Uncertainty Propagation Analysis in 3-D Shape Measurement Using Laser Range Finding

Quality evaluation of rolling processes in the metal industry involves an inspection of the shape of the outgoing products in real time during manufacturing. Shape measurement systems are usually based on 3-D reconstructions of the surface of rolled products. As surface properties are crucial, these systems favor contactless techniques. Using 3-D measurements of the surface of rolled products, several geometric properties can be analyzed. In this paper, we analyze how uncertainty is propagated in a contactless shape measurement system designed and developed by the authors and presented in previous published works. This measurement system is based on active triangulation, and it is able to provide inline width and flatness measurements of long, flat-rolled products in harsh industrial environments. The camera model used to calibrate the vision system is described, and it is used to estimate the uncertainty of the reprojected 3-D points on the scene. The system uses the reprojected 3-D points, and the speed of the product movement in the production line to reconstruct its surface. Thus, the uncertainty of the speed is also estimated. Finally, the propagation of both the uncertainty of the 3-D reprojection, and the uncertainty of the speed into the final width and flatness measurements is analyzed. This paper comprises a detailed uncertainty propagation analysis in 3-D shape measurements computed indirectly through functional relationships.