Digital and Discrete Deformation

Abstract

In science, changing one curve α into another curve β continuously is called deformation. To describe this action, we usually use a sequence of curves in a sketch: the beginning curve C ₀ is the original curve α and the final curve C ₁ indicates the targeting curve β. Therefore, deformation can be defined as a function f ^α(t) = C _t , where t ∈ [0, 1]. f ^α(t) and f ^α(t ₀) are getting closer (infinitively) when t → t ₀. Such a concept has essential importance since it relates to the topological equivalence and effect on entire modern mathematics. It also has great deal of impact in 3D image processing, what we call morphing one 2D/3D picture into another. In this chapter, we introduce the basic method of digital deformation and homotopic equivalence. We also give a brief overview of the fundamental groups and homology groups for digital objects. (Note The material in this chapter is much different than that of other chapters because it contains some graduate level material in the mathematical field of topology. In this book, the author tries to explain some profound concepts in an elementary way, which may not always be successful meaning that it is not always appreciated by some others.)