I tried 3D-printing a mechanical LUT on Julian Cerruti's Prusa Mendel the other day. The idea is that you position a probe in X and Y to set the input, then drop it down to see how far down it can fall before hitting the LUT to get the output.
It didn't go well, for several reasons.
First, I was trying to print at high resolution, both horizontally and vertically.
Vertically: my model had a 1/32 solid part at the bottom, and rose to a max height of 15/64 above that; these units were scaled to √2 centimeters, so the minimum thickness was √2 cm/32 = 0.044 cm = 0.44 mm, which worked okay (the first layer thickness was set to 0.35 mm, and subsequent layers 0.2 mm); the total thickness then would have been 17 √2 cm/32 = 3.76 mm. But note that 3.76mm is only 18 layers of plastic, which is close enough to the nominal LUT's desired resolution of 16 thicknesses that there could be aliasing problems. It would be better to double the thickness and align it precisely with the layers: two layers of solid, plus two layers per LUT level, for a total of 34 layers: (+ .15 (* 34 .2)) = 6.95 mm.
Horizontally: across the √2 cm width of the model, I had 15 intervals between grid lines, or 0.94 mm per interval. While the Mendel is perfectly capable of positioning the print head with precision well below a millimeter, the extruded plastic is substantially wider. Measurement of the infill lines on the bottom surface shows them to be spaced about 1mm to 1.5mm apart. Corner radius seems to be about 0.5mm. This means you can't practically make a positive feature with a total width below 1mm, with the software stack Julian's using, anyway (slic3r and pronterface, I think).
I think it's possible to achieve this with 1mm to 1.5mm horizontal spacing. It should be possible to make holes much narrower than 1mm, as long as they don't have to be too close together.
This brings me to the second problem: discontinuous layers. FDM machines typically print in horizontal slices to avoid crashing the print head into already-printed stuff, but they have various kinds of trouble when those horizontal slices contain discontinuities. At a minimum, they pull threads of plastic between the different places they visit; but also, the process of switching the pinch wheel between forward and reverse seems to be fairly undependable. So it might be desirable to add material in between grid points so that each layer is a single continuous piece. If the object is still a heightfield, this continuous-piece constraint simplifies to "no local maxima".
A third potential problem — which didn't show up in this case, but did in previous prints — is horizontal surfaces over infill. This requires the infill to be nice and solid to support the surface well; otherwise it can end up with holes in it.