Fitting Egg-Shapes to Discretized Object Boundaries

This paper presents a novel method for fitting egg-shapes to discrete sets of boundary points. Egg-shapes extend ellipses by assigning a positive weight to one of the two focal points. Fitting of egg-shapes thus requires optimization of 6 parameters. Our approach simplifies this to a 1D parameter space exploration. First, we utilize a least square algorithm to fit an ellipse to the boundary. While the desired egg-shape shares the orientation of the major axis and to a certain extent also the size of the ellipse, its fine-tuning to the boundary is more involved than merely adjusting the focal weight. To this end, we establish a relation between the eccentricity of the ellipse and the two shape-defining parameters of the closest egg-shapes. Subsequently, we utilize this relationship to iterate over a 1D space of closest egg-shape candidates while assessing their fitness to the boundary. Our results underscore the benefits of using egg-shapes over ellipses for representing a spectrum of real-world objects.