Solutions of Quadratic First-Order ODEs applied to Computer Vision Problems

TL;DR

This paper studies quadratic first-order ODEs in computer vision, proving existence and uniqueness of solutions related to planar curve reconstruction, and introduces the maximal depth function to identify the most significant solution.

Contribution

It provides new theoretical results on the existence, uniqueness, and extension of solutions for the planar-perspective equation, a key model in computer vision reconstruction problems.

Findings

Proves only two local solutions exist under regular initial conditions.

Introduces the maximal depth function to bound all solutions.

Establishes the uniqueness of the maximal-depth solution.

Abstract

This article is a study about the existence and the uniqueness of solutions of a specific quadratic first-order ODE that frequently appears in multiple reconstruction problems. It is called the \emph{planar-perspective equation} due to the duality with the geometric problem of reconstruction of planar-perspective curves from their modulus. Solutions of the \emph{planar-perspective equation} are related with planar curves parametrized with perspective parametrization due to this geometric interpretation. The article proves the existence of only two local solutions to the \emph{initial value problem} with \emph{regular initial conditions} and a maximum of two analytic solutions with \emph{critical initial conditions}. The article also gives theorems to extend the local definition domain where the existence of both solutions are guaranteed. It introduces the \emph{maximal depth function} as a function that upper-bound all possible solutions of the \emph{planar-perspective equation} and contains all its possible \emph{critical points}. Finally, the article describes the \emph{maximal-depth solution problem} that consists of finding the solution of the referred equation that has maximum the depth and proves its uniqueness. It is an important problem as it does not need initial conditions to obtain the unique solution and its the frequent solution that practical algorithms of the state-of-the-art give.