Are there shapes defined by 3 (or more) generative parameters whose mapping to psychological similarity space is known?
Consider the colour visual system. Take 3 monochromatic (in the physics sense) light sources of wavelength $420 \; nm$, $534 \; nm$, and $564 \; nm$; i.e. the peaks of spectral sensitivity for cones. Your 3 physical parameters are then 3 knobs $b$, $g$, $r$ that control the intensity of each light source as they shine on the same white surface. If you want more physical parameters, you can just add more knobs for other monochromatic lightsources, however 3 is sufficient. Colour perception is extremely well studied, so this comes as close as I can think of to a known mapping between between objective and psychological space. Unfortunately, you will run into some culture and language specific issues for judging the similarity of perceived colours.