Gaussian Density Function Research Articles

Lobachevsky spline functions and interpolation to scattered data

To investigate errors in astronomical measurements Lobachevsky introduced in 1842 an infinite sequence of univariate spline functions with equally spaced knots, whom classic B-splines are directly connected to. A remarkable property is the convergence of the sequences of the Lobachevsky splines and of their derivatives to the normal (or Gaussian) density function and to its derivatives, respectively. This fact suggests to consider Lobachevsky splines for applications to univariate and multivariate scattered interpolation. First, this paper attempts to gather the most significant properties of Lobachevsky splines, generally sparse in the literature, maintaining for convenience a probabilistic setting. Then, applications to interpolation are discussed and numerical experiments, which show an interesting approximation performance, are given.

Optimal Threshold Computing in Automatic Image Thresholding using Adaptive Particle Swarm Optimization

histogram which can be modeled as a mixture of two Gaussian density functions, estimating these densities inpractice is not simply feasible. The objective of this paper is to use adaptive particle swarm optimization (APSO) forthe suboptimal estimation of the means and variances of these two Gaussian density functions; then, the computationof the optimal threshold value is straightforward. The comparisons of experimental results in a wide range of complexbimodal images show that this proposed thresholding algorithm presents higher correct detection rate of object andbackground in comparison to the other methods including Otsu’s method and estimating the parameters of Gaussiandensity functions using genetic algorithm (GA). Meanwhile, the proposed thresholding method needs lower executiontime than the PSO-based method, while it shows a little higher correct detection rate of object and background, withlower false acceptance rate and false rejection rate.

Reduced-Reference Video Quality Assessment of Compressed Video Sequences

In this paper, a novel reduced-reference (RR) video quality assessment (VQA) is proposed by exploiting the spatial information loss and the temporal statistical characteristics of the interframe histogram. From the spatial perspective, an energy variation descriptor (EVD) is proposed to measure the energy change of each individual encoded frame, which results from the quantization process. Besides depicting the energy change, EVD can further simulate the texture masking property of the human visual system (HVS). From the temporal perspective, the generalized Gaussian density (GGD) function is employed to capture the natural statistics of the interframe histogram distribution. The city-block distance (CBD) is used to calculate the histogram distance between the original video sequence and the encoded one. For simplicity, the difference image between adjacent frames is employed to characterize the temporal interframe relationship. By combining the spatial EVD together with the temporal CBD, an efficient RR VQA is developed. Evaluation on the subjective quality video database demonstrates that the proposed method outperforms the representative RR video quality metric and the full-reference VQAs, such as peak signal-to-noise ratio and structure similarity index in matching subjective ratings. This means that the proposed metric is more consistent with the HVS perception. Furthermore, as only a small number of RR features are extracted for representing the original video sequence (each frame requires only one parameter for describing EVD and three parameters for recording GGD), the RR features can be embedded into the video sequences or transmitted through the ancillary data channel, which can be used in the video quality monitoring system.

Reduced-reference image quality assessment in reorganized DCT domain

In this paper, a novel reduced-reference (RR) image quality assessment (IQA) is proposed by depicting the subband statistical characteristics in the reorganized discrete cosine transform (RDCT) domain. First, the block-based DCT coefficients are reorganized into a three-level coefficient tree, resulting in ten RDCT subbands. For the intra RDCT subband characteristic, the coefficient distribution of each RDCT subband is modeled by the generalized Gaussian density (GGD) function. The city-block distance (CBD) is employed to measure the modeling error between the actual distribution and the fitted GGD curve. For the inter RDCT subband characteristic, the mutual information (MI) is utilized to depict the dependencies between coefficient pairs in related RDCT subbands. Moreover, a frequency ratio descriptor (FRD) calculated in the RDCT domain is proposed to depict how the image energy distributes among different frequency components. The FRD values computed from both the reference and distorted images are jointly considered to derive a novel mutual masking strategy for simulating the texture masking property of the human visual system (HVS). By considering the GGD modeling of intra RDCT subband, MI of inter RDCT subbands, and FRD of the image, the proposed RR IQA is developed. Experimental results demonstrate that a small number of RR features is sufficient to represent the reference image for the perceptual quality analysis. The proposed method can outperform the state-of-the-art RR IQAs, and even the full-reference (FR) PSNR and SSIM.

Application of 3D-wavelet statistics to video analysis

Video activity analysis is used in various video applications such as human action recognition, video retrieval, video archiving. In this paper, we propose to apply 3D wavelet transform statistics to natural video signals and employ the resulting statistical attributes for video modeling and analysis. From the 3D wavelet transform, we investigate the marginal and joint statistics as well as the Mutual Information (MI) estimates. We show that marginal histograms are approximated quite well by Generalized Gaussian Density (GGD) functions; and the MI between coefficients decreases when the activity level increases in videos. Joint statistics attributes are applied to scene activity grouping, leading to 87.3% accurate grouping of videos. Also, marginal and joint statistics features extracted from the video are used for human action classification employing Support Vector Machine (SVM) classifiers and 93.4% of the human activities are properly classified.

Improved Method Based on MPF for Multi-target Tracking

Particle filter algorithm is poor at consistently maintaining the multi-modality problem and remains particle degeneracy phenomenon. This shortcoming can be addressed through using Mixture Particle Filter (MPF) to model the target distribution as a non-parametric mixture model. According to computational complexity of the particle filter algorithm, Gaussian Particle Filter (GPF) is used. The particle set is obtained by sampling of Gaussian density function instead of re-sampling to improve the computing speed of particle filter algorithm. In this paper, combined with the advantages of GPF and MPF, GM-MPF algorithm is proposed to improve the real-time of MPF. The experimental results show that GM-MPF algorithm can effectively solve the problem of multi-modality, particle degradation and achieve real-time multi-target tracking.

Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams

Dirac’s delta functions enable simple and effective representations of point loads and singularities in a variety of structural problems, leading very often to elegant and otherwise unworkable closed-form solutions. This is the case of cracked beams under static loads, whose theoretical and practical significance has attracted in recent years the interest of many researchers. Nevertheless, analytical formulations currently available for this problem are not completely satisfactory, either in terms of computational efficiency, when the continuity conditions must be enforced with auxiliary equations, or in terms of physical consistency, when the singularities in the beam’s flexural rigidity are represented with Dirac’s delta functions having a questionable negative sign. These considerations motivate the present study, which offers a novel and physically-based modelling of slender Euler–Bernoulli beams and short Timoshenko beams with any number and severity of cracks, conducing in both cases to exact closed-form solutions. For validation purposes, a standard finite element code is used, along with two nascent deltas (uniform and Gaussian density functions) to describe a smeared increase in the bending flexibility around the abscissa of the crack.

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory: the jumping distance and the waiting time, are replaced by two new ones: the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement 〈x2(t)〉 ∼ tα is realized numerically and analysed theoretically, where the value of the power index α is in a region of 0 < α < 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.

A SWITCHED-ADAPTIVE QUANTIZATION TECHNIQUE USING μ-LAW QUANTIZERS

In this paper, we propose a switched-adaptive quantization technique suitable for high quality quantization of signals which, as well as speech signals, have short-term statistics modeled by Gaussian probability density function. Particularly, from the set of the 2k disposable quantizers, the technique we propose performs a two-stage selection of the quantizer which is designed for the signal statistic nearest to the one of the signal to be processed. In the first selection stage, the choice of the quantizer from the set of k disposable quantizers is managed by the switched technique. Further, in the second selection stage, the choice is made between the restricted and the unrestricted quantizers, regarding the fact whether the maximum amplitude of the samples within the current frame is lower or higher than the upper support region threshold of the restricted quantizer, respectively. In order to properly define the frame length, in this paper we are introducing the rate-quality compromise criterion, along which we are studying the gain in the signal to quantization noise ratio and in the compression, that are achievable with the proposed technique in reference to the G.711 recommendation and another technique we have recently proposed.

Separation and Tracking of Maneuvering Sources with ICA and Particle Filters using a New Switching Dynamic Model

The problem addressed in this work is to simultaneously separate multiple maneuvering sources and track their kinematics (position, velocity, and acceleration) in the working space. It is developed upon the incorporation of the nonstationary independent component analysis (ICA) and the nonlinear state estimator problems in a noisy environment. The sampling importance resampling (SIR) particle filter is exploited as the nonlinear state estimator to track current kinematics of the sources even though the state densities are non-Gaussian, and the observation equations are nonlinear. Given source kinematics, nonstationary ICA with a generalized Gaussian density function is used to separate each source signal. Also a novel scheme is proposed as a better alternative for the conventional interacting multiple model (IMM) algorithm to cover the unpredictable movement of the source over time. The proposed scheme deals with the uncertainty of each source motion by incorporating multiple dynamic models in the tracking process. The single best dynamic mode is identified at each time step for all the sources rather than tracking sources for several IMMs as in the IMM algorithm by finding the mode that tracks an indicator source with minimum root mean square error (RMSE). The method is strictly causal and can be used for online tracking. The algorithm performance has been verified by illustrating some simulation results.

A Universal Property of Axonal and Dendritic Arbors

Axonal and dendritic arbors can be characterized statistically by their spatial density function, a function that specifies the probability of finding a branch of a particular arbor at each point in a neural circuit. Based on an analysis of over a thousand arbors from many neuron types in various species, we have discovered an unexpected simplicity in arbor structure: all of the arbors we have examined, both axonal and dendritic, can be described by a Gaussian density function truncated at about two standard deviations. Because all arbors are characterized by density functions with this single functional form, only four parameters are required to specify an arbor's size and shape: the total length of its branches and the standard deviations of the Gaussian in three orthogonal directions. This simplicity in arbor structure can have implications for the developmental wiring of neural circuits.

Vector quantization for saturated SAR raw data compression

Spaceborne SAR involves the storage and transmission of large-size sampling data. Block adaptive quantization (BAQ) is now the most widely used onboard data compression algorithm due to its good tradeoff between system performance and complexity. However, when spaceborne SAR raw data is saturated, the performance of conventional BAQ deteriorates dramatically because its precondition of Gaussian distribution of raw data no longer holds. In order to solve this problem, an improved vector quantization (VQ) algorithm is proposed. This algorithm firstly introduces saturation modification to a conventional vector quantizer, obtains the saturation codebook based on Gaussian density function, and then obtains the new vector quantizer for the whole set of Saturation Degree (SD). This algorithm makes the vector quantizer match statistical model of data for the whole set of SD, so the performance of the compression is improved. The case of the 2D signal is explicitly computed. The performance of the proposed algorithm is verified by simulated and real data experiments.

A semi-analytical particle filter for identification of nonlinear oscillators

Particle filters find important applications in the problems of state and parameter estimations of dynamical systems of engineering interest. Since a typical filtering algorithm involves Monte Carlo simulations of the process equations, sample variance of the estimator is inversely proportional to the number of particles. The sample variance may be reduced if one uses a Rao–Blackwell marginalization of states and performs analytical computations as much as possible. In this work, we propose a semi-analytical particle filter, requiring no Rao–Blackwell marginalization, for state and parameter estimations of nonlinear dynamical systems with additively Gaussian process/observation noises. Through local linearizations of the nonlinear drift fields in the process/observation equations via explicit Ito–Taylor expansions, the given nonlinear system is transformed into an ensemble of locally linearized systems. Using the most recent observation, conditionally Gaussian posterior density functions of the linearized systems are analytically obtained through the Kalman filter. This information is further exploited within the particle filter algorithm for obtaining samples from the optimal posterior density of the states. The potential of the method in state/parameter estimations is demonstrated through numerical illustrations for a few nonlinear oscillators. The proposed filter is found to yield estimates with reduced sample variance and improved accuracy vis-à-vis results from a form of sequential importance sampling filter.

State Estimation for Stochastic Linear Hybrid Systems with Continuous-State-Dependent Transitions: An IMM Approach

Hybrid systems consist of both continuous state evolutions and discrete state (or mode) transitions. In many hybrid systems, mode transitions are governed by guard conditions that are dependent on the continuous state. These kind of mode transitions can be described by continuous-state-dependent mode transition probabilities. However, most multiple model Kalman filter based algorithms, such as the well-known interacting multiple model (IMM) algorithm, assume constant mode transition probabilities. We propose a hybrid estimation algorithm based on the IMM approach for the stochastic hybrid systems with continuous-state-dependent transitions. We first develop two models to describe the guard conditions for various stochastic hybrid systems. From the models of the guard conditions, we derive the corresponding continuous-state-dependent mode transition probabilities, which are expressed in terms of Gaussian probability density functions (pdfs) or Gaussian cumulative density functions (cdfs). We then propose a hybrid estimation algorithm for the stochastic hybrid systems based on the IMM algorithm. Like the IMM algorithm, our algorithm utilizes Gaussian distributions to represent the continuous state pdfs which are updated analytically at each time step. We also use analytical integration methods to reduce the computational cost of updating the mode probabilities. As a result, the proposed algorithm is computationally efficient and is applicable to high-dimensional problems. The performance of the proposed algorithm is illustrated with two examples in air traffic control tracking applications.

Spacecraft Attitude Estimation Using Adaptive Gaussian Sum Filter

This paper is concerned with improving the attitude estimation accuracy by implementing an adaptive Gaussian sum filter where the a posteriori density function is approximated by a sum of Gaussian density functions. Compared to the traditional Gaussian sum filter, this adaptive approach utilizes the Fokker-Planck-Kolmogorov residual minimization to update the weights associated with different components of the Gaussian mixture model. Updating the weights provides an accurate approximation of the a posteriori density function and thus superior estimates. Simulation results show that updating the weights during the propagation stage not only provides better estimates between the observations but also provides superior estimator performance where the measurements are ambiguous.

Multiple Subunit Fitting into a Low-Resolution Density Map of a Macromolecular Complex Using a Gaussian Mixture Model

Recently, electron microscopy measurement of single particles has enabled us to reconstruct a low-resolution 3D density map of large biomolecular complexes. If structures of the complex subunits can be solved by x-ray crystallography at atomic resolution, fitting these models into the 3D density map can generate an atomic resolution model of the entire large complex. The fitting of multiple subunits, however, generally requires large computational costs; therefore, development of an efficient algorithm is required. We developed a fast fitting program, “ gmfit”, which employs a Gaussian mixture model (GMM) to represent approximated shapes of the 3D density map and the atomic models. A GMM is a distribution function composed by adding together several 3D Gaussian density functions. Because our model analytically provides an integral of a product of two distribution functions, it enables us to quickly calculate the fitness of the density map and the atomic models. Using the integral, two types of potential energy function are introduced: the attraction potential energy between a 3D density map and each subunit, and the repulsion potential energy between subunits. The restraint energy for symmetry is also employed to build symmetrical origomeric complexes. To find the optimal configuration of subunits, we randomly generated initial configurations of subunit models, and performed a steepest-descent method using forces and torques of the three potential energies. Comparison between an original density map and its GMM showed that the required number of Gaussian distribution functions for a given accuracy depended on both resolution and molecular size. We then performed test fitting calculations for simulated low-resolution density maps of atomic models of homodimer, trimer, and hexamer, using different search parameters. The results indicated that our method was able to rebuild atomic models of a complex even for maps of 30 Å resolution if sufficient numbers (eight or more) of Gaussian distribution functions were employed for each subunit, and the symmetric restraints were assigned for complexes with more than three subunits. As a more realistic test, we tried to build an atomic model of the GroEL/ES complex by fitting 21-subunit atomic models into the 3D density map obtained by cryoelectron microscopy using the C7 symmetric restraints. A model with low root mean-square deviations (14.7 Å) was obtained as the lowest-energy model, showing that our fitting method was reasonably accurate. Inclusion of other restraints from biological and biochemical experiments could further enhance the accuracy.

Uncertainty Propagation for Nonlinear Dynamic Systems Using Gaussian Mixture Models

A Gaussian-mixture-model approach is proposed for accurate uncertainty propagation through a general nonlinear system. The transition probability density function is approximated by a finite sum of Gaussian density functions for which the parameters (mean and covariance) are propagated using linear propagation theory. Two different approaches are introduced to update the weights of different components of a Gaussian-mixture model for uncertainty propagation through nonlinear system. The first method updates the weights such that they minimize the integral square difference between the true forecast probability density function and its Gaussian-sum approximation. The second method uses the Fokker-Planck-Kohnogorov equation error as feedback to adapt for the amplitude of different Gaussian components while solving a quadratic programming problem. The proposed methods are applied to a variety of problems in the open literature and are argued to be an excellent candidate for higher-dimensional uncertainty-propagation problems.

A Generalized Likelihood Ratio Test for Detecting Land Mines Using Multispectral Images

In this letter, we address the problem of detecting anomalies in multispectral images that are due to the presence of land mines. The proposed detection scheme follows from a generalized likelihood ratio test (GLRT) involving a mixture of certain probability density functions. More specifically, the GLRT is based on the assumption that the pixel values of a subblock within a single spectral-plane image can be modeled as a mixture of two Gaussian density functions (model distribution of background pixels) and a uniform density function (model distribution of anomalous pixels when they are present). We extend the single spectral-plane GLRT to the multiple spectral-plane case where the subblocks are from multispectral images. To assess the proposed GLRT, which we call the Gaussian-uniform GLRT (GU-GLRT) algorithm, we applied the GU-GLRT and popular RX algorithms to images from a real multispectral image sequence and used receiver operating characteristic (ROC) curves as the figure of merit. In the experiments that we conducted, the GU-GLRT outperformed the RX algorithm in the sense that the area under the ROC curve was greatest for the GU-GLRT algorithm.

Random Forests of Phonetic Decision Trees for Acoustic Modeling in Conversational Speech Recognition

In this paper, we present a novel technique of constructing phonetic decision trees (PDTs) for acoustic modeling in conversational speech recognition. We use random forests (RFs) to train a set of PDTs for each phone state unit and obtain multiple acoustic models accordingly. We investigate several methods of combining acoustic scores from the multiple models, including maximum-likelihood estimation of the weights of different acoustic models from training data, as well as using confidence score of -value or relative entropy to obtain the weights dynamically from online data. Since computing acoustic scores from the multiple models slows down decoding search, we propose clustering methods to compact the RF-generated acoustic models. The conventional concept of PDT-based state tying is extended to RF-based state tying. On each RF tied state, we cluster the Gaussian density functions (GDFs) from multiple acoustic models into classes and compute a prototype for each class to represent the original GDFs. In this way, the number of GDFs in each RF tied state is decreased greatly, which significantly reduces the time for computing acoustic scores. Experimental results on a telemedicine automatic captioning task demonstrate that the proposed RF-PDT technique leads to significant improvements in word recognition accuracy.

Alpha stable modeling of human visual systems for digital halftoning in rectangular and hexagonal grids

Human visual system (HVS) modeling has become a critical component in the design of digital halftoning algorithms. Methods that exploit the characteristics of the HVS include the direct binary search (DBS) and optimized tone-dependent halftoning approaches. The spatial sensitivity of the HVS is low-pass in nature, reflecting the physiological characteristics of the eye. Several HVS models have been proposed in the literature, among them, the broadly used Näsänen’s exponential model, which was later shown to be constrained in shape. Richer models are needed to attain better halftone attributes and to control the appearance of undesired patterns. As an alternative, models based on the mixture of bivariate Gaussian density functions have been proposed. The mathematical characteristics of the HVS model thus play a key role in the synthesis of model-based halftoning. In this work, alpha stable functions, an elegant class of functions richer than mixed Gaussians, are exploited to design HVS models to be used in two different contexts: monochrome halftoning over rectangular and hexagonal sampling grids. In the two scenarios, alpha stable models prove to be more efficient than Gaussian mixtures, as they use less parameters to characterize the tails and bandwidth of the model. It is shown that a decrease in the model’s bandwidth leads to homogeneous halftone patterns, and conversely, models with heavier tails yield smoother textures. These characteristics, added to their simplicity, make alpha stable models a powerful tool for HVS characterization.