Numerical solution of scattering problems using a Riemann--Hilbert formulation

A fast and accurate numerical method for the solution of scalar and matrix Wiener--Hopf problems is presented. The Wiener--Hopf problems are formulated as Riemann--Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix Wiener--Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach

Master of Science in Computing

thesisThis document introduces the Soft Shadow Mip-Maps technique, which consists of three methods for overcoming the fundamental limitations of filtering-oriented soft shadows. Filtering-oriented soft shadowing techniques filter shadow maps with varying filter sizes determined by desired penumbra widths. Different varieties of this approach have been commonly applied in interactive and real-time applications. Nonetheless, they share some fundamental limitations. First, soft shadow filter size is not always guaranteed to be the correct size for producing the right penumbra width based on the light source size. Second, filtering with large kernels for soft shadows requires a large number of samples, thereby increasing the cost of filtering. Stochastic approximations for filtering introduce noise and prefiltering leads to inaccuracies. Finally, calculating shadows based on a single blocker estimation can produce significantly inaccurate penumbra widths when the shadow penumbras of different blockers overlap. We discuss three methods to overcome these limitations. First, we introduce a method for computing the soft shadow filter size for a receiver with a blocker distance. Then, we present a filtering scheme based on shadow mip-maps. Mipmap-based filtering uses shadow mip-maps to efficiently generate soft shadows using a constant size filter kernel for each layer, and linear interpolation between layers. Finally, we introduce an improved blocker estimation approach. With the improved blocker estimaiton, we explore the shadow contribution of every blocker by calculating the light occluded by potential blockers. Hence, the calculated penumbra areas correspond to the blockers correctly. Finally, we discuss how to select filter kernels for filtering. These approaches successively solve issues regarding shadow penumbra width calculation apparent in prior techniques. Our result shows that we can produce correct penumbra widths, as evident in our comparisons to ray-traced soft shadows. Nonetheless, the Soft Shadow Mip-Maps technique suffers from light bleeding issues. This is because our method only calculates shadows using the geometry that is available in the shadow depth map. Therefore, the occluded geometry is not taken into consideration, which leads to light bleeding. Another limitation of our method is that using lower resolution shadow mip-map layers limits the resolution of the shadow placement. As a result, when a blocker moves slowly, its shadow follows it with discrete steps, the size of which is determined by the corresponding mip-map layer resolution

Large-Scale Rendering Using Shadowmaps

Shadow mapovanie je najpoužívanejšia metóda ktorá sa využíva v real-time 3D grafike na tvorbu tieňov v lokálnych osvetlovacích modeloch. Táto práca krok-za-krokom vysvetľuje proces vytvárania shadow máp. Porovnané su metódy výpočtu hĺbkovej odchylky ako aj filtrovacie metódy, a zároveň je odvodený výpočet normálovej odchýlky pre filtrovacie kernely s premennou veľkosťou. Taktiež popíšeme proces ako efektívne obaliť frustum kamery kaskádovým frustumom. Popri tom vysvetlíme ako využiť moderné OpenGL API na zníženie výkonnostných nedostatkov.Shadow mapping is the most widely used method in real-time 3D graphics for producing shadows in local light models. This thesis step-by-step explains the process of creating shadow maps. Depth biasing as well as filtering methods are analysed, then the calculation of normal offset bias for variable sized kernels is derived. We describe the process of efficiently fitting stable cascade frustums to view frustum. Also shown is how to use modern OpenGL to reduce performance overhead.

Probing the galactic halo with ROSAT

We discuss the current status of ROSAT shadowing observations designed to search for emission from million degree gas in the halo of the Milky Way galaxy. Preliminary results indicate that million degree halo gas is observed in the 1/4 keV band in some directions, most notably toward the Draco cloud at (l,b) = (92 deg, +38 deg), but that the halo emission is patchy and highly anisotropic. Our current understanding of this halo emission is based on a small handful of observations which have been analyzed to date. Many more observations are currently being analyzed or are scheduled for observation within the next year, and we expect our understanding of this component of the galactic halo to improve dramatically in the near future

A Unified Quantitative Model of Vision and Audition

We have put forwards a unified quantitative framework of vision and audition, based on existing data and theories. According to this model, the retina is a feedforward network self-adaptive to inputs in a specific period. After fully grown, cells become specialized detectors based on statistics of stimulus history. This model has provided explanations for perception mechanisms of colour, shape, depth and motion. Moreover, based on this ground we have put forwards a bold conjecture that single ear can detect sound direction. This is complementary to existing theories and has provided better explanations for sound localization.Comment: 7 pages, 3 figure

Dynamical dimer correlations at bipartite and non-bipartite Rokhsar-Kivelson points

We determine the dynamical dimer correlation functions of quantum dimer models at the Rokhsar-Kivelson point on the bipartite square and cubic lattices and the non-bipartite triangular lattice. Based on an algorithmic idea by Henley, we simulate a stochastic process of classical dimer configurations in continuous time and perform a stochastic analytical continuation to obtain the dynamical correlations in momentum space and the frequency domain. This approach allows us to observe directly the dispersion relations and the evolution of the spectral intensity within the Brillouin zone beyond the single-mode approximation. On the square lattice, we confirm analytical predictions related to soft modes close to the wavevectors (pi,pi) and (pi,0) and further reveal the existence of shadow bands close to the wavevector (0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at (pi,pi,pi) is found, as predicted by the single mode approximation. The soft mode has a quadratic dispersion at very long wavelength, but crosses over to a linear behavior very rapidly. We believe this to be the remnant of the linearly dispersing "photon" of the Coulomb phase. Finally the triangular lattice is in a fully gapped liquid phase where the bottom of the dimer spectrum exhibits a rich structure. At the M point the gap is minimal and the spectral response is dominated by a sharp quasiparticle peak. On the other hand, at the X point the spectral function is much broader. We sketch a possible explanation based on the crossing of the coherent dimer excitations into the two-vison continuum.Comment: 16 pages, 7 figures, published versio