Anti-aliasing with stratified B-spline filters of arbitrary degree

A simple and elegant method is presented to perform anti-aliasing in raytraced images. The method uses stratified sampling to reduce the occurrence of artefacts in an image and features a B-spline filter to compute the final luminous intensity at each pixel. The method is scalable through the specification of the filter degree. A B-spline filter of degree one amounts to a simple anti-aliasing scheme with box filtering. Increasing the degree of the B-spline generates progressively smoother filters. Computation of the filter values is done in a recursive way, as part of a sequence of Newton-Raphson iterations, to obtain the optimal sample positions in screen space. The proposed method can perform both anti-aliasing in space and in time, the latter being more commonly known as motion blur. We show an application of the method to the ray casting of implicit procedural surfaces

Progressive refinement rendering of implicit surfaces

The visualisation of implicit surfaces can be an inefficient task when such surfaces are complex and highly detailed. Visualising a surface by first converting it to a polygon mesh may lead to an excessive polygon count. Visualising a surface by direct ray casting is often a slow procedure. In this paper we present a progressive refinement renderer for implicit surfaces that are Lipschitz continuous. The renderer first displays a low resolution estimate of what the final image is going to be and, as the computation progresses, increases the quality of this estimate at an interactive frame rate. This renderer provides a quick previewing facility that significantly reduces the design cycle of a new and complex implicit surface. The renderer is also capable of completing an image faster than a conventional implicit surface rendering algorithm based on ray casting

Directional seismic source signature deconvolution

Marine seismic source arrays are directional. Source directivity is used to attenuate coherent noise, but primary reflected data may be degraded. Source directivity is ignored in a standard processing sequence, so directional source signature deconvolution may be required. In the frequency-wavenumber (f-k) directional deconvolution method, a filter is calculated from far-field source signatures and is applied to the f-k transform of common-receiver gathers. Reflections on common-receiver gathers are often spatially aliased, and this causes practical problems with the technique. Directional deconvolution may also be performed in combination with prestack migration because the prestack Kirchhoff summation migration operator is a function of source take-off angle. The constant-offset section is deconvolved separately with a full range of filters for source signatures radiated in different directions; then the migration summation operator sums across the deconvolved sections, selecting the section which has been deconvolved for the correct source signature at each point. Physical model data, which were acquired over simple models using a directional source, are used to evaluate directional deconvolution assuming constant velocity. Reflector continuity and resolution are improved by using directional deconvolution. Directional deconvolution combined with prestack migration is extended to media in which the velocity varies with depth, and is applied to two datasets from the Southern North Sea. The second dataset, which has shallow steeply dipping reflectors, is improved by using directional deconvolution. Directional deconvolution may be combined with a Kirchhoff migration technique which assumes a linear velocity-depth model. Results are superior to conventional Kirchhoff migration because ray bending is honoured. Directional deconvolution cannot synthesise fully point-source equivalent data from data acquired with a source array without excessive noise amplification. Source arrays with a short in-line dimension should be used where possible. For data which have been acquired with a long source array, directional deconvolution is desirable

Digital Filters and Signal Processing

Digital filters, together with signal processing, are being employed in the new technologies and information systems, and are implemented in different areas and applications. Digital filters and signal processing are used with no costs and they can be adapted to different cases with great flexibility and reliability. This book presents advanced developments in digital filters and signal process methods covering different cases studies. They present the main essence of the subject, with the principal approaches to the most recent mathematical models that are being employed worldwide

Optimization techniques for computationally expensive rendering algorithms

Realistic rendering in computer graphics simulates the interactions of light and surfaces. While many accurate models for surface reflection and lighting, including solid surfaces and participating media have been described; most of them rely on intensive computation. Common practices such as adding constraints and assumptions can increase performance. However, they may compromise the quality of the resulting images or the variety of phenomena that can be accurately represented. In this thesis, we will focus on rendering methods that require high amounts of computational resources. Our intention is to consider several conceptually different approaches capable of reducing these requirements with only limited implications in the quality of the results. The first part of this work will study rendering of time-­¿varying participating media. Examples of this type of matter are smoke, optically thick gases and any material that, unlike the vacuum, scatters and absorbs the light that travels through it. We will focus on a subset of algorithms that approximate realistic illumination using images of real world scenes. Starting from the traditional ray marching algorithm, we will suggest and implement different optimizations that will allow performing the computation at interactive frame rates. This thesis will also analyze two different aspects of the generation of anti-­¿aliased images. One targeted to the rendering of screen-­¿space anti-­¿aliased images and the reduction of the artifacts generated in rasterized lines and edges. We expect to describe an implementation that, working as a post process, it is efficient enough to be added to existing rendering pipelines with reduced performance impact. A third method will take advantage of the limitations of the human visual system (HVS) to reduce the resources required to render temporally antialiased images. While film and digital cameras naturally produce motion blur, rendering pipelines need to explicitly simulate it. This process is known to be one of the most important burdens for every rendering pipeline. Motivated by this, we plan to run a series of psychophysical experiments targeted at identifying groups of motion-­¿blurred images that are perceptually equivalent. A possible outcome is the proposal of criteria that may lead to reductions of the rendering budgets

High-Order Numerical Methods in Lake Modelling

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

Interpretation of equatorial current meter data as internal waves

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution January 1987Garrett and Munk use linear dynamics to synthesize frequency-wavenumber energy spectra for internal waves (GM72, GM75, GM79). The GM internal wave models are horizontally isotropic, vertically symmetric, purely propagating, and universal in both time and space. This set of properties effectively eliminates all the interesting physics, since such models do not allow localized sources and sinks of energy. Thus an important step in understanding internal wave dynamics is to make measurements of deviations from the simple GM models. This thesis continues the search for deviations from the GM models. It has three advantages over earlier work: extensive data from an equatorial region, long time series (2 years), and relatively sophisticated linear internal wave models. Since the GM models are based on mid-latitude data, having data from an equatorial region which has a strong mean current system offers an opportunity to examine a region with a distinctly different basic state. The longer time series mean there is a larger statistical ensemble of realizations, making it possible to detect smaller internal wave signals. The internal wave models include several important extensions to the GM models: horizontal anisotropy and vertical asymmetry, resolution between standing modes and propagating waves, general vertical structure, and kinematic effects of mean shear flow. Also investigated are the effects of scattering on internal waves, effects that are especially strong on the equator because the buoyancy frequency variability is a factor of ten higher than at mid-latitudes. In the high frequency internal wave field considered (frequencies between .125 cph and .458 cph), several features are found that are not included in the GM models. Both the kinematic effects of a mean shear flow and the phase-locking that distinguishes standing modes from propagating waves are observed. There is a seasonal dependence in energy level of roughly 10% of the mean level. At times the wave field is zonally and vertically asymmetric, with resulting energy fluxes that are a small (4% to 10%) fraction of the maximum energy flux the internal wave field could support. The fluxes are, however, as big as many of the postulated sources of energy for the internal wave field.This work has been supported under grants from the National Science Foundation and the Office of Naval Research, grants numbered NSF-89076, ONR-88914, NSF-9l002, NSF-94971, and NSF-93661