Gaussian Scale Mixture Model Research Articles

Efficient Modeling of Spatial Extremes over Large Geographical Domains

Various natural phenomena exhibit spatial extremal dependence at short spatial distances. However, existing models proposed in the spatial extremes literature often assume that extremal dependence persists across the entire domain. This is a strong limitation when modeling extremes over large geographical domains, and yet it has been mostly overlooked in the literature. We here develop a more realistic Bayesian framework based on a novel Gaussian scale mixture model, with the Gaussian process component defined by a stochastic partial differential equation yielding a sparse precision matrix, and the random scale component modeled as a low-rank Pareto-tailed or Weibull-tailed spatial process determined by compactly-supported basis functions. We show that our proposed model is approximately tail-stationary and that it can capture a wide range of extremal dependence structures. Its inherently sparse structure allows fast Bayesian computations in high spatial dimensions based on a customized Markov chain Monte Carlo algorithm prioritizing calibration in the tail. We fit our model to analyze heavy monsoon rainfall data in Bangladesh. Our study shows that our model outperforms natural competitors and that it fits precipitation extremes well. We finally use the fitted model to draw inference on long-term return levels for marginal precipitation and spatial aggregates.

Deep Gaussian Scale Mixture Prior for Image Reconstruction.

Image reconstruction from partial observations has attracted increasing attention. Conventional image reconstruction methods with hand-crafted priors often fail to recover fine image details due to the poor representation capability of the hand-crafted priors. Deep learning methods attack this problem by directly learning mapping functions between the observations and the targeted images can achieve much better results. However, most powerful deep networks lack transparency and are nontrivial to design heuristically. This paper proposes a novel image reconstruction method based on the Maximum a Posterior (MAP) estimation framework using learned Gaussian Scale Mixture (GSM) prior. Unlike existing unfolding methods that only estimate the image means (i.e., the denoising prior) but neglected the variances, we propose characterizing images by the GSM models with learned means and variances through a deep network. Furthermore, to learn the long-range dependencies of images, we develop an enhanced variant based on the Swin Transformer for learning GSM models. All parameters of the MAP estimator and the deep network are jointly optimized through end-to-end training. Extensive simulation and real data experimental results on spectral compressive imaging and image super-resolution demonstrate that the proposed method outperforms existing state-of-the-art methods.

Joint Activity Detection and Channel Estimation for Intelligent-Reflecting-Surface-Assisted Wireless IoT Networks

Intelligent reflecting surfaces (IRSs) provide the ability to tune the wireless environments by introducing passive scattering elements between the base station (BS) or access points (APs) and users, and it is a promising technology for next generation wireless networks. In this paper, we consider joint activity detection and channel estimation in both centralized and distributed IRS assisted IoT networks. We determine the explicit distribution for the equivalent channel coefficients from the BS or APs to the users. It is shown that this distribution can be equivalently represented with a Gaussian scale mixture (GSM) model. With generalized approximate message passing (GAMP) algorithm, the received pilot signals at the BS or APs are decoupled into scalar Gaussian noise corrupted versions of the effective channel coefficients. Subsequently, minimum mean square error (MMSE) estimate of the effective channel coefficients and threshold detection rules are acquired. Finally, the optimal fusion rule is used to obtain the activity detection results of each user. As an additional low-complexity approach, we approximate the equivalent channel coefficients with a Gaussian approximation (GA) model. We derive the theoretical mean square error for the MMSE estimate of the effective channel coefficients using the GA model and the state evolution equations of the GAMP algorithm, under the assumption that the direct links from the BS to the users can be neglected due to unfavorable propagation conditions. Numerical results show that the Gaussian approximation for the distribution of the equivalent channel coefficients is accurate even for moderate number of elements at the IRS. And the theoretical mean square error for the MMSE estimate of the effective channel coefficients under the assumption that direct link is negligible is shown to overlap with the corresponding numerical simulations.

Identification of MISO Hammerstein system using sparse multiple kernel-based hierarchical mixture prior and variational Bayesian inference

The Hammerstein model is a cascade composition of a static memoryless nonlinear function followed by a linear time-invariant dynamical subsystem, which is capable of modeling a wide range of nonlinear dynamical systems. Model structural parameter selection (including the model order and the nonlinearity order) and sparse representation of the static nonlinear function are two issues that receive increasing interests in Hammerstein system identification. In this paper, a novel Bayesian sparse multiple kernel-based identification method (BSMKM) for multiple-input single-output (MISO) Hammerstein system is proposed to handle those issues, where the basis-function model and the finite impulse response model are used to represent the nonlinear part and the linear part respectively. Firstly, in order to jointly realize the model parameter estimation, the sparse representation of static nonlinear function (the nonlinearity order selection can also be realized indirectly) and the model order selection of linear dynamical system, a hierarchical prior distribution is constructed based on Gaussian scale mixture model and sparse multiple kernel, which can characterize both inter-group sparsity and intra-group correlation structure. Then, a full Bayesian method based on variational Bayesian inference is proposed to estimate all unknown model parameters, including finite impulse response coefficients, hyperparameters and noise variance. Finally, the performance of the proposed BSMKM identification method is evaluated by numerical experiments using both simulation and real data.

GSM Filter: A Learning Model of Measurement Noise Distributions for Bayesian Filtering

In this paper, a new universal Bayesian filter with a Gaussian scale mixture (GSM) model, called the Gaussian scale mixture (GSM) filter, for adaptively learning any measurement noise distribution of a state space system is presented. It is demonstrated that the discrete measurement noise distribution of the system takes on a Gaussian sum form when described by a GSM model. It implies that, as the Gaussian sum model does, the GSM model can also be used to approximate any measurement distribution as closely as desired, whatever the probability density function (pdf) of the measurement noise is. If its covariance is known as a priori or can be estimated, it is found that the scale parameter and matrix of the GSM model at each time step can be achieved adaptively to make the following Bayesian inference much simpler. Further analyses show that the GSM model's auxiliary random variable and the system's state posterior at each time step can be inferred by a variational Bayesian (VB) approach adaptively. At each time step, the posterior distribution of system states shown is a Gaussian whatever the measurement noise pdf of the system is. As a result, the Bayesian filtering for nonlinear systems with any measurement noise pdf can be done by our GSM filter, almost as simple as the general Gaussian filters for filtering a nonlinear Gaussian system. Both simulations and experiments are conducted to verify the effectiveness and correctness of the analytical results. The results on the performances of our algorithms fully outperforming those of the existing state of the art Bayesian filters are given.

Bayesian Correlation Filter Learning With Gaussian Scale Mixture Model for Visual Tracking

Correlation filters (CF), a popular tool for visual tracking, suffer from unwanted boundary effects due to the periodic assumption needed for FFT implementation. To address this issue, spatially regularized discriminative correlation filters (SRDCF) have been proposed by introducing a weighting matrix to the regularization term. However, the existing design of spatial weighting matrix is often heuristic and non-adaptive. Inspired by recent advances in joint discrimination and reliability learning for correlation tracking, we propose a principled Bayesian correlation filter learning method using Gaussian scale mixture (GSM) model. The key idea is to decompose each CF coefficient into the product of a positive scalar multiplier and a Gaussian random variable. Treating positive multipliers as weighting coefficients, GSM-based modeling of CFs leads to a spatially adaptive regularization strategy with improved capability of handling various appearance-related uncertainty factors (e.g., scale variation, out-of-plane rotation, and motion blur). Moreover, by imposing a sparse prior over the multipliers, we can jointly learn multipliers and CFs under a unified Bayesian estimation framework. Structured GSM model allows us to better exploit the spatial correlations among CFs and further improve the tracking performance. Experimental results on OTB-2013, OTB-2015, Temple Color-128, VOT-2016, and VOT-2017 show that our tracking method performs favorably when compared with current state-of-the-art methods.

Correlation filters based on spatial-temporal Gaussion scale mixture modelling for visual tracking

Correlation Filters (CFs) have shown outstanding performance in tracking, but are subject to unwanted boundary effects. Spatial regularization (SR) is widely used as an efficient method to alleviate the boundary effects. However, spatial regularization is almost handcrafted and fixed during tracking process, which cannot handle the diversity of objects and the complexity of motion. Furthermore, the rich spatio-temporal correlations among multiple targets of interest cannot be fully exploited. Herein, we propose a spatio-temporal Gaussian scale mixture model (ST-GSM) for correlation-filter-based visual tracking. In our Gaussian scale mixture (GSM) model, each correlation filter coefficient is decomposed into the product of a positive scalar multiplier with sparsity and a Gaussian random variable. The reliable components of the Gaussian random variable can be adaptively selected based on the positive multipliers, aiming at alleviating the notorious boundary effects. To exploit the temporal consistency between adjacent frames, nonzero-means GSM models are developed to characterize the temporal correlations. Specifically, the filter coefficient obtained in the previous frame is used as the mean prior for the current frame. The spatial correlations among filter coefficients have been considered in the structured GSM model, thereby further improving the tracking performance. Experimental results show that the proposed model can significantly improve the performance of CF-based trackers.

Image Reconstruction of Electrical Capacitance Tomography Based on an Efficient Sparse Bayesian Learning Algorithm

Image reconstruction is the main research problem of electrical capacitance tomography (ECT). In this article, a novel ECT image reconstruction algorithm based on an efficient sparse Bayesian learning (ESBL) algorithm is presented. This algorithm takes the Gaussian-scale mixture model as the prior distribution of the parameters to increase the flexibility of the model. Then, a surrogate function is used to replace the Gaussian likelihood function to reduce the computational complexity of the algorithm. Finally, the original cost function is equivalently converted into a concave–convex optimization problem through logarithm, and the block coordinate descent (BCD) method is used to solve the problem under the majorization–minimization (MM) framework. In order to verify the effectiveness of this algorithm, the Laplace distribution and the Student’s T distribution are used as the prior distribution of the parameters to achieve two specific implementations of this algorithm, and simulation and experiments are carried out. Compared with the sparse Bayesian learning (SBL) algorithm, the Laplace prior-based Bayesian compress sensing (LPBCS) algorithm, the total variation (TV) algorithm, and the Landweber algorithm, the presented EBSL algorithm with the Laplace prior distribution has better image quality and fairly good real-time performance.

A low light natural image statistical model for joint contrast enhancement and denoising

We study the problem of joint low light image contrast enhancement and denoising using a statistical approach. The low light natural image in the band pass domain is modeled by statistically relating a Gaussian scale mixture model for the pristine image, to the low light image, through a detail loss coefficient and Gaussian noise. The detail loss coefficient is statistically described using a posterior distribution with respect to its estimate based on a prior contrast enhancement algorithm. We then design our low light enhancement and denoising (LLEAD) method by computing the minimum mean squared error estimate of the pristine image band pass coefficients. We create the Indian Institute of Science low light image dataset of well-lit and low light image pairs to learn the model parameters and evaluate our enhancement method. We show through extensive experiments on multiple datasets that our method helps better enhance the contrast while simultaneously controlling the noise when compared to other state of the art joint contrast enhancement and denoising methods.

MRI reconstruction based on Bayesian group sparse representation

Compressed sensing (CS) theory speeds up the magnetic resonance imaging (MRI) by undersampling the k-space data. Utilizing a proper sparse representation for image and incorporating prior information are vital to yield a high-quality reconstruction in CS-MRI. In this work, the multivariate Gaussian scale mixture (GSM) model is developed to precisely characterize to the statistical properties of sparse coefficients of group formed by similar patches, and a Bayesian group sparse representation (BGSR) is derived from maximum a posterior (MAP) estimation. The efficient multiclass orthogonal dictionaries learning is further integrated in BGSR driven CS-MRI reconstruction to enable high sparsity as well as performance enhancement. The solution is obtained by a use of alternating direction method of multipliers (ADMM) iteration, and a closed-form solution for each subproblem is separately deduced. The experimental results show that the proposed method provides a superior visual quality and performance indexes over the regularization based CS-MRI methods. Hence, the proposed method can be utilized to further accelerate MRI and produce the highly accurate reconstruction for subsequent image processing as well as clinical diagnosis. Furthermore, this work can be extended to other imaging applications and provide some references for Bayesian probabilistic model based CS-MRI reconstruction.

Multivariate Statistical Modeling of Retinal Optical Coherence Tomography.

In this paper a new statistical multivariate model for retinal Optical Coherence Tomography (OCT) B-scans is proposed. Due to the layered structure of OCT images, there is a horizontal dependency between adjacent pixels at specific distances, which led us to propose a more accurate multivariate statistical model to be employed in OCT processing applications such as denoising. Due to the asymmetric form of the probability density function (pdf) in each retinal layer, a generalized version of multivariate Gaussian Scale Mixture (GSM) model, which we refer to as GM-GSM model, is proposed for each retinal layer. In this model, the pixel intensities in each retinal layer are modeled with an asymmetric Bessel K Form (BKF) distribution as a specific form of the GM-GSM model. Then, by combining some layers together, a mixture of GM-GSM model with eight components is proposed. The proposed model is then easily converted to a multivariate Gaussian Mixture model (GMM) to be employed in the spatially constrained GMM denoising algorithm. The Q-Q plot is utilized to evaluate goodness of fit of each component of the final mixture model. The improvement in the noise reduction results based on the GM-GSM model, indicates that the proposed statistical model describes the OCT data more accurately than other competing methods that do not consider spatial dependencies between neighboring pixels.

Accelerating Convolutional Neural Network via Structured Gaussian Scale Mixture Models: A Joint Grouping and Pruning Approach

Despite the great success of deep convolutional neural networks (DCNNs), their heavy computational complexity remains a key obstacle to the wide use in practical applications. To meet this challenge, DCNN pruning has been recently developed as a technique of compressing DCNNs to facilitate their applications in the real world. In this paper, we propose a hybrid network compression technique for exploiting the prior knowledge of network parameters by Gaussian scale mixture (GSM) models. Specifically, the collection of network parameters are characterized by GSM models and network pruning is formulated as a maximum a posteriori (MAP) estimation problem with a sparsity prior. The key novel insight brought by this work is that groups of parameters associated with the same channel are similar, which is analogous to the grouping of similar patches in natural images. Such observation inspires us to leverage powerful structured sparsity prior from image restoration to network compression - i.e., to develop a flexible filter- grouping strategy that not only promotes structured sparsity but also can be seamlessly integrated with the existing network pruning framework. Extensive experimental results on several popular DCNN models including VGGNet, ResNet and DenseNet have shown that the proposed GSM-based joint grouping and pruning method convincingly outperforms other competing approaches (including both pruning and non-pruning based methods).

Spatial-Temporal Gaussian Scale Mixture Modeling for Foreground Estimation

Subtracting the backgrounds from the video frames is an important step for many video analysis applications. Assuming that the backgrounds are low-rank and the foregrounds are sparse, the robust principle component analysis (RPCA)-based methods have shown promising results. However, the RPCA-based methods suffered from the scale issue, i.e., the ℓ1-sparsity regularizer fails to model the varying sparsity of the moving objects. While several efforts have been made to address this issue with advanced sparse models, previous methods cannot fully exploit the spatial-temporal correlations among the foregrounds. In this paper, we proposed a novel spatial-temporal Gaussian scale mixture (STGSM) model for foreground estimation. In the proposed STGSM model, a temporal consistent constraint is imposed over the estimated foregrounds through nonzero-means Gaussian models. Specifically, the estimates of the foregrounds obtained in the previous frame are used as the prior for these of the current frame, and nonzero means Gaussian scale mixture models (GSM) are developed. To better characterize the temporal correlations, the optical flow has been used to model the correspondences between foreground pixels in adjacent frames. The spatial correlations have also been exploited by considering that local correlated pixels should be characterized by the same STGSM model, leading to further performance improvements. Experimental results on real video datasets show that the proposed method performs comparably or even better than current state-of-the-art background subtraction methods.

A Full-Reference Image Quality Assessment for Multiply Distorted Image based on Visual Mutual Information

Abstract A full-reference image quality assessment (FR-IQA) method for multi-distortion based on visual mutual information (MD-IQA) is proposed to solve the problem that the existing FR-IQA methods are mostly applicable to single-distorted images, but the assessment result for multiply distorted images is not ideal. First, the reference image and the distorted image are preprocessed by steerable pyramid decomposition and contrast sensitivity function (CSF). Next, a Gaussian scale mixture (GSM) model and an image distorted model are respectively constructed for the reference images and the distorted images. Then, visual distorted models are constructed both for the reference images and the distorted images. Finally, the mutual information between the processed reference image and the distorted image is calculated to obtain the full-reference quality assessment index for multiply distorted images. The experimental results show that the proposed method has higher accuracy and better performance for multiply distorted images.

Toward Flotation Process Operation-State Identification via Statistical Modeling of Biologically Inspired Gabor Filtering Responses

This paper presents a froth image statistical modeling-based online flotation process operation-state identification method by introducing a biologically inspired Gabor wavelet transform in accordance with the physiological findings in the biological vision system. It derived the latent probabilistic density models of these biologically inspired Gabor filtering responses (GFRs) based on a versatile intermediate probability modeling frame, Gaussian scale mixture model. It has demonstrated that both the real and the imaginary representation of GFR obey a Laplace distribution. Accordingly, the amplitude representation of GFR obeys a Gamma distribution. Whereas the phase representation of GFR is an important yet frequently ignored aspect in Gabor-based signal analysis; it is demonstrated to be a periodic distribution and can be expressed by a von Mises-like distribution model. Successively, a local spline regression (LSR)-based classifier that the maps scattered statistical feature points of froth images directly to the operation-state labels smoothly is introduced for the operation-state recognition. Extensive confirmatory and comparative experiments on an industrial-scale bauxite flotation process demonstrate the effectiveness and superiority of the proposed method. Performance effects on different parameter settings, e.g., parameters of Gabor kernel and dimensionalities of multivariate statistical models, are further discussed.

Robust Foreground Estimation via Structured Gaussian Scale Mixture Modeling.

Recovering the background and foreground parts from video frames has important applications in video surveillance. Under the assumption that the background parts are stationary and the foreground are sparse, most of existing methods are based on the framework of robust principal component analysis (RPCA), i.e., modeling the background and foreground parts as a low-rank and sparse matrices, respectively. However, in realistic complex scenarios, the conventional norm sparse regularizer often fails to well characterize the varying sparsity of the foreground components. How to select the sparsity regularizer parameters adaptively according to the local statistics is critical to the success of the RPCA framework for background subtraction task. In this paper, we propose to model the sparse component with a Gaussian scale mixture (GSM) model. Compared with the conventional norm, the GSM-based sparse model has the advantages of jointly estimating the variances of the sparse coefficients (and hence the regularization parameters) and the unknown sparse coefficients, leading to significant estimation accuracy improvements. Moreover, considering that the foreground parts are highly structured, a structured extension of the GSM model is further developed. Specifically, the input frame is divided into many homogeneous regions using superpixel segmentation. By characterizing the set of sparse coefficients in each homogeneous region with the same GSM prior, the local dependencies among the sparse coefficients can be effectively exploited, leading to further improvements for background subtraction. Experimental results on several challenging scenarios show that the proposed method performs much better than most of existing background subtraction methods in terms of both performance and speed.

Infrared Aerial Small Target Detection with NSCT and Two-Dimensional Property Histogram

A novel algorithm is presented based on non-subsampled contourlet transform (NSCT) and two dimension property histogram in order to realize the aerial small target detection of infrared imaging under complex background. First, this method transforms the infrared image from space domain to NSCT domain. In high frequency bandpass domain, this method describes the sub-band coefficients according to Gaussian scale mixture model based on Bayesian estimation and estimates the center coefficient with the local neighbor’s in order to predict the high frequency background. On the other hand, this method predicts the low frequency background with self-adaption median filter in low frequency lowpass domain. Subsequently, the reversing NSCT is done and the complex background is estimated. By means of subtracting the estimated background image from the source image, the complex background is suppressed and the outstanding small target is acquired. Second, constructing the target’s property set according to the priori knowledge, this method defines the corresponding two-dimensional property histogram which is applied into calculating the segmenting threshold on basis of the maximum entropy method. Subsequently, the infrared image whose complex background is suppressed will be segmented into binary image by the threshold. Finally, infrared small target is detected by the pipeline filter algorithm which makes use of the relativity of the target movement between frames. The experimental results prove the presented method’s effectiveness which can detect the small target whose signal noise ratio (SNR) value is above 2 steadily.

Generalized Gaussian scale mixtures: A model for wavelet coefficients of natural images

We develop a Generalized Gaussian scale mixture (GGSM) model of the wavelet coefficients of natural and distorted images. The GGSM model, which is more general than and which subsumes the Gaussian scale mixture (GSM) model, is shown to be a better representation of the statistics of the wavelet coefficients of both natural as well as distorted images. We demonstrate the utility of the model by applying it to various image processing applications, including blind distortion identification and no reference image quality assessment (NR-IQA). Similar to the GSM model, the GGSM model is useful for motivating the use of local divisive energy normalization, especially when the wavelet coefficients are computed on distorted pictures. We show that the GGSM model can lead to improved performance in distortion-related applications, while providing a more principled approach to the statistical processing of distorted image signals. The software release of a GGSM-based NR-IQA approach called DIIVINE-GGSM is available online at http://live.ece.utexas.edu/research/quality/diivine-ggsm.zip for further experimentation.

Compressed Sensing With a Gaussian Scale Mixture Model for Limited View Photoacoustic Computed Tomography In Vivo.

Photoacoustic computed tomography using an ultrasonic array is an attractive noninvasive imaging modality for many biomedical applications. However, the potentially long data acquisition time of array-based photoacoustic computed tomography—usually due to the required time-multiplexing for multiple laser pulses—decreases its applicability for rapid disease diagnoses and the successive monitoring of physiological functions. Compressed sensing is used to improve the imaging speed of photoacoustic computed tomography by decreasing the amount of acquired data; however, the imaging quality can be limited when fewer measurements are used, as traditional compressed sensing considers only the sparsity of the signals in the imaging process. In this work, an advanced compressed sensing reconstruction framework with a Wiener linear estimation-based Gaussian scale mixture model was developed for limited view photoacoustic computed tomography. In this method, the structure dependencies of signals in the wavelet domain were incorporated into the imaging framework through the Gaussian scale mixture model, and an operator based on the Wiener linear estimation was designed to filter the reconstruction artifacts. Phantom and human forearm imaging were performed to verify the developed method. The results demonstrated that compressed sensing with a Wiener linear estimation-based Gaussian scale mixture model more effectively suppressed the reconstruction artifacts of sparse-sampling photoacoustic computed tomography and recovered photoacoustic images with a higher contrast-to-noise ratio and edge resolution than the traditional compressed sensing method. This work may promote the development of low-cost photoacoustic computed tomography techniques with rapid data acquisition and enhance the performance of photoacoustic computed tomography in various biomedical studies.

Image super-resolution via Gaussian scale patch group sparse representation

This passage puts forward a Gaussian scale patch group sparse representation method, to solve the shortage problem of traditional image super-resolution restoration schemes. Our image reconstruction method is focused on the optimisation of sparse representation method model, which brings the method and performance improvement to image sparse reconstruction. The overall framework of our approach is as follows. First of all, we utilised the nonlocal similar patches to extract the patch groups, and then using the simultaneous sparse coding to develop a nonlocal extension of Gaussian scale mixture model. In the end, we integrate the patch group model and Gaussian scale sparsity model into encoding framework. The experimental simulation results show that the proposed framework method can both maintain the clarity of the edge and also inhibit the bad artefacts. Our method often provides a higher subjective/objective quality of reconstructed images than other competing methods.