Christian Perwass

Pose Estimation of 3D Free-Form Contours

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 and translation) 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.

Implementation of a Clifford Algebra Co-Processor Design on a Field Programmable Gate Array

We present the design of a Clifford algebra co-processor and its implementation on a Field Programmable Gate Array (FPGA). To the best of our knowledge this is the first such design developed. The design is scalable in both the Clifford algebra dimension and the bit width of the numerical factors. Both aspects are only limited by the hardware resources. Furthermore, the signature of the underlying vector space can be changed without reconfiguring the FPGA. High calculation speeds are achieved through a pipeline architecture.

Increasing pose estimation performance using multi-cue integration

We have developed a system which integrates the information output from several pose estimation algorithms and from several views of the scene. It is tested in a real setup with a robotic manipulator. It is shown that integrating pose estimates from several algorithms increases the overall performance of the pose estimation accuracy as well as the robustness as compared to using only a single algorithm. It is shown that increased robustness can be achieved by using pose estimation algorithms based on complementary features, so called algorithmic multi-cue integration (AMC). Furthermore it is also shown that increased accuracy can be achieved by integrating pose estimation results from different views of the scene, so-called temporal multi-cue integration (TMC). Temporal multi-cue integration is the most interesting aspect of this paper

Uncertain Geometry with Circles, Spheres and Conics

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 statis- tical tools has become a standard. Geometric reasoning in projective geometry with uncertain geometric elements has been advocated by Kanatani in the early 90s, and recently made transparent and generalized to basic entities in projective geometry including transformations by Forstner and Heuel, exploiting the multilinearity of nearly all relations, such as incidence and identity, which results from the underlying Grassmann-Cayley 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 Clifiord algebra of conformal space, as introduced by Hestenes et al. (11, 16), circles can be constructed from three uncertain points in 3D-Euclidean 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 Clifiord algebra over the vector space of 2D-conics, which allows us to apply the same error propagation procedures as for the Clifiord algebra of conformal space.

Estimation of geometric entities and operators from uncertain data

In this text we show how points, point pairs, lines, planes, circles, spheres, and rotation, translation and dilation operators and their uncertainty can be evaluated from uncertain data in a unified manner using the Geometric Algebra of conformal space. This extends previous work by Forstner et al. [3] from points, lines and planes to non-linear entities and operators, while keeping the linearity of the estimation method. We give a theoretical description of our approach and show the results of some synthetic experiments.

Spherical Decision Surfaces Using Conformal Modelling

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, it will be shown that Multi-Layer Percerptrons (MLP) based on such neurons are similar to Radial Basis Function (RBF) networks. It is also found that such MLPs can give better results than RBF networks of the same complexity. The abilities of the proposed MLPs are demonstrated on some classical data for neural computing, as well as on real data from a particular computer vision problem.

Numerical Evaluation of Versors with Clifford Algebra

This paper has two main parts. In the first part we discuss multivector null spaces with respect to the geometric product. In the second part we apply this analysis to the numerical evaluation of versors in conformal space. The main result of this paper is an algorithm that attempts to evaluate the best transformation between two sets of 3D-points. This transformation may be pure translation or rotation, or any combination of them. This is, of course, also possible using matrix methods. However, constraining the resultant transformation matrix to a particular transformation is not always easy. Using Clifford algebra it is straightforward to stay within the space of the transformation we are looking for.

Pose Estimation of Free-Form Surface Models

In this article we discuss the 2D-3D pose estimation problem of 3D free-form surface models. In our scenario we observe free-form 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 two-parametric surface model which is represented by Fourier descriptors. It enables a low-pass description of the surface model, which is advantageously applied to the pose problem. To achieve the combination of such a signal-based model within the geometry of the pose scenario, the conformal geometric algebra is used and applied.

Free-Form Pose Estimation by Using Twist Representations

Abstract 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.

The inversion camera model

In this paper a novel camera model, the inversion camera model, is introduced, which encompasses the standard pinhole camera model, an extension of the division model for lens distortion, and the model for catadioptric cameras with parabolic mirror. All these different camera types can be modeled by essentially varying two parameters. The feasibility of this camera model is presented in experiments where object pose, camera focal length and lens distortion are estimated simultaneously.