I'd like to fromalise the idea of k-frames somewhat. Given any shape with a lot non-smooth points, we'd like a way to determine all the possible frames of the shape. Obviously we can't start from the minimal frame, because there are infinity ways to "fill in" a shape. So we have to start with the maximal frame and fid a sequence of operators that will each reduce the frame by one. The first operator is equivalent to the usual boundary operator. Ideally we'd like all the operators to be defined the same way.
I'm thinking we could use the characteristic function somehow (the function that returns 1 if x is in the shape and 0 otherwise). Differentiability would be involved and also manifold charts might come in handy.