Summary

We categorize (pseudo-)differential operators according to specific properties (

Introduction

The L-spline formalism provides a direct connection between splines and differential operators. Similarily, it is well known from wavelet theory that a wavelet analysis can be qualitatively interpreted as a multiscale differentiation process. Our goal in this research is

• to make this fundamental link more explicit—especially, the wavelet aspect of it;
• to provide a general framework for the construction of spline and wavelet approximation spaces associated with a given operator and determine the structure and approximation properties of these spaces;
• to take advantage of this framework to design new splines and wavelets with some desirable properties ( e.g. , controlled smoothness, fractional differentiation behavior, isotropy, directionality, among others).

Main Contribution

A high-level approach to splines is to define them via some suitable generalized differential operator L. Specifically, we say that the continuously defined function s ( x ) is a cardinal L-spline if and only if

L{ s ( x )}=∑ a [ k ]δ( x - k )

In essence, the operator L extracts the spline singularities (stream of Dirac impulses), while acting as an analog-to-digital converter: transformation of s ( x ) into a discrete sequence of weights a [ k ]. The simplest example is the derivative operator which is associated to piecewise-constant splines. The concept remains valid in higher dimensions and for arbitrary lattices.

Within this abstract framework, we have chosen to identify interesting classes of shift-invariant operators based on some general invariance principles. In the case of scale invariance, we have shown that the corresponding class of operators in 1D reduces to a two-parameter family of fractional derivatives. These generalized derivatives lead to the definition of an extended family of fractional splines that has the interesting property of being closed under fractional differentation. If we move to higher dimensions and add rotation invariance, we end up with fractional iterates of the Laplacian, the defining operators of polyharmonic splines. A nice side effect of the scale-invariance property is that the underlying spline functions satisfy scaling relations which makes them suitable for constructing wavelet bases.

Along the same line, we have shown that certain types of Box splines (on a hexagonal grid) could be associated with a special multidimensional iterated derivative operator. This formulation yields explicit formulas for the underlying B-splines, which were not available previously.

The proposed spline formalism also lends itself to the specification of wavelet basis functions that essentially behave like multiscale versions of the underlying operator. We have demonstrated the concept with the exponential-spline wavelets, which can be tuned to replicate the behavior of any ordinary (constant-coefficient) differential operator. We have also considered fractional iterates of the Laplacian which have led to the construction of new, nonseparable wavelets that are nearly isotropic, and act like smoothed versions of the Laplacian.