B Splines

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Learning Project Summary

• Portals: Learning Projects, Engineering and Technology
• School: Computer Science
• Department: Scientific Computing

Content summary

This learning project aims to provide as an introduction to the specialized spline functions known as B (or Basis) splines. B splines have varying applications, including numerical analysis.

Goals

• Define B Spline
• Differentiate types of B Splines
• Approximation using B Splines

Lessons

Lesson 0: Prerequisite

B splines derive from Splines, therefore an understanding of Splines in general is beneficial to completing this lesson. In short, a Spline function approximates another function by defining a set of polynomials

• $S_{0}(x),\ S_{1}(x),\ldots ,S_{n}(x)$

where each of these polynomials defines a specific piece of the resulting Spline. $S_{0}(x)$ might exist on the interval $[0,1]$, $S_{1}(x)$ might exist on the interval $[1,2]$, and so on. The result will be a piecewise approximation to some other exact function.

Lesson 1: Definition

A definition of B splines assumes:

• an infinite set of knots are defined at points along the x-axis (can be spaced uniformally or not), that is,
  • $\ldots <t_{-2}<t_{-1}<t_{0}<t_{1}<t_{2}\ \ldots$
  • $\lim _{i\rightarrow \infty }t_{i}=\infty =-\lim _{i\rightarrow \infty }t_{-i}$

Degree 0 (or constant)

With that in mind, we can now move on to the simplest of B splines, those of degree 0, which are defined as

• $B_{i}^{0}\left(x\right)={\begin{cases}1&t_{i}\leq x<t_{i+1}\\0&-\end{cases}}$

In other words, a degree 0 B spline is equal to 0 at all points except on the interval $\left[t_{i},t_{i+1}\right)$.

It should now be easy to see that a degree 0 Spline can be formed as a weighted linear combination of degree 0 B splines so that,

• $S=\ldots +b_{i-1}B_{i-1}^{0}+b_{i}B_{i}^{0}+b_{i+1}B_{i+1}^{0}+\ldots$
• $-\infty <=i<=\infty$

Degree 1 (or linear)

Logically, the next B spline are those of degree 1, defined as

• $B_{i}^{1}\left(x\right)={\begin{cases}0&x\geq t_{i+2}\ or\ x<t_{i}\\{\frac {x-t_{i}}{t_{i+1}-t_{i}}}&t_{i}\leq x<t_{i+1}\\{\frac {t_{i+2}-x}{t_{i+2}-t_{i+1}}}&t_{i+1}\leq x<t_{i+2}\end{cases}}$

This might seem difficult to visualize at first glance, but its actually quite easy. Just like $B_{i}^{0}$, it is 0 at quite nearly all points. However, we now have the two intervals, $\left[t_{i},t_{i+1}\right)$ and $\left[t_{i+1},t_{i+2}\right)$, at which $B_{i}^{1}\neq 0$.

On the first interval it is easy to see that, substituting $t_{i}$ and $t_{i+1}$ give 0 and 1, respectively. Thus, this function yields an upward sloping line, with a maximum height of 1. Similarly, the second interval yields a downward sloping line, starting from the point that the first interval terminates.

Again, similarly to $B_{i}^{0}$, it should now be easy to see that a degree 1 Spline can be formed as a weighted linear combination of degree 1 B splines so that,

• $S=\ldots +b_{i-1}B_{i-1}^{1}+b_{i}B_{i}^{1}+b_{i+1}B_{i+1}^{1}+\ldots$
• $-\infty <=i<=\infty$

Degree k (or quadratic and above)

The higher degree B splines, and actually including $B_{i}^{1}$, are defined as

• $B_{i}^{k}\left(x\right)=\left({\frac {x-t_{i}}{t_{i+k}-t_{i}}}\right)B_{i}^{k-1}\left(x\right)+\left({\frac {t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}}\right)B_{i+1}^{k-1}\left(x\right)$

Lesson 2: Approximation

We have seen how B splines can be used to construct general Spline. Now we will discuss a process for approximating a generic function by using B splines.

Schoenberg's Approximation

This specific approximation utilizes $B_{i}^{2}$ (or quadratic B splines) to approximate a function with $S^{2}$ (or a quadratic Spline). The approximation is defined as

• $S\left(x\right)=\ldots +f\left(\tau _{i-1}\right)B_{i-1}^{2}+f\left(\tau _{i}\right)B_{i}^{2}+f\left(\tau _{i+1}\right)B_{i+1}^{2}+\ldots$
• $-\infty <=i<=\infty$
• $\tau _{i}={\frac {1}{2}}\left(t_{i+1}+t_{i+2}\right)$, or the average of the next two knots

In real life, we would only approximate the function over a specific interval $\left[a,b\right]$.

Active participants

Active participants in this Learning Group

Mathematics since 00:11, 11 November 2013‎ --Marshallsumter (discuss • contribs) 23:48, 11 May 2019 (UTC)

Inactive participants

• Jlietz 18:04, 12 December 2006 (UTC)