Generalized Perpendicular Bisector and exhaustive discrete circle recognition

This paper presents a generalization of the notion of circumcenter as the intersection of perpendicular bisectors. We define Generalized Perpendicular Bisectors between two regions as an area where each point is the center of at least one circle crossing both regions. This allows us to determine all the possible discrete circle centers that cross a given set of pixels. The possible radii can then easily be determined. This exhaustive digital circle parameter computation is adapted to various types of circles/digitization schemes such as Naïve, Pythagorean and standard/supercover circles.

The intersection of the Generalized Perpendicular Bisector (GPB) of every pair of pixels of a set defines the Generalized Circumcenter (GC). All the circles that cross all pixels of the set have their center in GC. In this example, we have 3 pixels, the corresponding GPBs, with the intersection corresponding to the GC. We also show all the possible circles (in different shades of blue). The black dot and the black circle are one example of a circle centered in the GC cutting the three pixels.Figure optionsDownload as PowerPoint slideHighlights
► The Generalized Circumcenter is defined as intersection of Generalized Bisectors.
► A Generalized Circumcenter contains all centers of circles intersecting a pixel set.
► Exhaustive circle recognition for different types of discrete circles is achieved.