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Well-Conditioned Polynomial Representations for Mathematical Handwriting Recognition

Robert M. Corless, Deepak Singh Kalhan, Stephen M. Watt

TL;DR

The paper addresses robustly representing handwritten mathematical strokes as parametric polynomials and systematically analyzes how the choice of basis (Legendre, Chebyshev, Legendre-Sobolev, Chebyshev-Sobolev) and polynomial degree affect conditioning, accuracy, and computational cost. It derives a Sobolev-norm bound $||f - g||_s ≤ sqrt(n) (1 + μ ||D||) ||f - g||_∞$ and investigates the impact of the differentiation matrix on stability, supported by experiments on real handwriting data. Empirical results show Sobolev bases significantly restrain coefficient growth and stabilize higher-degree representations, with Chebyshev-Sobolov yielding the best recognition accuracy (~97.5–98% around degree 12) at the cost of higher computation time. The work provides actionable guidance for selecting bases and degrees to balance efficiency and robustness in digital ink analysis and mathematical handwriting recognition.

Abstract

Previous work has made use of a parameterized plane curve polynomial representation for mathematical handwriting, with the polynomials represented in a Legendre or Legendre-Sobolev graded basis. This provides a compact geometric representation for the digital ink. Preliminary results have also been shown for Chebyshev and Chebyshev-Sobolev bases. This article explores the trade-offs between basis choice and polynomial degree to achieve accurate modeling with a low computational cost. To do this, we consider the condition number for polynomial evaluation in these bases and bound how the various inner products give norms for the variations between symbols.

Well-Conditioned Polynomial Representations for Mathematical Handwriting Recognition

TL;DR

The paper addresses robustly representing handwritten mathematical strokes as parametric polynomials and systematically analyzes how the choice of basis (Legendre, Chebyshev, Legendre-Sobolev, Chebyshev-Sobolev) and polynomial degree affect conditioning, accuracy, and computational cost. It derives a Sobolev-norm bound and investigates the impact of the differentiation matrix on stability, supported by experiments on real handwriting data. Empirical results show Sobolev bases significantly restrain coefficient growth and stabilize higher-degree representations, with Chebyshev-Sobolov yielding the best recognition accuracy (~97.5–98% around degree 12) at the cost of higher computation time. The work provides actionable guidance for selecting bases and degrees to balance efficiency and robustness in digital ink analysis and mathematical handwriting recognition.

Abstract

Previous work has made use of a parameterized plane curve polynomial representation for mathematical handwriting, with the polynomials represented in a Legendre or Legendre-Sobolev graded basis. This provides a compact geometric representation for the digital ink. Preliminary results have also been shown for Chebyshev and Chebyshev-Sobolev bases. This article explores the trade-offs between basis choice and polynomial degree to achieve accurate modeling with a low computational cost. To do this, we consider the condition number for polynomial evaluation in these bases and bound how the various inner products give norms for the variations between symbols.

Paper Structure

This paper contains 10 sections, 2 theorems, 17 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Parametric Curve Representation
  3. Choice of Basis
  4. Experiments
  5. Experimental setting
  6. Coefficient Norms Across Polynomial Bases
  7. Computation Time Analysis of Polynomial Bases
  8. Recognition Rate
  9. Approximation of Handwritten symbol
  10. Conclusions

Key Result

Theorem 1

Let $f(x)$ and $g(x)$ be polynomials expressed in an orthogonal basis $\{P_i(x)\}$, with coefficient vectors $\mathbf{f}$ and $\mathbf{g}$, respectively. Suppose the polynomials and their derivatives are evaluated on the interval $[0,1]$, and let $\mathbf{D}$ be the differentiation matrix correspond where $\mu$ is the weight for the derivative term.

Figures (6)

  • Figure 1: Cumulative norms for degrees 5--10.
  • Figure 2: Cumulative norms for degrees 10--15.
  • Figure 3: Cumulative norms for degrees 15--20.
  • Figure 4: Computation time per sample as a function of polynomial degree for different polynomial bases (Legendre, Legendre-Sobolev, Chebyshev, Chebyshev-Sobolev).
  • Figure 5: Recognition rate (accuracy) as a function of polynomial degree for different polynomial bases, with error bars showing variability.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2