Extra Invariance Research Articles

Dynamical realizations of the Lifshitz group

Dynamical realizations of the Lifshitz group are studied within the group-theoretic framework. A generalization of the 1d conformal mechanics is constructed, which involves an arbitrary dynamical exponent z. A similar generalization of the Ermakov-Milne-Pinney equation is proposed. Invariant derivative and field combinations are introduced, which enable one to construct a plethora of dynamical systems enjoying the Lifshitz symmetry. A metric of the Lorentzian signature in (d+2)-dimensional spacetime and the energy-momentum tensor are constructed, which lead to the generalized Ermakov-Milne-Pinney equation upon imposing the Einstein equations. The method of nonlinear realizations is used for building Lorentzian metrics with the Lifshitz isometry group. In particular, a (2d+2)-dimensional metric is constructed, which enjoys an extra invariance under the Galilei boosts.

Extra Invariance of Group Actions

Given discrete groups \(\Gamma \subset \Delta \) we characterize \((\Gamma ,\sigma )\)-invariant spaces that are also invariant under \(\Delta \). This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal \((\Gamma ,\sigma )\)-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the well-known characterization of shift-invariant spaces in terms of the Fourier transform. As a consequence of our results, we give a solution for the problem of finding the \((\Gamma ,\sigma )\)-invariant space nearest—in the sense of least squares—to a given set of data.

Subspaces with extra invariance nearest to observed data

Given an arbitrary finite set of data F={f1,…,fm}⊂L2(Rd) we prove the existence and show how to construct a “small shift invariant space” that is “closest” to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of Rd containing Zd. This is important for example in situations where we need to deal with jitter error of the data. Here small means that our solution subspace should be generated by the integer translates of a small number of generators. An expression for the error in terms of the data is provided and we construct a Parseval frame for the optimal space.We also consider the problem of approximating F from generalized Paley–Wiener spaces of Rd that are generated by the integer translates of a finite number of functions. That is finitely generated shift invariant spaces that are translation invariant. We characterize these spaces in terms of multi-tile sets of Rd, and show the connections with recent results on Riesz basis of exponentials on bounded sets of Rd. Finally we study the discrete case for our approximation problem.

Gauge-SURF descriptors

In this paper, we present a novel family of multiscale local feature descriptors, a theoretically and intuitively well justified variant of SURF which is straightforward to implement but which nevertheless is capable of demonstrably better performance with comparable computational cost. Our family of descriptors, called Gauge-SURF (G-SURF), is based on second-order multiscale gauge derivatives. While the standard derivatives used to build a SURF descriptor are all relative to a single chosen orientation, gauge derivatives are evaluated relative to the gradient direction at every pixel. Like standard SURF descriptors, G-SURF descriptors are fast to compute due to the use of integral images, but have extra matching robustness due to the extra invariance offered by gauge derivatives. We present extensive experimental image matching results on the Mikolajczyk and Schmid dataset which show the clear advantages of our family of descriptors against first-order local derivatives based descriptors such as: SURF, Modified-SURF (M-SURF) and SIFT, in both standard and upright forms. In addition, we also show experimental results on large-scale 3D Structure from Motion (SfM) and visual categorization applications.

Locally Oriented Optical Flow Computation

This paper proposes the use of an adaptive locally oriented coordinate frame when calculating an optical flow field. The coordinate frame is aligned with the least curvature direction in a local window about each pixel. This has advantages to both fitting the flow field to the image data and in imposing smoothness constraints between neighboring pixels. In terms of fitting, robustness is obtained to a wider variety of image motions due to the extra invariance provided by the coordinate frame. Smoothness constraints are naturally propagated along image boundaries which often correspond to motion boundaries. In addition, moving objects can be efficiently segmented in the least curvature direction. We show experimentally the benefits of the method and demonstrate robustness to fast rotational motion, such as what often occurs in human motion.

Principal shift-invariant spaces with extra invariance nearest to observed data

Given a set of functions \({\mathcal{F}=\{f_1, \dots, f_m\}\subset L^2(\mathbb{R}),}\) we construct a principal shift-invariant space V nearest to \({\mathcal{F}}\) in the sense that V minimizes the expression $$\sum_{i=1}^{m}\|f_i-P_{V}f_i\|^2,$$ among all the principal shift-invariant spaces with an orthonormal generator which are also translation invariant, or among all the principal shift-invariant spaces with an orthonormal generator which are also \({\frac{1}{n}\mathbb{Z}}\) -invariant for some fixed \({n\in\mathbb{N}}\) .

Extra invariance of shift-invariant spaces on LCA groups

This article generalizes recent results in the extra invariance for shift-invariant spaces to the context of LCA groups. Let G be a locally compact abelian (LCA) group and K a closed subgroup of G. A closed subspace of L 2 ( G ) is called K-invariant if it is invariant under translations by elements of K. Assume now that H is a countable uniform lattice in G and M is any closed subgroup of G containing H. In this article we study necessary and sufficient conditions for an H-invariant space to be M-invariant. As a consequence of our results we prove that for each closed subgroup M of G containing the lattice H, there exists an H-invariant space S that is exactly M-invariant. That is, S is not invariant under any other subgroup M ′ containing H. We also obtain estimates on the support of the Fourier transform of the generators of the H-invariant space, related to its M-invariance.

Invariance of a Shift-Invariant Space in Several Variables

In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of $${\mathbb {R}^d}$$ , and prove the existence of shift-invariant spaces that are exactly invariant for each given subgroup. As an application we relate the extra invariance to the size of support of the Fourier transform of the generators of the shift-invariant space. This work extends recent results obtained for the case of one variable to several variables.

Adaptive beamformer orthogonal rejection test

Research in the area of signal detection in the presence of unknown interference has resulted in a number of adaptive detection algorithms. Examples of such algorithms include the adaptive matched filter (AMF), the generalized likelihood ratio test (GLRT), and the adaptive coherence estimator (ACE). Each of these algorithms results in a tradeoff between detection performance for matched signals and rejection performance for mismatch signals. This paper introduces a new detection algorithm we call the adaptive beamformer orthogonal rejection test (ABORT). Our test decides if an observation contains a multidimensional signal belonging to one subspace or if it contains a multidimensional signal belonging to an orthogonal subspace when unknown complex Gaussian noise is present. In our analysis, we use a statistical hypothesis testing framework to develop a generalized likelihood ratio decision rule. We evaluate the performance of this decision rule in both the matched and mismatched signal cases. Our results show that for constant power complex Gaussian noise, if the signal is matched to the steering vector, ABORT, GLRT, and AMF give approximately equivalent probability of detection, higher than that of ACE, which trades detection probability for an extra invariance to scale mismatch between training and test data. Of these four tests, ACE is most selective and, therefore, least tolerant of mismatch, whereas AMF is most tolerant of mismatch and, therefore, least selective, ABORT and GLRT offer compromises between these extremes, with ABORT more like ACE and with GLRT more like AMF.

On the automorphic theta representation for simply laced groups

We construct an automorphic realization of the minimal representation of a split, simply laced groupG, over a number field. The realization is by a residue, at a certain point, of an Eisenstein series, induced from the Borel subgroup. This residue representation is square integrable and defines the automorphic theta representation. It has “very few” Fourier coefficients, which turn out to have some extra invariance properties.

THE EXTRA GAUGE SYMMETRY OF STRING DEFORMATIONS IN ELECTROMAGNETISM WITH CHARGES AND DIRAC MONOPOLES

We point out that electromagnetism with Dirac magnetic monopoles harbors an extra local gauge invariance called monopole gauge invariance. The gauge transformations act on a gauge field of monopoles [Formula: see text] and are independent of the ordinary electromagnetic gauge invariance. The extra invariance expresses the physical irrelevance of the shape of the Dirac strings attached to the monopoles. The independent nature of the new gauge symmetry is illustrated by comparison with two other systems, superfluids and solids, which are not gauge-invariant from the outset but which nevertheless possess a precise analog of the monopole gauge invariance in their vortex and defect structure, respectively. The extra monopole gauge invariance is shown to be responsible for the Dirac charge quantization condition 2eg/ħc=integer, which can now be proved for any fixed particle orbits, i.e. without invoking fluctuating orbits which would correspond to the standard derivation using Schrödinger wave functions. The only place where quantum physics enters in our theory is by admitting the action to jump by 2πħ×integer without physical consequences when moving the string at fixed particle orbits.