Cubic B-spline

Cubic B-Splin es Cubic B-splines can also be used to solve differential equations (Refs. 105 and 266). 

Figure 7. Part of a typical set of cubic B-spline functions calculated using equi-spaced knots...

The quantitative method in Section 2.2 is used to determine the intrinsic magnetization intensity for each voxel. Cubic B-spline basis functions with a partition of 60 interior knots logarithmically spaced between 1 x 10 5 and 10 s are used to represent the relaxation distribution within each voxel. The optimal regularization parameter, A, of each voxel is found within the range between 1 x 10 5 and 5 x 10"18 s by using the UBPR9 criterion. 

A piecewise polynomial will typically have / = 1 at the places where the pieces meet, so that the cubic B-spline is C2+1 at its knots. It is, of course C°° over the open intervals between the knots. 

This particular scheme is called the cubic B-spline scheme, because its limit curve is indeed a cubic B-spline curve. 

By looking at the extent of influence of one control point after 0,l,2,oo refinements, in the cubic B-spline scheme we can see that the refined polygons converge towards the basis function, and the last non-zero entry converges towards the end of the support region. 

In the cubic B-spline scheme, for example, the support is 4 spans wide and each end-point is influenced by only 3 original control points. 

In the cubic B-spline scheme each basis function has four non-zero spans, and thence each span is influenced by 4 original control points. 

Suppose that a control point is being moved by 50 pixels interactively. In places where b(t) <0.01 this is unlikely to change the image of the curve. Even for the cubic B-spline the last one-third of a span at each end is well below this threshold, as is the entire end span of the quintic B-spline. For higher degrees even more will have no visible effect. 

S is called the Subdivision Matrix. It has an interesting structure. Every column is a copy of the mask, but successive columns have their copies shifted down by the arity compared with their left neighbours. For example, the subdivision matrix for the Cubic B-spline scheme is... 

We can also ask the support analysis how many points influence one span of the limit curve, the piece corresponding to one edge of the control polygon. This turns out to be one fewer. Call it m. The value will be 4 for the cubic B-spline. 

This view can be illustrated by seeing how the cubic B-spline approximants... 

An even more interesting picture emerges if we plot the second derivatives of the cubic B-spline approximations at successive steps. In the next figure each line is a plot of the B-spline second derivative at a dyadic place near to t = 1/3. The steps are highlighted by vertical bars. 

The first of these is to use instead of the polygon after a number of iterations, the B-spline which, as we saw in chapter 19 above, can be used in the limiting process to define the limit curve. With rendering engines which are capable of accepting cubics, such as PostScript, it is sensible to use a cubic B-spline for this purpose. 

For the cubic B-spline this can be achieved in an ad-hoc way by just retaining the first control point of the old polygon in the new polygon. This is very easily achieved and implemented, but it has the unfortunate effect that the curvature is always zero at the end of the limit curve. 

Make a cubic B-spline pass through all the data points [x(z),))(/), i = 1,..., zm], A cubic spline is a cubic function of position, defined on small regions between data points. It is constructed so the function and its first and second derivatives are continuous from one region to another. It usually makes a nice smooth curve through the points. The following commands create Figure B.3. 

The only requirement for the knot sequence is that it must be a non-decreasing sequence of numbers. When ti = ti+i it indicates a multiple knot and the segment Qj is reduced to a point. This is one of the great advantages with non-uniform B-splines since it offers great flexibility in the representation ol functions. For example [0, 0, 1, 1, 1, 1, 2, 3, 4, 4] is a valid sequence of knots. The knot value 0 has multiplicity of 2, knot value 1 has multiplicity of 4 and so on. The multiplicity is used to control the continuity of a point. The higher the multiplicity, the less smooth the spline function at this point becomes. A curve segment Qj in cubic B-splines is defined by four control points... 

How are the Bj k(t) functions computed An efficient way is to use the Cox-DeBoor algorithm [18,19] which is based on a recursive formulation of the Bj,k(t) functions. Using a cubic B-spline to illustrate, it is possible to write out the recursive steps explicitly ... 

We derive daily zero-coupon yield curves from five countries of the Eurozone (France, Germany, Italy, Spain, and the Netherlands) during the period from 2 January 2001 to 21 August 2002, using zero-coupon rates with 26 different maturities ranging from one month to 30 years. The yield curves are extracted from daily Treasury bond market prices by using a standard cubic B-splines method. Our input baskets are composed of... 

Molecular dynamics was performed at constant temperature with AMBER 4.1 all-atom force field [121] and Particle Mesh Ewald method (PME) was used for the calculation of electrostatic interactions [122]. This is a fast implementation of the Ewald summation method for calculating the full electrostatic energy of a unit cell in a macroscopic lattice of repeating images. The PME grid spacing was 1.0A. It was interpolated on a cubic B-spline, with the direct set tolerance set to 0.000001. Periodic boundary conditions were imposed in all directions. All solute-solute non-bonded interactions were calculated without jmy cut-off distance, while a non-bonded residue based cutoff distance of 9A was used for the solvent-solvent and for the solute-solvent interactions. The non-bonded pair list was updated every 20 steps and the... 

We will use uniform translates of the tensor product of cubic B-splines that are dilated according to a resolution parameter N. These translates constitute a basis for a finite dimensional subspace V of S. Since these basis functions have finite support the Gramian matrix B is sparse. Moreover, due to fundamental properties of cubic B-splines, V2 is included in V. 

We further consider the tensor product of the centered cubic B-spline... 

Fig. 3. Centered cubic B-spline tensor-product in two dimensions, N = 7.

Fig. 6. Cubic B-spline interpolation versus surface spline interpolation.

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