A digital blurred plane is a finite set of points with controlled thickness. This notion can be used to characterize a real world object surface, considered as planar at a large scale, but actually quite irregular at finer scale. In this paper, we introduce a new framework to detect a blurred plane in a point set, based on the analysis of two orthogonal height profiles using digital geometry tools. It is applied to the interactive extraction of relict charcoal platforms from an airborne LiDAR point cloud. This process was tested on two sets of geolocated platforms obtained by on-site prospecting or visual survey of the LiDAR digital terrain model. Experimental validations show that a large amount of them is successfully extracted and that their estimated extent closely matches the ground truth.