On a regular grid, the analysis of digital straight lines
(DSL for short) has been intensively studied for nearly half a century. In this article, we attend to multi-scale discrete geometry.
More precisely, we are interested in defining geometrical properties
on heterogeneous grids that are mappings of the Euclidean plane with
different sized isothetic squares. In some applications, such a
heterogeneous grid can be a hierarchical subdivision of a regular unit
grid. First of all, we define the objects in such a geometry
(heterogeneous digital objects, arcs, curves...). Based on these
definitions, we characterize the DSL on such grids and then, we
develop the algorithms to recognize segments and to decompose a curve
into maximal pieces of DSL. Finally, both algorithms are illustrated
and practical examples that have motivated this research are given.
The digital structure of 3D discrete volumes induces many difficulties in the exploitation and study of these objects due to the
huge volume of data stored. The general idea to solve those problems
is to transform the discrete surfaces of those volumes into polygonal
surfaces in a reversible way (the original object can be retrieved
from the polygonal surface). The aim of this article is to present a
first reversible and topologically correct solution using both Marching Cubes and digital plane segmentation processes.