B-form and B-Splines

The building blocks for the B-form of a spline are the B-splines.A B-Spline of Order 4, and the Four Cubic Polynomials from Which It Is Madeshows a picture of such a B-spline, the one with the knot sequence[0 1.5 2.3 4 5], hence of order 4, together with the polynomials whose pieces make up the B-spline. The information for that picture could be generated by the command

bspline([0 1.5 2.3 4 5])

To summarize: The B-spline with knotst(i)≤····≤ t(i+k)is positive on the interval(t(i)..t(i+k))and is zero outside that interval. It is piecewise-polynomial of orderkwith breaks at the sitest(i),...,t(i+k)。These knots may coincide, and the precise多重性governs the smoothness with which the two polynomial pieces join there.

B-Spline Knot Multiplicity

The rule is

knot multiplicity + condition multiplicity =order

For example, for a B-spline of order 3, a simple knot would mean twosmoothness conditions, i.e., continuity of function and first derivative, while a double knot would only leave one smoothness condition, i.e., just continuity, and a triple knot would leave no smoothness condition, i.e., even the function would be discontinuous.

All Third-Order B-Splines for a Certain Knot Sequence with Various Knot Multiplicitiesshows a picture of all the third-order B-splines for a certain mystery knot sequencet。The breaks are indicated by vertical lines. For each break, try to determine its multiplicity in the knot sequence (it is 1,2,1,1,3), as well as its multiplicity as a knot in each of the B-splines. For example, the second break has multiplicity 2 but appears only with multiplicity 1 in the third B-spline and not at all, i.e., with multiplicity 0, in the last two B-splines. Note that only one of the B-splines shown has all its knots simple. It is the only one having three different nontrivial polynomial pieces. Note also that you can tell the knot-sequence multiplicity of a knot by the number of B-splines whose nonzero part begins or ends there. The picture is generated by the following MATLAB^®statements, which use the commandspcolfrom this toolbox to generate the function values of all these B-splines at a fine netx。

t=[0,1,1,3,4,6,6,6]; x=linspace(-1,7,81); c=spcol(t,3,x);[l,m]=size(c); c=c+ones(l,1)*[0:m-1]; axis([-1 7 0 m]); hold on for tt=t, plot([tt tt],[0 m],'-'), end plot(x,c,'linew',2), hold off, axis off

Further illustrated examples are provided by the example ”Construct and Work with the B-form”. You can also use the GUIbspliguito study the dependence of a B-spline on its knots experimentally.

Choice of Knots for B-form

The rule “knot multiplicity + condition multiplicity = order” has the following consequence for the process of choosing a knot sequence for the B-form of a spline approximant. Suppose the splinesis to be of orderk, with basic interval [a。.b], and withinterior breaks ξ₂< ·· ·<ξ_l。Suppose, further, that, at ξ_i, the spline is to satisfy μ_ismoothness conditions, i.e.,

Then, the appropriateknot sequencetshould contain thebreak ξ_iexactlyk– μ_itimes,i=2,...,l。In addition, it should contain the two endpoints,aandb, of the basic interval exactlyktimes. This last requirement can be relaxed, but has become standard. With this choice, there is exactly one way to write each splineswith theproperties described as a weighted sum of the B-splines of orderkwith knots a segment of the knot sequencet。This is the reason for theBinB-spline: B-splines are, in Schoenberg's terminology,basicsplines.

For example, if you want to generate the B-form of a cubic spline on the interval [1 .. 3], with interior breaks 1.5, 1.8, 2.6, and with two continuous derivatives, then the following would be theappropriate knot sequence:

t = [1, 1, 1, 1, 1.5, 1.8, 2.6, 3, 3, 3, 3];

This is supplied byaugknt([1, 1.5, 1.8, 2.6, 3], 4)。If you wanted, instead, to allow for a corner at 1.8, i.e., a possiblejump in the first derivative there, you would triple the knot 1.8, i.e., use

t = [1, 1, 1, 1, 1.5, 1.8, 1.8, 1.8, 2.6, 3, 3, 3, 3];

and this is provided by the statement

t = augknt([1, 1.5, 1.8, 2.6, 3], 4, [1, 3, 1] );