Recovering Euclidean deformable models from stereo-motion

In this paper we present a novel structure from motion (SfM) approach able to infer 3D deformable models from uncalibrated stereo images. Using a stereo setup dramatically improves the 3D model estimation when the observed 3D shape is mostly deforming without undergoing strong rigid motion. Our approach first calibrates the stereo system automatically and then computes a single metric rigid structure for each frame. Afterwards, these 3D shapes are aligned to a reference view using a RANSAC method in order to compute the mean shape of the object and to select the subset of points on the object which have remained rigid throughout the sequence without deforming. The selected rigid points are then used to compute frame-wise shape registration and to extract the motion parameters robustly from frame to frame. Finally, all this information is used in a global optimization stage with bundle adjustment which allows to refine the frame-wise initial solution and also to recover the non-rigid 3D model. We show results on synthetic and real data that prove the performance of the proposed method even when there is no rigid motion in the original sequence

Integrating control systems defined on the frame bundles of the space forms

This paper considers left-invariant control systems defined on the orthonormal frame bundles of simply connected manifolds of constant sectional curvature, namely the space forms Euclidean space E-3, the sphere S-3 and Hyperboloid H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1, 3). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. In the Euclidean case the elements of the Lie algebra se(3) are often referred to as twists. For constant twist motions, the corresponding curves g(t) is an element of SE(3) are known as screw motions, given in closed form by using the well known Rodrigues' formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Oriented vehicles travelling in spherical space

This paper tackles the path planning problem for oriented vehicles travelling in the non-Euclidean 3-Dimensional space; spherical space S3. For such problem, the orientation of the vehicle is naturally represented by orthonormal frame bundle; the rotation group SO(4). Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to control systems defined on Lie groups. The oriented vehicles, in this case, are constrained to travel at constant speed in a forward direction and their angular velocities directly controlled. In this paper we identify controls that induce steady motions of these oriented vehicles and yield closed form parametric expressions for these motions. The paths these vehicles trace are defined explicitly in terms of the controls and therefore invariant with respect to the coordinate system used to describe the motion.

Integrable quadratic Hamiltonians on the Euclidean group of motions

In this paper, we discuss the problem of globally computing sub-Riemannian curves on the Euclidean group of motions SE(3). In particular, we derive a global result for special sub-Riemannian curves whose Hamiltonian satisfies a particular condition. In this paper, sub-Riemannian curves are defined in the context of a constrained optimal control problem. The maximum principle is then applied to this problem to yield an appropriate left-invariant quadratic Hamiltonian. A number of integrable quadratic Hamiltonians are identified. We then proceed to derive convenient expressions for sub-Riemannian curves in SE(3) that correspond to particular extremal curves. These equations are then used to compute sub-Riemannian curves that could potentially be used for motion planning of underwater vehicles.

Optimal steady motions for oriented vehicles

Motivated by the motion planning problem for oriented vehicles travelling in a 3-Dimensional space; Euclidean space E3, the sphere S3 and Hyperboloid H3. For such problems the orientation of the vehicle is naturally represented by an orthonormal frame over a point in the underlying manifold. The orthonormal frame bundles of the space forms R3,S3 and H3 correspond with their isometry groups and are the Euclidean group of motion SE(3), the rotation group SO(4) and the Lorentzian group SO(1; 3) respectively. Orthonormal frame bundles of space forms coincide with their isometry groups and therefore the focus shifts to left-invariant control systems defined on Lie groups. In this paper a method for integrating these systems is given where the controls are time-independent. For constant twist motions or helical motions, the corresponding curves g(t) 2 SE(3) are given in closed form by using the well known Rodrigues’ formula. However, this formula is only applicable to the Euclidean case. This paper gives a method for computing the non-Euclidean screw/helical motions in closed form. This involves decoupling the system into two lower dimensional systems using the double cover properties of Lie groups, then the lower dimensional systems are solved explicitly in closed form.

Fractal-to-Euclidean crossover of the isotropy restoration feature in a family of fractal resistor networks

Fractal with microscopic anisotropy shows a unique type of macroscopic isotropy restoration phenomenon that is absent in Euclidean space [M. T. Barlow et al., Phys. Rev. Lett. 75, 3042]. In this paper the isotropy restoration feature is considered for a family of two-dimensional Sierpinski gasket type fractal resistor networks. A parameter xi is introduced to describe this phenomenon. Our numerical results show that xi satisfies the scaling law xi similar to l(-alpha), where l is the system size and alpha is an exponent independent of the degree of microscopic anisotropy, characterizing the isotropy restoration feature of the fractal systems. By changing the underlying fractal structure towards the Euclidean triangular lattice through increasing the side length b of the gasket generators, the fractal-to-Euclidean crossover behavior of the isotropy restoration feature is discussed.

3-manifolds in Euclidean space from a contact viewpoint

We study the geometry of 3-manifolds generically embedded in R(n) by means of the analysis of the singularities of the distance-squared and height functions on them. We describe the local structure of the discriminant (associated to the distribution of asymptotic directions), the ridges and the flat ridges.

Construction of Hamiltonian-Minimal Lagrangian submanifolds in Complex Euclidean Space

We describe several families of Lagrangian submanifolds in complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving planar, spherical and hyperbolic curves, as well as Legendrian submanifolds of the odd-dimensional unit sphere.

Using the Euclidean Distance as a Mechanism Distance between Signatures to the Detect Spam and Phishing Scams

Spams and Phishing Scams are some of the abuse forms on the Internet that have grown up now. These abuses influence in user's routine of electronic mail and in the infrastructure of Internet communication. So, this paper proposes a new model messages filter based in Euclidian distance, beyond show the containment's methodologies currently more used. A new model messages filter, based in frequency's distribution of character present in your content and in signature generation is described. An architecture to combat Phishing Scam and spam is proposed in order to contribute to the containment of attempted fraud by mail.

Computational Aspects of the Calculation of Isometry Groups in General Relativity

We describe the ideas behind the package 'isometry', implemented in Maple to calculate isometry groups of dimensions 2, 3 and 4 in General Relativity. The package extends the functionality of previous programs written to perform invariant classification of space-times in General Relativity. Programming solutions used to surmount problems encountered with the calculation of eigenvectors and the determination of the signs of expressions are described. We also show how the package can be used to find the Killing vectors of a space-time.

Constraints on space-time manifold in Euclidean supergravity in terms of Dirac eigenvalues

We generalize a previous work on Dirac eigenvalues as dynamical variables of Euclidean supergravity. The most general set of constraints on the curvatures of the tangent bundle and on the spinor bundle of the space-time manifold, under which space-time admits Dirac eigenvalues as observables, are derived.

Euclidean 4d exact solitons in a Skyrme type model

We introduce a Skyrme type, four-dimensional Euclidean field theory made of a triplet of scalar fields n→, taking values on the sphere S2, and an additional real scalar field φ, which is dynamical only on a three-dimensional surface embedded in R4. Using a special ansatz we reduce the 4d non-linear equations of motion into linear ordinary differential equations, which lead to the construction of an infinite number of exact soliton solutions with vanishing Euclidean action. The theory possesses a mass scale which fixes the size of the solitons in way which differs from Derrick's scaling arguments. The model may be relevant to the study of the low energy limit of pure SU(2) Yang-Mills theory. © 2004 Elsevier B.V. All rights reserved.