Geometric Algebra: A computational framework for geometrical applications Part 1

Dorst, Leo; Mann, Stephen

doi:10.1109/mcg.2002.999785

Cited by 70 publications

(48 citation statements)

References 3 publications

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“…A brief introduction to the theory is given in the following sections. Interested readers may find further details in [2,4,5,14,17,18,22].…”

Section: Geometric Algebra and Computer Graphicsmentioning

confidence: 99%

“…Also known as Clifford algebra (from its main contributor, W. K. Clifford [3]), this type of algebra offers an integrated approach to geometric modeling and algorithms. It permits the specification of geometric objects at a coordinate-free, unified level, where points, lines, and planes become basic elements of computation and their transformations are executed directly [2,4,5].…”

Section: Introductionmentioning

confidence: 99%

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An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

et al. 2009

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“…A brief introduction to the theory is given in the following sections. Interested readers may find further details in [2,4,5,14,17,18,22].…”

Section: Geometric Algebra and Computer Graphicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

et al. 2009

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“…For a more detailed review please see [5] [6]. Scalars, vectors, bivectors and trivectors (sweeping a bivector a ∧ c along another vector c) represent 0, 1, 2 and 3 dimensional subspaces respectively within Geometric Algebra.…”

Section: Basic Definitionsmentioning

confidence: 99%

Color edge detection hardware based on geometric algebra

Mishra

Wilson

2006

3rd European Conference on Visual Media Production (CVMP 2006). Part of the 2nd Multimedia Conference 2006

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Modern techniques treat color images as separate monochrome images for processing.Partly, because there is no straightforward generalization of linear filters available for color. However the algorithms yield more accurate results when the correlation among color bands are exploited which shows fundamental difference to process the color images. Earlier work [8] reported the transformation of the color images using Quaternion Fourier Transforms and the realization of a holistic filter based on Quaternion convolution. Here, we discuss the rotor based convolution techniques, a generalization of the previous work, within a new mathematical framework in Geometric Algebra. Based thereupon, a novel hardware architecture is proposed. Experiments show the edge detection with this technique belong to a class of linear vector filter and is holistic in nature. It is tailored for image processing applications, providing an acceptable application performance requirements. The usefulness of the introduced approach was demonstrated by analyzing and implementing a computation intensive edge detection algorithm on this hardware.

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“…Based upon affine geometry, Goldman [5,6] and DeRose [3,4,16] pioneered a method of writing graphics programs that are independent of the choice of reference coordinate frames. The study on geometric algebra pursues a similar goal with various geometric primitives rather than just vectors and points [12,13].…”

Section: Introductionmentioning

confidence: 99%

Representing Rotations and Orientations in Geometric Computing

Lee

2008

IEEE Comput. Grap. Appl.

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Geometric computing with three-dimensional rotations and orientations is a fundamental issue in three-dimensional computer graphics. Our approach was inspired by affine geometry and coordinatefree geometric programming. The basic idea of affine geometry is to make a distinction between points and vectors. The relation between orientations and rotations is analogous to the relation between points and vectors. Similarly to affine geometry, we argue that rotations and orientations should be represented differently in geometric computing. We show that, in three-dimensional space, unit quaternions (or rotation matrices) cannot parameterize rotations without singularity, but rotation vectors can. Conversely, rotation vectors cannot parameterize orientations without singularity, but unit quaternions (or rotation matrices) can. From these observations, we suggest that orientations should be represented by unit quaternions (or rotation matrices) and rotations should be represented by three-dimensional vectors. Furthermore, a set of operations are identified for combining rotations and orientations. Those operations are geometrically meaningful and independent of the choice of reference coordinate frames.

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Geometric Algebra: A computational framework for geometrical applications Part 1

Cited by 70 publications

References 3 publications

An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

Color edge detection hardware based on geometric algebra

Representing Rotations and Orientations in Geometric Computing

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