Some illusions create the impression of contours where there are none in the real image. Best known are examples like the Ehrenstein illusion, where white space located at the place where perpendicular lines would cross, creates the illusion of a circular brighter-than-white disc at this location. In the Hermann grid, diagonals look darker (or brighter) than the background. Clearly, no linear operation is able to produce these effects.

In order to study the connection between the raw image and the perception it creates, first the effect has to be defined quantitatively by production of an image containing the illusion as real colour image, if possible. Sometimes this reification is not possible at all, as when the luminance is always increasing on a path as if in a Penrose_triangle graphic.

First, we can try to use a nonlinear function on the image values, e.g. $\tan$ function on the Laplacian of the luminance image. This will increase the contrast in the corners. In the Hermann grid the shadows in the crossings are reified this way. The diagonals need some more nonlinearity. The main problem with this solution is that it is purely local and does not explain why the illusion disappears in the Geier version, when the lines are curved.

We have to explain the Mach_bands also. The explanation in WP is only partially correct. The interesting feature of these bands is that they appear not only at small steps in luminance itself but also at steps in the gradient of luminance in continuous luminance profiles. See https://doi.org/10.3389/fnhum.2014.00843 . When steps are large the illusion disappears. So it cannot be explained by a simple nonlinear transformation of the luminance or its Fourier transform. F. Kingdom suggests that the mechanism creating this illusion (and perhaps other contrast illusions like Ehrenstein also) is local contrast normalisation (LCN).