A process for printing substrates includes assigning image dots of printer-ready copy to pixels of a multilevel map and assigning values to the pixels depending on the tonal value of the image dot for assigning n+1 different values, where n>1. Artifacts are prevented in multilevel printing processes by assigning threshold values of any threshold value matrix to the pixels of the multilevel map on which the printing is based with respect to their positions relative to the printing substrate, calculating tonal values of the image dot, fixing the values of the pixels depending on the threshold value and the tonal value of the assigned image dot, and printing the multilevel map on a printing substrate in a multilevel printing process, with different values representing different amounts of ink of a printed pixel. A screen assignment device assigns a value to a pixel of a multilevel map generated by an RIP.