Idealized computational models for auditory receptive fields

This paper presents a theory by which idealized models of auditory receptive fields can be derived in a principled axiomatic manner, from a set of structural properties to enable invariance of receptive field responses under natural sound transformations and ensure internal consistency between spectro-temporal receptive fields at different temporal and spectral scales. For defining a time-frequency transformation of a purely temporal sound signal, it is shown that the framework allows for a new way of deriving the Gabor and Gammatone filters as well as a novel family of generalized Gammatone filters, with additional degrees of freedom to obtain different trade-offs between the spectral selectivity and the temporal delay of time-causal temporal window functions. When applied to the definition of a second-layer of receptive fields from a spectrogram, it is shown that the framework leads to two canonical families of spectro-temporal receptive fields, in terms of spectro-temporal derivatives of either spectro-temporal Gaussian kernels for non-causal time or the combination of a time-causal generalized Gammatone filter over the temporal domain and a Gaussian filter over the logspectral domain. For each filter family, the spectro-temporal receptive fields can be either separable over the time-frequency domain or be adapted to local glissando transformations that represent variations in logarithmic frequencies over time. Within each domain of either non-causal or time-causal time, these receptive field families are derived by uniqueness from the assumptions. It is demonstrated how the presented framework allows for computation of basic auditory features for audio processing and that it leads to predictions about auditory receptive fields with good qualitative similarity to biological receptive fields measured in the inferior colliculus (ICC) and primary auditory cortex (A1) of mammals.Comment: 55 pages, 22 figures, 3 table

Separable time-causal and time-recursive spatio-temporal receptive fields

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, obtained by a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about parameterizing the intermediate temporal scale levels, analysing the resulting temporal dynamics and transferring the theory to a discrete implementation in terms of recursive filters over time.Comment: 12 pages, 2 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1404.203

The Centers of Early-Type Galaxies with HST III: Non-Parametric Recovery of Stellar Luminosity Distributions

We have non-parametrically determined the luminosity density profiles and their logarithmic slopes for 42 early-type galaxies observed with HST. Assuming that the isodensity contours are spheroidal, then the luminosity density is uniquely determined from the surface brightness data through the Abel equation. For nearly all the galaxies in our sample, the logarithmic slope of the luminosity density measured at 0.1" (the innermost reliable measurement with the uncorrected HST) is significantly different from zero; i.e. most elliptical galaxies have cusps. There are only two galaxies for which an analytic core cannot be excluded. The distribution of logarithmic slopes at 0.1" appears to be bimodal, confirming the conclusion of Lauer et al. (1995) that early-type galaxies can be divided into two types based on their surface-brightness profiles; i.e., those with cuspy cores and those whose steep power-law profiles continue essentially unchanged in to the resolution limit. The peaks in the slope distribution occur at -0.8 and -1.9. More than half of the galaxies have slopes steeper than -1.0. Taken together with the recent theoretical work of Merritt & Fridman, these results suggest that many (and maybe most) elliptical galaxies are either nearly axisymmetric or spherical near the center, or slowly evolve due to the influence of stochastic orbits.Comment: uuencoded compressed tarfile 21 pages with 6 fig, 1 tabl

Time-causal and time-recursive spatio-temporal receptive fields

We present an improved model and theory for time-causal and time-recursive spatio-temporal receptive fields, based on a combination of Gaussian receptive fields over the spatial domain and first-order integrators or equivalently truncated exponential filters coupled in cascade over the temporal domain. Compared to previous spatio-temporal scale-space formulations in terms of non-enhancement of local extrema or scale invariance, these receptive fields are based on different scale-space axiomatics over time by ensuring non-creation of new local extrema or zero-crossings with increasing temporal scale. Specifically, extensions are presented about (i) parameterizing the intermediate temporal scale levels, (ii) analysing the resulting temporal dynamics, (iii) transferring the theory to a discrete implementation, (iv) computing scale-normalized spatio-temporal derivative expressions for spatio-temporal feature detection and (v) computational modelling of receptive fields in the lateral geniculate nucleus (LGN) and the primary visual cortex (V1) in biological vision. We show that by distributing the intermediate temporal scale levels according to a logarithmic distribution, we obtain much faster temporal response properties (shorter temporal delays) compared to a uniform distribution. Specifically, these kernels converge very rapidly to a limit kernel possessing true self-similar scale-invariant properties over temporal scales, thereby allowing for true scale invariance over variations in the temporal scale, although the underlying temporal scale-space representation is based on a discretized temporal scale parameter. We show how scale-normalized temporal derivatives can be defined for these time-causal scale-space kernels and how the composed theory can be used for computing basic types of scale-normalized spatio-temporal derivative expressions in a computationally efficient manner.Comment: 39 pages, 12 figures, 5 tables in Journal of Mathematical Imaging and Vision, published online Dec 201

Steepening mass profiles, dark matter and environment of X-ray bright elliptical galaxies

We use a new non-parametric Bayesian approach to obtain the most probable mass distributions and circular velocity curves along with their confidence ranges, given deprojected density and temperature profiles of the hot gas surrounding X-ray bright elliptical galaxies. For a sample of six X-ray bright ellipticals, we find that all circular velocity curves are rising in the outer parts due to a combination of a rising temperature profile and a logarithmic pressure gradient that increases in magnitude. Comparing the circular velocity curves we obtain from X-rays to those obtained from dynamical models, we find that the former are often lower in the central ~10 kpc. This is probably due to a combination of: i) Non-thermal contributions of up to ~35% in the pressure (with stronger effects in NGC 4486), ii) multiple-temperature components in the hot gas, iii) incomplete kinematic spatial coverage in the dynamical models, and iv) mass profiles that are insufficiently general in the dynamical modelling. Complementing the total mass information from the X-rays with photometry and stellar population models to infer the dark matter content, we find evidence for massive dark matter haloes with dark matter mass fractions of ~35-80% at 2Re, rising to a maximum of 80-90% at the outermost radii. We also find that the six galaxies follow a Tully-Fisher relation with slope ~4 and that their circular velocities at 1Re correlate strongly with the velocity dispersion of the local environment. As a result, the galaxy luminosity at 1Re also correlates with the velocity dispersion of the environment. These relations suggest a close link between the properties of central X-ray bright elliptical galaxies and their environments (abridged).Comment: 20 pages, 11 figures, accepted for publication in MNRA

Distances between power spectral densities

We present several natural notions of distance between spectral density functions of (discrete-time) random processes. They are motivated by certain filtering problems. First we quantify the degradation of performance of a predictor which is designed for a particular spectral density function and then it is used to predict the values of a random process having a different spectral density. The logarithm of the ratio between the variance of the error, over the corresponding minimal (optimal) variance, produces a measure of distance between the two power spectra with several desirable properties. Analogous quantities based on smoothing problems produce alternative distances and suggest a class of measures based on fractions of generalized means of ratios of power spectral densities. These distance measures endow the manifold of spectral density functions with a (pseudo) Riemannian metric. We pursue one of the possible options for a distance measure, characterize the relevant geodesics, and compute corresponding distances.Comment: 16 pages, 4 figures; revision (July 29, 2006) includes two added section

On the equilibrium morphology of systems drawn from spherical collapse experiments

We present a purely theoretical study of the morphological evolution of self-gravitating systems formed through the dissipationless collapse of N-point sources. We explore the effects of resolution in mass and length on the growth of triaxial structures formed by an instability triggered by an excess of radial orbits. We point out that as resolution increases, the equilibria shift, from mildly prolate, to oblate. A number of particles N ~= 100000 or larger is required for convergence of axial aspect ratios. An upper bound for the softening, e ~ 1/256, is also identified. We then study the properties of a set of equilibria formed from scale-free cold initial mass distributions, ro ~ r^-g with 0 <= g <= 2. Oblateness is enhanced for initially more peaked structures (larger values of g). We map the run of density in space and find no evidence for a power-law inner structure when g <= 3/2 down to a mass fraction <~0.1 per cent of the total. However, when 3/2 < g <= 2, the mass profile in equilibrium is well matched by a power law of index ~g out to a mass fraction ~ 10 per cent. We interpret this in terms of less-effective violent relaxation for more peaked profiles when more phase mixing takes place at the centre. We map out the velocity field of the equilibria and note that at small radii the velocity coarse-grained distribution function (DF) is Maxwellian to a very good approximation.Comment: 16 page

Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

We present and analyze finite difference numerical schemes for the Allen Cahn/Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential. Both the first order and second order accurate temporal algorithms are considered. In the first order scheme, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term and the mobility explicitly. We provide a theoretical justification that, this numerical algorithm has a unique solution such that the positivity is always preserved for the logarithmic arguments. In particular, our analysis reveals a subtle fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays similarly bounded at the previous time step. Furthermore, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The unique solvability and the positivity-preserving property for the second order scheme are proved using similar ideas, in which the singular nature of the logarithmic term plays an essential role. For both the first and second order accurate schemes, we are able to derive an optimal rate convergence analysis, which gives the full order error estimate. The case with a non-constant mobility is analyzed as well. We also describe a practical and efficient multigrid solver for the proposed numerical schemes, and present some numerical results, which demonstrate the robustness of the numerical schemes