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Home » GATE Study Material » Mathematics » Numerical Analysis » Interpolation and Polynomial Approximation » B-Splines

B-Splines

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B-Splines

B-Splines

   

Cubic spline construction using the B-spline function.

    Under special circumstances a basis set of splines [Graphics:Images/B-SplinesMod_gr_1.gif] can be used to form a cubic B-spline function.  This concept makes the construction of a spline very easy, it is just at linear combination:   

        [Graphics:Images/B-SplinesMod_gr_2.gif]

All we need to do is solve for the  [Graphics:Images/B-SplinesMod_gr_3.gif] coefficients [Graphics:Images/B-SplinesMod_gr_4.gif].  And to make things even more appealing, the linear system to be solved has a tri-diagonal "appearance":   

        [Graphics:Images/B-SplinesMod_gr_5.gif][Graphics:Images/B-SplinesMod_gr_6.gif] = [Graphics:Images/B-SplinesMod_gr_7.gif].  

Caution must prevail when solving this underdetermined system of  [Graphics:Images/B-SplinesMod_gr_8.gif]  equations in  [Graphics:Images/B-SplinesMod_gr_9.gif] unknowns.  Two end conditions must be supplied for constructing the coefficients [Graphics:Images/B-SplinesMod_gr_10.gif] and [Graphics:Images/B-SplinesMod_gr_11.gif].  These end conditions are specially crafted to form either a natural cubic spline or a clamped cubic spline.  

    How can such an elegant construction possible ?  It's simple, you must have a uniform grid of points  [Graphics:Images/B-SplinesMod_gr_12.gif] on the interval  [Graphics:Images/B-SplinesMod_gr_13.gif].  The uniform spacing is [Graphics:Images/B-SplinesMod_gr_14.gif] and the interpolation nodes to be used are [Graphics:Images/B-SplinesMod_gr_15.gif] for  [Graphics:Images/B-SplinesMod_gr_16.gif].  The equally spaced abscissa's are

        [Graphics:Images/B-SplinesMod_gr_17.gif]  

The corresponding ordinates are  [Graphics:Images/B-SplinesMod_gr_18.gif]  and the data points are  [Graphics:Images/B-SplinesMod_gr_19.gif].  They are often times referred to as the knots because this is where we join the piecewise cubics, like pieces of string "knotted" together to form a larger piece of string.  If this is your situation, then the B-spline construction is for you.  

Caveat. If you have unequally spaced points, then this is construction does not apply and construction of the cubic spline require a more cumbersome algorithm because each piecewise cubic will need to be individually crafted in order to meet all the conditions for a cubic spline.  

 

The basic B-spline function.

    Construction of cubic B-spline interpolation can be accomplished by first considering the following basic function.

The function [Graphics:Images/B-SplinesMod_gr_20.gif] is a piecewise continuous on the interval [Graphics:Images/B-SplinesMod_gr_21.gif] , it is zero elsewhere.  In advanced courses this simple concept is glamorized by saying that [Graphics:Images/B-SplinesMod_gr_22.gif] is a function with "compact support."  That is, it is supported (or non-zero) only on a small set.  

 

[Graphics:Images/B-SplinesMod_gr_23.gif]

The graph of the function [Graphics:Images/B-SplinesMod_gr_24.gif] .

[Graphics:Images/B-SplinesMod_gr_25.gif]

[Graphics:Images/B-SplinesMod_gr_26.gif]

[Graphics:Images/B-SplinesMod_gr_27.gif] Global`B

 

 
[Graphics:Images/B-SplinesMod_gr_28.gif]
 
[Graphics:Images/B-SplinesMod_gr_29.gif]
 
[Graphics:Images/B-SplinesMod_gr_30.gif]
 
[Graphics:Images/B-SplinesMod_gr_31.gif]
 
[Graphics:Images/B-SplinesMod_gr_32.gif]
 
[Graphics:Images/B-SplinesMod_gr_33.gif]
 

 

 

Verify that  [Graphics:Images/B-SplinesMod_gr_34.gif] is a cubic spline.

Each part of  [Graphics:Images/B-SplinesMod_gr_35.gif] is piecewise cubic.
Are the functions  [Graphics:Images/B-SplinesMod_gr_36.gif],  [Graphics:Images/B-SplinesMod_gr_37.gif]  and   [Graphics:Images/B-SplinesMod_gr_38.gif] continuous for all  [Graphics:Images/B-SplinesMod_gr_39.gif] ?
Since  [Graphics:Images/B-SplinesMod_gr_40.gif]  is composed of the piecewise functions [Graphics:Images/B-SplinesMod_gr_41.gif],[Graphics:Images/B-SplinesMod_gr_42.gif],[Graphics:Images/B-SplinesMod_gr_43.gif],[Graphics:Images/B-SplinesMod_gr_44.gif],[Graphics:Images/B-SplinesMod_gr_45.gif],[Graphics:Images/B-SplinesMod_gr_46.gif], all that is necessary is to see if they join up properly at the nodes  [Graphics:Images/B-SplinesMod_gr_47.gif].  However, this will take 15 computations to verify.  This is where Mathematica comes in handy.  Follow the link below if you are interested in the proof.  

 

The above proof that  [Graphics:Images/B-SplinesMod_gr_48.gif]  is a cubic spline used the formulas [Graphics:Images/B-SplinesMod_gr_49.gif],[Graphics:Images/B-SplinesMod_gr_50.gif],[Graphics:Images/B-SplinesMod_gr_51.gif],[Graphics:Images/B-SplinesMod_gr_52.gif],[Graphics:Images/B-SplinesMod_gr_53.gif],[Graphics:Images/B-SplinesMod_gr_54.gif].
It is the analytic way to do things and illustrates "precise mathematical" reasoning.

If you trust graphs, then just look at the graphs of  [Graphics:Images/B-SplinesMod_gr_55.gif],  [Graphics:Images/B-SplinesMod_gr_56.gif],  and  [Graphics:Images/B-SplinesMod_gr_57.gif] and try to determine if they are continuous.  
However, there might be problems lurking about.  If you seek the mathematical truth you should look at the next cell link.  

 

Let's translate the B-spline over to the node [Graphics:Images/B-SplinesMod_gr_58.gif] and use the uniform step size [Graphics:Images/B-SplinesMod_gr_59.gif].  

 

[Graphics:Images/B-SplinesMod_gr_60.gif] [Graphics:Images/B-SplinesMod_gr_61.gif]

Now form the linear combination for the spline.

[Graphics:Images/B-SplinesMod_gr_62.gif] [Graphics:Images/B-SplinesMod_gr_63.gif] [Graphics:Images/B-SplinesMod_gr_64.gif] [Graphics:Images/B-SplinesMod_gr_65.gif]

At each of the nodes [Graphics:Images/B-SplinesMod_gr_66.gif] for  [Graphics:Images/B-SplinesMod_gr_67.gif] computation will reveal that

[Graphics:Images/B-SplinesMod_gr_68.gif] [Graphics:Images/B-SplinesMod_gr_69.gif] [Graphics:Images/B-SplinesMod_gr_70.gif] [Graphics:Images/B-SplinesMod_gr_71.gif] [Graphics:Images/B-SplinesMod_gr_72.gif] [Graphics:Images/B-SplinesMod_gr_73.gif] [Graphics:Images/B-SplinesMod_gr_74.gif] [Graphics:Images/B-SplinesMod_gr_75.gif] [Graphics:Images/B-SplinesMod_gr_76.gif] [Graphics:Images/B-SplinesMod_gr_77.gif]

If the B-spline is to go through the points  [Graphics:Images/B-SplinesMod_gr_78.gif] for  [Graphics:Images/B-SplinesMod_gr_79.gif], then the following equations must hold true

    [Graphics:Images/B-SplinesMod_gr_80.gif]   for   [Graphics:Images/B-SplinesMod_gr_81.gif].  

For the natural cubic spline, we want the second derivatives to be zero at the left endpoint  [Graphics:Images/B-SplinesMod_gr_82.gif].  

Therefore we must have   [Graphics:Images/B-SplinesMod_gr_83.gif].  

Computation will reveal that

    [Graphics:Images/B-SplinesMod_gr_84.gif]

To construct the natural cubic spline, we must have

    [Graphics:Images/B-SplinesMod_gr_85.gif].

We can solve this equation for the spline coefficient [Graphics:Images/B-SplinesMod_gr_86.gif]

[Graphics:Images/B-SplinesMod_gr_87.gif] [Graphics:Images/B-SplinesMod_gr_88.gif]

For the natural cubic spline, we want the second derivatives to be zero at the right endpoint  [Graphics:Images/B-SplinesMod_gr_89.gif].   

Therefore we must have   [Graphics:Images/B-SplinesMod_gr_90.gif].  

Computation will reveal that

    [Graphics:Images/B-SplinesMod_gr_91.gif]

To construct the natural cubic spline, we must have

    [Graphics:Images/B-SplinesMod_gr_92.gif].

We can solve this equation for the spline coefficient [Graphics:Images/B-SplinesMod_gr_93.gif]

[Graphics:Images/B-SplinesMod_gr_94.gif] [Graphics:Images/B-SplinesMod_gr_95.gif]

The above construction shows how to calculate all the coefficients [Graphics:Images/B-SplinesMod_gr_96.gif].

[Graphics:Images/B-SplinesMod_gr_97.gif] [Graphics:Images/B-SplinesMod_gr_98.gif] [Graphics:Images/B-SplinesMod_gr_99.gif] [Graphics:Images/B-SplinesMod_gr_100.gif]

 

[Graphics:Images/B-SplinesMod_gr_101.gif]
[Graphics:Images/B-SplinesMod_gr_102.gif]
[Graphics:Images/B-SplinesMod_gr_103.gif]
[Graphics:Images/B-SplinesMod_gr_104.gif]
[Graphics:Images/B-SplinesMod_gr_105.gif]
[Graphics:Images/B-SplinesMod_gr_106.gif]
[Graphics:Images/B-SplinesMod_gr_107.gif]
 

 

 

Method I.  B-spline construction using equations.

Illustration using 7 knots.

The following example uses  n = 6.  There are  n+3 = 9  equations to solve and  n+1 = 7  data points or knots.  First set up the 9 equations to be solved.  

[Graphics:Images/B-SplinesMod_gr_108.gif] [Graphics:Images/B-SplinesMod_gr_109.gif] [Graphics:Images/B-SplinesMod_gr_110.gif]



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