How to tell computationally that a volume of points constitutes a manifold

Question

Suppose that a space ${\mathbb R}^{r}$ contains a set of points which we want to consider as enclosing a volume within the space, or perhaps a volume in a submanifold (e.g., the sphere $S^{2}$ within 3-space). How do we detect computationally that a set of points constitutes a manifold? Some cases such as regularly spaced rectangular arrays are easy, but I will be looking at irregularly distributed points, and they won't necessarily look "nice". Any literature on this subject? Thanks.