Two-dimensional Gaussian Function Research Articles

Planar-basis pseudopotential method and planar Wannier functions for surfaces and heterostructures

Planar basis functions are used in self-consistent and empirical pseudopotential calculations for surfaces and heterostructures. The basis functions are products of two-dimensional plane waves (in the x− y plane) and Gaussian functions in the third ( z) direction. With proper linear combinations of these basis functions, generalized planar Wannier functions are constructed, and the application of them for solving large-size GaAs/AlAs superlattices is demonstrated. It is shown that the results obtained from using the Wannier functions are essentially identical to those obtained with the fully convergent plane-wave basis.

Automatic Image Annotation Using Adaptive Color Classification

We describe a system which automatically annotates images with a set of prespecified keywords, based on supervised color classification of pixels intoNprespecified classes using simple pixelwise operations. The conditional distribution of the chrominance components of pixels belonging to each class is modeled by a two-dimensional Gaussian function, where the mean vector and the covariance matrix for each class are estimated from appropriate training sets. Then, a succession of binary hypothesis tests with image-adaptive thresholds has been employed to decide whether each pixel in a given image belongs to one of the predetermined classes. To this effect, a universal decision threshold is first selected for each class based on receiver operating characteristics (ROC) curves quantifying the optimum “true positive” vs “false positive” performance on the training set. Then, a new method is introduced for adapting these thresholds to the characteristics of individual input images based on histogram cluster analysis. If a particular pixel is found to belong to more than one class, a maximuma posterioriprobability (MAP) rule is employed to resolve the ambiguity. The performance improvement obtained by the proposed adaptive hypothesis testing approach over using universal decision thresholds is demonstrated by annotating a database of 31 images.

E.p.r. study of the orientation of polymer segments induced by cold drawing of a low density polyethylene

The orientation of low density polyethylene (LDPE) chain segments induced by cold drawing was determined by electron paramagnetic resonance spectra measured for samples probed by tempyl stearate or alkyl radicals formed by γ-irradiation. We characterized the induced orientation in the amorphous phase using a two-dimensional Gaussian distribution function ( σ = 12°), axially symmetrical about the draw direction. A simultaneously induced alignment of the crystallite c-axes with the draw direction, accompanied by a random orientation of the crystallite a-axes and b-axes, was also observed. We propose that the molecular morphology of cold-drawn LDPE can be described in terms of specifically oriented crystallites emanating chain segments packed in the condis state.

Electron paramagnetic resonance of Fe3+ ions in borate glass: computer simulations

Computer simulations of Fe3+ electron paramagnetic resonance spectra at X (9.5 GHz) and Q (34 GHz) bands in the alkali borate glass Li2O-2B2O3 doped with Fe2O3 have been carried out using an approach based on the eigenfield method applied to the 'rhombic' spin Hamiltonian, which contains only the Zeeman and quadrupole fine-structure terms. In order to account for the structural disorder in the glass, two different distribution densities of fine-structure parameters D and E have been tried: a two-dimensional Gaussian function of D and lambda = mod E/D mod , and the 'Czjzek function', analogous to the one used in Mossbauer-effect studies. In simulating the experimental spectra, care has been taken to fit not only to the most prominent features arising at gef approximately=4.3 (at X and Q bands) and gef approximately=2.0 (at Q band), but also to an obvious plateau of the derivative of the absorption, which extends down to the magnetic field corresponding to gef approximately=9.7 (at both bands). As a result, the Czjzek function can be ruled out. The agreement between the experimental and computer-simulated spectra found with the Gaussian distribution density suggests the existence, besides orthorhombic symmetry sites (with lambda approximately=1/3), of a considerable number of Fe3+ sites with axial or feebly rhombic distortions ( lambda <or=0.08). The relatively high mean value of the axial fine-structure parameter D is consistent with a highly distorted environment of Fe3+ ions in the glass.

Analysis of receptor clustering on cell surfaces by imaging fluorescent particles

Fluorescently labeled low density lipoproteins (LDL) and influenza virus particles were bound to the surface of human fibroblasts and imaged with a cooled slow-scan CCD camera attached to a fluorescence microscope. Particles were also imaged after attachment to polylysine-coated microscope slides. The digital images were analyzed by fitting data points in the region of fluorescent spots by a two-dimensional Gaussian function, thus obtaining a measure of spot intensity with correction for local background. The intensity distributions for particles bound to polylysine slides were mainly accounted for by particle size distributions as determined by electron microscopy. In the case of LDL, the intensity distributions for particles bound to fibroblasts were considerably broadened, indicative of clustering. The on-cell intensity distributions were deconvolved into 1-particle, 2-particle, 3-particle, etc. components using the data obtained with LDL bound to polylysine-coated slides as an empirical measure of the single particle intensity distribution. This procedure yielded a reasonably accurate measure of the proportion of single particles, but large errors were encountered in the proportions of larger cluster sizes. The possibility of studying the dynamics of clustering was investigated by binding LDL to cells at 4 degrees C and observing changes in the intensity distribution with time after warming to 20 degrees C.

Planar-basis pseudopotential calculations of the Si(001)2 x 1 surface with and without hydrogen passivation.

A planar basis is introduced for self-consistent pseudopotential calculations. The basis is made of products of two-dimensional plane waves (in x-y plane) and Gaussian functions in the z direction. This basis is particularly suited for describing electronic states of heterostructures including surfaces, interfaces, and superlattices. The method is applied to the Si(001)2\ifmmode\times\else\texttimes\fi{}1 surface with symmetric dimer reconstruction and the effects of hydrogen passivation are studied. The work function and dissociation energy of hydrogen from the Si(001)2\ifmmode\times\else\texttimes\fi{}1 surface are calculated. The results for hydrogen dissocation are in agreement with previous experimental findings.

Optical Surface Roughness Measurement Using a Two-Dimensional Gaussian Function.

A new noncontact optical technique for rapidly measuring the surface roughness was proposed. The surfaces of ground specimens were illuminated with a LED light. The reflected light intensity distribution was measured with a CCD optical sensor, and it was approximated by a two-dimensional (2-D) Gaussian function. The standard deviations of the 2-D Gaussian function, σx and σy, with respect to the x and y axes were calculated for evaluating the broadness of the reflected intensity distribution curve in the two directions. The standard deviation σx with respect to the axis perpendicular to the ground direction increased with increasing the center line-averaged surface roughness Ra, giving the experimental relationship Ra = 21.9σ2x-6.8σx+0.6. On the other hand, the standard deviation σy with respect to the ground direction remained unchanged, independent of the surface roughness Ra. This technique allows optical measurement of the surface roughness in two perpendicularly intersecting directions simultaneously.

Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera. Low-density lipoprotein and influenza virus receptor mobility at 4 degrees C.

A fluorescence imaging system, based on using a cooled slow-scan CCD camera, has been developed for tracking receptors on the surfaces of living cells. The technique is applicable to receptors for particles such as lipoproteins and viruses that can be labeled with a few tens of fluorophores. The positions of single particles in each image are determined to within 25 nm by fitting the fluorescence distribution to a two-dimensional Gaussian function. This procedure also provides an accurate measure of intensity, which is used as a tag for automated tracking of particles from frame to frame. The method is applied to an investigation of the mobility of receptors for LDL and influenza virus particles on human dermal fibroblasts at 4 degrees C. In contrast to previous studies by FRAP (fluorescence recovery after photo-bleaching), it is found that receptors have a low but measurable mobility at 4 degrees C. Analysis of individual particle tracks indicates that whilst some receptors undergo random diffusion, others undergo directed motion (flow) or diffusion restricted to a domain. A procedure is proposed for subdividing receptors according to their different types of motion and hence determining their motional parameters. The finding that receptors are not completely immobilised at 4 degrees C is significant for studies of receptor distributions performed at this temperature.

Statistical analysis of collagen alignment in ligaments by scale-space analysis

Injuries to ligaments in the knee are common in sports and other physical activities. Some clinical methods are available for qualitatively evaluating the degree of ligament injury and healing. It is, however, desirable to objectively assess the healing of ligaments and to predicate optimal treatment on quantitative measurements of their structure. Information such as areas of coverage and spatial orientations of collagen fibrils, for example, may provide important information about the internal structure of ligament tissues. Since normal ligament tissues are made up of collagen fibrils which are highly organized, they can be considered as oriented piecewise linear patterns. In this paper, we propose a computational technique for statistical analysis of collagen alignment in ligament images using the scale-space approach. In this method, a ligament image is preprocessed by a sequence of filters which are second derivatives of two-dimensional Gaussian functions with different scales. This gives a set of zero-crossing maps (the scale space) from which a stability map is generated. Significant linear patterns are captured by analyzing the stability map. The directional information in terms of orientation distributions of the collagen fibrils in the image and the area covered by the fibrils in specific directions are extracted for statistical analysis. Examples illustrating the performance of this method with scanning electron microscope images of the collagen fibrils in healing rabbit medial collateral ligaments are presented in this paper.

Collicular ensemble coding of saccades based on vector summation

The superior colliculus in the monkey contains a topographically organized representation of the target in its upper layers and saccade-related activity in its deeper layers. Since collicular movement fields are quite large, a considerable region of the colliculus is active whenever a saccade is made. We have modelled the collicular role in saccade generation based on the idea, proposed earlier in the literature, that each movement cell causes a movement tendency in the direction of the external world point which it represents in the collicular map. The model is organized as follows: (1) An anisotropic logarithmic mapping transforms retinal coordinates into collicular coordinates. (2) A two-dimensional Gaussian function describes the spatial extent of the movement-related activity in the deeper layers. (3) An efferent mapping function specifies how the direction and the size of the movement contribution of each colliculus neuron depends on its location and its firing rate. (4) The total saccade is the vector sum of the individual cell contributions. This very simple model (seven fixed parameters) has been used to simulate metrical properties of saccades: (a) in response to visual targets; (b) in response to electrical stimulation in one colliculus, and (c) after a colliculus lesion. Model performance appears to be remarkably realistic but cannot account for some border effects and responses to double stimulation. Suggestions on how the model can be improved and extended will be presented.

Combined optimization of image-gathering optics and image-processing algorithm for edge detection

This paper investigates the relationships between the image-gathering and image-processing systems for minimum mean-squared error estimation of scene characteristics. A stochastic optimization problem is formulated in which the objective is to determine a spatial characteristic of the scene rather than a feature of the already blurred, sampled, and noisy image data. An analytical solution for the optimal characteristic image processor is developed. The Wiener filter for the sampled image case is obtained as a special case, where the desired characteristic is scene restoration. Optimal edge detection is investigated by using ∇2G as the desired characteristic, where G is a two-dimensional Gaussian distribution function. It is shown that the optimal edge detector compensates for the blurring introduced by the image-gathering optics, and, notably, that it is not circularly symmetric. The lack of circular symmetry is largely due to the geometric effects of the sampling lattice used in image acquisition. The optimal image-gathering optical transfer function is also investigated and the results of a sensitivity analysis are shown.

Count-dependent filter for smoothing bivariate FCM histograms.

A data-smoothing filter has been developed that permits the improvement in accuracy of individual elements of a bivariate flow cytometry (FCM) histogram by making use of data from adjacent elements, a knowledge of the two-dimensional measurement system point spread function (PSF), and the local count density. For FCM data, the PSF is assumed to be a set of two-dimensional Gaussian functions with a constant coefficient of variation for each axis. A set of space variant smoothing kernels are developed from the basic PSF by adjusting the orthogonal standard deviations of each Gaussian smoothing kernel according to the local count density. This adjustment in kernel size matches the degree of smoothing to the local reliability of the data. When the count density is high, a small kernel is sufficient. When the density is low, however, a broader kernel should be used. The local count density is taken from a region defined by the measurement PSF. The smoothing algorithm permits the reduction in statistical fluctuations present in bivariate FCM histograms due to the low count densities often encountered in some elements. This reduction in high-frequency spatial noise aids in the visual interpretation of the data. Additionally, by making more efficient use of smaller samples, systematic errors due to system drift may be minimized.

Source range information loss in waveguides

In a previous study [E. C. Shang, C. S. Clay, and Y. Y. Wang, J. Acoust. Soc. Am. 78, 172 (1985)] a new method of passive source ranging in a layered waveguide was proposed. This paper investigates the reduction (loss) in range information due to phase fluctuations and to an inaccurate description of the environment (waveguide mismatch). The effects of phase fluctuations are investigated using a two-dimensional Gaussian distribution function. A numerical normal mode code is used to study the effects of waveguide mismatching. The loss in range information caused by incorrect descriptions of both the bottom sediment type and the sound speed profile in the water column is calculated. Examples are given for a water depth of 100 m and frequencies in the range from 100–500 Hz.

Visuomotor fields of the superior colliculus: A quantitative model

Electrophysiological and electrical stimulation studies in the monkey have disclosed that both the retinal surface and the metrics of saccades are topographically represented in the superior colliculus. This mapping of sensory and motor space onto the colliculus is nonhomogeneous in that the central region is over-represented in both the visual and the motor map. Single unit studies have revealed that visual receptive fields of collicular neurons are typically quite large and are characterized by a skewed (asymmetrical) sensitivity profile. Analyses by McIlwain [ J. Neurophysiol. 38, 219–230 (1975) ] in the cat have suggested that this skewness property reflects mainly the spatial distortion inherent in the afferent mapping. In this paper we describe a quantitative model, based on a logarithmic mapping function combined with a Gaussian connectivity function in the colliculus, which can account for the extent and the shape of collicular receptive fields. Collicular neurons in the deeper layers have movement-related bursts of activity for saccades in a limited amplitude and direction range related to their location in the collicular map. These movement fields, like visual receptive fields, may be quite extensive and typically have a skewed profile. In our model, an efferent-mapping function is defined, which relates the locus of a population of recruited cells to the metrics of the ensuing saccade. The parameters of this function, which was taken to be identical with the afferent mapping function, were estimated from Robinson's [ Vision Res. 12, 1795–1808 (1972) ] electrical stimulation data. Based on the assumption that the population-activity profile resembles a two-dimensional Gaussian function, the shape and the size of movement fields can then be described with just 2 or 3 free parameters. Electrophysiological data recorded from a small sample of collicular visuomotor neurons were used to illustrate the procedure, which we designed to enable application of our model to the experimental data. The best fit was obtained when the mapping function was slightly anisotropic. Suggestions on how the model could be improved and extended are offered in the Discussion.

A quantum-mechanical model of coupled proton and neutron asymmetries and the fast charge equilibration process in deep-inelastic collisions

A quantum-mechanical model of coupled harmonic oscillators in collective coordinates of proton and neutron asymmetries is solved analytically, in a non-perturbative formalism, for the charge and neutron distributions in deep-inelastic reactions. The time evolution of the mean values (centroids) and the variances (widths) is studied for the initial distribution taken to be a two-dimensional Gaussian function. Model calculations are presented that show the detailed dependence of the relevant quantities on the variables of our model. Fits to the experimental data are made for the reactions 129Xe on 116Sn and 124Sn.

Charge and mass exchange inFe56-induced reactions at 8.3 MeV/nucleon

Projectilelike fragments from reactions of 465-MeV $^{56}\mathrm{Fe}$ projectiles with targets of $^{56}\mathrm{Fe}$, $^{165}\mathrm{Ho}$, $^{209}\mathrm{Bi}$, and $^{238}\mathrm{U}$ have been measured as a function of energy loss, and neutron and proton number. The mass and charge of each fragment were uniquely determined with discrete resolution. Data were analyzed as a function of energy loss by fitting a two-dimensional Gaussian function to the neutron-proton distributions. The fit parameters provide a convenient basis for comparison with theoretical predictions. In addition, centroids and variances derived from more detailed onedimensional Gaussian fits to isobaric charge distributions are presented. Results are given for measured data as well as for the primary fragment distributions after correction for the dominant neutron evaporation. The centroid data show that in the asymmetric reactions proton transfer from projectile to target nucleus is preferred while the average fragment neutron number is largely preserved. This produces a charge equilibration process which is dependent on energy loss within the combined systems. The neutron and proton variances indicate an early dominant neutron flow followed by a constant relative neutron and proton exchange rate at energy losses above about 30 MeV. The degree of correlation of the neutron-proton distributions increases with energy loss and is found to be strongly related to the potential energy surface. Consistently observed negative correlation coefficients at low energy losses suggest the possible importance of processes other than particle exchange in the early stages of the interaction. Variances for constant $N$,$Z$, and $A$ are found to saturate at energy losses and with magnitudes dependent on the target nucleus.

K+]o clearance in cortex: a new analytical model.

1. It has been suggested that passive diffusion is the principal mechanism of the clearance of locally elevated extracellular potassium, K+o, in the cerebral cortex. This concept was based on the assumption that elevated K+ could be modeled as a point source. In the present study, the functional role of passive diffusion was reevaluated in the anesthetized cat cortex following local electrical stimulation. 2. The initial spatial distribution of extracellular potassium activity, [K+]o, elevated by monopolar stimulation, could be modeled by a two-dimensional Gaussian function at and below the 500-micron cortical depth. Azimuthal symmetry around the stimulating electrode was assumed and cylindrical spatial coordinates were used. 3. The observed clearance of transiently elevated [K+]o as a function of space and time was much more rapid than that predicted by an analytical model consisting of the homogeneous diffusion equation whose initial condition was the Gaussian spatial distribution of [K+]o at the onset of the clearance process. 4. It is concluded that passive diffusion does not significantly contribute to the rapid clearance of locally elevated extracellular potassium in the cortex. Active uptake of potassium by cortical cells should be more seriously considered as being primarily responsible for the potassium clearance.