setting t to zero, we hve t minimum, 0 = φ (x, f, f (0 = g + (x, f, f f f g dx (6 The second term in the integrnd involves g (x. We cn use integrtion by prts, u dv = uv v du, with u = / f nd dv = g dx, f g dx = f g b ( d b dx f g dx = ( d dx f g dx (7 where the lst equlity follows from g( = g(b = 0. Combining this with eqution (6, we hve [ 0 = f d ( ] dx f g dx (8 This eqution is true no mtter which function direction g we choose, which forces f d ( dx f = 0 (9 This is referred to s the Euler-Lgrnge differentil eqution. As n exmple, let us construct the function f(x for which the rclength integrl is minimum on the intervl [x 0, x 1 ]. Let the function vlues t the endpoints be y 0 nd y 1. The integrnd is F (x, f, f = 1 + (f 2. The Euler-Lgrnge differentil eqution is 0 = f d dx ( f = 0 d dx ( f [1 + (f 2 ] 1/2 = f [1 + (f 2 ] 3/2 (10 The eqution is stisfied when f (x = 0 for ll x, which mens f(x = y 0 + (y 1 y 0 (x x 0 /(x 1 x 0. This is exctly wht we expect the shortest-length curve connecting two points is line segment. 3

4 2.2 Functionls of f, f, nd f The sme ide of directionl derivtive pplies when the integrnd depends on the function nd its firstnd second-order derivtives, F (x, f, f, f. When computing the directionl derivtive, we use function g(x for which g( = g(b = 0 nd g ( = g (b = 0. The equivlent of eqution (6 is 0 = f g + f g + f g dx (11 The second term in the integrnd is integrted by prts once. The third term is integrted by prts twice, nd uses g( = g(b = g ( = g (b = 0 to eliminte the nonintegrl terms tht occur. The result is 0 = [ f d ( ( ] dx f + d2 dx 2 f g dx (12 Once gin, this eqution is true no mtter the choice of g which forces f d ( ( dx f + d2 dx 2 f = 0 (13 The introduction of f llows us to hndle the bending energy integrl. 2.3 Cubic Splines nd Green s Functions The dt points re (x i, y i for 1 i m. We require tht f(x i = y i for ll i. The bending energy is E[f] = [f (x] 2 dx (14 A complicting fctor is tht the integrl is over the entire rel line, so the clculus of vritions rgument must be extended to hndle this. Effectively, we need to work with distributions. In this cse, think of this s introducing Dirc delt functions into the problem. Recll tht the Dirc delt function hs the substitution property φ( = φ(xδ(x dx. In the nottion for the clculus of vritions, the integrnd is F (x, f, f, f = (f 2 ; tht is, F depends only on the second derivtive of f. Eqution (13 must be stisfied, f (4 (x = 0, where the left-hnd side is the fourth-order derivtive of f. If it were the cse tht f hs continuous fourth-order derivtive, then f would hve to be cubic polynomil. However, it is then not possible to stisfy ll the conditions f(x i = y i unless the dt points do ll lie on the sme cubic grph. This requires us to tret f (4 (x = 0 in distributionl sense the fourth derivtive is zero for ll x except t the points x i where the fourth derivtive is discontinuous. We cn construct Green s function G(x, s tht is the solution to 4 G/ x 4 = δ(x s, where δ(x is the Dirc delt function. The clssicl solution is G(x, s = 1 12 x s 3 (15 Observe tht G hs derivtive discontinuity t x = s. The function f tht minimizes eqution (14 is liner combintion of the G(x, s with the s-vlues set to the x i where the derivtive discontinuities must 4

5 occur. Also notice tht ny liner polynomil is in the kernel of E[f], the set of functions for which E[f] = 0; thus, we need to ccount for this. The form of f is x x i 3 f(x = i G(x, x i + b 0 + b 1 x = i + b 0 + b 1 x (16 12 This eqution hs m + 2 unknown vlues, the i nd b j, but we hve only m constrints f(x i = y i. The remining two come from n orthogonlity condition tht is mentioned in [3]. Specificlly, the liner polynomil b 0 +b 1 x is in the orthogonl complement of the function spce tht contins the Green s functions. This mnifests itself s m i = 0 nd m ix i = Euler-Lgrnge Equtions for Multivrite f Consider functions of the form f(x 1,..., x n. The function F is of the form F (x 1,..., x n, f, f x1,..., f xn, where f xi = f/ x i. The equivlent of eqution (9 is n f ( d = 0 (17 dx i f xi Second-order derivtives f xix j = 2 f/ x i x j my lso be included in F. The equivlent of eqution (13 is n f ( d n n ( d d + = 0 (18 dx i f xi dx i dx j j=1 f xix j 3 Thin-Plte Splines in n Dimensions In 2D, the motivtion for thin-plte splines is to exctly interpolte the smple points nd to minimize the bending energy of the surfce tht does so. The presenttion here is for n dimensions, lthough for n 3, the physicl motivtion does not pply. The bending energy is n n E[f] = fx 2 ix j (19 R n j=1 where f xix j re the second-order derivtives of f nd where the integrtion is over the entire set of rel-vlued n-tuples. Eqution (18 becomes the bihrmonic eqution n n 0 = 2 f = f xix ix jx j (20 j=1 where f xix jx k x l re the fourth-order derivtives of f. The constnt fctor 2 obtined from differentition is discrded. The bihrmonic eqution involves two pplictions to f of the Lplcin opertor = n k=1 2 / x 2 k. Just s for the cubic spline, f need only hve fourth-order prtil derivtives tht re continuous lmost everywhere (except t the dt points. We need Green s function G(x, s tht is solution to the nonhomogeneous bihrmonic eqution, 2 G(x, s = δ(x s; x R n, s R n (21 5

6 where δ(z is the Dirc delt function with singulrity t z = 0. The bihrmonic opertor hs derivtives with respect to the x vrible. The Green s function is chosen to be of the form G(x, s = u( x s. The function u(r is solution to the nonhomogeneous bihrmonic eqution δ(r = 2 u(r = u (4 (r + 2(n 1 r u (3 (r + (n 1(n 3 r 2 u (2 (n 1(n 3 (r r 3 u (1 (r (22 for r [0,, where u (k (r is the k-th order derivtive of u(r. The generl solution is c 0 + c 1 r 2 + c 2 ln r + c 3 r 2 ln r, n = 2 u(r = c 0 + c 1 r 2 + c 2 ln r + c 3 r 2, n = 4 c 0 + c 1 r 2 + c 2 r 2 n + c 3 r 4 n, ll other n (23 Let B(s, ε be the n-dimensionl bll with center s nd rdius ε. Let S(s, ε be the surfce of the bll. Integrte eqution (21 over the bll region, where dv is n infinitesiml volume element. ds is n infinitesiml surfce element nd N re unit-length outer-pointing surfce normls, 1 = δ(x s dv property of the Dirc delt function B(s,ε = B(s,ε 2 G(x, s dv eqution (21 = ( G(x, s N ds surfce integrl using the divergence theorem S(s,ε (24 = ( u(r N ds setting r = x s S(0,ε The Lplcin of u(r is u(r = 4c 1 + 4c 3 (1 + ln r, n = 2 8c 1 + 2c 2 r 2, n = 4 2nc 1 + 2c 3 (4 nr 2 n, ll other n (25 nd the grdient of the Lplcin of u(r is 4c 3 r 2 x, n = 2 ( u(r = 4c 2 r 4 x, n = 4 2c 3 (4 n(2 nr n x, ll other n (26 The term x occurs in the surfce integrl of eqution (24 nd the surfce is bll of rdius ε, so x = εn nd r = ε. Thus, 4c 3 ε 1, n = 2 1 = ( u(r N ds = SurfceAre(S(0, ε 4c 2 ε 3, n = 4 (27 S(0,ε 2c 3 (4 n(2 nε 1 n, ll other n The surfce re of the bll in n dimensions is 2π n/2 ε n 1 /Γ(n/2. The surfce integrl eqution hs ll ε terms cncel, so the c-constnts re determined by the eqution. For n = 2, we hve c 3 = 1/(8π. For 6

7 n = 4, we hve c 2 = 1/(8π 2. For ll other n, we hve c 3 = Γ(n/2/(4π n/2 (4 n(2 n. The terms of u(r whose constnts re not determined by the construction re discrded becuse they hve no effect (in the distributionl sense in determining solutions to the minimiztion problem. In summry, the Green s functions re r 4 n ln r, n = 2 or n = 4 G(r = α (28 r 4 n, otherwise where α = 1/(8π for n = 2, α = 1/(8π 2 for n = 4 nd α = Γ(n/2 2/(16π n/2 for ll other n. Observe tht when n = 1, α = 1/12 s shown in eqution (15. The minimizer f is liner combintion of the Green s function with the rgument s set to the x i of the dt points. There is lso liner polynomil term; this term is in the kernel of E[f]. The function is n f(x = i G(x, x i + b 0 + b j x j (29 where x j is the jth component of vrible x. NOTE: This is of prcticl vlue for interpoltion in dimensions 1, 2, nd 3 becuse G(x, x i re bounded functions; however, for dimensions 4 nd lrger, G(x, x i hs singulrity t x i, so the interpolting function is not defined t the smple points. This is clerly not desired. In prctice, u(r is chosen so tht G(x, s = u( x s is bounded nd 2 G(x, s = 0; tht is, G is no longer the fundmentl solution of the bihrmonic eqution becuse the Dirc delt function is omitted. The simplest bounded choice for n 4 is u(r = r 2. Sometime the bsis function for the 2-dimensionl cse is chosen, u(r = r 2 ln r, but then G is no longer solution to the bihrmonic eqution. Choosing different rdil bsis function for dimensions n 4 produces f(x tht is not generlly the minimizer for E[f] for the specified smple points. Define y to be the m 1 vector whose components re the dt point y i vlues. Define to be the m 1 vector whose components re the coefficients i. Define b to be the (n vector whose components re the b j. The constrints y i = f(x i led to the system of equtions j=1 y = M + Nb (30 where M is the m m mtrix whose entries re M ij = G(x i, x j nd where N is the m (n+1 mtrix whose rows re [1 x T i ]. An orthogonlity condition tht comes from the functionl nlysis in [3] is N T = 0. The equtions hve solution = M 1 (y Nb, b = (N T M 1 N 1 N T M 1 y (31 Of course, b is computed first. The minimum bending energy is T M. When is zero, this qudrtic form is zero this is the cse when f is liner function whose grph is hyperplne (no bending of the surfce. 4 Smoothed Thin-Plte Splines The smoothed functionl is mentioned in eqution (2, which my be rewritten s ( m E[f] = (f(x y i 2 δ(x x i + λ D 2 f(x 2 dx = F (f(x, D 2 f(x dx (32 R n R n 7

8 where δ(x is the Dirc delt function of multivrite input. The Euler-Lgrnge differentil eqution for the integrnd F (f, D 2 f is computed using eqution (18, (f(x y i δ(x x i + λ 2 f(x = 0 (33 where 2 is the bihrmonic opertor. The fctor of 2 tht occurs during the differentition hs been discrded. Using the Green s functions defined previously, the solution to the differentil eqution is of the form ( n f(x = w j G(x, x j + b 0 + b k x k (34 j=1 where the w i nd b i re the unknown prmeters to be determined. The vlues x k re the components of x. For the function f(x in eqution (34, observe tht j=1 k=1 2 f(x = w j 2 G(x, x j = w j δ(x x j (35 where the second equlity is bsed on G being the Green s function for the bihrmonic eqution (21. Substituting this into eqution (33 nd fctoring out the Dirc delt function leds to the distributionl eqution (f(x y i + λw i δ(x x i = 0 (36 Let B(s, ε be the n-dimensionl bll with center s nd rdius ε < min i0 i 1 x i0 x i1. At smple point x j, integrte eqution (36 over the bll B(x j, ε. The only smple point in this bll is x j, so f(x j y j + λw j, i = j 0 = (f(x y i + λw i δ(x x i dx = (37 B(x j,ε 0 i j where the substitution property of the Dirc delt fucntion ws used. Eqution (37 is rewritten s ( n y j = f(x j + λw j = w i G(x j, x i + b 0 + b k x k, 1 j m (38 where we hve computed f(x j using eqution (34 with x (j k the kth component of x j. Writing this in vector nd mtrix form, we hve the mtrix system j=1 k=1 y = (M + λiw + Nb, N T w = 0 (39 The mtrix M hs size m m with entry M rc = G(x r, x c. The mtrix I is the m m identity mtrix. The vector w hs size m 1 nd stores the w i coefficients. The mtrix N hs size m (n + 1 with rows [1 x T r ]. The vector b hs size (n nd stores the b k coefficients. The eqution N T w = 0 is n orthogonlity condition similr to wht ws used for thin-plte splines without smoothing. The solution is ( 1 w = (M + λi 1 (y Nb, b = N T (M + λi N 1 N T (M + λi 1 y (40 The minimum of the functionl is λw T (M + λiw. As λ increses, the vlue is symptotic to the discrete summtion (first term of the functionl. 8

9 References [1] J. Duchon, Interpoltion des fonctions de deux vribles suivnt le principe de l flexion des plques minces, RAIRO Anlyse Numérique, vol. 10, pp. 5-12, [2] J. Duchon, Splines minimizing rottion-invrint semi-norms in Sobolev spces, Lecture Notes in Mthemtics, vol. 57, pp , [3] Jen Meinguet, Multivrite interpoltion t rbitrry points mde simple, Journl of Applied Mthemtics nd Physics (ZAMP, vol. 30, pp , [4] G. Whb, Spline models for observtionl dt, CBMS-NSF Regionl Conference Series in Applied Mthemtics, Society for Industril nd Applied Mthemtics (SIAM, 180 pges,

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