Slides are available by following the link.
In a digital image, any shape is represented by a set of pixels, and rigid transformations are often applied to such a digital shape (or to the entire image) in many applications of digital image computing. However, in such a discrete space, congruence is not preserved, in general, between an initial digital shape and its transformed one after some rigid motion. This is because rigid transformations in the discrete space (rigid transformation followed by a digitization) are not isometric, i.e. distances are not preserved, contrary to those in the Euclidean space. Indeed, not only congruence but many properties are lost in a discrete space, such as topology and even bijection.
In this talk, we point out such problems induced by the discrete nature of the pixel world, and present our recent contribution to the topology preservation of digital shapes under their rigid transformations, without interpolation (namely, in a purely discrete way). It should be noted that"topology" is one of necessary notions in order to define "curvatures" on digital shape boundary. We also discuss desirable approaches for computing discrete curvatures in such discrete situations.
This is joint work with Phuc Ngo, Nicolas Passat and Hugues Talbot.