Constructors of Periodic One-Dimensional Splines

How to create a periodic one-dimensional spline that interpolates data samples.

Periodic One-Dimensional Splines

The class PeriodicSpline1D of the splinekit library handles continuously defined real functions \(f:{\mathbb{R}}\rightarrow{\mathbb{R}},x\mapsto f(x).\) As maintained by this class in the library, these functions have a few remarkable properties.

  • The function \(f\) is periodic, with positive integer period \(K\in{\mathbb{N}}+1\) such that \(\forall x\in{\mathbb{R}}:f(x)=f(x+K).\)

  • The function \(f\) is a uniform polynomial spline of nonnegative degree \(n\in{\mathbb{N}}\) and shift \(\delta x\in{\mathbb{R}},\) which we write as

\[f(x)=\sum_{k\in{\mathbb{Z}}}\,c[{k\bmod K}]\,\beta^{n}(x-\delta x -k).\]

There, \({\mathbf{c}}\) is a vector \(\left(c[k]\right)_{k=0}^{K-1}\) of \(K\) arbitrary coefficients and \(\beta^{n}\) is a polynomial B-spline. (B-splines have a finite support and the infinite sum is necessarily well-behaved.) Because the function \(f\) is a weighted sum of B-splines shifted by \(\delta x+k,\) it inherits from these B-splines important properties such as \(n\)-times differentiablility and \(\left(n-1\right)\)-times continuous differentiablility. Moreover, because B-splines have the highest order of approximation for their support, the function \(f\) is guaranteed to offer a faithful (in a mathematically precise sense) and computationally efficient representation of data. Finally, the determination of \(f(x)\) at any \(x\in{\mathbb{R}}\) happens in finitely many arithmetic steps.

Defining Parameters

The functions we consider are characterized by parameters that can be chosen freely and independently.

  • Period \(K\)

  • Degree \(n\)

  • Delay \(\delta x\)

  • Spline coefficients \({\mathbf{c}}\)

A large variety of functions can be synthesized by the tuning of these parameters; in particular, by the tuning of the spline coefficients. However, it rarely occurs in practice that these coefficients be known beforehand. It is much more common that one is given a vector \(\left(y[k]\right)_{k=0}^{K-1}=\left(y_{q}\right)_{q=1}^{K}={\mathbf{y}}\in{\mathbb{R}}^{K}\) of \(K\) samples, and that it is desired that the synthesized spline interpolates the samples, a requirement that we write as

\[\left(f(k)\right)_{k=0}^{K-1}=\left(y[k]\right)_{k=0}^{K-1}.\]

Regularizations

Alternatively, it is also sometimes desirable that

\[\left(f(k)\right)_{k=0}^{K-1}\approx\left(y[k]\right)_{k=0}^{K-1},\]

with the approximation being such that it balances the requirement of interpolation with some a priori requirement on the continuously defined \(f.\) The splinekit library offers two mechanisms that result in (desirable) approximate interpolation.

  • The first mechanism acknowleges the fact that exact interpolation cannot be achieved (for generic data) in the very specific case of an even period \(K\in2\,{\mathbb{N}}+2\) combined with a half-integer delay \(\delta x\in{\mathbb{Z}}+1/2.\) In this unstable edge case, stability is recovered by the addition of a constant corrective term to all even samples and by the subtraction of this term from all odd samples. The corrective term is chosen so that the residual difference between \(\left(f(k)\right)_{k=0}^{K-1}\) and \(\left(y[k]\right)_{k=0}^{K-1}\) is minimized in a discrete least-squares sense under the constraint that \(f\) is well-defined. This mechanism is activated by setting regularized = True in the PeriodicSpline1D.from_samples constructor. It can be thought of as a doctoring of the samples that would enforce that the component at the highest discrete frequency of their discrete Fourier transform vanishes.

  • The second mechanism is made available in the PeriodicSpline1D.from_smoothed_samples constructor. It acknowledges the fact that the samples may exhibit a local variability that, sometimes, one is inclined to attribute to noise. In this case, the user can choose to restore smoothness to \(f\) beyond the smoothness inherited by the B-splines, at some cost in the exactitude of interpolation. More precisely, if we let \(\left(\lambda[m]\right)_{m=0}^{n}\) be a vector that represents the weight of variational regularization, then the criterion being minimized is

\[J=\sum_{k=0}^{K-1}\,\left(f(k)-y[k]\right)^{2}+\sum_{m=0}^{n}\,\lambda[m]\,\int_{0}^{K}\,\left(\frac{{\mathrm{d}}^{m}f(x)}{{\mathrm{d}}x^{m}}\right)^{2}\,{\mathrm{d}}x.\]

From Noiseless Samples

We now propose a few lines of code that create and display a spline that interpolates random samples.

Jupyter Lab notebook

Interpolation

From Noisy Samples

We now propose a few lines of code that create and display a smooth spline that performs the approximate interpolation of random samples.

Jupyter Lab notebook

Smoothed interpolation