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

The article proves the existence of a maximum of two possible solutions to the initial value problem composed by the planar-perspective equation and an initial condition. This initial value problem has a geometric interpretation. Solutions are curves than pass trough the initial condition which is point of the plane. 1 Description of the problem Let C be a regular curve in the plane R parameterized by the perspective parameterization X(t) = ρ(t)(t, 1), where t ∈ I ⊂ R is an interval and ρ : I 7−→ R is the objective function. A complete description of parametric curves is available in any generic differential geometric book as Kreyszig (1991). Let U(t) be a function of class U(t) ∈ C(I,R) defined in the interval I except for a finite number of points A where the function is of class U(t) ∈ C∞(A,R) and where the first-order derivative of the ρ function is null. The function U(t) is positive function in I and it has the geometric interpretation of being the square of the modulus of the velocity vector of the curve C. It means that U(t) = ||~v(t)||. This function U(t) is known. Consider the next non-linear firstorder ordinary differential equation (ODE ) (strictly speaking, the equation is a differential algebraic equation DAE): ε dρ dt 2 + ρ = U, (1) where ρ(t) is the function that we have to find which is positive or null ρ(t) ≥ 0. The function ε = (1 + t) is a polynomial of degree 4. The variable t is the independent variable. We consider the Initial Value Problem (IVP) (also called the Cauchy problem) composed by the equation 1 (also called planar-perspective reconstruction equation) with the initial condition ρ(t0) = ρ0 in a neighborhood t0 ∈ J ⊂ I. The formal description of an initial value problem appear in the book Tenenbaum and Pollard (1985).