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A Visualization Model Based on the Mathematics of Fiber Bundles
Abstract
In this report, we describe a visualization model based on the mathematics of fiber bundles. In a companion report (SAND 88-8972), we introduced the visualization management system (ViMS), a new approach to the development of software for visualization in scientific computing (ViSC). A visualization management system implements a visualization model which specifies the class of geometric objects, the graphic representations of the objects and the operations provided by the system. In the model described here, the geometric objects are sections of fiber bundles. We give a brief, intuitive description of the mathematics of fiber bundles, emphasizing aspects relevant to our visualization model. We describe three important classes of operations on fiber bundles. We develop a flexible scheme for constructing graphic representations of fiber bundles and a simple but useful visualization taxonomy. 7 refs., 5 figs., 1 tab.
... The fiber bundles are geometric as well as topological objects, which can be simulated and visualized in computer models. The computer visualization of fiber bundles as geometric objects has opened up a wide array of applications in various domains of physical sciences as well as computational sciences [14]. In the topological spaces of fiber bundles, the determination of equivalence between fiber bundles is a challenging task. ...
... I = R, z a + z b = z a+b ∈ C, µ a×I + µ b×I = µ (a+b)×I = (z a+b , I), (µ a×I + µ b×I ) + µ c×I = µ a×I + (µ b×I + µ c×I ). (14) Moreover, at the origin x 0 of (X, τ X ) the fiber µ 0×I is an identity fiber (i.e., original fiber) because, ∀µ a×I ∈ Γ CR , µ a×I + µ 0×I = µ 0×I + µ a×I = µ a×I . Furthermore, if we consider fibers such that ∀µ a×I ∈ Γ CR , ∃µ −a×I ∈ Γ CR maintaining µ a×I + µ −a×I = µ −a×I + µ a×I = µ 0×I then µ −a×I is the inverse of µ a×I . ...
A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non-compact) in the non-uniformly scalable quasinormed topological (C, R) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non-compact fiber space. The finite translation is continuous on complex planar subspace under non-compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non-commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non-compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological (C, R) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma-sections as compared to the fiber bundles on manifolds.
... The mathematics of fiber bundles provides a framework to model data for scientific visualization [BP89,Ben04]. This concept considers data sets based on the properties of their "base space" versus its "fiber space". ...
We present a novel methodological approach for the interactive editing of big point clouds. Based on the mathematics of fiber bundles, the proposed approach to model a data structure that is efficient for visualization, modification and I/O including an unlimited multi-level set of editing states useful for expressing and maintaining multiple undo histories. Backed by HDF5 as high performance file format, this data structure naturally allows persistent storage for the history of modification actions, an unique new feature of our approach. The challenges of visually based manual editing of big point clouds are discussed and a proper rendering solution is presented. The implemented solution and its features as consequences of the underlying methodology is compared with two major mainstream applications providing point-cloud editing tools as well.
... It casts all data into a hierarchy of five plus two optional levels ). The premise is that nature is best described by a differentiable manifold (Butler and Pendley, 1989;Benger et al., 2011). This scheme covers both point clouds with attributes per point in a natural way as well as waveforms as extended information per point or per laser shot. ...
Topo-bathymetric LiDAR data captured with modern systems are complex. The primary information received is a time-dependent amplitude variation of the reflected light. These so-called waveforms are processed into singular points with 3D coordinates by advanced on-board devices of the LiDAR sensor (online waveform processing), but the full-waveform (FWF) information may be available as well. However, available software tools are often insufficient to manage all required processing steps. Common file formats do not allow to store originally recorded sensor parameters together with subsequently processed parameters in one database or file. The FWF, however, can contain information to better cover the terrain below dense vegetation, and to improve aerial coverage of the water ground. Thus, we extended the software suite HydroVISH with respect to an integrated FWF processing pipeline. Employing the open-source Hierarchical Data Format V5 (HDF5) with the F5 layout thereby allows for efficient data storage and handling throughout the processing chain. The potential benefit of performing a comprehensive FWF analysis can be assessed via the simultaneous visualization of the complete FWF information on all points in an interactive display environment. Next, the valuable point information is extracted using various provided FWF processing tools, such as Richardson–Lucy deconvolution or Gaussian decomposition. For topo-bathymetric data, the correct point classification of the terrain above and below water as well as the actual water surface is crucial to correctly calculate the refraction correction for points beneath the water surface. Finally, we also outline the implemented classification approach for the terrain and water surface.
... Our data model of choice is the Fiber Bundle HDF5 "F5" model , which casts all data into a hierarchy of five (plus two optional) levels [3]. The premise is that nature is best described by a differentiable manifold [4]. This data model is very suitable for I/O and is directly compatible with GPU data structures such as vertex and index buffer objects. ...
... 3.1. Mathematical Inheritance Relationships D. Butler & Pendley envisioned a generic data model for scientific data based on fiber and vector bundles [16], advocating that their "mathematical hierarchy can be considered a prescription for a library of abstract classes" [17]. They considered vector bundles as carrier for linear algebra operations, a vector field is then a section of the vector bundle (the "macro level") where for each point of the underlying manifold one particular vector value (the "micro level") is given. ...
http://rdcu.be/vdRh
Geometric algebra (GA) is a promising approach to address interoperability bottlenecks that are particularly prominent in the big data era. Similar to how GA unites and simplifies otherwise distinct mathematical branches it may also help to unite software via common interfaces between otherwise distinct applications. The promising potential of GA would be best exhibited by the ability of seamless integration into existing, complex applications. To achieve this vision various constraints have to be considered. Particularly having C++ in focus, we discuss the “wish list” that an optimal C++ implementation should provide. We find that to cover the various constraints an hybrid approach benefiting from multiple programming paradigms, ranging from generic to object-oriented programming, will be needed. C++ is a very suitable platform providing all these capabilities and promising approaches like Generative Programming and Active Libraries provide technology highly desirable for universally promoting GA to an extensive range of application domains.
... Butler and Pendley envisioned a generic data model for scientific data based on fiber and vector bundles [7], advocating that their "mathematical hierarchy can be considered a prescription for a library of abstract classes" [6]. They considered vector bundles as carrier for linear algebra operations, a vector field is then a section of the vector bundle (the "macro level") where for each point of the underlying manifold one particular vector value (the "micro level") is given. ...
Geometric algebra (GA) is a promising approach to address interoperability bottlenecks that are particularly prominent in the big data era. Similar to how GA unites and simplifies otherwise distinct mathematical branches it may also help to unite software via common interfaces between otherwise distinct applications. The promising potential of GA would be best exhibited by the ability of seamless integration into existing, complex applications. To achieve this vision various constraints have to be considered. Particularly having C++ in focus, we discuss the “wish list” that an optimal C++ implementation should provide. We find that to cover the various constraints an hybrid approach benefiting from multiple programming paradigms, ranging from generic to object-oriented programming, will be needed. C++ is a very suitable platform providing all these capabilities and promising approaches like Generative Programming and Active Libraries provide technology highly desirable for universally promoting GA to an extensive range of application domains.
... Notations to describe the data space including data context and data content are, for example, the framework of Galhardas et al. (1998) that is based on first-order logic, as well as the E-notation by Brodlie (1992) and its extension into the domino notation (Brodlie and Noor 2007). Older variants are the fiber bundle-based notation by Butler and Pendley (1989) and the L-notation by Bergeron and Grinstein (1989). File formats with metadata capabilities are, for example, CDF, HDF, NetCDF, XDF, or XSIL. ...
Visualization has become an important ingredient of data analysis, supporting users in exploring data and confirming hypotheses. At the beginning of a visual data analysis process, data characteristics are often assessed in an initial data profiling step. These include, for example, statistical properties of the data and information on the data’s well-formedness, which can be used during the subsequent analysis to adequately parametrize views and to highlight or exclude data items. We term this information data descriptors, which can span such diverse aspects as the data’s provenance, its storage schema, or its uncertainties. Gathered descriptors encapsulate basic knowledge about the data and can thus be used as objective starting points for the visual analysis process. In this article, we bring together these different aspects in a systematic form that describes the data itself (e.g. its content and context) and its relation to the larger data gathering and visual analysis process (e.g. its provenance and its utility). Once established in general, we further detail the concept of data descriptors specifically for tabular data as the most common form of structured data today. Finally, we utilize these data descriptors for tabular data to capture domain-specific data characteristics in the field of climate impact research. This procedure from the general concept via the concrete data type to the specific application domain effectively provides a blueprint for instantiating data descriptors for other data types and domains in the future.
... Secondly, the MVTree structure is limited to regular geometric primitives, therefore, the continuous geographic space and manifold objects representation will be the important work in the future research. Some meaningful data models like fiber bundles model [3] and manifolds structure would be beneficial references. ...
Resolving the conflicts between the high dimensionality of geometry representation and the linear organization and storage of geometrical objects in computer memories plays a key role in spatial data structure constructions. In this paper, a new data structure MVTree is developed based on geometric algebra to support the unified organization and computation of geometrical primitives. The MVTree is a tree-like data structure which has a dimensional hierarchical structure generated by outer product. Multidimensional geometrical primitives is represented as the combination of blades stored in the nodes of MVTrees. The geometric computation between different geometrical objects is operated with GA operators with a judgement-based hierarchical computation. Applications of the MVTree are demonstrated by a topological relation computation and a Delaunay-TIN intersection. The results suggest that the MVTree structure can support unified representation of arbitrary-dimensional geometrical primitives, and can integrate data organization and computation in a unitary structure as well. The application of the new MVTree data structure can not only reduce the complexity of data architectures but also inherit the power of geometric algebra computing to improve the processing ability of computer graphic software.
A fascinating survey of pre-computer visualization is given by Collins [15]. He concludes that, whilst there is little in computer-generated visualization that is new in terms of techniques, computers do nonetheless make visualization a possibility for large quantities of data. Some useful insights into the factors contributing to the rapid development of computer visualization in the late 1980s are given by Brodlie in [3].
Multi-sensor analysis is a novel scientific approach in remote sensing science. The basic idea is to enable users to combine various satellite mission data (called scenes) into a common data set. This combination produces millions of high-resolution time series (one time series for each pixel) from which users want to extract potentially interesting spatio-temporal patterns. A challenge of multi-sensor analysis is that users often experience difficulties interpreting the extracted patterns. We use Visual Analytics (VA) to help users understand these patterns. We learned from our interdisciplinary cooperation in the GeoMultiSens project that VA has to support the assessment and selection of scenes suitable for the current application scenario and question to achieve this goal. The contribution of this paper is twofold. First, we describe how we devised a VA approach that supports users in the assessment and selection of remote sensing data based on a user and task analysis. We demonstrate how our VA approach helps users to select and assess scenes to study forest cover change in Europe between 2010 and 2016. The study of forest cover change is an important scientific scenario because the loss of forest cover has negative effects on the environment, such as undermining the capacity of ecosystems to maintain fresh water, loosing the ability to regulate the climate, and poorer air quality. Second, we discuss the Scientific Data Explorer, our research vision for VA to enable users to effectively develop VA approaches for a variety of scientific scenarios.
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