Filtering in a unit quaternion space for model-based object tracking

Ude, Aleš

doi:10.1016/s0921-8890(99)00014-7

Cited by 48 publications

(40 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although being a transformation by itself, it subsumes rotation and translation. To distinguish between two rigid body motions, a distance measure on the manifold has to be defined [7,42]. But this is no simple task in general.…”

Section: The Stratification Hierarchy and Pose Estimationmentioning

confidence: 99%

“…The second advantage of the approach is that the error measures are formalized in the 3D Euclidean space and are directly connected to a spatial distance measure. This is in contrast to other approaches, where the minimization of estimating errors of the rigid body motion has to be computed directly on the manifold of the geometric transformation [7,42]. The third argument is that the depth dependence of the 3D constraints can be adapted in each situation.…”

Section: Principles Of Solving the Pose Estimation Problemmentioning

confidence: 99%

See 1 more Smart Citation

Pose Estimation in Conformal Geometric Algebra Part II: Real-Time Pose Estimation Using Extended Feature Concepts

Rosenhahn

Sommer

2005

J Math Imaging Vis

View full text Add to dashboard Cite

Abstract. 2D-3D 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 2D-3D 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 (e.g. 3D projection rays or 3D reconstructed planes) and express constraints in the 3D space. It turns out that the stratification hierarchy [11] introduced by Faugeras is involved in the scenario. But since the stratification hierarchy is based on pure point concepts a new algebraic embedding is required when dealing with higher order entities. The conformal geometric algebra (CGA) [24] is well suited to solve this problem, since it subsumes the involved mathematical spaces. Operators are defined to switch entities between the algebras of the conformal space and its Euclidean and projective subspaces. This leads to another interpretation of the stratification hierarchy, which is not restricted to be based solely on point concepts. This work summarizes the theoretical foundations needed to deal with the pose problem. Therefore it contains mainly basics of Euclidean, projective and conformal geometry. Since especially conformal geometry is not well known in computer science, we recapitulate the mathematical concepts in some detail. We believe that this geometric model is useful also for many other computer vision tasks and has been ignored so far. Applications of these foundations are presented in Part II [36].

show abstract

Section: The Stratification Hierarchy and Pose Estimationmentioning

confidence: 99%

Section: Principles Of Solving the Pose Estimation Problemmentioning

confidence: 99%

Pose Estimation in Conformal Geometric Algebra Part II: Real-Time Pose Estimation Using Extended Feature Concepts

Rosenhahn

Sommer

2005

J Math Imaging Vis

View full text Add to dashboard Cite

show abstract

“…Such compact equations subsume the pose estimation problem at hand: find the best motor M which satisfies the constraint. But in contrast to other approaches, where the minimization of errors has to be computed directly on the manifold of the geometric transformations (Chiuso and Picci, 1998;Ude, 1999), in our approach a distance in the Euclidean space constitutes the error measure. To change our constraint equation from the conformal to the Euclidean space, the equations are rescaled without loosing linearity within our unknowns.…”

Section: Pose Estimation In Stratified Spacesmentioning

confidence: 99%

Pose Estimation of 3D Free-Form Contours

Rosenhahn

Perwass

Sommer

2004

Int J Comput Vision

View full text Add to dashboard Cite

In this article we discuss the 2D-3D pose estimation problem of 3D free-form contours. In our scenario we observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means to estimate the relative position and orientation (containing a rotation R and translation T ) of the 3D object to the reference camera system. The fusion of modeling free-form contours within the pose estimation problem is achieved by using the conformal geometric algebra. The conformal geometric algebra is a geometric algebra which models entities as stereographically projected entities in a homogeneous model. This leads to a linear description of kinematics on the one hand and projective geometry on the other hand. To model free-form contours in the conformal framework we use twists to model cycloidal curves as twist-depending functions and interpret n-times nested twist generated curves as functions generated by 3D Fourier descriptors. This means, we use the twist concept to apply a spectral domain representation of 3D contours within the pose estimation problem. We will show that twist representations of objects can be numerically efficient and easily be applied to the pose estimation problem. The pose problem itself is formalized as implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion. Several experiments visualize the robustness and real-time performance of our algorithms.

show abstract

“…Such compact equations subsume the pose estimation problem at hand: find the best motor M which satisfies the constraint. However, in contrast to other approaches, where the minimization of errors has to be computed directly on the manifold of the geometric transformations [8], [47], in our approach a distance in the Euclidean space constitutes the error measure. To change our constraint equation from the conformal to the Euclidean space, the equations are rescaled without loosing linearity within our unknowns.…”

Section: Pose Estimation In Stratified Spacesmentioning

confidence: 99%

Free-Form Pose Estimation by Using Twist Representations

2003

View full text Add to dashboard Cite

In this article we discuss the 2D-3D pose estimation problem of 3D free-form contours. We observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means estimating the relative position and orientation of the 3D object to the reference camera system. While cycloidal curves are derived as orbits of coupled twist transformations, we apply a spectral domain representation of 3D contours as an extension of cycloidal curves. Their Fourier descriptors are also related to twist representations. A twist is an element of se(3) and is a pair containing two 3D vectors. In a matrix representation, its exponential leads to an element of SE(3) and therefore to a rigid motion. We show that twist representations of objects can numerically efficiently and easily be applied to the free-form pose estimation problem. The pose problem itself is formalized as an implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion.

show abstract

Filtering in a unit quaternion space for model-based object tracking

Cited by 48 publications

References 12 publications

Pose Estimation in Conformal Geometric Algebra Part II: Real-Time Pose Estimation Using Extended Feature Concepts

Pose Estimation in Conformal Geometric Algebra Part II: Real-Time Pose Estimation Using Extended Feature Concepts

Pose Estimation of 3D Free-Form Contours

Free-Form Pose Estimation by Using Twist Representations

Contact Info

Product

Resources

About