3D billiards: visualization of regular structures and trapping of chaotic trajectories

The dynamics in three-dimensional billiards leads, using a Poincar\'e section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincar\'e recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.Comment: 20 pages, 11 figures. For videos of 3D phase-space slices and time-resolved animations see http://www.comp-phys.tu-dresden.de/supp

Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains

We describe a multidomain spectral-tau method for solving the three-dimensional helically reduced wave equation on the type of two-center domain that arises when modeling compact binary objects in astrophysical applications. A global two-center domain may arise as the union of Cartesian blocks, cylindrical shells, and inner and outer spherical shells. For each such subdomain, our key objective is to realize certain (differential and multiplication) physical-space operators as matrices acting on the corresponding set of modal coefficients. We achieve sparse banded realizations through the integration "preconditioning" of Coutsias, Hagstrom, Hesthaven, and Torres. Since ours is the first three-dimensional multidomain implementation of the technique, we focus on the issue of convergence for the global solver, here the alternating Schwarz method accelerated by GMRES. Our methods may prove relevant for numerical solution of other mixed-type or elliptic problems, and in particular for the generation of initial data in general relativity.Comment: 37 pages, 3 figures, 12 table

Interaction of cortical networks mediating object motion detection by moving observers

Published in final edited form as: Exp Brain Res. 2012 August ; 221(2): 177–189. doi:10.1007/s00221-012-3159-8.The task of parceling perceived visual motion into self- and object motion components is critical to safe and accurate visually guided navigation. In this paper, we used functional magnetic resonance imaging to determine the cortical areas functionally active in this task and the pattern connectivity among them to investigate the cortical regions of interest and networks that allow subjects to detect object motion separately from induced self-motion. Subjects were presented with nine textured objects during simulated forward self-motion and were asked to identify the target object, which had an additional, independent motion component toward or away from the observer. Cortical activation was distributed among occipital, intra-parietal and fronto-parietal areas. We performed a network analysis of connectivity data derived from partial correlation and multivariate Granger causality analyses among functionally active areas. This revealed four coarsely separated network clusters: bilateral V1 and V2; visually responsive occipito-temporal areas, including bilateral LO, V3A, KO (V3B) and hMT; bilateral VIP, DIPSM and right precuneus; and a cluster of higher, primarily left hemispheric regions, including the central sulcus, post-, pre- and sub-central sulci, pre-central gyrus, and FEF. We suggest that the visually responsive networks are involved in forming the representation of the visual stimulus, while the higher, left hemisphere cluster is involved in mediating the interpretation of the stimulus for action. Our main focus was on the relationships of activations during our task among the visually responsive areas. To determine the properties of the mechanism corresponding to the visual processing networks, we compared subjects’ psychophysical performance to a model of object motion detection based solely on relative motion among objects and found that it was inconsistent with observer performance. Our results support the use of scene context (e.g., eccentricity, depth) in the detection of object motion. We suggest that the cortical activation and visually responsive networks provide a potential substrate for this computation.This work was supported by NIH grant RO1NS064100 to L.M.V. We thank Victor Solo for discussions regarding models of functional connectivity and our subjects for participating in the psychophysical and fMRI experiments. This research was carried out in part at the Athinoula A. Martinos Center for Biomedical Imaging at the Massachusetts General Hospital, using resources provided by the Center for Functional Neuroimaging Technologies, P41RR14075, a P41 Regional Resource supported by the Biomedical Technology Program of the National Center for Research Resources (NCRR), National Institutes of Health. This work also involved the use of instrumentation supported by the NCRR Shared Instrumentation Grant Program and/or High-End Instrumentation Grant Program; specifically, grant number S10RR021110. (RO1NS064100 - NIH; National Center for Research Resources (NCRR), National Institutes of Health; S10RR021110 - NCRR)Accepted manuscrip

Shape-Based Models for Interactive Segmentation of Medical Images

Accurate image segmentation is one of the key problems in computer vision. In domains such as radiation treatment planning, dosimetrists must manually trace the outlines of a few critical structures on large numbers of images. Considerable similarity can be seen in the shape of these regions, both between adjacent slices in a particular patient and across the spectrum of patients. Consequently we should be able to model this similarity and use it to assist in the process of segmentation. Previous work has demonstrated that a constraint-based 2D radial model can capture generic shape information for certain shape classes, and can reduce user interaction by a factor of three over purely manual segmentation. Additional simulation studies have shown that a probabilistic version of the model has the potential to further reduce user interaction. This paper describes an implementation of both models in a general-purpose imaging and graphics framework and compares the usefulness of the models on several shape classes