Automatic Multivector Differentiation and Optimization

Tingelstad, Lars; Egeland, Olav

doi:10.1007/s00006-016-0722-6

Cited by 12 publications

(6 citation statements)

References 35 publications

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“…Fortunately, as shown, the exponential map for bivectors B ∈ B in (4) has a closed-form solution; see e.g. [27] for an implementation in C++.…”

Section: The Exponential Mapmentioning

confidence: 99%

“…In [27,28], we employed the exponential map in optimization of rigid body motions, parameterized using motors in 3-D conformal geometric algebra, from observations of geometric objects such as points, lines, circles, and planes. In [6], the authors used observations from lines to estimate motors using the so called motor extended Kalman filter.…”

Section: Introductionmentioning

confidence: 99%

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Motor Parameterization

Tingelstad¹,

Egeland²

2018

Adv. Appl. Clifford Algebras

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In this paper, we consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization; retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement.

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“…Fortunately, as shown, the exponential map for bivectors B ∈ B in (4) has a closed-form solution; see e.g. [27] for an implementation in C++.…”

Section: The Exponential Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Motor Parameterization

Tingelstad¹,

Egeland²

2018

Adv. Appl. Clifford Algebras

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“…2.2. Cylinders are fitted with nonlinear least squares methods using the optimization tool [17], which is a software library that enables optimization through automatic differentiation of conformal entities [18] and the Ceres Solver [1] by Google. The cost-function to be minimized is defined as…”

Section: Fittingmentioning

confidence: 99%

“…The point cloud contains a sphere with radius ρ = 0.02 m and 109 inlier points, which is to be detected. The point cloud fits inside an octree with a root node with sides of 2.56 m. According to (18) a resolution of h res = 0.01 m suggest a total of 2,396,745 nodes, and setting k = 2 results in 4,793,490 iterations. In this case, only 10,118 nodes are occupied with more than four points, which is required to construct a sphere, resulting in a total of 20,236 iterations.…”

Section: Performance Considerationsmentioning

confidence: 99%

Object Detection in Point Clouds Using Conformal Geometric Algebra

Sveier

Kleppe

Tingelstad

et al. 2017

Adv. Appl. Clifford Algebras

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Abstract. This paper presents an approach for detecting primitive geometric objects in point clouds captured from 3D cameras. Primitive objects are objects that are well defined with parameters and mathematical relations, such as lines, spheres and ellipsoids. RANSAC, a robust parameter estimator that classifies and neglects outliers, is used for object detection. The primitives considered are modeled, filtered and fitted using the conformal model of geometric algebra. Methods for detecting planes, spheres and cylinders are suggested. Least squares fitting of spheres and planes to point data are done analytically with conformal geometric algebra, while a cylinder is fitted by defining a nonlinear cost function which is optimized using a nonlinear least squares solver. Furthermore, the suggested object detection scheme is combined with an octree sampling strategy that results in fast detection of multiple primitive objects in point clouds.

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“…, non-linear optimization, sensitivity analysis, robotics, machine learning, computer graphics, and computer vision (see, e.g. , Abdelhafez, Schuster & Koch, 2019 ; Dawood, 2014 , Dawood & Megahed, 2019 ; Fries, 2019 , Sommer, Pradalier & Furgale, 2016 , and Tingelstad & Egeland, 2017 ).…”

Section: Introductionmentioning

confidence: 99%

Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation

Dawood

Megahed

2023

PeerJ Computer Science

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Acquiring reliable knowledge amidst uncertainty is a topical issue of modern science. Interval mathematics has proved to be of central importance in coping with uncertainty and imprecision. Algorithmic differentiation, being superior to both numeric and symbolic differentiation, is nowadays one of the most celebrated techniques in the field of computational mathematics. In this connexion, laying out a concrete theory of interval differentiation arithmetic, combining subtlety of ordinary algorithmic differentiation with power and reliability of interval mathematics, can extend real differentiation arithmetic so markedly both in method and objective, and can so far surpass it in power as well as applicability. This article is intended to lay out a systematic theory of dyadic interval differentiation numbers that wholly addresses first and higher order automatic derivatives under uncertainty. We begin by axiomatizing a differential interval algebra and then we present the notion of an interval extension of a family of real functions, together with some analytic notions of interval functions. Next, we put forward an axiomatic theory of interval differentiation arithmetic, as a two-sorted extension of the theory of a differential interval algebra, and provide the proofs for its categoricity and consistency. Thereupon, we investigate the ensuing structure and show that it constitutes a multiplicatively non-associative S-semiring in which multiplication is subalternative and flexible. Finally, we show how to computationally realize interval automatic differentiation. Many examples are given, illustrating automatic differentiation of interval functions and families of real functions.

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Automatic Multivector Differentiation and Optimization

Cited by 12 publications

References 35 publications

Motor Parameterization

Motor Parameterization

Object Detection in Point Clouds Using Conformal Geometric Algebra

Automatic differentiation of uncertainties: an interval computational differentiation for first and higher derivatives with implementation

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