Center Projection Vortices in Continuum Yang-Mills Theory

The maximal center gauge, combined with center projection, is a means to associate Yang-Mills lattice gauge configurations with closed center vortex world-surfaces. This technique allows to study center vortex physics in lattice gauge experiments. In the present work, the continuum analogue of the maximal center gauge is constructed. This sheds new light on the meaning of the procedure on the lattice and leads to a sketch of an effective vortex theory in the continuum. Furthermore, the manner in which center vortex configurations generate the Pontryagin index is investigated. The Pontryagin index is built up from self-intersections of the vortex world-surfaces, where it is crucial that the surfaces be globally non-oriented.Comment: 64 latex pages, 3 ps figures included via eps

Anisotropic Diffusion Partial Differential Equations in Multi-Channel Image Processing : Framework and Applications

We review recent methods based on diffusion PDE's (Partial Differential Equations) for the purpose of multi-channel image regularization. Such methods have the ability to smooth multi-channel images anisotropically and can preserve then image contours while removing noise or other undesired local artifacts. We point out the pros and cons of the existing equations, providing at each time a local geometric interpretation of the corresponding processes. We focus then on an alternate and generic tensor-driven formulation, able to regularize images while specifically taking the curvatures of local image structures into account. This particular diffusion PDE variant is actually well suited for the preservation of thin structures and gives regularization results where important image features can be particularly well preserved compared to its competitors. A direct link between this curvature-preserving equation and a continuous formulation of the Line Integral Convolution technique (Cabral and Leedom, 1993) is demonstrated. It allows the design of a very fast and stable numerical scheme which implements the multi-valued regularization method by successive integrations of the pixel values along curved integral lines. Besides, the proposed implementation, based on a fourth-order Runge Kutta numerical integration, can be applied with a subpixel accuracy and preserves then thin image structures much better than classical finite-differences discretizations, usually chosen to implement PDE-based diffusions. We finally illustrate the efficiency of this diffusion PDE's for multi-channel image regularization - in terms of speed and visual quality - with various applications and results on color images, including image denoising, inpainting and edge-preserving interpolation

Generalized Quark Transversity Distribution of the Pion in Chiral Quark Models

The transversity generalized parton distributions (tGPDs) of the the pion, involving matrix elements of the tensor bilocal quark current, are analyzed in chiral quark models. We apply the nonlocal chiral models involving a momentum-dependent quark mass, as well as the local Nambu--Jona-Lasinio with the Pauli-Villars regularization to calculate the pion tGPDs, as well as related quantities following from restrained kinematics, evaluation of moments, or taking the Fourier-Bessel transforms to the impact-parameter space. The obtained distributions satisfy the formal requirements, such as proper support and polynomiality, following from Lorentz covariance. We carry out the leading-order QCD evolution from the low quark-model scale to higher lattice scales, applying the method of Kivel and Mankiewicz. We evaluate several lowest-order generalized transversity form factors, accessible from the recent lattice QCD calculations. These form factors, after evolution, agree properly with the lattice data, in support of the fact that the spontaneously broken chiral symmetry is the key element also in the evaluation of the transversity observables.Comment: 17 pages, 17 figures, regular pape

Distribution Functions of the Nucleon and Pion in the Valence Region

We provide an experimental and theoretical perspective on the behavior of unpolarized distribution functions for the nucleon and pion on the valence-quark domain; namely, Bjorken-x \gtrsim 0.4. This domain is key to much of hadron physics; e.g., a hadron is defined by its flavor content and that is a valence-quark property. Furthermore, its accurate parametrization is crucial to the provision of reliable input for large collider experiments. We focus on experimental extractions of distribution functions via electron and muon inelastic scattering, and from Drell-Yan interactions; and on theoretical treatments that emphasize an explanation of the distribution functions, providing an overview of major contemporary approaches and issues. Valence-quark physics is a compelling subject, which probes at the heart of our understanding of the Standard Model. There are numerous outstanding and unresolved challenges, which experiment and theory must confront. In connection with experiment, we explain that an upgraded Jefferson Lab facility is well-suited to provide new data on the nucleon, while a future electron ion collider could provide essential new data for the mesons. There is also great potential in using Drell-Yan interactions, at FNAL, J-PARC and GSI, to push into the large-x domain for both mesons and nucleons. We argue furthermore that explanation, in contrast to modeling and parametrization, requires a widespread acceptance of the need to adapt theory: to the lessons learnt already from the methods of nonperturbative quantum field theory; and a fuller exploitation of those methods.Comment: Review article: 133 double-spaced pages, 44 figures, 6 table

Center Vortices and the Gribov Horizon

We show how the infinite color-Coulomb energy of color-charged states is related to enhanced density of near-zero modes of the Faddeev-Popov operator, and calculate this density numerically for both pure Yang-Mills and gauge-Higgs systems at zero temperature, and for pure gauge theory in the deconfined phase. We find that the enhancement of the eigenvalue density is tied to the presence of percolating center vortex configurations, and that this property disappears when center vortices are either removed from the lattice configurations, or cease to percolate. We further demonstrate that thin center vortices have a special geometrical status in gauge-field configuration space: Thin vortices are located at conical or wedge singularities on the Gribov horizon. We show that the Gribov region is itself a convex manifold in lattice configuration space. The Coulomb gauge condition also has a special status; it is shown to be an attractive fixed point of a more general gauge condition, interpolating between the Coulomb and Landau gauges.Comment: 19 pages, 17 EPS figures, RevTeX4; v2: added references, corrected caption of fig. 11; v3: new data for higher couplings, clarifications on color-Coulomb potential in deconfined phase, version to appear in JHE

Geodesic Active Fields:A Geometric Framework for Image Registration

Image registration is the concept of mapping homologous points in a pair of images. In other words, one is looking for an underlying deformation field that matches one image to a target image. The spectrum of applications of image registration is extremely large: It ranges from bio-medical imaging and computer vision, to remote sensing or geographic information systems, and even involves consumer electronics. Mathematically, image registration is an inverse problem that is ill-posed, which means that the exact solution might not exist or not be unique. In order to render the problem tractable, it is usual to write the problem as an energy minimization, and to introduce additional regularity constraints on the unknown data. In the case of image registration, one often minimizes an image mismatch energy, and adds an additive penalty on the deformation field regularity as smoothness prior. Here, we focus on the registration of the human cerebral cortex. Precise cortical registration is required, for example, in statistical group studies in functional MR imaging, or in the analysis of brain connectivity. In particular, we work with spherical inflations of the extracted hemispherical surface and associated features, such as cortical mean curvature. Spatial mapping between cortical surfaces can then be achieved by registering the respective spherical feature maps. Despite the simplified spherical geometry, inter-subject registration remains a challenging task, mainly due to the complexity and inter-subject variability of the involved brain structures. In this thesis, we therefore present a registration scheme, which takes the peculiarities of the spherical feature maps into particular consideration. First, we realize that we need an appropriate hierarchical representation, so as to coarsely align based on the important structures with greater inter-subject stability, before taking smaller and more variable details into account. Based on arguments from brain morphogenesis, we propose an anisotropic scale-space of mean-curvature maps, built around the Beltrami framework. Second, inspired by concepts from vision-related elements of psycho-physical Gestalt theory, we hypothesize that anisotropic Beltrami regularization better suits the requirements of image registration regularization, compared to traditional Gaussian filtering. Different objects in an image should be allowed to move separately, and regularization should be limited to within the individual Gestalts. We render the regularization feature-preserving by limiting diffusion across edges in the deformation field, which is in clear contrast to the indifferent linear smoothing. We do so by embedding the deformation field as a manifold in higher-dimensional space, and minimize the associated Beltrami energy which represents the hyperarea of this embedded manifold as measure of deformation field regularity. Further, instead of simply adding this regularity penalty to the image mismatch in lieu of the standard penalty, we propose to incorporate the local image mismatch as weighting function into the Beltrami energy. The image registration problem is thus reformulated as a weighted minimal surface problem. This approach has several appealing aspects, including (1) invariance to re-parametrization and ability to work with images defined on non-flat, Riemannian domains (e.g., curved surfaces, scalespaces), and (2) intrinsic modulation of the local regularization strength as a function of the local image mismatch and/or noise level. On a side note, we show that the proposed scheme can easily keep up with recent trends in image registration towards using diffeomorphic and inverse consistent deformation models. The proposed registration scheme, called Geodesic Active Fields (GAF), is non-linear and non-convex. Therefore we propose an efficient optimization scheme, based on splitting. Data-mismatch and deformation field regularity are optimized over two different deformation fields, which are constrained to be equal. The constraint is addressed using an augmented Lagrangian scheme, and the resulting optimization problem is solved efficiently using alternate minimization of simpler sub-problems. In particular, we show that the proposed method can easily compete with state-of-the-art registration methods, such as Demons. Finally, we provide an implementation of the fast GAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, so as to automatically delineate the corresponding regions on a subject brain given its feature map. In a leave-one-out cross-validation study on 39 brain surfaces with 35 manually delineated gyral regions, we show that the pairwise subject-atlas registration with the proposed spherical registration scheme significantly improves the individual alignment of cortical labels between subject and atlas brains, and, consequently, that the estimated automatic parcellations after label fusion are of better quality

Integrability of two-loop dilatation operator in gauge theories

We study the two-loop dilatation operator in the noncompact SL(2) sector of QCD and supersymmetric Yang-Mills theories with N=1,2,4 supercharges. The analysis is performed for Wilson operators built from three quark/gaugino fields of the same helicity belonging to the fundamental/adjoint representation of the SU(3)/SU(N_c) gauge group and involving an arbitrary number of covariant derivatives projected onto the light-cone. To one-loop order, the dilatation operator inherits the conformal symmetry of the classical theory and is given in the multi-color limit by a local Hamiltonian of the Heisenberg magnet with the spin operators being generators of the collinear subgroup of full (super)conformal group. Starting from two loops, the dilatation operator depends on the representation of the gauge group and, in addition, receives corrections stemming from the violation of the conformal symmetry. We compute its eigenspectrum and demonstrate that to two-loop order integrability survives the conformal symmetry breaking in the aforementioned gauge theories, but it is violated in QCD by the contribution of nonplanar diagrams. In SYM theories with extended supersymmetry, the N-dependence of the two-loop dilatation operator can be factorized (modulo an additive normalization constant) into a multiplicative c-number. This property makes the eigenspectrum of the two-loop dilatation operator alike in all gauge theories including the maximally supersymmetric theory. Our analysis suggests that integrability is only tied to the planar limit and it is sensitive neither to conformal symmetry nor supersymmetry.Comment: 70 pages, 10 figure