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Euclidean and Affine Curve Reconstruction

Jose Agudelo, Brooke Dippold, Ian Klein, Alex Kokot, Eric Geiger, Irina Kogan

TL;DR

This work addresses reconstructing planar curves from curvature data that are invariant under the groups $SE(2)$ and $SA(2)$. It develops a Euclidean reconstruction via successive integrations and an affine reconstruction via Picard iterations, providing explicit formulas, convergence results, and near-miss distance bounds: $d(\C_1, g\C_2) \leq C \delta$ when $\kappa_1$ and $\kappa_2$ are $\delta$-close in appropriate norms, and a parallel bound for the affine case. The study leverages moving-frame invariants ($\kappa$ and $\mu$) and derives both theoretical guarantees and practical algorithms, including constant-curvature conics and power-series methods for analytic curvatures. The results have practical relevance for curve matching and shape analysis in computer vision, and point toward extensions to projective geometry and space curves using the generalized moving-frame framework.

Abstract

We consider practical aspects of reconstructing planar curves with prescribed Euclidean or affine curvatures. These curvatures are invariant under the special Euclidean group and the equi-affine groups, respectively, and play an important role in computer vision and shape analysis. We discuss and implement algorithms for such reconstruction, and give estimates on how close reconstructed curves are relative to the closeness of their curvatures in appropriate metrics. Several illustrative examples are provided.

Euclidean and Affine Curve Reconstruction

TL;DR

This work addresses reconstructing planar curves from curvature data that are invariant under the groups and . It develops a Euclidean reconstruction via successive integrations and an affine reconstruction via Picard iterations, providing explicit formulas, convergence results, and near-miss distance bounds: when and are -close in appropriate norms, and a parallel bound for the affine case. The study leverages moving-frame invariants ( and ) and derives both theoretical guarantees and practical algorithms, including constant-curvature conics and power-series methods for analytic curvatures. The results have practical relevance for curve matching and shape analysis in computer vision, and point toward extensions to projective geometry and space curves using the generalized moving-frame framework.

Abstract

We consider practical aspects of reconstructing planar curves with prescribed Euclidean or affine curvatures. These curvatures are invariant under the special Euclidean group and the equi-affine groups, respectively, and play an important role in computer vision and shape analysis. We discuss and implement algorithms for such reconstruction, and give estimates on how close reconstructed curves are relative to the closeness of their curvatures in appropriate metrics. Several illustrative examples are provided.
Paper Structure (12 sections, 11 theorems, 105 equations, 16 figures)

This paper contains 12 sections, 11 theorems, 105 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Congruence and symmetry of the planar curves
  4. Euclidean moving frame and invariants
  5. Affine moving frame and invariants
  6. Norms and distances
  7. Convergence
  8. Euclidean reconstruction
  9. Affine reconstruction
  10. Conclusion
  11. Appendix
  12. Acknowledgement:

Key Result

Lemma 7

Let $\{f_n\}_{n=1}^\infty$ be a sequence of real valued functions on a domain $P$uniformly convergent to a function $f$ on $P$. Assume further that each of the functions $f_n$, and also $f$ achieves, its maximum value on $P$, then

Figures (16)

  • Figure 1: A special Euclidean transformation is a composition of a rotation and a translation.
  • Figure 2: A special affine transformation is a composition of a unimodular linear transformation and a translation.
  • Figure 3: The $SE(2)$-action preserves the lengths of vectors and the angle between them.
  • Figure 4: The $SA(2)$-action preserves the area of the parallelogram defined by the affine tangent and normal vectors, but not their lengths or the angle between them.
  • Figure 5: Lemma \ref{['lem-closed']} guarantees that the left curve is closed, but does not make any assertion about the right curve.
  • ...and 11 more figures

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3: Planar curve
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Corollary 8
  • proof
  • ...and 21 more