Springer's lines and Hermann's grid

Hermann's grid is a complete matrix of black squares; Springer's figure has alternate squares whited out. Both figures generate illusory elements, bright and dark lines in the white spaces, with different elements seen from different distances. The visibility of these different elements is shown to be related to the detectability of sinusoidal gratings of wavenumbers corresponding to certain critical components of the Fourier decomposition of the figure. Evidence from the detection of gratings shows that near- and sub- (but not supra-) threshold contrasts are subject to a fourth power compression which shows, in the present figures, as a defect in the percept, equivalent to the deletion of (a part of) the corresponding Fourier components. There are some residual problems in understanding the perception of Hermann's grid.