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A First Look at Chebyshev-Sobolev Series for Digital Ink

Deepak Singh Kalhan, Stephen M. Watt

TL;DR

The paper addresses robust, resolution-independent digital ink representation by modeling ink traces as parametric plane curves and introducing a Chebyshev-Sobolev series framework. It defines a Chebyshev-Sobolev inner product and constructs the associated polynomials $S_{\lambda i}$ to project handwriting traces onto a stable coefficient space, enabling efficient, model-based matching that scales with degree $d$ rather than sample count. A practical matching algorithm computes arc-length reparameterizations, derives coefficients for $C_X(s)$ and $C_Y(s)$ in the Chebyshev-Sobolev basis, centers the representation by dropping the $T_0$ term, and compares traces via Euclidean distance in coefficient space. Experimental results on InkML and UCI pendigits show that Chebyshev-Sobolev representations can outperform Legendre-Sobolev baselines, with strong performance in both representation accuracy and $k$-NN handwriting recognition, suggesting improved robustness and efficiency for digital ink tasks.

Abstract

Considering digital ink as plane curves provides a valuable framework for various applications, including signature verification, note-taking, and mathematical handwriting recognition. These plane curves can be obtained as parameterized pairs of approximating truncated series (x(s), y(s)) determined by sampled points. Earlier work has found that representing these truncated series (polynomials) in a Legendre or Legendre-Sobolev basis has a number of desirable properties. These include compact data representation, meaningful clustering of like symbols in the vector space of polynomial coefficients, linear separability of classes in this space, and highly efficient calculation of variation between curves. In this work, we take a first step at examining the use of Chebyshev-Sobolev series for symbol recognition. The early indication is that this representation may be superior to Legendre-Sobolev representation for some purposes.

A First Look at Chebyshev-Sobolev Series for Digital Ink

TL;DR

The paper addresses robust, resolution-independent digital ink representation by modeling ink traces as parametric plane curves and introducing a Chebyshev-Sobolev series framework. It defines a Chebyshev-Sobolev inner product and constructs the associated polynomials to project handwriting traces onto a stable coefficient space, enabling efficient, model-based matching that scales with degree rather than sample count. A practical matching algorithm computes arc-length reparameterizations, derives coefficients for and in the Chebyshev-Sobolev basis, centers the representation by dropping the term, and compares traces via Euclidean distance in coefficient space. Experimental results on InkML and UCI pendigits show that Chebyshev-Sobolev representations can outperform Legendre-Sobolev baselines, with strong performance in both representation accuracy and -NN handwriting recognition, suggesting improved robustness and efficiency for digital ink tasks.

Abstract

Considering digital ink as plane curves provides a valuable framework for various applications, including signature verification, note-taking, and mathematical handwriting recognition. These plane curves can be obtained as parameterized pairs of approximating truncated series (x(s), y(s)) determined by sampled points. Earlier work has found that representing these truncated series (polynomials) in a Legendre or Legendre-Sobolev basis has a number of desirable properties. These include compact data representation, meaningful clustering of like symbols in the vector space of polynomial coefficients, linear separability of classes in this space, and highly efficient calculation of variation between curves. In this work, we take a first step at examining the use of Chebyshev-Sobolev series for symbol recognition. The early indication is that this representation may be superior to Legendre-Sobolev representation for some purposes.
Paper Structure (14 sections, 23 equations, 6 figures)

This paper contains 14 sections, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Parameterized Curves in an Orthogonal Basis
  4. Legendre Inner Product and Legendre Basis
  5. Legendre-Sobolev Inner Product
  6. Chebyshev-Sobolev Inner Product
  7. Chebyshev Inner Product
  8. Chebyshev-Sobolev Inner Product
  9. A Matching Algorithm
  10. Experiments on Digital Ink
  11. Handwriting representation using Chebyshev-Sobolev series
  12. Representation error of approximated curve using Chebyshev-Sobolev series
  13. Handwriting recognition using $k$ nearest neighbours
  14. Conclusions

Figures (6)

  • Figure 1: Handwriting representation test on Ink ML dataset
  • Figure 2: Handwriting representation test on UCI pen digits dataset
  • Figure 3: Representation error on UCI pen digits dataset
  • Figure 4: Representation error on Ink ML dataset
  • Figure 5: Representation error using Chebyshev-Sobolev series
  • ...and 1 more figures