Irreducibility of Random Hilbert Schemes

We prove that a random Hilbert scheme that parametrizes the closed subschemes with a fixed Hilbert polynomial in some projective space is irreducible and nonsingular with probability greater than $0.5$. To consider the set of nonempty Hilbert schemes as a probability space, we transform this set into a disjoint union of infinite binary trees, reinterpreting Macaulay's classification of admissible Hilbert polynomials. Choosing discrete probability distributions with infinite support on the trees establishes our notion of random Hilbert schemes. To bound the probability that random Hilbert schemes are irreducible and nonsingular, we show that at least half of the vertices in the binary trees correspond to Hilbert schemes with unique Borel-fixed points.

Automatic Identification of Large Fragments in a Pile of Broken Rock Using a Time-of-Flight Camera

This paper presents a solution to part of the problem of making robotic or semi-robotic digging equipment less dependant on human supervision. A method is described for identifying rocks of a certain size that may affect digging efficiency or require special handling. The process involves three main steps. First, by using range and intensity data from a time-of-flight (TOF) camera, a feature descriptor is used to rank points and separate regions surrounding high scoring points. This allows a wide range of rocks to be recognized because features can represent a whole or just part of a rock. Second, these points are filtered to extract only points thought to belong to the large object. Finally, a check is carried out to verify that the resultant point cloud actually represents a rock. Results are presented from field testing on piles of fragmented rock. Note to Practitioners—This paper presents an algorithm to identify large boulders in a pile of broken rock as a step towards an autonomous mining dig planner. In mining, piles of broken rock can contain large fragments that may need to be specially handled. To assess rock piles for excavation, we make use of a TOF camera that does not rely on external lighting to generate a point cloud of the rock pile. We then segment large boulders from its surface by using a novel feature descriptor and distinguish between real and false boulder candidates. Preliminary field experiments show promising results with the algorithm performing nearly as well as human test subjects.