On the classical discrete grid, the analysis of digital straight lines (DSL for short) has been intensively studied for nearly half a century. In this article, we are interested in a discrete geometry on irregular grids. More precisely, our goal is to define geometrical properties on irregular isothetic grids that are tilings of the Euclidean plane with different sized axis parallel rectangles. On these irregular isothetic grids, we define digital straight lines with recognition algorithms and a process to reconstruct an invertible polygonal representation of an irregular discrete curve.