Cubic spline interpolation

Transcription

1 Cubic spline interpolation In the following, we want to derive the collocation matrix for cubic spline interpolation. Let us assume that we have equidistant knots. To fulfill the Schoenberg-Whitney condition that N in (u i ) 0, for n=3 we set u i =i+2 for all i. The spline shall be given in B-spline representation, i.e., with n= : ESM4A - Numerical Methods 363

2 Cubic spline interpolation Cubic B-splines: For the collocation matrix, we need to evaluate the B- splines at the knots. Obviously, N i3 (u j ) 0 only for j=i-1, j=i, and j=i+1. And, N i3 (u i-1 ) = N i3 (u i+1 ) : ESM4A - Numerical Methods 364

3 Cubic spline interpolation We use the de Boor algorithm to obtain N i3 (u i-1 ) = 1/6, N i3 (u i ) = 4/6, and N i3 (u i+1 ) = 1/6. (Exercise) : ESM4A - Numerical Methods 365

4 Cubic spline interpolation Putting it all together, we obtain the collocation matrix The matrix is sparse. More precisely, it is a tridiagonal matrix. The linear equation system can be solved efficiently (see Section 2.3). Thisiswhywepickedbasissplineswithminimal support : ESM4A - Numerical Methods 366

5 Affine invariance Looking at the first and the last row of the collocation matrix, we observe that they do not sum up to one. Hence, the solution is not affinely invariant. Affine invariance, however, is often desired, especially in geometric modeling. How can we fix it? : ESM4A - Numerical Methods 367

6 Modified cubic spline interpolation One idea is to adjust the knot sequence, i.e., make it non-equidistant at the endpoints. Suggestion: instead of knot sequence (2,3,4,5,,m-1,m,m+1,m+2) use knot sequence (3,3½,4,5,,m-1,m,m+½,m+1). Hence, for the collocation matrix, we need to evaluate the B-splines at values 3½ and m+½ : ESM4A - Numerical Methods 368

7 Modified cubic spline interpolation We have: N 03 (3½) = N 33 (3½) = 1/48, N 13 (3½) = N 23 (3½) = 23/48. The collocation matrix becomes Now all rows sum up to one (partition of unity of B- splines over [3,m-3]). The Schoenberg-Whitney condition is still satisfied : ESM4A - Numerical Methods 369

8 Example Given: points (p 0,, p 4 ) = (1,2,2,0,1) knots (u 0,, u 4 ) = (3,3½,4,4½,5) Collocation matrix: Solving for the control points. Result: (c 0,, c 4 ) = (5/3, 0, 13/3, -16/3, 23) : ESM4A - Numerical Methods 370

9 Example Scaling of delivers : ESM4A - Numerical Methods 371

10 Example Summing of delivers s(u) : ESM4A - Numerical Methods 372

11 Example Interpolation is, of course, restricted to interval [3,5]. For example, at u=u 0 =3, we obtain: s(u 0 ) = s(3) = 5/3 N 03 (3) + 0 N 13 (3) + 13/3 N 23 (3) = 5/3 1/ /6 + 13/3 4/6 = 1 = c : ESM4A - Numerical Methods 373

12 Clamped cubic spline interpolation Clamped spline interpolation is another way to obtain affine invariance. The idea is to only interpolate points p i at knots u i = i+2 for u 1 =3,, u m-1 =m+1. In addition, we interpolate the derivatives at the endpoints u 1 and u m-1. I.e., we have additional interpolation conditions s (u 1 )= d 1 and s (u m-1 )=d m : ESM4A - Numerical Methods 374

13 Clamped cubic spline interpolation To do so, we need to compute s (u). The derivative of a B-spline of degree n can be derived to be (Exercise). Hence, we directly obtain with first backward differences : ESM4A - Numerical Methods 375

14 Clamped cubic spline interpolation In order to fullfill s (u 1 ) = d 1, we have to look at the B-splines of degree 2. Recall: For our equidistant knots, we obtain N 12 (u 1 ) = N 22 (u 1 ) = ½ Hence, s (u 1 ) = ½ a 1 + ½ a 2 = ½ (c 1 c 0 ) + ½ (c 2 - c 1 ) = ½ ( c 0 + c 2 ) : ESM4A - Numerical Methods 376

15 Clamped cubic spline interpolation The collocation matrix becomes : ESM4A - Numerical Methods 377

16 Clamped cubic spline interpolation Clamped cubic spline: : ESM4A - Numerical Methods 378

17 Periodic cubic spline interpolation For some applications, one is interested in periodic splines, i.e., s(u) = s(u+m). This implies that c i+m = c i. In particular, c m = c 0 (and p m = p 0 ). We have the interpolation conditions s(u i ) = p i for i=0,,m-1 and s(u m ) = p : ESM4A - Numerical Methods 379

18 Periodic cubic spline interpolation Collocation matrix: : ESM4A - Numerical Methods 380

19 Periodic cubic spline interpolation Periodic spline: : ESM4A - Numerical Methods 381