Inverse of B-Spline Sequences
Illustration of the B-spline inverse sequence \(\left(b^{-1}\right)\) and its periodized version.
Usefulness
B-spline inverse sequences are a crucial ingredient of spline processing. They play a particularly important role when one desires to build a uniform spline that interpolates given discrete data. The splinekit library provides access to such sequences, both in non-periodic and periodic cases.
Non-Periodic Case
B-splines are continuously defined functions \(\beta^{n}:{\mathbb{R}}\rightarrow{\mathbb{R}},\) where the superscript \(n\in{\mathbb{N}}\) is the nonnegative integer degree of a B-spline. Since B-splines map any real argument \(x\in{\mathbb{R}}\) to \(\beta^{n}(x)\in{\mathbb{R}},\) they can also map any integer argument \(k\in{\mathbb{Z}}\) to \(\beta^{n}(k).\) This corresponds to the integer samples of the B-spline and forms the sequence \(\left(b^{n}[k]\right)_{k\in{\mathbb{Z}}}=\left(\beta^{n}(k)\right)_{k\in{\mathbb{R}}\cap{\mathbb{Z}}}.\) This sequence has a finite support; more precisely, it turns out that
One question that arises now is whether the sequence \(b^{n}\) possesses a discrete-convolution inverse sequence. Let us notate such inverse sequence as \(\left(b^{n}\right)^{-1}.\) What we are primarily asking is whether there exists \(\left(b^{n}\right)^{-1}\) such that
where the notation \({\mathbf{[\![}}\cdot\,{\mathbf{]\!]}}\) is that of the Iverson bracket. (From a mathematical point of view, let us observe that the convolution is always well-defined because of the finite-support property of \(b^{n}.\))
The answer to our primary question is yes. Indeed, for \(n>1\) there are infinitely many such sequences, parameterized by \(2\,\left\lfloor\frac{n}{2}\right\rfloor\) free numbers. Among all of them, however, only one has the finite energy \(\sum_{k\in{\mathbb{Z}}}\,\left(\left(b^{n}\right)^{-1}[k]\right)^{2}\in{\mathbb{R}}.\) We show now this sequence for a few degrees.
Jupyter Lab notebook
Periodic Case
Consider now that the convolution is a periodic one, with positive integer period \(K.\) We’re asking about the existence of a vector \(\left(\left(b_{K}^{n}\right)^{-1}[k]\right)_{k=0}^{K-1}\) such that
where the periodization of \(b^{n}\) defines the vector \(\left(b_{K}^{n}[k]\right)_{k=0}^{K-1}=\left(\sum_{p\in{\mathbb{Z}}}\,b^{n}[p\,K+k]\right)_{k=0}^{K-1}.\) It can be verified that the periodized version of \(\left(b^{n}\right)^{-1}\) described by the vector
satisfies our periodic-convolution requirement. We show now this sequence for a few periods and degrees.
Jupyter Lab notebook