B-Splines

2.4 Interpolation by Spline Functions

2.4.4 B-Splines

it follows from the above result fork= 1 that

|f(x)−S_∆(x)| ≤ ⁷₄L∆³· ¹₂∆=⁷₈L∆⁴. x∈[a, b].

Clearly, (2.4.3.3) implies that for sequences

∆_m={a=x^(m)₀ < x^(m)₁ <· · ·< x^(m)_n

m =b}

of partitions with∆_m→0 the corresponding cubic spline functions S_∆m and their first two derivatives converge tof and its corresponding deriva- tives uniformly on [a, b]. If in addition

sup

m,j

∆_m

|x^(m)_j+1−x^(m)_j | ≤K <+∞,

even the third derivativefis uniformly approximated byS_∆

m, a usually discontinuous sequence of step functions.

consisting of B-splines. As a consequence, B-splines provide the framework for an efficient and numerically stable calculation of splines.

In order to define B-splines, we introduce the functionf_x: IR→IR f_x(t) := (t−x)₊:= max(t−x,0) =

t−x fort > x, 0 fort≤x, and its powersf_x^r,r≥0, in particular forr= 0,

f_x⁰(t) :=

1 fort > x, 0 fort≤x.

The function f_x^r(.) is composed of two polynomials of degree≤ r: the 0- polynomial P₀(t) := 0 for t ≤xand the polynomial P₁(t) := (t−x)^r for t > x. Note thatf_x^r depends on a real parameter x, and the definition of f_x^r(t) is such that it is a function ofxwhich is continuous from the right for each fixed t. Clearly, for r ≥ 1, f_x^r(t) is (r−1) times continuously differentiable both with respect tot and with respect tox.

Recall further that under certain conditions the generalized divided difference f[t_i, t_i+1, . . . , t_i+r] of a real function f(t), f: IR → IR is well defined for any segment t_i ≤ t_i+1 ≤ · · · ≤ t_i+r of real numbers, even if thet_j are not mutually distinct [see Section 2.1.5, in particular (2.1.5.4)].

The only requirement is that f be (s_j −1)-times differentiable att =t_j, j=i,i+ 1,. . .,i+r, if the value oft_j occurs s_j times in the subsegment t_i ≤t_i+1≤ · · · ≤t_j terminating witht_j. In this case, by (2.1.5.7)

(2.4.4.1)

f[t_i, . . . , t_i+r] := f^(r)(t_i)

r! , if t_i=t_i+r,

f[t_i, . . . , t_i+r] := f[t_i+1, . . . , t_i+r]−f[t_i, . . . , t_i+r−1]

t_i+r−t_i , otherwise.

It follows by induction overrthat the divided differences of the functionf are linear combinations of its derivatives at the pointst_j:

(2.4.4.2) f[t_i, t_i+1, . . . , t_i+r] = i+r j=i

α_jf^(sj−1)(t_j), where, in particular,α_i+r= 0.

Now letr≥1 be an integer andt={t_i}_i∈ZZ any infinite nondecreasing unbounded sequence of realst_i with

inf t_i=−∞, sup t_i= +∞ andt_i< t_i+rfor alli∈ZZ.

Then thei-th B-spline of orderrassociated withtis defined as the following function inx:

(2.4.4.3) B_i,r,t(x) := (t_i+r−t_i)f_x^r−1[t_i, t_i+1, . . . , t_i+r],

for which we also write sometimes more brieflyB_i orB_i,r [see Figure 2 for simple examples].

Clearly, B_i,r,t(x) is well defined for all x=t_i, t_i+1, . . ., t_i+r and, by (2.4.4.2), (2.4.4.3) is a linear combination of the functions

(2.4.4.4a) (t_j−x)^r−s₊ ^j

t=tj

, i≤j≤i+r,

if the value oft_j occurs s_j times within the subsegmentt_i ≤t_i+1 ≤ · · · ≤ t_j. If in addition, the value of t_j occurs n_j times within the full segment t_i ≤t_i+1 ≤ · · · ≤t_i+r, then the definition of s_j shows that for each index σwith 1≤σ≤n_j there exists exactly one integerl=l(σ) satisfying

t_l=t_j, s_l=σ, i≤l≤i+r.

ThusB_i,r,t is also a linear combination of

(2.4.4.4b) (t_j−x)^s₊, wherer−n_j ≤s≤r−1,i≤j≤i+r.

Hence the function B_i,r,t coincides with a polynomial of degree at most r−1 on each of the open intervals in the following set

(−∞, t_i)∪ {(t_j, t_j+1), i≤j < i+r&t_j < t_j+1} ∪(t_i+r,+∞).

That is,B_i,r,t(x) is a piecewise polynomial function in xof order r with respect to the partition of the real axis given by thet_k withk∈T_i,r, where

T_i,r:={j|i≤j < i+r &t_j< t_j+1} ∪ {i+r}.

Only at the knotsx=t_k,k∈T_i,r, the functionB_i,r,t(x) may have jump discontinuities, but it is continuous from the right, because the functions (t_j−x)^s₊ occuring in (2.4.4.4) are functions ofxthat are continuous from the right everywhere, even whens= 0. Also by (2.4.4.4) for givent= (t_j), the B-splineB_i,r(x)≡B_i,r,t(x) is (r−n_j−1) times differentiable atx=t_j, if t_j occurs n_j times within t_i, t_i+1, . . ., t_i+r (ifn_j =r then B_i,r(x) has a jump discontinuity at x = t_j). Hence the order of differentiability of B_i,r,t(x) atx=t_j is governed by the number of repetitions of the value of t_j.

We note several important properties of B-Splines:

(2.4.4.5) Theorem.(a)B_i,r,t(x) = 0forx∈[t_i, t_i+r].

(b)B_i,r,t(x)>0 fort_i < x < t_i+r.

(c) For all x∈IR

i

B_i,r,t(x) = 1,

and the sum contains only finitely many nonzero terms.

By that theorem, the functions B_i,r ≡ B_i,r,t are nonnegative weight functions with sum 1 and support [t_i, t_i+r],

suppB_i,r :={x|B_i,r(x)= 0}= [t_i, t_i+r], they form a “partition of unity.”

Example.Forr= 5 and a sequencetwith· · · ≤t₀ < t₁=t₂ =t₃< t₄ =t₅<

t₆≤ · · ·, the B-SplineB₁,5=B₁,5,tof order 5 is a piecewiese polynomial function of degree 4 with respect to the partitiont₃ < t₅ < t₆ of IR. Forx=t₃,t₅,t₆ it has continuous derivatives up to the orders 1, 2, and 3, respectively, asn₃ = 3, n₅= 2, andn₆= 1.

t₀ t₁ t₂ t₃ t₄ t₅

1 ...

... .......................................

B_0,1,t B_2,2,t

...... .......................

...

...

...

...

...

...

Fig. 2.Some simple B-Splines.

Proof.(a) Forx < t_i≤t≤t_i+r,f_x^r−1(t) = (t−x)^r−1 is a polynomial of degree (r−1) int, which has a vanishingrth divided difference

f_x^r−1[t_i, t_i+1, . . . , t_i+r] = 0 =⇒ B_i,r(x) = 0

by Theorem (2.1.3.10). On the other hand, if t_i ≤ t ≤ t_i+r < x then f_x^r−1(t) := (t−x)^r−1₊ = 0, is trivially true, so that againB_i,r(x) = 0.

(b) Forr= 1 andt_i< x < t_i+1, the assertion follows from the definition B_i,1(x) =,

(t_i+1−x)⁰₊−(t_i−x)⁰₊-

= 1−0 = 1,

and forr >1, from recursion (2.4.5.2) for the functionsN_i,r:=B_i,r/(t_i+r− t_i), which will be derived later on.

(c) Assume firstt_j < x < t_j+1. Then by (a), B_i,r(x) = 0 for all i, r withi+r≤j and alli≥j+ 1, so that

i

B_i,r(x) = j i=j−r+1

B_i,r(x).

Now, (2.4.4.1) implies

B_i,r(x) =f_x^r−1[t_i+1, t_i+2, . . . , t_i+r]−f_x^r−1[t_i, t_i+r, . . . , t_i+r−1].

Therefore,

i

B_i,r(x) =f_x^r−1[t_j+1, . . . , t_j+r]−f_x^r−1[t_j−r+1, . . . , t_j] = 1−0.

The last equality holds because the function f_x^r−1(t) = (t−x)^r−1 is a polynomial of degree (r−1) in t fort_j < x < t_j+1 ≤t ≤t_j+r, for which f_x^r−1[t_j+1, . . . , t_j+r] = 1 by (2.1.4.3), and the function f_x^(r−1)(t) = (t− x)^r−1₊ = 0 vanishes for t_j−r+1 ≤ t ≤ t_j < x < t_j+1. For x = t_j, the assertion follows because of the continuity ofB_i,r from the right,

B_i,r(t_j) = lim

y↓tj

B_i,r(y).

We now return to the spaceS_∆,kof splines of degreek≥0 and construct a sequence t= (t_j) so that the B-splinesB_i,k+1,t(x) of orderk+ 1 form a basis ofS_∆,k. For this purpose, we associate the partition

∆={a=x₀< x₁<· · ·< x_n=b}

with any infinite unbounded nondecreasing sequencet= (t_j)_j∈ZZ satisfying t_j< t_j+k+1 for allj,

t_−k≤ · · · ≤t₀:=x₀< t₁:=x₁<· · ·< t_n:=x_n. In the following only its finite subsequence

t_−k, t_−k+1, . . . , t_n+k

will matter. Then a (special case of a) famous result of Curry and Schoen- berg (1966) is

(2.4.4.6) Theorem.The restrictionsB_i,k+1,t|[a, b]of then+k B-splines B_i≡B_i,k+1,t,i=−k,−k+ 1,. . .,n−1, to[a, b]form a basis of S_∆,k. Proof.EachB_i,−k≤i≤n−1, agrees with a polynomial of degree≤k on the subintervals [x_j, x_j+1] of [a, b], and has derivatives up to the order k−n_j = k−1 in the interior knots t_j =x_j, j = 1, 2, . . ., n−1, of ∆ since theset_j occur only once int,n_j = 1. This showsB_i|[a, b]∈S_∆,k for alli=−k,. . .,n−1. On the other hand we have seen dimS_∆,k=n+k.

Hence it remains to show only that the functionsB_i,−k≤i≤n−1, are linearly independent on [a, b].

Assume "_n−1

i=−ka_iB_i(x) = 0 for all x∈[a, b]. Consider the first subin- terval [x₀, x₁]. Then

0 i=−k

a_iB_i(x) = 0 forx∈[x₀, x₁]

because B_i|[x₀, x₁] = 0 fori ≥1 by Theorem (2.4.4.5)(a). We now prove thatB_−k,. . .,B₀are linearly independent on [x₀, x₁], which impliesa_−k = a_−k+1=· · ·=a₀= 0.

We claim that forx∈[x₀, x₁]

span{B_−k(x), . . . , B₀(x)}= span{(t₁−x)^k+1−s1, . . . ,(t_k+1−x)^k+1−sk+1}, if the number t_j occurs exactly s_j times in the sequence t_−k ≤t_−k+1 ≤

· · · ≤t_j. This is easily seen by induction since by (2.4.4.2), (2.4.4.4a) each B_i(x),−k≤i≤0, has the form

B_i(x) =α₁(t₁−x)^k+1−s1+· · ·+α_i+k+1(t_i+k+1−x)^k+1−si+k+1, whereα_i+k+1= 0.

Therefore it suffices to establish the linear independence of the functions (t₁−x)^k−(s1−1), . . . , (t_k+1−x)^{k−(sk+1−1)}

on [x₀, x₁], or equivalently, of the normalized functions (2.4.4.7) p^(sj−1)(t_j−x), j= 1,2, . . . , k+ 1, defined by

p(t) := 1 k!t^k.

To prove this, assume for some constantsc₁,c₂,. . .,c_k+1

k+1

j=1

c_jp^(sj−1)(t_j−x) = 0 for allx∈[x₀, x₁].

Then by Taylor expansion the polynomial

k+1

j=1

c_j k

l=0

p^(l+sj−1)(t_j)(−x)^l

l! = 0 on [x₀, x₁]

vanishes on [x₀, x₁] and therefore must be the zero polynomial, so that for alll= 0, 1,. . . k

k+1

j=1

c_jp^(l+sj−1)(t_j) = 0.

But this means that

k+1

j=1

c_jΠ^(sj−1)(t_j) = 0,

whereΠ(t) is the row vector Π(t) : =

p^(k)(t), p^(k−1)(t), . . . , p(t)

=

1, t, . . . ,t^k k!

.

However the vectors Π^(sj−1)(t_j), j = 1, 2, . . ., k+ 1, are linearly independent [see Corollary (2.1.5.5)] because of the unique solvability of the Hermite interpolation problem with respect to the sequencet₁≤t₂ ≤

· · · ≤t_k+1. Hencec₁=· · ·=c_k+1= 0 and therefore alsoa_i= 0 fori=−k,

−k+ 1,. . ., 0.

Repeating the same reasoning for each subinterval [x_j, x_j+1], j= 1, 2, . . ., n−1, of [a, b] we find a_i = 0 for all i =−k, −k+ 1, . . ., n−1, so that all B_i, −k ≤ i ≤ n−1, are linearly independent on [a, b] and their restrictions to [a, b] form a basis ofS_∆,k. 2.4.5 The Computation of B-Splines

B-splines can be computed recursively. The recursions are based on a re- markable generalization of the Leibniz formula for the derivatives of the product of two functions. Indeed, in terms of divided differences, we find the following.

(2.4.5.1) Product Rule for Divided Differences.Supposet_i≤t_i+1≤

· · · ≤t_i+k. Assume further that the function f(t) = g(t)h(t) is a product of two functions that are sufficiently often differentiable at t =t_j, j =i, . . .,i+k, so thatg[t_i, t_i+1, . . . , t_i+k]andh[t_i, t_i+1, . . . , t_i+k] are defined by (2.4.4.1).Then

f[t_i, t_i+1, . . . , t_i+k] =

i+k

r=i

g[t_i, t_i+1, . . . , t_r]h[t_r, t_r+1, . . . , t_i+k].

Proof.From (2.1.5.8) the polynomials i+k

r=i

g[t_i, . . . , t_r](t−t_i)· · ·(t−t_r−1)

and i+k

s=i

h[t_s, . . . , t_i+k](t−t_s+1)· · ·(t−t_i+k)

interpolate the functions g andh, respectively, at the pointst =t_i, t_i+1, . . .,t_i+k (in the sense of Hermite interpolation, see Section 2.1.5, if thet_j are not mutually distinct). Therefore, the product polynomial

F(t) :=

i+k r=i

g[t_i, . . . , t_r](t−t_i)· · ·(t−t_r−1)

·

i+k

s=i

h[t_s, . . . , t_i+k](t−t_s+1)· · ·(t−t_i+k)

also interpolates the functionf(t) att=t_i,. . .,t_i+k. This product can be written as the sum of two polynomials

F(t) =

i+k

r,s=i

· · ·=

r≤s

· · ·+

r>s

· · ·=P₁(t) +P₂(t).

Since each term of the second sum"

r>s is a multiple of the polynomial 4_i+k

j=i(t−t_j), the polynomialP₂(t) interpolates the 0-function att=t_i,. . ., t_i+k. Therefore, the polynomialP₁(t), which is of degree≤k, interpolates f(t) at t=t_i, . . ., t_i+k. Hence,P₁(t) is the unique (Hermite-) interpolant off(t) of degree≤k.

By (2.1.5.8) the highest coefficient of P₁(t) is f[t_i, . . . , t_i+k]. A com- parison of the coefficients of t^k on both sides of the sum representation P₁(t) ="

r≤s . . . ofP₁ proves the desired formula f[t_i, . . . , t_i+k] =

i+k r=i

g[t_i, . . . , t_r]h[t_r, . . . , t_i+k].

Now we use (2.4.5.1) to derive a recursion for the B-splines B_i,r(x) ≡ B_i,r,t(x) (2.4.4.3). To do this, it will be convenient to renormalize the func- tionsB_i,r(x), letting

N_i,r(x) := B_i,r(x)

t_i+r−t_i ≡f_x^r−1[t_i, t_i+1, . . . , t_i+r], for which the following simple recursion holds:

Forr≥2 andt_i< t_i+r:

(2.4.5.2) N_i,r(x) = x−t_i

t_i+r−t_i N_i,r−1(x) + t_i+r−x

t_i+r−t_i N_i+1,r−1(x).

Proof.Suppose firstx=t_jfor allj. We apply rule (2.4.5.1) to the product f_x^r−1(t) = (t−x)^r−1₊ = (t−x)(t−x)^r−2₊ =g(t)f_x^r−2(t).

Noting thatg(t) is a linear polynomial intfor which by (2.1.4.3) g[t_i] =t_i−x, g[t_i, t_i+1] = 1, g[t_i, . . . , t_j] = 0 forj > i+ 1, we obtain

f_x^r−1[t_i, . . . , t_i+r] = (t_i−x)f_x^r−2[t_i, . . . , t_i+r] + 1·f_x^r−2[t_i+1, . . . , t_i+r]

= (t_i−x)

t_i+r−t_i(f_x^r−2[t_i+1, . . . , t_i+r]−f_x^r−2[t_i, . . . , t_i+r−1]) + 1·f_x^r−2[t_i+1, . . . , t_i+r]

= x−t_i

t_i+r−t_if_x^r−2[t_i, . . . , t_i+r−1] + t_i+r−x

t_i+r−t_if_x^r−2[t_i+1, . . . , t_i+r], and this proves (2.4.5.2) forx=t_i,. . .,t_i+r. The result is furthermore true for allxsince all B_i,r(x) are continuous from the right andt_i< t_i+r The proof of (2.4.4.5), (b) can now be completed: By (2.4.5.2), the valueN_i,r(x) is a convex linear combination ofN_i,r−1(x) andN_i+1,r−1(x) for t_i < x < t_i+r with positive weights λ_i(x) = (x−t_i)/(t_i+r−t_i) > 0, 1−λ_i(x)>0. AlsoN_i,r(x) andB_i,r(x) have the same sign, and we already know thatB_i,1(x) = 0 for x /∈[t_i, t_i+1] andB_i,1(x)>0 fort_i < x < t_i+1. Induction overrusing (2.4.5.2) shows thatB_i,r(x)>0 fort_i< x < t_i+r.

The formula

(2.4.5.3) B_i,r(x) = x−t_i

t_i+r−1−t_i B_i,r−1(x) + t_i+r−x

t_i+r−t_i+1 B_i+1,r−1(x) is equivalent to (2.4.5.2), and representsB_i,r(x) directly as a positive linear combination of B_i,r−1(x) and B_i+1,r−1(x). It can be used to compute the values of all B-splinesB_i,r(x) =B_i,r,t(x) for a given fixed value ofx.

To show this, letxbe given. Then there is at_j ∈twitht_j≤x < t_j+1. By (2.4.4.5)a) we know B_i,r(x) = 0 for all i, r with x ∈ [t_i, t_i+r], i.e., for i ≤ j −r and for i ≥ j + 1. Therefore, in the following tableau of B_i,r :=B_i,r(x), theB_i,r vanish at the positions denoted by 0:

(2.4.5.4)

0 0 0 0 . . .

0 0 0 B_j−3,4 . . .

0 0 B_j−2,3 B_j−2,4 . . . 0 B_j−1,2 B_j−1,3 B_j−1,4 . . . B_j,1 B_j,2 B_j,3 B_j,4 . . .

0 0 0 0 . . .

... ... ... ...

By definition B_j,1 = B_j,1(x) = 1 for t_j ≤ x < t_j+1, which determines the first column of (2.4.5.4). The remaining columns can be computed consecutively using recursion (2.4.5.3): Each elementB_i,r can be derived from its two left neighbors,B_i,r−1 andB_i+1,r−1

B_i,r−1 → B_i,r B_i+1,r−1

This method is numerically very stable because only nonnegative multiples of nonnegative numbers are added together.

Example.Forti=i,i= 0, 1,. . . andx= 3.5∈[t₃, t₄] the following tableau of valuesBi,r=Bi,r(x) is obtained.

r = 1 2 3 4

i = 0 0 0 0 1/48

i = 1 0 0 1/8 23/48

i = 2 0 1/2 6/8 23/48

i = 3 1 1/2 1/8 1/48

i = 4 0 0 0 0

For instance,B₂,4 is obtained from B₂,4 =B₂,4(3.5) =3.5−2

5−2 ·6

8+6−3.5 6−3 ·1

8= 23 48.

We now consider the interpolation problem for spline functions, namely the problem of finding a splineS that assumes prescribed values at given locations. We may proceed as follows. Assume thatr≥1 is an integer and t= (t_i)_1≤i≤N+r a finite sequence of real numbers satisfying

t₁≤t₂≤ · · · ≤t_N+r

andt_i< t_i+r fori= 1, 2, . . .,N. Denote byB_i(x)≡B_i,r,t(x),i= 1,. . ., N, the associated B-splines, and by

S_r,t = N i=1

α_iB_i(x)|α_i∈IR 5

,

the vector space spanned by the B_i, i = 1, . . ., N. Further, assume that we are givenN pairs (ξ_j, f_j),j= 1, . . .,N, of interpolation points with

ξ₁< ξ₂<· · ·< ξ_N.

These are the data for the interpolation problem of finding a functionS∈ Sr,t satisfying

(2.4.5.5) S(ξ_j) =f_j, j= 1, . . . , N.

Since anyS∈ S_r,t can be written as a linear combination of theB_i,i= 1, . . .,N, this is equivalent to the problem of solving the linear equations (2.4.5.6)

N i=1

α_iB_i(ξ_j) =f_j, j= 1, . . . , N.

The matrix of this system

A=





B₁(ξ₁) . . . B_N(ξ₁)

... ...

B₁(ξ_N) . . . B_N(ξ_N)





has a special structure:A is a band matrix, because by (2.4.4.5) the func- tionsB_i(x) =B_i,r,t(x) have support [t_i, t_i+r], so that within thejth row of Aall elementsB_i(ξ_j) witht_i+r< ξ_jort_i> ξ_j are zero. Therefore, each row ofA contains at most relements different from 0, and these elements are in consecutive positions. The componentsB_i(ξ_j) ofAcan be computed by the recursion described previously. The system (2.4.5.6), and thereby the interpolation problem, is uniquely solvable if A is nonsingular. The non- singularity ofAcan be checked by means of the following simple criterion due to Schoenberg and Whitney (1953), which is quoted without proof.

(2.4.5.7) Theorem.The matrix A= (B_i(ξ_j))of (2.4.5.6) is nonsingular if and only if all its diagonal elementsB_i(ξ_i)= 0are nonzero.

It is possible to show [see Karlin (1968)] that the matrix A is totally positive in the following sense: allr×rsubmatricesB ofAof the form

B= (a_ip,jq)^r_p,q=1 with r≥1, i₁< i₂<· · ·< i_r, j₁< j₂<· · ·< j_r, have a nonnegative determinant, det(B) ≥ 0. As a consequence, solving (2.4.5.6) for nonsingular A by Gaussian elimination without pivoting is numerically stable [see de Boor and Pinkus (1977)]. Also the band structure ofAcan be exploited for additional savings.

For further properties of B-Splines, their applications, and algorithms the reader is referred to the literature, in particular to de Boor (1978), where one can also find numerousfortranprograms.