Rewinder for making paper logs of paper material including a detection system with optical means adapted to detect, in a succession of predetermined detection times, a succession of diameters assumed in such times by a log being formed in a winding station and a programmable electronic unit which compares the diameters measured by the optical means with a succession of corresponding preset diameters and to calculate a sequence of differences. The processing unit receives a signal relating to the value of the actual diameter of the completed log and determines a parameter related to the trend over time of the differences. Elements for adjusting the thickness of the paper web automatically adjust said thickness if the parameter is between predetermined values.