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First order joint differential projective invariants

Leonid Bedratyuk

TL;DR

This work provides a complete algebraic solution for first-order joint projective invariants under the diagonal action of $PGL(3,\mathbb{R})$ on $n$ points in the plane. It constructs an explicit minimal generating set for the field of absolute invariants with proven algebraic independence and shows that the full invariant field is a simple extension of this absolute field by adjoining a single primitive relative invariant $z_n$ of weight $-1/g$ (where $g=\gcd(n,3)$). The authors give explicit forms of the invariants for small $n$ (up to $n=6$) and provide a general inductive construction for $n>6$, together with a closed-form $z_n$; their approach hinges on gradient-lines and dual-plane geometry via determinants $\Delta_{ijk}$. The results offer a principled, algebraic foundation for robust, first-order projective features with potential applications in discrete image analysis and computer vision, while acknowledging practical challenges in computing high-dimensional projective invariants and proposing future numerical strategies.

Abstract

We present a complete algebraic description of the field of first-order joint projective invariants for configurations of \( n \) points in the plane, under the natural diagonal action of the projective group \( PGL(3,\mathbb{R}) \). For \( n > 1 \), we construct an explicit minimal generating set for the field of absolute invariants and prove its algebraic independence. We further determine the structure of the full field of invariants as a simple algebraic extension of field of absolute invariants, generated by a single primitive relative invariant of weight~$-1$, for which we provide a closed-form expression valid for all \( n > 1 \).

First order joint differential projective invariants

TL;DR

This work provides a complete algebraic solution for first-order joint projective invariants under the diagonal action of on points in the plane. It constructs an explicit minimal generating set for the field of absolute invariants with proven algebraic independence and shows that the full invariant field is a simple extension of this absolute field by adjoining a single primitive relative invariant of weight (where ). The authors give explicit forms of the invariants for small (up to ) and provide a general inductive construction for , together with a closed-form ; their approach hinges on gradient-lines and dual-plane geometry via determinants . The results offer a principled, algebraic foundation for robust, first-order projective features with potential applications in discrete image analysis and computer vision, while acknowledging practical challenges in computing high-dimensional projective invariants and proposing future numerical strategies.

Abstract

We present a complete algebraic description of the field of first-order joint projective invariants for configurations of points in the plane, under the natural diagonal action of the projective group \( PGL(3,\mathbb{R}) \). For , we construct an explicit minimal generating set for the field of absolute invariants and prove its algebraic independence. We further determine the structure of the full field of invariants as a simple algebraic extension of field of absolute invariants, generated by a single primitive relative invariant of weight~, for which we provide a closed-form expression valid for all .

Paper Structure

This paper contains 19 sections, 8 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous coordinates and projective geometry of $\mathbb{P}^2$
  4. The plane projective group $PGL(3,\mathbb{R})$
  5. Prolongation of the action to derivatives
  6. Projective Differential Invariants
  7. Joint First-Order Projective Differential Invariants
  8. The Field of Absolute Invariants
  9. Geometric Setup and Notation
  10. The Case $n=3$
  11. The case $n=4$
  12. The cases $n=5$ and $n=6$
  13. The case $n>6$
  14. What happens in the case $n=2$?
  15. Field of Relative Invariants
  16. ...and 4 more sections

Key Result

Theorem 1

Let $T$ be the projective transformation. Then the partial derivatives of the transformed function $\tilde{u}$ are given by:

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Definition
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 5 more