Beyond Gr\"obner Bases: Basis Selection for Minimal Solvers

Many computer vision applications require robust estimation of the underlying geometry, in terms of camera motion and 3D structure of the scene. These robust methods often rely on running minimal solvers in a RANSAC framework. In this paper we show how we can make polynomial solvers based on the action matrix method faster, by careful selection of the monomial bases. These monomial bases have traditionally been based on a Gr\"obner basis for the polynomial ideal. Here we describe how we can enumerate all such bases in an efficient way. We also show that going beyond Gr\"obner bases leads to more efficient solvers in many cases. We present a novel basis sampling scheme that we evaluate on a number of problems

Spectrum of Hot O At the Exobases of the Terrestrial Planets

The distribution of energetic O produced in dissociative recombination of O2+ at the exobases of the terrestrial planets is important in determining the structure of the outer hot O coronas and for the escape flux of O from Mars. Using recently measured values for the branching ratios of the energetically allowed channels in O2+ dissociative recombination, along with models of the vibrational distribution of O2+, we compute the velocity distribution of hot O atoms produced at the exobases of Mars, Venus, and Earth. We take into account the effects of the ion and electron temperatures, the rotational energy of the ions, and the decrease of the dissociative recombination cross section with relative velocity of the ion and electron. We find that distinct peaks appear at the energies associated with the various channels in the Martian spectrum but that for Venus, the features are less distinct, and for Earth, the distribution appears as a single, but broad and asymmetrical peak. For Mars, we compute the fractions of atoms produced with speeds greater than the escape velocity for 16O, 17O, and 18O of 0.68, 0.64, and 0.58, respectively. Thus there is a small isotope effect inherent in the dissociative recombination mechanism that has not been taken into account in models of Martian atmospheric evolution or in estimates of the total inventory of oxygen that is implied by the observed lack of enhancement of the heavier isotopes relative to the terrestrial values