VERSOR: Spatial Computing with Conformal Geometric Algebra (original) (raw)
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Conformal Geometry, Euclidean Space and Geometric Algebra
Uncertainty in Geometric Computations, 2002
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to programming complicated geometrical operations. But there is a fundamental weakness in this approach-the Euclidean distance between points is not handled in a straightforward manner. Here we discuss a solution to this problem, based on conformal geometry. The language of geometric algebra is best suited to exploiting this geometry, as it handles the interior and exterior products in a single, unified framework. A number of applications are discussed, including a compact formula for reflecting a line off a general spherical surface.
Geometric algebra and its application to computer graphics
2004
Early in the development of computer graphics it was realized that projective geometry is suited quite well to represent points and transformations. Now, maybe another change of paradigm is lying ahead of us based on Geometric Algebra. If you already use quaternions or Lie algebra in additon to the well-known vector algebra, then you may already be familiar with some of the algebraic ideas that will be presented in this tutorial. In fact, quaternions can be represented by Geometric Algebra, next to a number of other algebras like complex numbers, dual-quaternions, Grassmann algebra and Grassmann-Cayley algebra. In this half day tutorial we will emphasize that Geometric Algebra
Gaalop—High Performance Parallel Computing Based on Conformal Geometric Algebra
Geometric Algebra Computing, 2010
We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computing based on conformal geometric algebra. The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. In order to achieve this goal, our focus is on parallel target platforms like FPGA (field-programmable gate arrays) or the CUDA technology from NVIDIA. We describe the concepts, the current status, as well as the future perspectives of Gaalop dealing with optimized software implementations, hardware implementations as well as mixed solutions. An inverse kinematics algorithm of a humanoid robot is described as an example.
Inverse Kinematics Computation in Computer Graphics and Robotics Using Conformal Geometric Algebra
Advances in Applied Clifford Algebras, 2008
We focus on inverse kinematics applications in computer graphics and robotics based on Conformal Geometric Algebra. Here, geometric objects like spheres and circles that are often needed in inverse kinematics algorithms are simply represented by algebraic objects. We present algorithms for the inverse kinematics of a human arm like kinematic chain and for the grasping of robots and virtual humans. The main benefits of using geometric algebra in the virtual reality software Avalon are the easy, compact and geometrically intuitive formulation of the algorithms and the immediate computation of quaternions.
ConformalALU: A Conformal Geometric Algebra Coprocessor for Medical Image Processing
IEEE Transactions on Computers, 2015
Medical imaging involves important computational geometric problems, such as image segmentation and analysis, shape approximation, three-dimensional (3D) modeling, and registration of volumetric data. In the last few years, Conformal Geometric Algebra (CGA), based on five-dimensional (5D) Clifford Algebra, is emerging as a new paradigm that offers simple and universal operators for the representation and solution of complex geometric problems. However, the widespread use of CGA has been so far hindered by its high dimensionality and computational complexity. This paper proposes a simplified formulation of the conformal geometric operations (reflections, rotations, translations, and uniform scaling) aimed at a parallel hardware implementation. A specialized coprocessing architecture (ConformalALU) that offers direct hardware support to the new CGA operators, is also presented. The ConformalALU has been prototyped as a complete System-on-Programmable-Chip (SoPC) on the Xilinx ML507 FPGA board, containing a Virtex-5 FPGA device. Experimental results show average speedups of one order of magnitude for CGA rotations, translations, and dilations with respect to the geometric algebra software library Gaigen running on the general-purpose PowerPC processor embedded in the target FPGA device. A suite of medical imaging applications, including segmentation, 3D modeling and registration of medical data, has been used as testbench to evaluate the coprocessor effectiveness. Index Terms-Conformal geometric algebra, five-dimensional clifford algebra, computational geometry, embedded coprocessors, systemson-programmable-chip, FPGA-based prototyping, medical imaging, segmentation, 3D modeling, Volume registration, Growing Neural Gas, marching spheres, iterative closest point (ICP), thin-plate spline robust point matching (TPS-RPM) Ç 1 INTRODUCTION M EDICAL imaging plays an important role in current medical research and clinical practice. Efficient algorithms are required to solve complex geometric problems arising in medical image processing, such as segmentation, shape extraction, three-dimensional (3D) modeling and registration of medical data. A key problem in medical computation is the reconstruction of 3D shapes (of organs, bones, tumors, etc.) from two-dimensional (2D) slices derived from Magnetic Resonance (MR) or Computed Tomography (CT) scans [1], [2], [3], [4]. This is a typical geometric problem that consists in finding a proper surface connecting a set of contour data points. Efficient geometric tools are also required for medical image registration that consists in finding a proper geometrical transformation that aligns different views of the same image taken in different moments or by diverse acquisition modalities [5], [6], [7]. Computational geometry deals with finding solutions to geometric problems that arise in medical imaging as well as in other application domains, such as computer graphics, robotics, computer vision, and
2004
CLUCalc is a user friendly frontend to these libraries. It is used in the "Interactive Introduction. .. " and is available for download from [27]. In CLUCalc you can type your equations in a simple script language, called CLUScript and visualize the results immediately with OpenGL graphics. The program comes with a manual in HTML form and a number of example scripts. There is also an online version of the manual under: http://www.perwass.de/CLU/CLUCalcDoc/ CLUCalc should serve as a good accompaniment to this script, helping you to understand the concepts behind Geometric algebra visually. The CLUScripts used in chapter three can also be downloaded through the following link: www.dgm.informatik.tu-darmstadt.de/staff/dietmar/ By the way, CLUCalc was also used to create all of the 2d and 3d graphics in this script. You can use it for the same purpose, illustrating your publications or web-pages, from the version 3.0 onwards, which is now available. Some other features of CLUCalc v3.0.0 are: • render and display LaTeX text and formulas to annotate your graphics, or to create slides for presentations, • prepare presentations with user interactive 3D-graphics included in your slides, • draw 2D-surfaces, including the surface generated by a set of circles, • do structured programming with if-clauses and loops, • do error propagation in Clifford algebra, • and much more...
Engineering Graphics in Geometric Algebra
Geometric Algebra Computing, 2010
We illustrate the suitability of geometric algebra for representing structures and developing algorithms in computer graphics, especially for engineering applications. A number of example applications are reviewed. Geometric algebra unites many underpinning mathematical concepts in computer graphics such as vector algebra and vector fields, quaternions, kinematics and projective geometry, and it easily deals with geometric objects, operations and transformations. Not only are these properties important for computational engineering, but also for the computational point-of-view they provide. We also include the potential of geometric algebra for optimizations and highly efficient implementations.
Navigation functions in Conformal Geometric Algebra
IX Latin American Robotics Symposium and IEEE Colombian Conference on Automatic Control, 2011 IEEE, 2011
Conformal Geometric Algebra (CGA) can greatly improve controllers by simplifying the necessary equations and by its ability to apply geometric operations to more complicated geometric entities. In this paper we extend a singularity free CGA-based angular and linear velocity controller with navigation functions. The first navigation function ensures that the object being tracked is always within the camera's field of view. The second navigation function is the ability of the controller to avoid collisions with other objects. These navigation functions can be easily added to the CGA-based controller, experimentally ensured the desired goals and proven stable by Lyapunov.
Analysis of Point Clouds - Using Conformal Geometric Algebra
This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra. In this framework it is easy to compute with osculating circles for the description of local curvature. Also methods for the fitting of spheres as well as bounding spheres are presented. In a nutshell, this paper provides a starting point for shape analysis based on this new, geometrically intuitive and promising technology.
Recent Advances in Computational Conformal Geometry
Communications in Information and Systems, 2009
Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. Holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower quality. Curvature flow method is nonlinear and requires higher quality triangulations, but it is more flexible. The conformal geometric methods have been broadly applied in many engineering fields, such as computer graphics, vision, geometric modeling and medical imaging. The algorithms are robust for surfaces scanned from real life, general for surfaces with different topologies. The efficiency and efficacy of the algorithms are demonstrated by the experimental results.