In the early days of Computer Graphics primary attention had to be given to the mathematics of simple geometric elements like points, lines and planes Homogeneous coordinates and space transformation using matrices also received considerable attention This is much less true today, since most of this mathematics has now become standard, is available in most computer graphics texts, is usually incorporated as part of many of the available graphics software packages and is even being incorporated into some of the more sophisticated graphics workstation processors In this tutorial therefore the emphasis has been shifted to the mathematics of more complex geometric entities such as polygonal regions, curves and surfaces Emphasis has also been placed on algorithms for processing these shapes and in particular generating various kinds of images of these geometric shapes

Mathematical elements for computer graphics

Stochastic sampling techniques, in particular Poisson and fittered sampling, are developed and analyzed. These approaches allow the construction of alias-free approximations to continuous functions using discrete calculations. Stochastic sampling scatters high frequency information into broadband noise rather than generating the false patterne produced by regular sampling. The type of randomness used in the sampling process controls the spectral character of the noise. The average sampling rate and the function being sampled determine the amount of noise that is produced. Stochastic sampling is applied adaptively so that a greater number of samples are taken where the function varies most. An estimate is used to determine how many samples to take over a given region. Noise reducing filters are used to increase the efficacy of a given sampling rate. The filter width is adaptively controlled to further improve performance. Stochastic sampling can be applied spatiotemporally as well as to other aspects of scene simulation. Ray tracing is one example of an image synthesis approach that can be antialiased by stochastic sampling.

Antialiasing through stochastic sampling

A novel model-based approach to 3D hand tracking from monocular video is presented. The 3D hand pose, the hand texture, and the illuminant are dynamically estimated through minimization of an objective function. Derived from an inverse problem formulation, the objective function enables explicit use of temporal texture continuity and shading information while handling important self-occlusions and time-varying illumination. The minimization is done efficiently using a quasi-Newton method, for which we provide a rigorous derivation of the objective function gradient. Particular attention is given to terms related to the change of visibility near self-occlusion boundaries that are neglected in existing formulations. To this end, we introduce new occlusion forces and show that using all gradient terms greatly improves the performance of the method. Qualitative and quantitative experimental results demonstrate the potential of the approach.

/pdf/model-based-3d-hand-pose-estimation-from-monocular-video-2b1afr8vel.pdf

Model-Based 3D Hand Pose Estimation from Monocular Video

In the early days of Computer Graphics primary attention had to be given to the mathematics of simple geometric elements like points, lines and planes. Homogeneous coordinates and space transformation using matrices also received considerable attention. This is much less true today, since most of this mathematics has now become standard, is available in most computer graphics texts, is usually incorporated as part of many of the available graphics software packages and is even being incorporated into some of the more sophisticated graphics workstation processors. In this tutorial therefore the emphasis has been shifted to the mathematics of more complex geometric entities such as polygonal regions, curves and surfaces. Emphasis has also been placed on algorithms for processing these shapes and in particular generating various kinds of images of these geometric shapes.

Mathematical Elements for Computer Graphics.

The benefits of the virtual pixel maps technique in the support of high-quality rendering operations are described. A dedicated frame buffer with a fixed number of bits per pixel is inappropriate for implementing high-quality rendering techniques within the framework of a graphics computer system. The virtual pixel maps feature is an elegant abstraction for solving problems that inherently require a large number of bits per pixel. A system implementation is presented to illustrate the types of hardware rendering algorithms that benefit from the concept of virtual pixel maps. Two specific algorithms, namely, transparency and antialiasing, are used as examples, but many other algorithms could easily be adapted to such an environment. >

Transparency and antialiasing algorithms implemented with the virtual pixel maps technique

Three antialiasing techniques were applied to a scene of moderate complexity. Early results suggest that prefiltering is still the most computationally effective method.

A Comparison of Antialiasing Techniques

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A Comparison of Antialiasing Techniques