M-Scale Relation of B-Splines
Illustration of the multiresolution embedding of the B-spline \(\beta.\)
A polynomial B-spline is a real function notated \(\beta^{n}:{\mathbb{R}}\rightarrow{\mathbb{R}},x\mapsto\beta^{n}(x),\) where its nonnegative integer degree \(n\in{\mathbb{N}}\) is typeset as a superscript (not a power). This function has many remarkable properties; here, we focus on multiresolution ones. In particular, it can be established that B-splines satisfy the M-scale equality
where \(M\in{\mathbb{N}}+1\) is a positive integer that gives the scale parameter.
This equality, also called the M-scale relation, tells us that a B-spline at its nominal scale (the left-hand-side member) can be expressed as a finite weighted sum of shifted B-splines of same degree at scale \(M\) (the right-hand-side member). Indeed, the free variable \(\color{blue}x\) is multiplied by \(\color{blue}M\) on the right, but not on the left. The weights are the coefficients \(h^{n}_{M}[k]/M^{n}\) while the shifts \((\frac{\left(M-1\right)\,\left(n+1\right)}{2}-k)\) are integer-valued when at least one of \(M\) or \(n\) is odd.
The M-scale relation has numerous applications. It is invaluable in the development of algorithms that rely on a multiresolution approach to gradually establish the solution of certain problems. It also has fundamental links with wavelets, which allows for the decomposition of a signal into scale-related local contributions.
In the following Jupyter Lab notebook, we show the contribution of each term of the right-hand-side of the M-scale equality.
Jupyter Lab notebook