Dietmar Hildenbrand

Tutorial: Geometric computing in computer graphics using conformal geometric algebra

Early in the development of computer graphics it was realized that projective geometry was well suited for the representation of transformations. Now, it seems that another change of paradigm is lying ahead of us based on geometric computing using conformal geometric algebra. Due to its geometric intuitiveness, elegance and simplicity, the underlying conformal geometric algebra appears to be a promising mathematical tool for computer graphics and animations. In this tutorial paper we introduce into the basics of the conformal geometric algebra and show its advantages based on two computer graphics applications. First, we will present an algorithm for the inverse kinematics of a robot that you are able to comprehend without prior knowledge of geometric algebra. We expect that here you will obtain the basic knowledge for developing your own algorithm afterwards. Second, we will show how easy it is in conformal geometric algebra, to fit the best suitable object in a set of points, whether it is a plane or a sphere.

Gaalop—High Performance Parallel Computing Based on Conformal Geometric Algebra

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, current status, and future perspectives of Gaalop dealing with optimized software implementations, hardware implementations, and mixed solutions. An inverse kinematics algorithm of a humanoid robot is described as an example.

Acceleration and Energy Efficiency of a Geometric Algebra Computation using Reconfigurable Computers and GPUs

Geometric algebra (GA) is a mathematical framework that allows the compact description of geometric relationships and algorithms in many fields of science and engineering. The execution of these algorithms, however, requires significant computational power that made the use of GA impractical for many real-world applications. We describe how a GA-based formulation of the inverse kinematics problem from computer animation and robotics can be accelerated using reconfigurable FPGA-based computing and using a graphics processing unit (GPU). The practical evaluation covers not only the sheer compute performance, but also the energy efficiency.

Inverse kinematics of a humanoid robot based on conformal geometric algebra using optimized code generation

This paper presents a solution to solve the inverse kinematics for the legs of a humanoid robot using conformal geometric algebra. We geometrically intuitively develop the algorithm with the freely available CLUCalc software and optimize it with the help of the computer algebra system Maple and the Clifford package for geometric algebras. We describe our Gaalop code generator which produces executable C code with just elementary expressions leading to a very efficient implementation.

FPGA-accelerated color edge detection using a Geometric-Algebra-to-Verilog compiler

Geometric Algebra (GA) is a branch of mathematics that generalizes complex numbers and quaternions. One of the advantages of the framework is, that it allows intuitive description and manipulation of geometric objects. While even complex operations can be described concisely, the actual evaluation of these GA expressions is extremely compute intensive. However, it has significant fine-grained parallelism, which makes it a profitable target for hardware implementation. In this paper, we present the automatic acceleration of a color edge-detection algorithm from a GA description. Using our Gaalop GA compiler with its Verilog back-end, we can show speed-ups of over 1000x even compared to a recent GA processor ASIC.

Accelerating high-level engineering computations by automatic compilation of Geometric Algebra to hardware accelerators

Geometric Algebra (GA), a generalization of quaternions, is a very powerful form for intuitively expressing and manipulating complex geometric relationships common to engineering problems. The actual evaluation of GA expressions, though, is extremely compute intensive due to the high-dimensionality of data being processed. On standard desktop CPUs, GA evaluations take considerably longer than conventional mathematical formulations. GPUs do offer sufficient throughput to make the use of concise GA formulations practical, but require power far exceeding the budgets for most embedded applications. While the suitability of low-power reconfigurable accelerators for evaluating specific GA computations has already been demonstrated, these often required a significant manual design effort. We present a proof-of-concept compile flow combining symbolic and hardware optimization techniques to automatically generate accelerators from the abstract GA descriptions without user intervention that are suitable for high-performance embedded computing.

Geometric Algebra Computers

How should a computer for Geometric Algebra be designed? This chapter investigates different computing architectures with the goal of implementing Geometric Algebra algorithms with as high a performance as possible.

A hybrid approach for computing products of high-dimensional geometric algebras

Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics solutions. For example, the Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. This paper presents a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution for precomputed code with fast execution and runtime computations with low memory requirement. More specifically, the proposed method combines a precomputed table approach with a recursive method using binary trees. Some rules are defined to select the most appropriate choice, according to the dimension of the algebra and the type of multivectors involved in the product. The resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional space. This paper details the integration of this hybrid method as a plug-in into Gaalop, which is a very advanced optimizing code generator. This paper also presents some benchmarks to show the performances of our method, especially in high dimensional spaces.

Differential Methods for Multi-Dimensional Visual Data Analysis

Images in scientific visualization are the end-product of the data processing. Starting from higher-dimensional datasets, such as scalar-, vector-, tensor- fields given on 2D, 3D, 4D domains, the o ...

Conformal Geometric Algebra

In this book, we focus on 5D Conformal Geometric Algebra (CGA). The “conformal” comes from the fact that it handles conformal transformations easily. These transformations leave angles invariant.