Part of the CMB polarization signal in the direction of galaxy clusters is produced by Thomson scattering of the CMB temperature quadrupole. In principle this allows measurement of the CMB power spectrum harmonic C 2 z with higher accuracy (at z > 0) than the cosmic variance limit imposed by sample variance on one CMB sky. However the observed signals are statistically correlated if the comoving separation between the clusters is small enough. Thus one cannot reduce the sample variance by more than roughly the number of separate regions available which produce uncorrelated signals, as first pointed out by Kamionkowski and Loeb. In this paper we analyze in detail the procedure outlined by Kamionkowski and Loeb, computing the correlation of the polarization signals by considering the variation of the spherical harmonic expansion coefficients of the temperature anisotropy on our past light cone. Given a hypothetical set of Stokes parameter measurements of the CMB polarization in the directions of galaxy clusters, distributed at random on a given redshift shell, we show how to construct an estimator of the angular power spectrum harmonic C 2 at that redshift. We then compare the variance of this estimator with the cosmic variance of the CMB multipole on our sky which probes the same scale. We find that in fact the cosmic variance is not reducible below the single sky CMB value using the cluster method. Thus this method is not likely to be of use for reconstruction of the primordial power spectrum. However the method does yield a measurement of C 2 as a function of redshift with increasing accuracy at higher redshift, and thus potentially a probe of the mechanism which may have suppressed the quadrupole. We also examine to what extent the redshift dependence of C 2 can be used to probe the time changing potential anisotropy as the universe evolves into the vacuum dominated phase (the late-time integrated Sachs-Wolfe effect). We find that this effect is not observable in the time dependence of C 2 since it is swamped by cosmic variance, but there is an observable signature in the correlation functions of the Stokes parameters.
For the rendering of multiple scattering effects in participating media, methods based on the diffusion approximation are an extremely efficient alternative to Monte Carlo path tracing. However, in sufficiently transparent regions, classical diffusion approximation suffers from non-physical radiative fluxes which leads to a poor match to correct light transport. In particular, this prevents the application of classical diffusion approximation to heterogeneous media, where opaque material is embedded within transparent regions. To address this limitation, we introduce flux-limited diffusion, a technique from the astrophysics domain. This method provides a better approximation to light transport than classical diffusion approximation, particularly when applied to heterogeneous media, and hence broadens the applicability of diffusion-based techniques. We provide an algorithm for flux-limited diffusion, which is validated using the transport theory for a point light source in an infinite homogeneous medium. We further demonstrate that our implementation of flux-limited diffusion produces more accurate renderings of multiple scattering in various heterogeneous datasets than classical diffusion approximation, by comparing both methods to ground truth renderings obtained via volumetric path tracing.Comment: Accepted in Computer Graphics Foru
Figure 1: Scenes from Wrath of the Titans (a) and Prometheus (b) involve extremely dense volumetric objects. We adapt the Transmittance Function Mapping algorithm for high quality interactive previsualization and tuning of those media. Images: (a) c 2012 Warner Bros. Entertainment, (b) c 2012 20 th Century Fox. AbstractParticipating media are an unavoidable part of todays visual effects. The computation of compelling lighting effects within clouds or smoke remains challenging, both in terms of memory occupancy and computational power. Also, the fine tuning and layout of production-quality scenes requires efficient techniques for fast previsualization of the results. The Transmittance Function Maps provide an efficient solution for real-time previsualization of relatively wispy media such as clouds. However, this technique cannot support the extremely high densities encountered within the pyroclastic clouds of Wrath of the Titans, or in the sandstorm of Prometheus. We propose an adaptation of the Transmittance Function Mapping technique for the interactive previsualization of extremely dense, production-quality participating media. Based on a dual ray marching approach, our technique provides significant quality improvements while preserving real-time performance.
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