From circles to generalized conics: : Enriching the properties of 2D shape representation and description

Abstract

Like every type of human activity, research is intertwined with knowledge and experience. It aims at systematic exploration of phenomena in order to expand views and possibilities of solving a particular problem. In computer vision, the conventional way of measuring the distance between two objects is to find a pair of closest points belonging to them. When using the Euclidean metric, the equidistant set of a point is a circle. This thesis presents the alternative solutions (with an emphasis on 2D space), involving the implications for shape representation and description. They rely on other types of equidistant sets: conics (ellipse and hyperbola) and generalized conics (multifocal ellipse and hyperbola). The first solution rests on the fact that a circle is a special case of an ellipse, implying a pair of coinciding focal points. Among the variety of ellipse properties, a constant distance sum to the pair of focal points enables defining a metric. It measures the distance between a point and a line segment bounded by the focal points. This metric defines an increment in the line segment length when moving from one focal point to another through the point of interest. The immediate advantages over the classical approach are computational efficiency and independence of the line segment discretization. The second solution exhibits the key property of multifocal ellipse – each of its points has the same distance sum to the set of focal points. This concept alternatively defines the distance from a point to the collection of points. Such an interpretation is valuable in optimization problems, which can, in turn, benefit from efficient image processing techniques for solving their tasks. The third solution reflects a necessity in image processing techniques like skeletonization not only to find the distance to an object but also to find a set of points that are equidistant from a pair of objects. By definition, a multifocal hyperbola contains the points that have a constant difference between the distance sums to the pair of point sets. Assuming the focal point to be any geometric shape, the multifocal hyperbola with the associated zero distance value is an equidistant set to the pair of objects. The central notion behind this thesis is a generalization. Starting with a circle, a special case of an ellipse, it considers a generalized conic – a further conceptual extension. This transformation is reflected in the analysis of the geometric properties of these curves: from the conventional facts to the innovative findings. Such an approach enables explaining the existing and proposed methodologies through a prism of the single theoretical framework.