I have been reading literature about polynomial interpolation (Lagrange) where the principles are described around nodes. The literature I have read about spline interpolation, however, talks only about knots. From this, it is not entirely clear to me what the difference is. It is my impression that nodes are the known points between which you wish to interpolate while knots are the points where the spline segments of a spline curve meet. Is this correct? It is my impression that the nodes and knots typically coincide which could explain why the literature I have read does not seem to mention nodes in relation to splines?

Update: In particular, I have studied B-splines in Unser, Splines - A Perfect Fit for Signal and Image Processing. This paper, which is written in a wonderfully intuitive way for a signal processing engineer like me to understand, only mentions knots - not nodes. The paper, however, talks about samples of the signal $s(k)$ we wish to interpolate (where the knots are $k$). Comparing to my understanding of nodes from Lagrange interpolation, it seems the signal samples $s(k)$ play the role of those nodes and so, the knots $k$ coincide with the locations of these samples. Is this the correct interpretation?

Various other texts seem to talk about nodes and knots as two distinctly different things and some also talk about spline control points which seem to not have been necessary to introduce in Unser's paper?

  • $\begingroup$ Did my answer clear things up, or is anything still unclear? $\endgroup$ – cfh Apr 23 '15 at 9:09
  • $\begingroup$ @cfh I think I understand it better now. $\endgroup$ – Thomas Arildsen Apr 23 '15 at 11:52

Let's say you have a vector space (which you want to interpolate in) $\Phi$ of dimension $n$.

Then, the nodes would be $n$ points $(\xi_1, \ldots, \xi_n)$ in the domain of your basis functions which satisfy the following: for arbitrary given values $(y_1, \ldots, y_n)$, you can find a unique $f \in \Phi$ such that $$ f(\xi_i) = y_i \qquad \forall i=1,\ldots,n. $$ This property is called unisolvency of the points $(\xi_i)$ for the space $\Phi$. You can call any set which is unisolvent a set of interpolation nodes, since you can do interpolation in them. This is a general concept, no matter how the space looks like.

Knots are something which is particular to the way splines are constructed. For a sequence of knots, $(t_1, \ldots, t_m)$, a spline is a function which is polynomial when restricted to each nonempty knot span $(t_i, t_{i+1})$ and satisfies some additional continuity assumptions in the knots.

So knots and nodes are not the same. For a spline space, however, it is common to choose the interpolation nodes as certain averages of the knots. See for instance the definition of Gréville abscissae, which are just particular interpolation nodes, here.

However, choosing the Gréville abscissae is not the only possible choice. In fact, the Schönberg-Whitney theorem states that, if you choose the $i$-th interpolation node $\xi_i$ to lie in the interior of the support of the $i$-th B-spline function, then the resulting nodes $(x_i)$ are unisolvent.

  • $\begingroup$ The Gréville abscissae definition of nodes does not seem to cover the interpretation of spline nodes I have acquired from reading the paper mentioned in my (now extended) question. I am still not sure how to understand nodes in the context of splines. To put it another way: what is it called when a spline is calculated to pass through values observed at the knot locations? $\endgroup$ – Thomas Arildsen Apr 22 '15 at 12:54
  • $\begingroup$ @ThomasArildsen: A spline typically has more knots than basis functions, so you can't do that. It would be an overdetermined problem. It only works for linear splines. $\endgroup$ – cfh Apr 22 '15 at 12:56
  • $\begingroup$ I'm looking at the paper now to understand what's going on. $\endgroup$ – cfh Apr 22 '15 at 12:59
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    $\begingroup$ @ThomasArildsen: Ok, I understand the confusion now. The paper talks about splines of maximum continuity defined over the entire real line. In this case, there is indeed one knot per basis function (both can be enumerated in $\mathbb Z$). The missing piece to the puzzle is the Schönberg-Whitney theorem, which I have now added to my answer. I hope everything is clear now. $\endgroup$ – cfh Apr 22 '15 at 13:09
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    $\begingroup$ The usual way to define splines on a finite interval is to use so-called open knot vectors, that is, you use repeated knots at the start and end of the interval. The resulting B-splines then look like this: support.sas.com/documentation/cdl/en/statug/63033/HTML/default/… $\endgroup$ – cfh Apr 23 '15 at 12:23

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