Developments in texture-based vector field visualisation techniques

Vector field visualisation is one of the classic sub-fields of scientific data visualisation. The need for effective visualisation of flow data arises in many scientific domains ranging from medical sciences to aerodynamics. Though there has been much research on the topic, the question of how to communicate flow information effectively in real, practical situations is still largely an unsolved problem. This is particularly true for complex 3D flows. In this presentation we give a brief introduction and background to vector field visualisation and comment on the effectiveness of the most common solutions. We will then give some examples of current development on texture-based techniques, and given practical examples of their use in CFD research and hydrodynamic applications.

Bilateral smoothing of gradient vector field and application to image segmentation

Medical image segmentation finds application in computer-aided diagnosis, computer-guided surgery, measuring tissue volumes, locating tumors, and pathologies. One approach to segmentation is to use active contours or snakes. Active contours start from an initialization (often manually specified) and are guided by image-dependent forces to the object boundary. Snakes may also be guided by gradient vector fields associated with an image. The first main result in this direction is that of Xu and Prince, who proposed the notion of gradient vector flow (GVF), which is computed iteratively. We propose a new formalism to compute the vector flow based on the notion of bilateral filtering of the gradient field associated with the edge map - we refer to it as the bilateral vector flow (BVF). The range kernel definition that we employ is different from the one employed in the standard Gaussian bilateral filter. The advantage of the BVF formalism is that smooth gradient vector flow fields with enhanced edge information can be computed noniteratively. The quality of image segmentation turned out to be on par with that obtained using the GVF and in some cases better than the GVF.

Divergence-free vector-field and reduction

By the Lie symmetry group, the reduction for divergence-free vector-fields (DFVs) is studied, and the following results are found. A n-dimensional DFV can be locally reduced to a (n - 1)-dimensional DFV if it admits a one-parameter symmetry group that is spatial and divergenceless. More generally, a n-dimensional DFV admitting a r-parameter, spatial, divergenceless Abelian (commutable) symmetry group can be locally reduced to a (n - r)-dimensional DFV.

Interactive Vector Field Feature Identification

We introduce a flexible technique for interactive exploration of vector field data through classification derived from user-specified feature templates. Our method is founded on the observation that, while similar features within the vector field may be spatially disparate, they share similar neighborhood characteristics. Users generate feature-based visualizations by interactively highlighting well-accepted and domain specific representative feature points. Feature exploration begins with the computation of attributes that describe the neighborhood of each sample within the input vector field. Compilation of these attributes forms a representation of the vector field samples in the attribute space. We project the attribute points onto the canonical 2D plane to enable interactive exploration of the vector field using a painting interface. The projection encodes the similarities between vector field points within the distances computed between their associated attribute points. The proposed method is performed at interactive rates for enhanced user experience and is completely flexible as showcased by the simultaneous identification of diverse feature types.

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Hamiltonian constraint analysis of vector field theories with spontaneous Lorentz symmetry breaking

Recent investigations of various quantum-gravity theories have revealed a variety of possible mechanisms that lead to Lorentz violation. One of the more elegant of these mechanisms is known as Spontaneous Lorentz Symmetry Breaking (SLSB), where a vector or tensor field acquires a nonzero vacuum expectation value. As a consequence of this symmetry breaking, massless Nambu-Goldstone modes appear with properties similar to the photon in Electromagnetism. This thesis considers the most general class of vector field theories that exhibit spontaneous Lorentz violation-known as bumblebee models-and examines their candidacy as potential alternative explanations of E&M, offering the possibility that Einstein-Maxwell theory could emerge as a result of SLSB rather than of local U(1) gauge invariance. With this aim we employ Dirac's Hamiltonian Constraint Analysis procedure to examine the constraint structures and degrees of freedom inherent in three candidate bumblebee models, each with a different potential function, and compare these results to those of Electromagnetism. We find that none of these models share similar constraint structures to that of E&M, and that the number of degrees of freedom for each model exceeds that of Electromagnetism by at least two, pointing to the potential existence of massive modes or propagating ghost modes in the bumblebee theories.

Periodic orbits for a class of reversible quadratic vector field on R-3

For a class of reversible quadratic vector fields on R-3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U-2. More specifically, we prove that for all n is an element of N, there exists epsilon(n) > 0 such that the reversible quadratic polynomial differential systemx = a(0) + a(1y) + a(3y)(2) + a(4Y)(2) + epsilon(a(2x)(2) + a(3xz)),y = b(1z) + b(3yz) + epsilon b(2xy),z = c(1y) +c(4az)(2) + epsilon c(2xz)in R-3, with a(0) < 0, b(1)c(1) < 0, a(2) < 0, b(2) < a(2), a(4) > 0, c(2) < a(2) and b(3) is not an element of (c(4), 4c(4)), for epsilon is an element of (0, epsilon(n)) has at least n periodic orbits near the heteroclinic loop. (c) 2007 Elsevier B.V. All rights reserved.

NONDEGENERATE UMBILICS, THE PATH FORMULATION and GRADIENT BIFURCATION PROBLEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Renormalization of the axial-vector current in QCD

Following the method of Ioffe and Smilga, the propagation of the baryon current in an external constant axial-vector field is considered. The close similarity of the operator-product expansion with and without an external field is shown to arise from the chiral invariance of gauge interactions in perturbation theory. Several sum rules corresponding to various invariants both for the nucleon and the hyperons are derived. The analysis of the sum rules is carried out by two independent methods, one called the ratio method and the other called the continuum method, paying special attention to the nondiagonal transitions induced by the external field between the ground state and excited states. Up to operators of dimension six, two new external-field-induced vacuum expectation values enter the calculations. Previous work determining these expectation values from PCAC (partial conservation of axial-vector current) are utilized. Our determination from the sum rules of the nucleon axial-vector renormalization constant GA, as well as the Cabibbo coupling constants in the SU3-symmetric limit (ms=0), is in reasonable accord with the experimental values. Uncertainties in the analysis are pointed out. The case of broken flavor SU3 symmetry is also considered. While in the ratio method, the results are stable for variation of the fiducial interval of the Borel mass parameter over which the left-hand side and the right-hand side of the sum rules are matched, in the continuum method the results are less stable. Another set of sum rules determines the value of the linear combination 7F-5D to be ≊0, or D/(F+D)≊(7/12). .AE

The vector fields admitting one-parameter spatial symmetry group and their reduction

For a n-dimensional vector fields preserving some n-form, the following conclusion is reached by the method of Lie group. That is, if it admits an one-parameter, n-form preserving symmetry group, a transformation independent of the vector field is constructed explicitly, which can reduce not only dimesion of the vector field by one, but also make the reduced vector field preserve the corresponding ( n - 1)-form. In partic ular, while n = 3, an important result can be directly got which is given by Me,ie and Wiggins in 1994.

Estimating Sensor Motion from Wide-Field Optical Flow on a Log-Dipolar Sensor

Log-polar image architectures, motivated by the structure of the human visual field, have long been investigated in computer vision for use in estimating motion parameters from an optical flow vector field. Practical problems with this approach have been: (i) dependence on assumed alignment of the visual and motion axes; (ii) sensitivity to occlusion form moving and stationary objects in the central visual field, where much of the numerical sensitivity is concentrated; and (iii) inaccuracy of the log-polar architecture (which is an approximation to the central 20°) for wide-field biological vision. In the present paper, we show that an algorithm based on generalization of the log-polar architecture; termed the log-dipolar sensor, provides a large improvement in performance relative to the usual log-polar sampling. Specifically, our algorithm: (i) is tolerant of large misalignmnet of the optical and motion axes; (ii) is insensitive to significant occlusion by objects of unknown motion; and (iii) represents a more correct analogy to the wide-field structure of human vision. Using the Helmholtz-Hodge decomposition to estimate the optical flow vector field on a log-dipolar sensor, we demonstrate these advantages, using synthetic optical flow maps as well as natural image sequences.

UNIT VECTOR FIELDS ON ANTIPODALLY PUNCTURED SPHERES: BIG INDEX, BIG VOLUME

We establish in this paper a lower bound for the volume of a unit vector field (v) over right arrow defined ou S(n) \ {+/-x}, n = 2,3. This lower bound is related to the sum of the absolute values of the indices of (v) over right arrow at x and -x.

Vector Theory of Spontaneous Lorentz Violation

Classical electromagnetism predicts two massless propagating modes, which are known as the two polarizations of the photon. On the other hand, if the Lorentz symmetry of classical electromagnetism is spontaneously broken, the new theory will still have two massless Nambu-Goldstone modes resembling the photon. If the Lorentz symmetry is broken by a bumblebee potential that allows for excitations out of the minimum, then massive modes arise. Furthermore, in curved spacetime, such massive modes will be created through a process other than the usual Higgs mechanism because of the dependence of the bumblebee potential on both the vector field and the metric tensor. Also, it is found that these massive modes do not propagate due to the extra constraints.

Animated feather coats using field lines

The tedious task of manually placing feathers on computer animated objects involves aligning feathers, ensuring that they do not insect each other or penetrate the surface, deforming every feather to match the local surface features, and ensuring that the feather coat is consistent when the underlying object is animated.

We present a technique for generating a feather coat over an object. Feather orientation is specified quickly and easily, feathers are deformed while ensuring collision prevention, and the coat can be animated. We create a vector field in the space surrounding the body object and deform feathers to align with the field lines. The non-intersection property of the field lines ensures that feather intersections are avoided. We provide a formulation of a suitable vector field and demonstrate that it is capable of producing realistic feather coats. The process can easily be integrated into the work-flow of standard modelling and animation processes.

We show examples of feather coat creation on a range of objects, proving that field line based placement of feather coats provides the desired functionality for feather modelling and animation.