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Old Wine in New Bottles: A new algebraic framework for computational geometry
 In E. BayroCorrochano and G. Sobczyk (eds), Advances in Geometric Algebra with Applications in Science and Engineering. (Birkhauser
, 2001
"... My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computeraided geometry, robotics, computer vision and the like. A detailed description and analysis of the ..."
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Cited by 24 (4 self)
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My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computeraided geometry, robotics, computer vision and the like. A detailed description and analysis of the
Pose estimation in conformal geometric algebra. Part II: Realtime pose estimation using extended feature concepts
 Journal of Mathematical Imaging and Vision
, 2005
"... Abstract. 2D3D pose estimation means to estimate the relative position and orientation of a 3D object with respect to a reference camera system. This work has its main focus on the theoretical foundations of the 2D3D pose estimation problem: We discuss the involved mathematical spaces and their in ..."
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Cited by 21 (15 self)
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Abstract. 2D3D pose estimation means to estimate the relative position and orientation of a 3D object with respect to a reference camera system. This work has its main focus on the theoretical foundations of the 2D3D pose estimation problem: We discuss the involved mathematical spaces and their interaction within higher order entities. To cope with the pose problem (how to compare 2D projective image features with 3D Euclidean object features), the principle we propose is to reconstruct image features (e.g. points or lines) to one dimensional higher entities
Adaptive Pose Estimation for Different Corresponding Entities
 Pattern Recognition, 24th DAGM Symposium
, 2002
"... This paper concerns the 2D3D pose estimation problem for different corresponding entities. Many articles concentrate on specific types of correspondences (mostly point, rarely line correspondences). Instead, in this work we are interested to relate the following image and model types simultaneo ..."
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Cited by 17 (8 self)
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This paper concerns the 2D3D pose estimation problem for different corresponding entities. Many articles concentrate on specific types of correspondences (mostly point, rarely line correspondences). Instead, in this work we are interested to relate the following image and model types simultaneously: 2D point/3D point, 2D line/3D point, 2D line/3D line, 2D conic/3D circle, 2D circle/3D sphere. Furthermore, to handle also articulated objects, we describe kinematic chains in this context in a similar manner. We further discuss the use of weighted constraint equations, and different numerical solution approaches.
Applications of conformal geometric algebra in computer vision and graphics
 6th International Workshop IWMM 2004
, 2005
"... Abstract. This paper introduces the mathematical framework of conformal geometric algebra (CGA) as a language for computer graphics and computer vision. Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for existing interpolation methods to ..."
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Cited by 14 (0 self)
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Abstract. This paper introduces the mathematical framework of conformal geometric algebra (CGA) as a language for computer graphics and computer vision. Specifically it discusses a new method for pose and position interpolation based on CGA which firstly allows for existing interpolation methods to be cleanly extended to pose and position interpolation, but also allows for this to be extended to higherdimension spaces and all conformal transforms (including dilations). In addition, we discuss a method of dealing with conics in CGA and the intersection and reflections of rays with such conic surfaces. Possible applications for these algorithms are also discussed. 1
Gaigen: a Geometric Algebra Implementation Generator
 FD03] FONTIJNE D., DORST L.: MODELING 3D EUCLIDEAN GEOMETRY. IEEE COMPUTER GRAPHICS AND APPLICATIONS 23, 2 (MARCHAPRIL 2003
, 2001
"... This paper describes an approach to implementing geometric algebra. The goal of the implementation was to create an efficient, general implementation of geometric algebras of relatively low dimension, based on an orthogonal basis of any signature, for use in applications like computer graphics, comp ..."
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Cited by 12 (3 self)
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This paper describes an approach to implementing geometric algebra. The goal of the implementation was to create an efficient, general implementation of geometric algebras of relatively low dimension, based on an orthogonal basis of any signature, for use in applications like computer graphics, computer vision, physics and robotics. The approach taken is to let the user specify the properties of the geometric algebra required, and to automatically generate source code accordingly. The resulting source code consist of three layers, of which the lower two are automatically generated. The top layer hides the implementation and optimization details from the user and provides a dimension independent, object oriented interface to using geometric algebra in software, while the lower layers implement the algebra efficiently. Coordinates of multivectors are stored in a compressed form, which does not store coordinates of grade parts that are known to be equal to. Optimized implementations of products can be automatically generated according to a profile analysis of the user application. We present benchmarks that compare the performance of this approach to other GA implementations available to us and demonstrate the impact of various settings our code generator offers.
Monocular Pose Estimation of Kinematic Chains
 Applications of Geometric Algebra in Computer Science and Engineering
, 2002
"... ABSTRACT In this paper conformal geometric algebra is used to formalize an algebraic embedding for the problem of monocular pose estimation of kinematic chains. The problem is modeled on the base of several geometric constraint equations. In conformal geometric algebra the resulting equations are co ..."
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Cited by 12 (6 self)
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ABSTRACT In this paper conformal geometric algebra is used to formalize an algebraic embedding for the problem of monocular pose estimation of kinematic chains. The problem is modeled on the base of several geometric constraint equations. In conformal geometric algebra the resulting equations are compact and clear. To solve the equations we linearize and iterate the equations to approximate the pose and the kinematic chain parameters. 1.1
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 fu ..."
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Cited by 9 (4 self)
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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.
The hypersphere neuron
 In 11th European Symposium on Artificial Neural Networks, ESANN 2003, Bruges
, 2003
"... Abstract. In this paper a special higher order neuron, the hypersphere neuron, is introduced. By embedding Euclidean space in a conformal space, hyperspheres can be expressed as vectors. The scalar product of points and spheres in conformal space, gives a measure for how far a point lies inside or o ..."
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Cited by 6 (6 self)
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Abstract. In this paper a special higher order neuron, the hypersphere neuron, is introduced. By embedding Euclidean space in a conformal space, hyperspheres can be expressed as vectors. The scalar product of points and spheres in conformal space, gives a measure for how far a point lies inside or outside a hypersphere. It will be shown that a hypersphere neuron may be implemented as a perceptron with two bias inputs. By using hyperspheres instead of hyperplanes as decision surfaces, a reduction in computational complexity can be achieved for certain types of problems. Furthermore, in this setup, a reliability measure can be associated with data points in a straight forward way. 1
Pose estimation of freeform surface models
 In 25. Symposium für Mustererkennung, DAGM 2003
, 2003
"... Abstract. In this article we discuss the 2D3D pose estimation problem of 3D freeform surface models. In our scenario we observe freeform surface models in an image of a calibrated camera. Pose estimation means to estimate the relative position and orientation of the 3D object to the reference cam ..."
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Cited by 5 (3 self)
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Abstract. In this article we discuss the 2D3D pose estimation problem of 3D freeform surface models. In our scenario we observe freeform surface models in an image of a calibrated camera. Pose estimation means to estimate the relative position and orientation of the 3D object to the reference camera system. The object itself is modelled as a twoparametric surface model which is represented by Fourier descriptors. It enables a lowpass description of the surface model, which is advantageously applied to the pose problem. To achieve the combination of such a signalbased model within the geometry of the pose scenario, the conformal geometric algebra is used and applied. 1
Uncertain geometry with circles, spheres and conics
 Geometric Properties from Incomplete Data
"... Abstract. Spatial reasoning is one of the central tasks in Computer Vision. It always has to deal with uncertain data. Projective geometry has become the working horse for modelling multiple view geometry, while modelling uncertainty with statistical tools has become a standard. Geometric reasoning ..."
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Abstract. Spatial reasoning is one of the central tasks in Computer Vision. It always has to deal with uncertain data. Projective geometry has become the working horse for modelling multiple view geometry, while modelling uncertainty with statistical tools has become a standard. Geometric reasoning in projective geometry with uncertain geometric elements has been advocated by Kanatani in the early 90’s, and recently made transparent and generalized to basic entities in projective geometry including transformations by Förstner and Heuel, exploiting the multilinearity of nearly all relations, such as incidence and identity, which results from the underlying GrassmannCayley algebra (cf. [21, 8, 7]). This paper generalizes geometric reasoning under uncertainty towards circles, spheres and conics, which play a role in many computer vision applications. In particular it will be shown how within the Clifford algebra of conformal space, as introduced by Hestenes et al. [11, 16], circles can be constructed from three uncertain points in 3DEuclidean space, while propagating the covariance matrices of the points. This then enables us to obtain and visualize the uncertainty of the resulting circle. We also introduce the Clifford algebra over the vector space of 2Dconics, which allows us to apply the same error propagation procedures as for the Clifford algebra of conformal space.