Finite Automata Encoding Piecewise Polynomials

TL;DR

This paper introduces a novel finite automata encoding method for piecewise polynomial functions, extending previous linear-only encoding to functions with arbitrary degrees of smoothness, leveraging hierarchical tensor product B-splines.

Contribution

It proposes a new representation enabling finite automata to encode piecewise polynomial functions of any smoothness degree, expanding their applicability in computational geometry.

Findings

Finite automata can encode piecewise polynomial functions with arbitrary smoothness.

The new encoding extends the class of functions representable by finite automata.

Automata-based methods solve computational problems in hierarchical tensor product B-splines.

Abstract

Finite automata are used to encode geometric figures, functions and can be used for image compression and processing. The original approach is to represent each point of a figure in $\mathbb{R}^n$ as a convolution of its $n$ coordinates written in some base. Then a figure is said to be encoded as a finite automaton if the set of convolutions corresponding to the points in this figure is accepted by a finite automaton. The only differentiable functions which can be encoded as a finite automaton in this way are linear. In this paper we propose a representation which enables to encode piecewise polynomial functions with arbitrary degrees of smoothness that substantially extends a family of functions which can be encoded as finite automata. Such representation naturally comes from the framework of hierarchical tensor product B-splines, which are piecewise polynomials widely utilized in numerical computational geometry. We show that finite automata provide a suitable tool for solving computational problems arising in this framework when the support of a function is unbounded.