Compressive sensing (CS) theory demonstrates that by using uniformly-random sampling, rather than uniformly-spaced sampling, higher quality image reconstructions are often achievable. Considering ...that the structure of sampling protocols has such a profound impact on the quality of image reconstructions, we formulate a new sampling scheme motivated by physiological receptive field structure, localized random sampling, which yields significantly improved CS image reconstructions. For each set of localized image measurements, our sampling method first randomly selects an image pixel and then measures its nearby pixels with probability depending on their distance from the initially selected pixel. We compare the uniformly-random and localized random sampling methods over a large space of sampling parameters, and show that, for the optimal parameter choices, higher quality image reconstructions can be consistently obtained by using localized random sampling. In addition, we argue that the localized random CS optimal parameter choice is stable with respect to diverse natural images, and scales with the number of samples used for reconstruction. We expect that the localized random sampling protocol helps to explain the evolutionarily advantageous nature of receptive field structure in visual systems and suggests several future research areas in CS theory and its application to brain imaging.
•A novel scheme for multi-material Maxwell's equations with general linear dispersive effects.•High-order accurate treatment of interfaces between dispersive materials using compatibility ...conditions.•A single stage, three level, fourth-order accurate scheme is developed with a large CFL-one time-step.•Composite overlapping grids are used for complex geometries with boundary conforming grids.•Interface equations are solved efficiently with a novel hierarchical approach.
A high-order accurate scheme for solving the time-domain dispersive Maxwell's equations and material interfaces is described. Maxwell's equations are solved in second-order form for the electric field. A generalized dispersive material (GDM) model is used to represent a general class of linear dispersive materials and this model is implemented in the time-domain with the auxiliary differential equation (ADE) approach. The interior updates use our recently developed second-order and fourth-order accurate single-stage three-level space-time finite-difference schemes, and this paper extends these schemes to treat interfaces between different dispersive materials. Composite overlapping grids are used to treat complex geometry, with Cartesian grids generally covering most of the domain and local conforming grids representing curved boundaries and interfaces. Compatibility conditions derived from the interface jump conditions and governing equations are used to derive accurate numerical interface conditions that define values at ghost points. Although some compatibility conditions couple the equations for the ghost points in tangential directions due to mixed-derivatives, it is shown how to decouple the equations to avoid solving a larger system of equations for all ghost points on the interface. The stability of the interface approximations is studied with mode analysis and it is shown that the schemes retain close to a CFL-one time-step restriction. Numerical results are presented in two and three space dimensions to confirm the accuracy and stability of the schemes. The schemes are verified using exact solutions for a planar interface, a disk in two dimensions, and a solid sphere in three dimensions.
A high-order accurate scheme for solving the time-domain Maxwell's equations with a generalized dispersive material model is described. The equations for the electric field are solved in second-order ...form, and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations (ADE). Numerical methods are developed for both second-order and fourth-order accuracy in space and time. The equations are discretized using finite-differences, and advanced in time with a single-stage, three-level, space–time scheme which remains stable up to the usual explicit CFL restriction, as proven using mode analysis. Because the equations are treated in their second-order form, there is no need for grid staggering, and instead a collocated grid is used. Composite overlapping grids are used to treat complex geometries with boundary-conforming grids, and a high-order upwind dissipation is added to ensure robust and stable approximations on overlapping grids. Numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes.
•A novel scheme for dispersive Maxwell's equations in second-order form using auxiliary differential equations.•Arbitrary dispersive effects are treated through a recently developed generalized dispersive material (GDM) model.•A three time-level, fourth-order accurate and memory efficient scheme is derived that has a large CFL “one” time-step.•Composite overlapping grids are used for complex geometries with accurate treatment of curved boundaries.•The scheme is verified using newly developed exact solutions for the dispersive equations.
From among the waves whose dynamics are governed by the nonlinear Schrödinger equation, we find a robust, spatiotemporally disordered family, in which waves initialized with increasing amplitudes, on ...average, over long timescales, effectively evolve as ever more weakly coupled collections of plane waves. In particular, the relative amount of energy contained in their coupling decays to zero with increasing wave amplitude.
In Slovenia the False ringlet Coenonympha oedippus uniquely occurs both on wet (Ljubljana Marsh and surroundings) and dry grasslands (Slovenian Istria, Karst, Gorica Hills). Natural hazards that ...threaten its habitats include fires and floods; the frequency of their occurrence in the former and existing habitats of C. oedippus was determined using the χ2-test. We showed that habitats on wet grasslands are less threatened by fire than those on dry grasslands. Among the latter, habitats in the Karst and Slovenian Istria are the most threatened. Habitats of C. oedippus are threatened by flooding only in Slovenian Istria and Ljubljana Marsh. Considering the extent of fire and flood risk and fragmentation of C. oedippus habitats in Slovenia, we assume that such natural hazards may lead to local extinction of the species.
In the area of the Orehek karst intrinsic vulnerability maps of groundwater and the Korentan karst spring, hazard maps and risk to contamination map have been made using Slovene Approach. For this ...purpose cartographic and other published data on geological, geomorphological, pedological, hydrological and meteorological characteristics have been gathered and extensive field inventory of the missing geomorphological and pedological characteristics has been made. Publically available land use and census data have been examined and checked in the field, manure heaps and illegal waste dumps have been mapped. The final thematic maps are excellent tools for national and local authorities when planning water protection and land use. The presented study in the catchment of an actual water source is an example of good practice of the revival of the in the past abandoned water sources that can in the future serve as a substitution water source in case of main water source overexploitation or contamination.
Early olfactory pathway responses to the presentation of an odor exhibit remarkably similar dynamical behavior across phyla from insects to mammals, and frequently involve transitions among ...quiescence, collective network oscillations, and asynchronous firing. We hypothesize that the time scales of fast excitation and fast and slow inhibition present in these networks may be the essential element underlying this similar behavior, and design an idealized, conductance-based integrate-and-fire model to verify this hypothesis via numerical simulations. To better understand the mathematical structure underlying the common dynamical behavior across species, we derive a firing-rate model and use it to extract a slow passage through a saddle-node-on-an-invariant-circle bifurcation structure. We expect this bifurcation structure to provide new insights into the understanding of the dynamical behavior of neuronal assemblies and that a similar structure can be found in other sensory systems.
Integrate-and-fire (I&F) neuronal networks are ubiquitous in diverse image processing applications, including image segmentation and visual perception. While conventional I&F network image processing ...requires the number of nodes composing the network to be equal to the number of image pixels driving the network, we determine whether I&F dynamics can accurately transmit image information when there are significantly fewer nodes than network input-signal components. Although compressive sensing (CS) theory facilitates the recovery of images using very few samples through linear signal processing, it does not address whether similar signal recovery techniques facilitate reconstructions through measurement of the nonlinear dynamics of an I&F network. In this paper, we present a new framework for recovering sparse inputs of nonlinear neuronal networks via compressive sensing. By recovering both one-dimensional inputs and two-dimensional images, resembling natural stimuli, we demonstrate that input information can be well-preserved through nonlinear I&F network dynamics even when the number of network-output measurements is significantly smaller than the number of input-signal components. This work suggests an important extension of CS theory potentially useful in improving the processing of medical or natural images through I&F network dynamics and understanding the transmission of stimulus information across the visual system.
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