Optimal Design of Multiple Description Lattice Vector Quantizers

In the design of multiple description lattice vector quantizers (MDLVQ), index assignment plays a critical role. In addition, one also needs to choose the Voronoi cell size of the central lattice v, the sublattice index N, and the number of side descriptions K to minimize the expected MDLVQ distortion, given the total entropy rate of all side descriptions Rt and description loss probability p. In this paper we propose a linear-time MDLVQ index assignment algorithm for any K >= 2 balanced descriptions in any dimensions, based on a new construction of so-called K-fraction lattice. The algorithm is greedy in nature but is proven to be asymptotically (N -> infinity) optimal for any K >= 2 balanced descriptions in any dimensions, given Rt and p. The result is stronger when K = 2: the optimality holds for finite N as well, under some mild conditions. For K > 2, a local adjustment algorithm is developed to augment the greedy index assignment, and conjectured to be optimal for finite N. Our algorithmic study also leads to better understanding of v, N and K in optimal MDLVQ design. For K = 2 we derive, for the first time, a non-asymptotical closed form expression of the expected distortion of optimal MDLVQ in p, Rt, N. For K > 2, we tighten the current asymptotic formula of the expected distortion, relating the optimal values of N and K to p and Rt more precisely.Comment: Submitted to IEEE Trans. on Information Theory, Sep 2006 (30 pages, 7 figures

Early soft X-ray to UV emission from double neutron star mergers: implications from the long-term radio and X-ray emissions of GW 170817

Recent long-term radio follow-up observations of GW 170817 reveals a simple power-law rising light curve, with a slope of $t^{0.78}$, up to 93 days after the merger. The latest X-ray detection at 109 days is also consistent with such a temporal slope. Such a shallow rise behavior requires a mildly relativistic outflow with a steep velocity gradient profile, so that slower material with larger energy catches up with the decelerating ejecta and re-energizes it. It has been suggested that this mildly relativistic outflow may represent a cocoon of material. We suggest that the velocity gradient profile may form during the stage that the cocoon is breaking out of the merger ejecta, resulted from shock propagation down a density gradient. The cooling of the hot relativistic cocoon material immediately after it breaks out should have produced soft X-ray to UV radiation at tens of seconds to hours after the merger. The soft X-ray emission has a luminosity of $L_{\rm X}\sim 10^{45}{\rm erg s^{-1}}$ over a period of tens of seconds for a merger event like GW 170817. The UV emission shows a rise initially and peaks at about a few hours with a luminosity of $L_{\rm UV}\sim 10^{42} {\rm erg s^{-1}}$. The soft X-ray transients could be detected by future wide-angle X-ray detectors, such as the Chinese mission Einstein Probe. This soft X-ray/UV emission would serve as one of the earliest electromagnetic counterparts of gravitation waves from double neutron star mergers and could provide the earliest localization of the sources.Comment: 5 pages, 2 figures, ApJL in press, discussions on the reverse shock emission in the refreshed shock scenario for the long-term radio and X-ray emissions are adde

Massively Parallel Ray Tracing Algorithm Using GPU

Ray tracing is a technique for generating an image by tracing the path of light through pixels in an image plane and simulating the effects of high-quality global illumination at a heavy computational cost. Because of the high computation complexity, it can't reach the requirement of real-time rendering. The emergence of many-core architectures, makes it possible to reduce significantly the running time of ray tracing algorithm by employing the powerful ability of floating point computation. In this paper, a new GPU implementation and optimization of the ray tracing to accelerate the rendering process is presented