A Covariant Information-Density Cutoff in Curved Space-Time

In information theory, the link between continuous information and discrete information is established through well-known sampling theorems. Sampling theory explains, for example, how frequency-filtered music signals are reconstructible perfectly from discrete samples. In this Letter, sampling theory is generalized to pseudo-Riemannian manifolds. This provides a new set of mathematical tools for the study of space-time at the Planck scale: theories formulated on a differentiable space-time manifold can be completely equivalent to lattice theories. There is a close connection to generalized uncertainty relations which have appeared in string theory and other studies of quantum gravity.Comment: 4 pages, RevTe

Stochastic discrete scale invariance: Renormalization group operators and Iterated Function Systems

International audienceWe revisit here the notion of discrete scale invariance. Initially defined for signal indexed by the positive reals, we present a generalized version of discrete scale invariant signals relying on a renormalization group approach. In this view, the signals are seen as fixed point of a renormalization operator acting on a space of signal. We recall how to show that these fixed point present discrete scale invariance. As an illustration we use the random iterated function system as generators of random processes of the interval that are dicretely scale invariant

Discrete-Time continuous-dilation construction of linear scale-invariant systems and multi-dimensional self-similar signals

This dissertation presents novel models for purely discrete-time self-similar processes and scale- invariant systems. The results developed are based on the definition of a discrete-time scaling (dilation) operation through a mapping between discrete and continuous frequencies. It is shown that it is possible to have continuous scaling factors through this operation even though the signal itself is discrete-time. Both deterministic and stochastic discrete-time self-similar signals are studied. Conditions of existence for self-similar signals are provided. Construction of discrete-time linear scale-invariant (LSI) systems and white noise driven models of self-similar stochastic processes are discussed. It is shown that unlike continuous-time self-similar signals, a wide class of non-trivial discrete-time self-similar signals can be constructed through these models. The results obtained in the one-dimensional case are extended to multi-dimensional case. Constructions of discrete-space self-similar ran dom fields are shown to be potentially useful for the generation, modeling and analysis of multi-dimensional self-similar signals such as textures. Constructions of discrete-time and discrete-space self-similar signals presented in the dissertation provide potential tools for applications such as image segmentation and classification, pattern recognition, image compression, digital halftoning, computer vision, and computer graphics. The other aspect of the dissertation deals with the construction of discrete-time continuous-dilation wavelet transform and its existence condition, based on the defined discrete-time continuous-dilation scaling operator

FINDING EEG SPACE-TIME-SCALE LOCALIZED FEATURES USING MATRIX-BASED PENALIZED DISCRIMINANT ANALYSIS

International audienceThis paper proposes a new method for constructing and selecting of discriminant space-time-scale features for electroencephalogram (EEG) signal classification, suitable for Error Related Potentials (ErrP)detection in brain-computer interface (BCI). The method rests on a new variant of matrix-variate Linear Discriminant Analysis (LDA), and differs from previously proposed approaches in mainly three ways. First, a discrete wavelet expansion is introduced for mapping time-courses to time-scale coefficients, yielding time-scale localized features. Second, the matrix-variate LDA is modified in such a way that it yields an interesting duality property, that makes interpretation easier. Third, a space penalization is introduced using a surface Laplacian, so as to enforce spatial smoothness. The proposed approaches, termed D-MLDA and D-MPDA are tested on EEG signals, with the goal of detecting ErrP. Numerical results show that D-MPDA outperforms D-MLDA and other matrix-variate LDA techniques. In addition this method produces relevant features for interpretation in ErrP signals

Stochastic discrete scale invariance: renormalization group operators and iterated function systems

Abstract We revisit here the notion of discrete scale invariance. Initially defined for signal indexed by the positive reals, we present a generalized version of discrete scale invariant signals relying on a renormalization group approach. In this view, the signals are seen as fixed point of a renormalization operator acting on a space of signal. We recall how to show that these fixed point present discrete scale invariance. As an illustration we use the random iterated function system as generators of random processes of the interval that are dicretely scale invariant

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for publication in MNRAS. Version 3 (identical to version 2) posted for code release announcement - "Steerable scale discretised wavelets on the sphere" - S2DW code available for download at http://www.mrao.cam.ac.uk/~jdm57/software.htm

On the equivalence of soft wavelet shrinkage, total variation diffusion, total variation regularization, and SIDEs

Soft wavelet shrinkage, total variation (TV) diffusion, total variation regularization, and a dynamical system called SIDEs are four useful techniques for discontinuity preserving denoising of signals and images. In this paper we investigate under which circumstances these methods are equivalent in the 1-D case. First we prove that Haar wavelet shrinkage on a single scale is equivalent to a single step of space-discrete TV diffusion or regularization of two-pixel pairs. In the translationally invariant case we show that applying cycle spinning to Haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of TV diffusion. We prove that space-discrete TV difusion and TV regularization are identical, and that they are also equivalent to the SIDEs system when a specific force function is chosen. Afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the Laplacian pyramid of the signal. We analyse possibilities to avoid Gibbs-like artifacts for multiscale Haar wavelet shrinkage by scaling the thesholds. Finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and PDE / variational approaches. These methods are based on iterated shift-invariant wavelet shrinkage at multiple scales with scaled thresholds

Event Texture Search for Phase Transitions in Pb+Pb Collisions

NA44 uses a 512 channel Si pad array covering $1.5 <\eta < 3.3$ to study charged hadron production in 158 A GeV Pb+Pb collisions at the CERN SPS. We apply a multiresolution analysis, based on a Discrete Wavelet Transformation, to probe the texture of particle distributions event-by-event, allowing simultaneous localization of features in space and scale. Scanning a broad range of multiplicities, we search for signals of clustering and of critical behavior in the power spectra of local density fluctuations. The data are compared with detailed simulations of detector response, using heavy ion event generators, and with a reference sample created via event mixing. An upper limit is set on the probability and magnitude of dynamical fluctuations

Natural Alignment in the Two Higgs Doublet Model

As the LHC Higgs data persistently suggest the couplings of the observed 125 GeV Higgs boson to be consistent with the Standard Model (SM) expectations, any extended Higgs sector must lead to the so-called SM alignment limit, where one of the Higgs bosons behaves exactly like that of the SM. In the context of the Two Higgs Doublet Model (2HDM), this alignment is often associated with either decoupling of the heavy Higgs sector or accidental cancellations in the 2HDM potential. We present a novel symmetry justification for `natural' alignment without necessarily decoupling or fine-tuning. We show that there exist only three different symmetry realizations of the natural alignment scenario in 2HDM. We identify the 2HDM parameter space satisfying the natural alignment condition up to the Planck scale. We also analyze new collider signals for the heavy Higgs sector in the natural alignment limit, which dominantly lead to third-generation quarks in the final state and can serve as a useful observational tool during the Run-II phase of the LHC.Comment: 15 pages, based on a plenary presentation given by A. Pilaftsis at the Fifth Symposium on Prospects in the Physics of Discrete Symmetries (2016, Warsaw, Poland), (significant text overlap with arXiv:1408.3405, arXiv:1503.09140 and arXiv:1510.08790