VERSOR: Spatial Computing with Conformal Geometric Algebra (original) (raw)

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

Articulating Space: Geometric Algebra for Parametric Design -- Symmetry, Kinematics, and Curvature

To advance the use of geometric algebra in practice, we develop computational methods for parameterizing spatial structures with the conformal model. Three discrete parameterizations – symmetric, kinematic, and curvilinear – are employed to generate space groups, linkage mechanisms, and rationalized surfaces. In the process we illustrate techniques that directly benefit from the underlying mathematics, and demonstrate how they might be ap- plied to various scenarios. Each technique engages the versor – as opposed to matrix – representation of transformations, which allows for structure-preserving operations on geometric primitives. This covariant methodology facilitates constructive design through geometric reasoning: incidence and movement are expressed in terms of spatial variables such as lines, circles and spheres. In addition to providing a toolset for generating forms and transformations in computer graphics, the resulting expressions could be used in the design and fabrication of machine parts, tensegrity systems, robot manipulators, deployable structures, and freeform architectures. Building upon existing CGA-specific algorithms, these methods participate in the advancement of geometric thinking, leveraging an intuitive spatial logic that can be creatively applied across disciplines, ranging from time-based media to mechanical and structural engineering, or reformulated in higher dimensions.

Conformal Geometric Algebra for Robotic Vision

Journal of Mathematical Imaging and Vision, 2005

In this paper the authors introduce the conformal geometric algebra in the field of visually guided robotics. This mathematical system keeps our intuitions and insight of the geometry of the problem at hand and it helps us to reduce considerably the computational burden of the problems. As opposite to the standard projective geometry, in conformal geometric algebra we can deal simultaneously with incidence algebra operations (meet and join) and conformal transformations represented effectively using spinors. In this regard, this framework appears promising for dealing with kinematics, dynamics and projective geometry problems without the need to resort to different mathematical systems (as most current approaches do). This paper presents real tasks of perception and action, treated in a very elegant and efficient way: body-eye calibration, 3D reconstruction and robot navigation, the computation of 3D kinematics of a robot arm in terms of spheres, visually guided 3D object grasping making use of the directed distance and intersections of lines, planes and spheres both involving conformal transformations. We strongly believe that the framework of conformal geometric algebra can be, in general, of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems.

Applications of Geometric Algebra in Computer Science and Engineering

Applications of Geometric Algebra in Computer Science and Engineering, 2002

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Computational conformal geometry applied in engineering fields

2007

Computational conformal geometry is an interdisciplinary field, combining modern geometry theories from pure mathematics with computational algorithms from computer science. Computational conformal geometry offers many powerful tools to handle a broad range of geometric problems in engineering fields. This work summarizes our research results in the past years. We have introduced efficient and robust algorithms for computing conformal structures of surfaces acquired from the real life, which are based on harmonic maps, holomorphic differential forms and surface Ricci flow. We have applied conformal geometric algorithms in computer graphics, computer vision, geometric modeling and medical imaging.

Conformal Geometric Objects with Focus on Oriented Points

2011

In this paper, we explore the geometric objects of conformal geometric algebra based on their IPNS (inner product null space) representation in some detail. Spheres of dimension 1 , 2 and three are objects of conformal geometric algebra. Usually, points in conformal geometric algebra are represented as ordinary spheres with zero radius, but what about circles with zero radius? We expect many practical applications of these points with additional orientation information.