The 3D pose estimation problem - aligning pairs of noisy 3D point clouds - is a problem with a wide variety of real-world applications. Here we focus on the use of quaternion-based neural network approaches to this problem and apparent anomalies that have arisen in previous efforts to resolve them. In addressing these anomalies, we draw heavily from the extensive literature on closed-form methods to solve this problem. We suggest that the major concerns that have been put forward could be resolved using a simple multi-valued training target derived from rigorous theoretical properties of the rotation-to-quaternion map of Bar-Itzhack. This multi-valued training target is then demonstrated to have good performance for both simulated and ModelNet targets. We provide a comprehensive theoretical context, using the quaternion adjugate, to confirm and establish the necessity of replacing single-valued quaternion functions by quaternions treated in the extended domain of multiple-charted manifolds.

This intermediate-level tutorial provides an introduction to the visualization of quaternions and recent developments concerning quaternion applications to computer graphics and scientific visualization. We begin by covering the fundamental nature of quaternions, explaining where they become important in computer graphics, and describing the basic tools needed to apply them. We then pursue several recent topics for which quaternions are relevant and note updated approaches to selected graphics and visualization applications.

In the analysis of spatially-referenced time-dependent data, gaining an understanding of the spatio-temporal distributions and relationships among the attributes in the data can be quite difficult. We present a visualization technique that addresses some of the challenges involved in visually exploring and analyzing the distributions of geo-spatial time-varying data. We have developed a pictorial representation that is based on the standard space-time cube metaphor and provides in a single display the overview and details of a large number of time-varying quantities. Our approach involves three-dimensional graphical widgets that intuitively represent profiles of the time-varying quantities and can be plotted on a geographic map to expose interesting spatio-temporal distributions of the data. We show how combining our visualization technique with standard data exploration features can assist in the exploration of salient patterns in a data set. The visualization approach described here supports expeditious exploration of multiple data sets; this in turn assists the process of building initial hypotheses about the attributes in a data set and enhances the user's ability to pose and explore interesting questions about the data.

This paper describes GL4D, an interactive system for visualizing 2-manifolds and 3-manifolds embedded in four Euclidean dimensions and illuminated by 4D light sources. It is a tetrahedron-based rendering pipeline that projects geometry into volume images, an exact parallel to the conventional triangle-based rendering pipeline for 3D graphics. Novel features include GPU-based algorithms for real-time 4D occlusion handling and transparency compositing; we thus enable a previously impossible level of quality and interactivity for exploring lit 4D objects. The 4D tetrahedrons are stored in GPU memory as vertex buffer objects, and the vertex shader is used to perform per-vertex 4D modelview transformations and 4D-to-3D projection. The geometry shader extension is utilized to slice the projected tetrahedrons and rasterize the slices into individual 2D layers of voxel fragments. Finally, the fragment shader performs per-voxel operations such as lighting and alpha blending with previously computed layers. We account for 4D voxel occlusion along the 4D-to-3D projection ray by supporting a multi-pass back-to-front fragment composition along the projection ray; to accomplish this, we exploit a new adaptation of the dual depth peeling technique to produce correct volume image data and to simultaneously render the resulting volume data using 3D transfer functions into the final 2D image. Previous CPU implementations of the rendering of 4D-embedded 3-manifolds could not perform either the 4D depth-buffered projection or manipulation of the volume-rendered image in real-time; in particular, the dual depth peeling algorithm is a novel GPU-based solution to the real-time 4D depth-buffering problem. GL4D is implemented as an integrated OpenGL-style API library, so that the underlying shader operations are as transparent as possible to the user.

With recent advances in the measurement technology for allsky astrophysical imaging, our view of the sky is no longer limited to the tiny visible spectral range over the 2D Celestial sphere. We now can access a third dimension corresponding to a broad electromagnetic spectrum with a wide range of allsky surveys; these surveys span frequency bands including long wavelength radio, microwaves, very short X-rays, and gamma rays. These advances motivate us to study and examine multiwavelength visualization techniques to maximize our capabilities to visualize and exploit these informative image data sets. In this work, we begin with the processing of the data themselves, uniformizing the representations and units of raw data obtained from varied detector sources. Then we apply tools to map, convert, color-code, and format the multiwavelength data in forms useful for applications. We explore different visual representations for displaying the data, including such methods as textured image stacks, the horseshoe representation, and GPU-based volume visualization. A family of visual tools and analysis methods is introduced to explore the data, including interactive data mapping on the graphics processing unit (GPU), the mini-map explorer, and GPU-based interactive feature analysis.

Modern astronomical instruments produce enormous amounts of three-dimensional data describing the physical Universe. The currently available data sets range from the solar system to nearby stars and portions of the Milky Way Galaxy, including the interstellar medium and some extrasolar planets, and extend out to include galaxies billions of light years away. Because of its gigantic scale and the fact that it is dominated by empty space, modeling and rendering the Universe is very different from modeling and rendering ordinary three-dimensional virtual worlds at human scales. Our purpose is to introduce a comprehensive approach to an architecture solving this visualization problem that encompasses the entire Universe while seeking to be as scale-neutral as possible. One key element is the representation of model-rendering procedures using power scaled coordinates (PSC), along with various PSC-based techniques that we have devised to generalize and optimize the conventional graphics framework to the scale domains of astronomical visualization. Employing this architecture, we have developed an assortment of scale-independent modeling and rendering methods for a large variety of astronomical models, and have demonstrated scale-insensitive interactive visualizations of the physical Universe covering scales ranging from human scale to the Earth, to the solar system, to the Milky Way Galaxy, and to the entire observable Universe.

Visualization of uncertainty or error in astrophysical data is seldom available in simulations of astronomical phenomena, and yet almost all rendered attributes possess some degree of uncertainty due to observational error. Uncertainties associated with spatial location typically vary signicantly with scale and thus introduce further complexity in the interpretation of a given visualization. This paper introduces effective techniques for visualizing uncertainty in large-scale virtual astrophysical environments. Building upon our previous transparently scalable visualization architecture, we develop tools that enhance the perception and comprehension of uncertainty across wide scale ranges. Our methods include a unified color-coding scheme for representing log-scale distances and percentage errors, an ellipsoid model to represent positional uncertainty, an ellipsoid envelope model to expose trajectory uncertainty, and a magic-glass design supporting the selection of ranges of log-scale distance and uncertainty parameters, as well as an overview mode and a scalable WIM tool for exposing the magnitudes of spatial context and uncertainty.

Just as we can work with two-dimensional floor plans to communicate 3D architectural design, we can exploit reduced-dimension shadows to manipulate the higher-dimensional objects generating the shadows. In particular, by taking advantage of physically reactive 3D shadow-space controllers, we can transform the task of interacting with 4D objects to a new level of physical reality. We begin with a teaching tool that uses 2D knot diagrams to manipulate the geometry of 3D mathematical knots via their projections; our unique 2D haptic interface allows the user to become familiar with sketching, editing, exploration, and manipulation of 3D knots rendered as projected imageson a 2D shadow space. By combining graphics and collision-sensing haptics, we can enhance the 2D shadow-driven editing protocol to successfully leverage 2D pen-and-paper or blackboard skills. Building on the reduced-dimension 2D editing tool for manipulating 3D shapes, we develop the natural analogy to produce a reduced-dimension 3D tool for manipulating 4D shapes. By physically modeling the correct properties of 4D surfaces, their bending forces, and their collisions in the 3D haptic controller interface, we can support full-featured physical exploration of 4D mathematical objects in a manner that is otherwise far beyond the experience accessible to human beings. As far as we are aware, this paper reports the first interactive system with force-feedback that provides "4D haptic visualization" permitting the user to model and interact with 4D cloth-like objects.

Navigating through large-scale virtual environments such as simulations of the astrophysical Universe is difficult. The huge spatial range of astronomical models and the dominance of empty space make it hard for users to travel across cosmological scales effectively, and the problem of wayfinding further impedes the user's ability to acquire reliable spatial knowledge of astronomical contexts. We introduce a new technique called the scalable world-in-miniature (WIM) map as a unifying interface to facilitate travel and wayfinding in a virtual environment spanning gigantic spatial scales: Power-law spatial scaling enables rapid and accurate transitions among widely separated regions; logarithmically mapped miniature spaces offer a global overview mode when the full context is too large; 3D landmarks represented in the WIM are enhanced by scale, positional, and directional cues to augment spatial context awareness; a series of navigation models are incorporated into the scalable WIM to improve the performance of travel tasks posed by the unique characteristics of virtual cosmic exploration. The scalable WIM user interface supports an improved physical navigation experience and assists pragmatic cognitive understanding of a visualization context that incorporates the features of large-scale astronomy.

We present a multimodal paradigm for exploring topological surfaces embedded in four dimensions; we exploit haptic methods in particular to overcome the intrinsic limitations of 3D graphics images and 3D physical models. The basic problem is that, just as 2D shadows of 3D curves lose structure where lines cross, 3D graphics projections of smooth 4D topological surfaces are interrupted where one surface intersects another. Furthermore, if one attempts to trace real knotted ropes or a plastic models of self-intersecting surfaces with a fingertip, one inevitably collides with parts of the physical artifact. In this work, we exploit the free motion of a computer-based haptic probe to support a continuous motion that follows the local continuity of the object being explored. For our principal test case of 4D-embedded surfaces projected to 3D, this permits us to follow the full local continuity of the surface as though in fact we were touching an actual 4D object. We exploit additional sensory cues to provide supplementary or redundant information. For example, we can use audio tags to note the relative 4D depth of illusory 3D surface intersections produced by projection from 4D, as well as providing automated refinement of the tactile exploration path to eliminate jitter and snagging, resulting in a much cleaner exploratory motion than a bare uncorrected motion. Visual enhancements provide still further improvement to the feedback: by opening a view-direction-defined cutaway into the interior of the 3D surface projection, we allow the viewer to keep the haptic probe continuously in view as it traverses any touchable part of the object. Finally, we extend the static tactile exploration framework using a dynamic mode that links each stylus motion to a change in orientation that creates at each instant a maximal-area screen projection of a neighborhood of the current point of interest. This minimizes 4D distortion and permits true metric sizes to be deduced locally at any point. All these methods combine to reveal the full richness of the complex spatial relationships of the target shapes, and to overcome many expected perceptual limitations in 4D visualization.

Stars form in dense clouds of interstellar gas and dust. The residual dust surrounding a young star scatters and diffuses its light, making the star's "cocoon" of dust observable from Earth. The resulting structures, called reflection nebulae, are commonly very colorful in appearance due to wavelength-dependent effects in the scattering and extinction of light. The intricate interplay of scattering and extinction cause the color hues, brightness distributions, and the apparent shapes of such nebulae to vary greatly with viewpoint. We describe an interactive visualization tool for realistically rendering the appearance of arbitrary 3D dust distributions surrounding one or more illuminating stars. Our rendering algorithm is based on the physical models used in astrophysics research. The tool can be used to create virtual fly-throughs of reflection nebulae for interactive desktop visualizations, or to produce scientifically accurate animations for educational purposes, e.g., in planetarium shows. The algorithm is also applicable to investigate on-the-fly the visual effects of physical parameter variations, exploiting visualization technology to help gain a deeper and more intuitive understanding of the complex interaction of light and dust in real astrophysical settings.

We propose a novel 2D representation for 3D visibility sorting, the Binary-Space-Partitioned Image (BSPI), to accelerate real-time image-based rendering. BSPI is an efficient 2D realization of a 3D BSP tree, which is commonly used in computer graphics for time-critical visibility sorting. Since the overall structure of a BSP tree is encoded in a BSPI, traversing a BSPI is comparable to traversing the corresponding BSP tree. BSPI performs visibility sorting efficiently and accurately in the 2D image space by warping the reference image triangle-by-triangle instead of pixel-by-pixel. Multiple BSPIs can be combined to solve "disocclusion," when an occluded portion of the scene becomes visible at a novel viewpoint. Our method is highly automatic, including a tensor voting preprocessing step that generates candidate image partition lines for BSPIs, filters the noisy input data by rejecting outliers, and interpolates missing information. Our system has been applied to a variety of real data, including stereo, motion, and range images.

We explore a family of specific dynamical methods that support the contrasting goals of presence and independence in collaborative virtual environments. We propose the basic tasks of "tethering" (keeping a collaborator close to a group or leader), and of "reattachment" (returning to a collaborative virtual activity after a period of independent exploration). We first present a taxonomy of methods and parameters associated with tethering and reattachment, and then describe a formative evaluation study.

We approach the problem of exploring a virtual space by exploiting positional and camera-model constraints on navigation to provide extra assistance that focuses the user's explorational wanderings on the task objectives. Our specific design incorporates not only task-based constraints on the viewer's location, gaze, and viewing parameters, but also a personal "glide" that serves two important functions: keeping the user oriented in the navigation space, and "pointing" to interesting subject areas as they are approached. The guide's cues may be ignored by continuing in motion, but if the user stops, the gaze shifts automatically toward whatever the guide was interested in. This design has the serendipitous feature that it automatically incorporates a nested collaborative paradigm simply by allowing any given viewer to be seen as the "guide" of one or more viewers following behind; the leading automated guide (we tend to select a guide dog for this avatar) can remind the leading live human guide of interesting sites to point out, while each real human collaborator down the chain has some choices about whether to follow the local leader's hints. We have chosen VRML as our initial development medium primarily because of its portability, and we have implemented a variety of natural modes for leading and collaborating, including ways for collaborators to attach to and detach from a particular leader.

We propose a general paradigm for computing optimal coordinate frame fields that may be exploited to visualize curves and surfaces. Parallel transport framings, which work well for open curves, generally fail to have desirable properties for cyclic curves and for surfaces. We suggest that minimal quaternion measure provides an appropriate heuristic generalization of parallel transport. Our approach differs from minimal tangential acceleration approaches due to the addition of "sliding ring" constraints that fix one frame axis, but allow an axial rotational freedom whose value is varied in the optimization process. Our fundamental tool is the quaternion Gauss map, a generalization to quaternion space of the tangent map for curves and of the Gauss map for surfaces. The quaternion Gauss map takes 3D coordinate frame fields for curves and surfaces into corresponding curves and surfaces constrained to the space of possible orientations in quaternion space. Standard optimization tools provide application specific means of choosing optimal, e.g., length- or area-minimizing, quaternion frame fields in this constrained space.

We attack the problem of image based rendering with occlusions and general camera motions by using distorted multiperspective images; such images provide multiple viewpoint photometry similar to the paintings of cubist artists. We take scene geometry, in contrast, to be embodied in mappings of viewing rays from their original 3D intercepts into the warped multiperspective image space. This approach allows us to render approximations of scenes with occlusions using time dense and spatially sparse sequences of camera rays, which is a significant improvement over the storage requirements of an equivalent animation sequence. Additional data compression can be achieved using sparse time keyframes as well. Interpolating the paths of sparse time key rays correctly in image space requires singular interpolation functions with spatial discontinuities. While there are many technical questions yet to be resolved, the employment of these singular interpolation functions in the multiperspective image space appears to be of potential interest for generating general viewpoint scene renderings with minimal data storage.

Navigation through 3D spaces is required in many interactive graphics and virtual reality applications. The authors consider the subclass of situations in which a 2D device such as a mouse controls smooth movements among viewpoints for a "through the screen" display of a 3D world. Frequently, there is a poor match between the goal of such a navigation activity, the control device, and the skills of the average user. They propose a unified mathematical framework for incorporating context-dependent constraints into the generalized viewpoint generation problem. These designer-supplied constraint modes provide a middle ground between the triviality of a single camera animation path and the confusing excess freedom of common unconstrained control paradigms. They illustrate the approach with a variety of examples, including terrain models, interior architectural spaces, and complex molecules.

Proposes an interactive method for exploring topological spaces based on the natural local geometry of the space. Examples of spaces appropriate for this visualization approach occur in abundance in mathematical visualization, surface and volume visualization problems, and scientific applications such as general relativity. Our approach is based on using a controller to choose a direction in which to "walk" a manifold along a local geodesic path. The method automatically generates orientation changes that produce a maximal viewable region with each step of the walk. The proposed interaction framework has many natural properties to help the user develop a useful cognitive map of a space and is well-suited to haptic interfaces that may be incorporated into desktop virtual reality systems.

Curves in space are difficult to perceive and analyze, especially when they form dense sets as in typical 3D flow and volume deformation applications. We propose a technique that exposes essential properties of space curves by attaching an appropriate moving coordinate frame to each point, reexpressing that moving frame as a unit quaternion, and supporting interaction with the resulting quaternion field. The original curves in 3-space are associated with piecewise continuous 4-vector quaternion fields, which map into new curves lying in the unit 3-sphere in 4-space. Since 4-space clusters of curves with similar moving frames occur independently of the curves' original proximity in 3-space, a powerful analysis tool results. We treat two separate moving-frame formalisms, the Frenet frame and the parallel-transport frame, and compare their properties. We describe several flexible approaches for interacting with and exploiting the properties of the 4D quaternion fields.>

The generalized Hough transform (GHough) is a useful technique for detecting and locating 2D shapes. However, GHough requires a 4D accumulator array to detect objects of unknown scale and orientation. In this paper, we propose an extension of GHough, the virtual line segment-based Hough transform (VHough) that requires much less storage than GHough to accurately determine the scale and orientation of an object instance. VHough takes O(N/sup 2/) time, where N is the number of edge pixels in an image, but requires only 2D accumulator array for the detection of arbitrarily rotated and scaled objects. We present an experimental result to show that VHough is well-suited to recognition tasks when no a priori knowledge about parameters is available.

We describe the theoretical and practical visualization issues solved in the implementation of an interactive real-time four-dimensional geometry interface for the CAVE, an immersive virtual reality environment. While our specific task is to produce a "virtual geometry" experience by approximating physically correct rendering of manifolds embedded in four dimensions, the general principles exploited by our approach reflect requirements common to many immersive virtual reality applications, especially those involving volume rendering. Among the issues we address are the classification of rendering tasks, the specialized hardware support required to attain interactivity, specific techniques required to render 4D objects, and interactive methods appropriate for our 4D virtual world application.>

Flow fields, geodesics, and deformed volumes are natural sources of families of space curves that can be characterized by intrinsic geometric properties such as curvature, torsion, and Frenet frames. By expressing a curve's moving Frenet coordinate frame as an equivalent unit quaternion, we reduce the number of components that must be displayed from nine with six constraints to four with one constraint. We can then assign a color to each curve point by dotting its quaternion frame with a 4D light vector, or we can plot the frame values separately as a curve in the three-sphere. As examples, we examine twisted volumes used in topology to construct knots and tangles, a spherical volume deformation known as the Dirac string trick, and streamlines of 3D vector flow fields.>

Making accurate computer graphics representations of surfaces and volumes (2-manifolds and 3-manifolds) embedded in four-dimensional space typically involves complex and time-consuming computations. In order to make simulated worlds that help develop human intuition about the fourth dimensions, we need techniques that permit real-time, interactive manipulation of the most sophisticated depictions available. We propose the following new methods that bring us significantly closer to this goal: an approach to high-speed 4D illuminated surface rendering incorporating 4D shading and occlusion coding; a procedure for rapidly generating 2D screen images of tessellated 3-manifolds illuminated by 4D light. These methods are orders of magnitude faster than previous approaches, enabling the real-time manipulation of high-resolution 4D images on commercial graphics hardware.>

Scalar functions of three variables, w=f(x, y, z), are common in many types of scientific and medical applications. Such 3D scalar fields can be understood as elevation maps in four dimensions, with three independent variables (x, y, z) and a fourth, dependent, variable w that corresponds to the elevations. It is shown how techniques developed originally for the display of 3-manifolds in 4D Euclidean space can be adapted to visualize 3D scalar fields in a variety of ways.>

Techniques for visualizing mathematical objects in four-dimensional (4D) space that exploit four-dimensional lighting effects are explored. The geometry of image production, stereography, and shadows in 4D is analyzed. Alternatives for smooth and specular shaded rendering of curves, surfaces, and solids in 4D are examined and a new approach that systematically converts curves or surfaces into uniquely renderable solids in 4D space by attaching spheres or circles to each point is proposed. Analogs of 3D shading methods are used to produce volume renderings that distinguish objects whose 3D projections from 4D are identical. Analyzing the procedures needed to justify and evaluate a system as this for teaching humans to 'see' in four dimensions leads to the proposal of a generally applicable four-step visualization paradigm.>

The authors describe some mathematical approaches and computer graphics techniques for illustrating concepts related to Fermat's last theorem. They present a selection of visualization methods, and describe observations made in the process of creating a three-minute computer animated videotape dealing with some elementary aspects of Fermat's last theorem, a problem in number theory. The approach to the representation of the different concepts presented in the video was influenced by many factors: the available hardware, real and perceived constraints of the available software, constraints imposed by the video medium, and a number of peculiarities and features of the mathematical domain itself. The authors describe the experiences with the software systems that played a part in these efforts, some specific successful visualization techniques, and some unexpected mathematical insights.>