Complete $SE(3)$ invariants for a comeagre set of $C^3$ compact orientable surfaces in $\mathbb{R}^3$

We introduce invariants for compact $C^1$-orientable surfaces (with boundary) in $\mathbb{R}^3$ up to rigid transformations. Our invariants are certain degree four polynomials in the moments of the delta function of the surface. We give an effective and numerically stable inversion algorithm for retrieving the surface from the invariants, which works on a comeagre subset of $C^3$-surfaces.