Recent experimental results on human vision show that low fractal dimensions curves are highly capable to evocate namable objects. In other terms, regular curves are recognized by human vision as object edges. In this report, a regularity measure of discrete lines geometry is presented. This scalar measure is based on a ratio between lines lengths at different scale. It is analyzed in the framework of brownian motion theory. The measure at a given scale is always computed from the maximum precision image, so that is doesn't introduce any sub-resolution assumption. A scale choice determines the quantity of global information vs. local information one wants to measure. We show how this quantitative measure leads to a relevant shape information. To illustrate this, an image segmentation application example is realized. The segmentation is based essentially on geometry criteria. The segmentation process uses a region growing process which depends on a single parameter that can be fixed in a natural way comparing boundaries regularity to a geometric model regularity. We present experimental results performed on real-scene images, including indoor and outdoor images.