Local Degeneracies Research Articles

A Closed-Form, Pairwise Solution to Local Non-Rigid Structure-from-Motion.

A recent trend in Non-Rigid Structure-from-Motion (NRSfM) is to express local, differential constraints between pairs of images, from which the surface normal at any point can be obtained by solving a system of polynomial equations. While this approach is more successful than its counterparts relying on global constraints, the resulting methods face two main problems: First, most of the equationsystems they formulate are of high degree and must be solved using computationally expensive polynomial solvers. Some methods use polynomial reduction strategies to simplify the system, but this adds some phantom solutions. In any event, an additional mechanism is employed to pick the best solution, which adds to the computation without any guarantees on the reliability of the solution. Second, these methods formulate constraints between a pair of images. Even if there is enough motion between them, they may suffer from local degeneracies that make the resulting estimates unreliable without any warning mechanism. %Unfortunately, these systems are of high degree with up to five real solutions. Hence, a computationally expensive strategy is required to select a unique solution. Furthermore, they suffer from degeneracies that make the resulting estimates unreliable, without any mechanism to identify this situation. In this paper, we solve these problems for isometric/conformal NRSfM. We show that, under widely applicable assumptions, we can derive a new system of equationsin terms of the surface normals, whose two solutions can be obtained in closed-form and can easily be disambiguated locally. Our formalism also allows us to assess how reliable the estimated local normals are and to discard them if they are not. Our experiments show that our reconstructions, obtained from two or more views, are significantly more accurate than those of state-of-the-art methods, while also being faster. %In this paper, we show that, under widely applicable assumptions, we can derive a new system of equationsin terms of the surface normals, whose two solutions can be obtained in closed-form and can easily be disambiguated locally. Our formalism also allows us to assess how reliable the estimated local normals are and to discard them if they are not. Our experiments show that our reconstructions, obtained from two or more views, are significantly more accurate than those of state-of-the-art methods, while also being faster.

On the relationship between local affinities of a Boolean function and some types of its degeneracy

Abstract The paper studies the relationships between local affinities of a Boolean function and some notions of its degeneracy (differential degeneracy, algebraic degeneracy, affine splittability). New relations connecting the parameters of local affinities and degeneracies are obtained. Relationships between some types of degeneracy of Boolean functions are found.

Filament plots for data visualization

The efficiency of modern computer graphics allows us to explore collections of space curves simultaneously with “drag-to-rotate” interfaces. This inspires us to replace “scatterplots of points” with “scatterplots of curves” to simultaneously visualize relationships across an entire dataset. Since spaces of curves are infinite dimensional, scatterplots of curves avoid the “lossy” nature of scatterplots of points. In particular, if two points are close in a scatterplot of points derived from high-dimensional data, it does not generally follow that the two associated data points are close in the data space. Andrews plots provide scatterplots of curves that perfectly preserve Euclidean distances, but simultaneous visualization of these graphs over an entire dataset produces “visual clutter” because graphs of functions generally overlap in 2D. We mitigate this “visual clutter” issue by constructing computationally inexpensive 3D extensions of Andrews plots. First, we construct optimally smooth 3D Andrews plots by considering linear isometries from Euclidean data spaces to spaces of planar parametric curves. We rigorously parametrize the linear isometries that produce (on average) optimally smooth curves over a given dataset. This parameterization of optimal isometries reveals many degrees of freedom, and (using recent results on generalized Gauss sums) we identify a particular member of this set which admits an asymptotic “tour” property that avoids certain local degeneracies as well. Finally, we construct unit-length 3D curves (filaments) from Bishop frames induced by 3D Andrews plots. We conclude with examples of filament plots for several standard datasets, 1 illustrating how filament plots avoid “visual clutter”.

An Algebraic Brascamp\u2013Lieb Inequality

The Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions f_jin L^{q_j}(mathbb {R}^{n_j}), for j=1,ldots ,m, under some corresponding linear maps B_j. This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where B_j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

Lens models under the microscope: comparison ofHubble Frontier Fieldcluster magnification maps

Using the power of gravitational lensing magnification by massive galaxy clusters, the Hubble Frontier Fields provide deep views of six patches of the high redshift Universe. The combination of deep Hubble imaging and exceptional lensing strength has revealed the greatest numbers of multiply-imaged galaxies available to constrain models of cluster mass distributions. However, even with $\mathcal{O}(100)$ images per cluster, the uncertainties associated with the reconstructions are not negligible. The goal of this paper is to show the diversity of model magnification predictions. We examine 7 and 9 mass models of Abell 2744 and MACS J0416, respectively, submitted to the Mikulski Archive for Space Telescopes for public distribution in September 2015. The dispersion between model predictions increases from 30% at common low magnifications ($\mu\sim2$) to 70% at rare high magnifications ($\mu\sim40$). MACS J0416 exhibits smaller dispersions than Abell 2744 for $2<\mu<10$. We show that magnification maps based on different lens inversion techniques typically differ from each other by more than their quoted statistical errors. This suggests that some models underestimate the true uncertainties, which are primarily due to various lensing degeneracies. Though the exact mass sheet degeneracy is broken, its generalized counterpart is not broken at least in Abell 2744. Other, local degeneracies are also present in both clusters. Our comparison of models is complementary to the comparison of reconstructions of known synthetic mass distributions. By focusing on observed clusters, we can identify those that are best constrained, and therefore provide the clearest view of the distant Universe.

Exploring photometric redshifts as an optimization problem: an ensemble MCMC and simulated annealing-driven template-fitting approach

Using a grid of $\sim 2$ million elements ($\Delta z = 0.005$) adapted from COSMOS photometric redshift (photo-z) searches, we investigate the general properties of template-based photo-z likelihood surfaces. We find these surfaces are filled with numerous local minima and large degeneracies that generally confound rapid but "greedy" optimization schemes, even with additional stochastic sampling methods. In order to robustly and efficiently explore these surfaces, we develop BAD-Z [Brisk Annealing-Driven Redshifts (Z)], which combines ensemble Markov Chain Monte Carlo (MCMC) sampling with simulated annealing to sample arbitrarily large, pre-generated grids in approximately constant time. Using a mock catalog of 384,662 objects, we show BAD-Z samples $\sim 40$ times more efficiently compared to a brute-force counterpart while maintaining similar levels of accuracy. Our results represent first steps toward designing template-fitting photo-z approaches limited mainly by memory constraints rather than computation time.

Observed versus modelledu-,g-,r-,i-,z-band photometry of local galaxies – evaluation of model performance

We test how well available stellar population models can reproduce observed u,g,r,i,z-band photometry of the local galaxy population (0.02<=z<=0.03) as probed by the SDSS. Our study is conducted from the perspective of a user of the models, who has observational data in hand and seeks to convert them into physical quantities. Stellar population models for galaxies are created by synthesizing star formations histories and chemical enrichments using single stellar populations from several groups (Starburst99, GALAXEV, Maraston2005, GALEV). The role of dust is addressed through a simplistic, but observationally motivated, dust model that couples the amplitude of the extinction to the star formation history, metallicity and the viewing angle. Moreover, the influence of emission lines is considered (for the subset of models for which this component is included). The performance of the models is investigated by: 1) comparing their prediction with the observed galaxy population in the SDSS using the (u-g)-(r-i) and (g-r)-(i-z) color planes, 2) comparing predicted stellar mass and luminosity weighted ages and metallicities, specific star formation rates, mass to light ratios and total extinctions with literature values from studies based on spectroscopy. Strong differences between the various models are seen, with several models occupying regions in the color-color diagrams where no galaxies are observed. We would therefore like to emphasize the importance of the choice of model. Using our preferred model we find that the star formation history, metallicity and also dust content can be constrained over a large part of the parameter space through the use of u,g,r,i,z-band photometry. However, strong local degeneracies are present due to overlap of models with high and low extinction in certain parts of color space.

The exact solutions for the interaction V(r) = αr 2d − 2 − βr d − 2 by Nikiforov–Uvarov method

The exact solutions for the two- and N-dimensional Schrodinger equation have been rederived for the potential V(r) = αr2d − 2 − βrd − 2 by Nikiforov–Uvarov method. Specific results are presented for (i) the hydrogen atom and (ii) an isotropic harmonic oscillator. The dimensionality of the problem is seen to enter into these relations in such a way that one can immediately verify the corresponding three-dimensional results. The local accidental degeneracies are also explained for the two- and N-dimensional problems.

A comparison between high‐resolution central and Godunov‐based schemes for the black‐oil simulation

PurposeThis paper aims to perform a comparative study between capabilities of two numerical schemes from two main branches of numerical methods for solving hyperbolic conservation equations.Design/methodology/approachThe accuracy and performance of a newly developed high‐resolution central scheme vs a higher‐order Godunov‐based method are evaluated in the context of black‐oil reservoir simulations. Both methods are modified enabling study of applications that are not strictly hyperbolic and exhibit local linear degeneracies in their wave structure.FindingsThe numerical computations show that while both schemes produce results with virtually the same accuracy, the Godunov method reproduces slightly more accurate results at the expense of calculation of eigen‐structures.Research limitations/implicationsThe paper investigates only one dimensional problems, but the idea can be easily extended to multi‐dimensional problems.Practical implicationsThe paper shows the strengths and weaknesses of two practical numerical methods.Originality/valueSuch comparative study has not been published elsewhere and in particular, the performance of numerical methods on simulating hysteresis effect in hydrocarbon reservoirs has not been investigated in detail before.

Phase retrieval from speckle images

In ground-based astronomy, the inverse problem of phase retrieval from speckle images is a means to calibrate static aberrations for correction by active optics. It can also be used to sense turbulent wavefronts. However, the number of local minima drastically increases with the turbulence strength, mainly because of phase wrapping ambiguities. Multifocal phase diversity has been considered to overcome some ambiguities of the phase retrieval problem. We propose an effective algorithm for phase retrieval from a single focused image. Our algorithm makes use of a global optimization strategy and an automatically tuned smoothness prior to overcome local minima and phase degeneracies. We push the limit of D/r(0)=4 achieved by Irwan and Lane [J. Opt. Soc. Am. A.15, 2302 (1998)] up to D/r(0)=11, which is a major improvement owing to the drastic increase in the problem complexity. We estimate the performances of our approach from consistent simulations for different turbulence strengths and noise levels (down to 1500 photons per image). We also investigate the benefit of temporal correlation.

A Higher-Order Godunov Method for Multidimensional Ideal Magnetohydrodynamics

The authors present a higher-order Godunov method for the solution of the two- and three-dimensional equations of ideal magnetohydrodynamics (MHD). This work is based both on a suitable operator-split approximation to the full multidimensional equations, and on a one-dimensional Riemann solver. This Riemann solver is sufficiently robust to handle the nonstrictly hyperbolic nature of the MHD equations and the presence of local linear degeneracies. Results from a set of test problems show that this operator-split methodology has no problems handling any of the three MHD waves, yet resolves shocks to three or four computational zones. The advantages and limitations of this method are discussed.

Path-integral solutions for a class of 2D systems with local degeneracies

The potential V(r)=αr 2d-2 -βr d-2 is studied via the path integral approach. The Green function is calculated for E=0 and for any value of d in polar coordinates as well as in coordinates generalizing the Levi-Civita transformation. It is shown that this Green function is the sum of a discrete but finite part and of a continuous part. Limiting cases will be investigated

Hidden symmetries of two-dimensional systems with local degeneracies

The energy spectrum of a nonrelativistic particle in radial potentials of the form V(r) = αr 2d−2 − βr d−2 shows one level with a singularly high multiplicity. The two-dimensional problems are focused upon and the constants of motion that generate the symmetries responsible for this local accidental degeneracy are determined. The conserved operators realize an SU(2) algebra and the bound states are seen to fall into either finite, semibounded, or unbounded representations of this algebra. An SO(3,2) module encompassing the eigenstates associated to the whole set of potentials is also identified. It is further shown that there exists a second coordinate system besides the polar one in which the Schrödinger equation separates. Finally, it is indicated how the two-dimensional Coulomb and harmonic oscillator problems arise as special cases.

Singularity analysis and representation of planar parallel manipulators

In this paper, analytical expressions describing the singularity loci of simple planar three-degree-of-freedom parallel manipulators are derived. They are obtained using the analytical form of the Jacobian matrix, which is known to be singular when the manipulator is in a degenerate configuration. The expressions derived are then used to superimpose graphical representations of the manipulators' singularity loci and workspace. Both types of local degeneracies that can occur for parallel manipulators are studied. However, architecture singularities, which are rather trivial for the type of manipulators studied here, are assumed to be avoided by a proper choice of the manipulator's kinematic parameters. It is shown that, given certain assumptions on the geometry and for a given orientation of the end-effector, the singularity locus in the ( x, y) plane will always be a hyperbola. Moreover, when the passive revolute joints on the base and on the platform are respectively placed symmetrically, the hyperbola degenerates into a straight line. All these results are illustrated with examples of planar parallel manipulators. Some cases for which the singularities are located outside of the workspace are also presented. The analytic description of the singularity loci of manipulators in the Cartesian space can be used for analysis and optimum kinematic design. The expressions derived here have been integrated in a CAD package for parallel manipulators which provides a quick graphical representation of the singularity loci.

Higher order Godunov methods for general systems of hyperbolic conservation laws

We describe an extension of higher order Godunov methods to general systems of hyperbolic conservation laws. This extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields. The method constructs an approximation of the Riemann problem from local wave information. A generalization of the Engquist-Osher flux for systems is then used to compute a numerical flux based on this approximation. This numerical flux replaces the Godunov numerical flux in the algorithm, thereby eliminating the need for a global Riemann problem solution. The additional modifications to the Godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail. The method is applied to some simple model problems for which the global analytic structure is known. The method is also applied to the black-oil model for multiphase flow in petroleum reservoirs.

A local mode model for tetrahedral molecules

A three parameter coupled Morse oscillator model fitted to the known stretching vibrational energy levels is used to predict the energies of all stretching overtone bands with total quantum number ν ⩽ 5> for the molecules CH4, SiH4, GeH4 and their deuterium and 13C isotopic substituents. Close local mode degeneracies, with splittings of 3 cm-1 or less, are predicted at the highest energies for all molecules except CD4. Such degeneracies prevail at all excitation levels for SiH4 and GeH4, but those for ν ⩽ 3> are removed by deuterium substitution. The overtone splitting patterns of different molecules and the effects of isotopic substitution are rationalized by means of a correlation diagram, in which the ratio of bond anharmonicity to interbond coupling strength is used to rank the molecules on a scale running from local mode to harmonic normal mode limits. The ranking by local mode character is GeH4 > SiH4 > GeD4 > CH4 > SiD4 > CD4. A model intensity calculation for CH4 confirms the local mode picture wi...

Local mode degeneracies in the vibrational spectrum of H 2O

The development of progressively close local mode degeneracies in higher overtone stales is shown to be clearly demonstrable in the stretching vibrational spectrum of H 2O and to be relatively little affected by the state of the bending vibration. Calculated vibrational term values derived from the Sorbie-Murrell potential surface are in good agreement with the experimental spectrum.