Alyn P. Rockwood

Computing singularities of 3D vector fields with geometric algebra

Critical points of a vector field are key to their characterization. Their positions as well as their indexes are crucial for understanding vector fields. Considerable work exists in 2D, but less is available for 3D or higher dimensions. Geometric algebra is a derivative of Clifford algebra that not only enables a succinct definition of the index of a critical point in higher dimension; it also provides insight and computational pathways for calculating the index. We describe the problems in terms of geometric algebra and present an octree based solution using the algebra for finding critical points and their index in a 3D vector field.

Enabling view-dependent progressive volume visualization on the grid

We present a method for enabling progressive client-server volume visualization of data from the computing grid. Rendering is performed on clients, while servers on the grid provide wavelet-compressed volume data.

A haptic interface for the explorable virtual human

Haptic technology enhances a graphic user interface by allowing the user to interact in three dimensions with models on the screen, with force feedback giving the perception of touch. Recently several groups developed Internet based haptic systems that allow users in different parts of the world to interact with the same models simultaneously [Hikichi et al. 2002; Peterson 2002]. Although this system operates in real time, delay jitter caused by time required to transfer data inhibits the ability to apply realistic qualities to the individual models, as the lag in time is compensated by increasing the apparent weight of the model. Therefore, those users with greater lag times perceive the models as heavier than those with smaller lags [Hikichi et al. 2002]. While the required refresh rate of 1kHz suggests at least a portion of the interaction must be accomplished locally even on these other systems, transform updates must be handled at a server level. The haptic interface for the Explorable Virtual Human (EVH) we present allows the user to interact with various models taken from the Visual Human data set and the 100 micron knee. Our approach implements the PHANToM haptic device and can either incorporate single point collision as determined by the General Haptic Open Software Toolkit (GHOST®) SDK or calculate collision and force feedback information based on the shape and size of the haptic cursor and various properties of the models being viewed. In addition, collision detection allows deformations for those models representing soft tissues. Because the interaction occurs in real time, both the graphics and haptics run on the local computer while the browser enables the large data set to be managed by the server.

Re-usable implicit functions

Implicit functions play a major role in modeling shapes. [Bloo] lists many implicit models such as super-quadrics, blended surfaces, convolution surfaces, metaballs, alpha patches and others. The zero-set of an implicit function f (where f(x) = 0) is typically the set of points employed as the model. The non-zero set is, however, also important as many operations on implicit models depend on the non-zero part. Operations on implicit models such as booleans, blending, shape texturing, or local parameterizations rely on smoothly varying space in the non-zero region to produce reasonable results. Many of the operations tend to degrade the space: thus booleans and blending create non-differentiabilities [Rock]. Some implicit functions like the Weierstrass function (a fractal) begin with undesirable characteristics. We present a method that redefines the non-zero space, without affecting the zero set; thus making operations more successful. It makes implicit modeling operations reusable and in some cases makes difficult implicit models usable. We modify space by finding the minimal energy function off the zero set, one that leaves the zero set unchanged. We illustrate in 2D. Consider the bi-variate function shown in the following figure on the left.