Description of shape in a vector form
There's lots of work in the computer vision community on "shape descriptors".  This might or might not be useful.  See e.g., https://en.wikipedia.org/wiki/Shape_analysis_(digital_geometry), https://en.wikipedia.org/wiki/Shape_context, https://en.wikipedia.org/wiki/Image_moment, https://en.wikipedia.org/wiki/Spectral_shape_analysis.