Application of the Fisher-Rao metric to ellipse detection

The parameter space for the ellipses in a two dimensional image is a five dimensional manifold, where each point of the manifold corresponds to an ellipse in the image. The parameter space becomes a Riemannian manifold under a Fisher-Rao metric, which is derived from a Gaussian model for the blurring of ellipses in the image. Two points in the parameter space are close together under the Fisher-Rao metric if the corresponding ellipses are close together in the image. The Fisher-Rao metric is accurately approximated by a simpler metric under the assumption that the blurring is small compared with the sizes of the ellipses under consideration. It is shown that the parameter space for the ellipses in the image has a finite volume under the approximation to the Fisher-Rao metric. As a consequence the parameter space can be replaced, for the purpose of ellipse detection, by a finite set of points sampled from it. An efficient algorithm for sampling the parameter space is described. The algorithm uses the fact that the approximating metric is flat, and therefore locally Euclidean, on each three dimensional family of ellipses with a fixed orientation and a fixed eccentricity. Once the sample points have been obtained, ellipses are detected in a given image by checking each sample point in turn to see if the corresponding ellipse is supported by the nearby image pixel values. The resulting algorithm for ellipse detection is implemented. A multiresolution version of the algorithm is also implemented. The experimental results suggest that ellipses can be reliably detected in a given low resolution image and that the number of false detections can be reduced using the multiresolution algorithm

Astrometric Microlensing Constraints on a Massive Body in the Outer Solar System with Gaia

A body in Solar orbit beyond the Kuiper belt exhibits an annual parallax that exceeds its apparent proper motion by up to many orders of magnitude. Apparent motion of this body along the parallactic ellipse will deflect the angular position of background stars due to astrometric microlensing ("induced parallax"). By synoptically sampling the astrometric position of background stars over the entire sky, constraints on the existence (and basic properties) of a massive nearby body may be inferred. With a simple simulation, we estimate the signal-to-noise for detecting such a body -- as function of mass, heliocentric distance, and ecliptic latitude -- using the anticipated sensitivity and temporal cadences from Gaia (launch 2011). A Jupiter-mass (M_Jup) object at 2000 AU is detectable by Gaia over the whole sky above 5-sigma, with even stronger constraints if it lies near the ecliptic plane. Hypotheses for the mass (~3M_Jup), distance (~20,000 AU) and location of the proposed perturber ("Planet X") which gives rise to long-period comets may be testable.Comment: 17 pages, 6 figures. Figures revised, new figure added, minor text revisions. Accepted to ApJ, to appear in the Dec 10, 2005 issue (v635

Constraining Attacker Capabilities Through Actuator Saturation

For LTI control systems, we provide mathematical tools - in terms of Linear Matrix Inequalities - for computing outer ellipsoidal bounds on the reachable sets that attacks can induce in the system when they are subject to the physical limits of the actuators. Next, for a given set of dangerous states, states that (if reached) compromise the integrity or safe operation of the system, we provide tools for designing new artificial limits on the actuators (smaller than their physical bounds) such that the new ellipsoidal bounds (and thus the new reachable sets) are as large as possible (in terms of volume) while guaranteeing that the dangerous states are not reachable. This guarantees that the new bounds cut as little as possible from the original reachable set to minimize the loss of system performance. Computer simulations using a platoon of vehicles are presented to illustrate the performance of our tools

Probing dark energy with the next generation X-ray surveys of galaxy clusters

We present forecasts on the capability of future wide-area high-sensitivity X-ray surveys of galaxy clusters to yield constraints on the parameters defining the Dark Energy (DE) equation of state (EoS). Our analysis is carried out for future X-ray surveys which have enough sensitivity to provide accurate measurements of X-ray mass proxies and Fe-line based redshifts for about 2x10^4 clusters. We base our analysis on the Fisher Matrix formalism, by combining information on the cluster number counts and power spectrum, also including, for the first time in the analysis of the large scale cluster distribution, the effect of linear redshift-space distortions (RSDs). This study is performed with the main purpose of dissecting the cosmological information provided by geometrical and growth tests, which are both included in the analysis of number counts and clustering of galaxy clusters. We compare cosmological constraints obtained by assuming different levels of prior knowledge of the parameters which define the observable-mass X-ray relation. This comparison further demonstrates the fundamental importance of having a well calibrated observable-mass relation and, most importantly, its redshift evolution. Such a calibration can be achieved only by having at least $\sim 10^3$ net photon counts for each cluster included in the survey. We show that RSDs in the power spectrum analysis carry important cosmological information also when traced with galaxy clusters and the DE FoM increases by a factor of 8. Besides confirming the potential that large cluster surveys have in constraining the nature of DE, our analysis emphasizes that a full exploitation of the cosmological information carried by such surveys requires not only a large statistic but also a robust measurement of the mass proxies and redshifts for a significant fraction of the cluster sample, derived from the same X-ray survey data.Comment: 16 pages, 14 figures,published on MNRA

Empirical Constraints on the Oblateness of an Exoplanet

We show that the gas giant exoplanet HD 189733b is less oblate than Saturn, based on Spitzer Space Telescope photometry of seven transits. The observable manifestations of oblatenesswould have been slight anomalies during the ingress and egress phases, as well as variations in the transit depth due to spin precession. Our nondetection of these effects gives the first empirical constraints on the shape of an exoplanet. The results are consistent with the theoretical expectation that the planetary rotation period and orbital period are synchronized, in which case the oblateness would be an order of magnitude smaller than our upper limits. Conversely, if HD 189733b is assumed to be in a synchronous, zero-obliquity state, then the data give an upper bound on the quadrupole moment of the planet (J2 < 0.068 with 95% confidence) that is too weak to constrain the interior structure of the planet. An Appendix describes a fast algorithm for computing the transit light curve of an oblate planet, which was necessary for our analysis.Comment: 14 pages, accepted for publication in The Astrophysical Journa

Spitzer Observations of Interstellar Object 1I/`Oumuamua

1I/`Oumuamua is the first confirmed interstellar body in our Solar System. Here we report on observations of `Oumuamua made with the Spitzer Space Telescope on 2017 November 21--22 (UT). We integrated for 30.2~hours at 4.5 micron (IRAC channel 2). We did not detect the object and place an upper limit on the flux of 0.3 uJy (3sigma). This implies an effective spherical diameter less than [98, 140, 440] meters and albedo greater than [0.2, 0.1, 0.01] under the assumption of low, middle, or high thermal beaming parameter eta, respectively. With an aspect ratio for `Oumuamua of 6:1, these results correspond to dimensions of [240:40, 341:57, 1080:180] meters, respectively. We place upper limits on the amount of dust, CO, and CO2 coming from this object that are lower than previous results; we are unable to constrain the production of other gas species. Both our size and outgassing limits are important because `Oumuamua's trajectory shows non-gravitational accelerations that are sensitive to size and mass and presumably caused by gas emission. We suggest that `Oumuamua may have experienced low-level post-perihelion volatile emission that produced a fresh, bright, icy mantle. This model is consistent with the expected eta value and implied high albedo value for this solution, but, given our strict limits on CO and CO2, requires another gas species --- probably H2O --- to explain the observed non-gravitational acceleration. Our results extend the mystery of `Oumuamua's origin and evolution