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In computer graphics a projection describes the mapping of scene geometry to the screen. While linear projections such as perspective and orthographic projection are common, increasing applications are being found for nonlinear projections, which do not necessarily map straight lines in the scene to straight lines on the screen. Nonlinear projections occur in reflections and refractions on curved surfaces, in art, and in visualisation. This thesis presents a new nonlinear projection technique called a trilinear projection that is based on the trilinear interpolation of surface normals used in Phong shading. Trilinear projections can be combined to represent more complicated nonlinear projections. Nonlinear projections have previously been implemented with ray tracing, where rays are generated by the nonlinear projections and traced into the scene. However for performance reasons, most current graphics software uses scanline rendering, where a scene point is imaged on a screen as a function of the projection parameters. The techniques developed in this thesis are of this nature. This thesis presents several algorithms used in trilinear projection: 1. An algorithm to analytically determine which screen locations image a given scene point. 2. An algorithm that correctly connects projected vertices. Each scene point may be imaged multiple times, which means a projected scene triangle may form from one to four different shapes of from two to nine vertices. Once connected, the projected shapes may be rendered with standard scanline algorithms. 3. An algorithm to more accurately render the curved edges between projected vertices. 4. A scene-space edge-clipping algorithm that handles continuity issues for projected shapes across composite projections. The trilinear projection technique is demonstrated in two different application areas: visualisation, and reflections and refractions. Specifically, various nonlinear projections that are congruent with pre-existing visualisation techniques are implemented with trilinear projections and a method for approximating the reflections and refractions on curved surfaces with trilinear projections is presented. Finally, the performance characteristics of the trilinear projection is explored over various parameter ranges and compared with a naive ray tracing approach.

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<p>Shading is a technique that is used in computer graphics to make faceted objects appear smooth and more realistic. In the research presented in this thesis we have investigated how shading can be generated as efficiently as possible without sacrificing quality.</p><p>In the classical approach to high quality shading proposed by Phong, the illumination equation is computed per pixel using an interpolated normal. The normals at the vertices are bi-linearly interpolated over the polygon to obtain a normal per pixel. Correct shading requires normalization of these normals, which is computationally demanding involving a square root. In our research we have shown how this normalization can be eliminated through the use of spherical interpolation and the Chebyshev recurrence formula, reducing the calculation to a few single arithmetic operations per pixel.</p><p>Still a substantial setup operation is needed for each scanline. We have studied how also this can be made more efficient, with some limited progress so far. An alternative approach is to do the most of the setup on polygon level and incrementally compute the setup needed per scanline. In particular, we have studied quadratic shading approaches, i.e. fitting second degree surfaces to the polygons. The most successful approach has been through what we have called X-shading, where the setup is calculated by using an efficient approximation for the mid-edge normals. This setup is about four times faster than previously known methods.</p><p>In the process of studying shading methods we have also made some contributions to improving bump-mapping and simulation of different kinds of light sources.</p><p>The developed methods will be of interest in future generations of computer graphics software and hardware systems, ranging from high end systems to generate realistic movies and 3D games, to handheld devices such as mobile phones with graphics displays.</p>

This thesis presents an efficient computational voxelization approach that utilizes the graphics pipeline. Our approach is hybrid in that it performs a precise gap-free computational voxelization, employs fixed-function components of the GPU, and utilizes the stages of the graphics pipeline to improve parallelism. This approach makes use of the latest features of OpenGL and fully supports both conservative and thin voxelization. In contrast to other computational voxelization approaches, this approach is implemented entirely in OpenGL, and achieves both triangle and fragment parallelism through its use of both the geometry and fragment shaders. A novel approach utilizing the graphics pipeline to complement geometric triangle intersection computations is presented. By exploiting features of the existing graphics pipeline we are able to rapidly compute accurate scene voxelization in a manner that integrates well with existing OpenGL applications, is robust across many different models, and eschews the need for complex work/load-balancing schemes. / Graduation date: 2013

Thesis (M.E.E.)--University of Delaware, 2006. / Principal faculty advisor: Gonzalo R. Arce, Dept. of Electrical and Computer Engineering. Includes bibliographical references.

Celtic art contains mysterious and fascinating aesthetic elements including complex knot work motifs. The problem is that creating and exploring these motifs require substantial human effort. One solution to this problem is to create a process that collaboratively uses interactive and procedural methods within a computer graphic environment. Spline models of Celtic knot work can be interactively modeled and used as input into procedural shaders. Procedural shaders are computer programs that describe surface, light, and volumetric appearances to a renderer. The control points of spline models can be used to drive shading procedures such as the coloring and displacement of surface meshes. The result of this thesis provides both an automated and interactive process that is capable of producing complex interlaced structures such as Celtic knot work within a three-dimensional environment.

Shading is a technique that is used in computer graphics to make faceted objects appear smooth and more realistic. In the research presented in this thesis we have investigated how shading can be generated as efficiently as possible without sacrificing quality. In the classical approach to high quality shading proposed by Phong, the illumination equation is computed per pixel using an interpolated normal. The normals at the vertices are bi-linearly interpolated over the polygon to obtain a normal per pixel. Correct shading requires normalization of these normals, which is computationally demanding involving a square root. In our research we have shown how this normalization can be eliminated through the use of spherical interpolation and the Chebyshev recurrence formula, reducing the calculation to a few single arithmetic operations per pixel. Still a substantial setup operation is needed for each scanline. We have studied how also this can be made more efficient, with some limited progress so far. An alternative approach is to do the most of the setup on polygon level and incrementally compute the setup needed per scanline. In particular, we have studied quadratic shading approaches, i.e. fitting second degree surfaces to the polygons. The most successful approach has been through what we have called X-shading, where the setup is calculated by using an efficient approximation for the mid-edge normals. This setup is about four times faster than previously known methods. In the process of studying shading methods we have also made some contributions to improving bump-mapping and simulation of different kinds of light sources. The developed methods will be of interest in future generations of computer graphics software and hardware systems, ranging from high end systems to generate realistic movies and 3D games, to handheld devices such as mobile phones with graphics displays.

Fluid simulations for computer graphics applications have attracted the attention of many researchers and practitioners due to the enhanced realism that natural phenomena simulation adds to graphical applications. Vortex methods are receiving increasing attention from the computer graphics community for simple and direct modeling of complex flow phenomena such as turbulence. However, vortex methods have not been developed yet to the level of other techniques for fluid simulation in computer graphics. In this work we present a novel simulation framework to model inviscid flows using Lagrangian vortex particle methods. We introduce novel stable methods to solve the vorticity flow equations that produce highly detailed visual fluid simulations. We incorporate the full interplay of solids and fluids in our framework. The coupling between free-form solids, represented by arbitrary surface meshes and fluids simulated with vortex methods, leads to visually rich simulations. Previous vortex simulators only focus on modeling the solid as a boundary for the flow. We model solid boundaries using an extended potential flow at the solid surface coupled with a boundary layer simulation. This allows the accurate simulation of two processes of visual interest. The first is the introduction of surface vorticity in the main flow as turbulence (vortex shedding). The second is the motion of the solid induced by fluid forces, which is calculated from the dynamics of vorticity in the flow and the rate of vorticity creation at solid surfaces. We demonstrate high quality results of our methods simulating flows around solid objects and solid object propulsion due to flows. This work ameliorates one of the important omissions in the development of vortex methods for computer graphics, which is the simulation of two-way coupling of solids and fluids.

Realistic computer generated images are computed by combining geometric effects, reflectance models for several captured and phenomenological materials, and real-world lighting according to mathematical models of physical light transport. Several important lighting phenomena should be considered when targeting realistic image simulation. A combination of soft and hard shadows, which arise from the interaction of surface and light geometries, provide necessary shape perception cues for a viewer. A wide variety of realistic materials, from physically-captured reflectance datasets to empirically designed mathematical models, modulate the virtual surface appearances in a manner that can further dissuade a viewer from considering the possibility of computational image synthesis over that of reality. Lastly, in many important cases, light reflects off many different surfaces before entering the eye. These secondary effects can be critical in grounding the viewer in a virtual world, since the human visual system is adapted to the physical world, where such effects are constantly in play. Simulating each of these effects is challenging due to their individual underlying complexity. The net complexity is compounded when several effects are combined. This thesis will investigate real-time approaches for simulating these effects under stringent performance and memory constraints, and with varying degrees of interactivity. In order to make these computations tractable given these added constraints, I will use data and signal analysis techniques to identify predictable patterns in the different spatial and angular signals used during image synthesis. The results of this analysis will be exploited with several analytic and data-driven mathematical models that are both efficient, and yield accurate approximations with predictable and controllable error.