Fitting Quadratic Curves to Data Points

Abstract

Fitting quadratic curves (a.k.a. conic sections, or conics) to data points (digitized images) is a fundamental task in image processing and computer vision. This problem reduces to minimization of a certain function over the parameter space of conics. Here we undertake a thorough investigation of that space and the properties of the objective function on it. We determine under what conditions that function is continuous and differentiable. We identify its discontinuities and other singularities and determined what effect those have on the performance of minimization algorithms. Our analysis shows that algebraic parameters of conics are more suitable for minimization procedures than more popular geometric parameters, for a number of reasons. First, the space of parameters is naturally compact, thus their estimated values cannot grow indefinitely causing divergence. Second, with algebraic parameters minimization procedures can move freely and smoothly between conics of different types allowing shortcuts and faster convergence. Third, with algebraic parameters one avoids known issues occurring when the fitting conic becomes a circle. To support our conclusions we prove a dozen of mathematical theorems and provide a plenty of illustrations.

Peer Review #2 of "Ellipse packing in two-dimensional cell tessellation: a theoretical explanation for Lewis’s law and Aboav-Weaire’s law (v0.1)"

“…coordinates of the vertices of each polygonal cell ( Fig. 1A) (Chernov et al 2014). For the SiO 2 film, the vertices of polygonal cells were formed by the silicon (Si) atoms.…”

“…(A) Coordinates of the vertices of a polygonal cell and fitted ellipse. We plotted the ellipse using software R plus package Conics (Chernov et al 2014). (B) A diagram shows semimajor-axis (a), semi-minor-axis (b), angle (δ) between line VC and X-axis, angle (θ) of tilt of the major, distance (D VC ) between the center of the ellipse and vertex of polygonal cell , Manuscript to be reviewed Parameters of polygonal cells of 2D structures.…”

Determining ellipses from low-resolution images with a comprehensive image formation model

“…Methods which explicitly decrease the distance between the points and the ellipse curve are considered geometric methods. The quintessential geometric method is orthogonal distance regression, which minimises the orthogonal distance from a point to the curve [2]- [8]. Algebraic methods, on the other hand, try to ensure that the data points satisfy an ellipse implicit equation as accurately as possible.…”

A fast approach for perceptually-based fitting strokes into elliptical arcs

“…An excellent discussion of the state of the art in ellipse fitting can be found in [5]. A distinction is made between Maximum Likelihood, geometric and algebraic approaches [6].…”