Image modeling and denoising with orientation-adapted Gaussian scale mixtures.

We present a probabilistic model for natural images that is based on mixtures of Gaussian scale mixtures and a simple multiscale representation. We show that it is able to generate images with interesting higher-order correlations when trained on natural images or samples from an occlusion-based model. More importantly, our multiscale model allows for a principled evaluation. While it is easy to generate visually appealing images, we demonstrate that our model also yields the best performance reported to date when evaluated with respect to the cross-entropy rate, a measure tightly linked to the average log-likelihood. The ability to quantitatively evaluate our model differentiates it from other multiscale models, for which evaluation of these kinds of measures is usually intractable.

This paper considers a hierarchical Bayesian analysis of regression models using a class of Gaussian scale mixtures. This class provides a robust alternative to the common use of the Gaussian distribution as a prior distribution in particular for estimating the regression function subject to uncertainty about the constraint. For this purpose, we use a family of rectangular screened multivariate scale mixtures of Gaussian distribution as a prior for the regression function, which is flexible enough to reflect the degrees of uncertainty about the functional constraint. Specifically, we propose a hierarchical Bayesian regression model for the constrained regression function with uncertainty on the basis of three stages of a prior hierarchy with Gaussian scale mixtures, referred to as a hierarchical screened scale mixture of Gaussian regression models (HSMGRM). We describe distributional properties of HSMGRM and an efficient Markov chain Monte Carlo algorithm for posterior inference, and apply the proposed model to real applications with constrained regression models subject to uncertainty.

Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the "nonnegative garrote" thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of O(N log N) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity.

We describe an efficient generalized expectation maximization algorithm for estimating the spectral features of a noise source corrupting an observed image. We use a statistical model for images decomposed in an overcomplete oriented pyramid. Each neighborhood of clean pyramid coefficients is modeled as a Gaussian scale mixture, whereas the noise is assumed Gaussian. Combining this GEM technique with a previous Bayesian denoise estimator, we obtain a full blind denoising algorithm, able to deal with homogeneous, Gaussian or mesokurtotic, noise sources of arbitrary covariance. Results demonstrate the high performance of the method for a wide range of corruption sources.

Wavelet-based demosaicing techniques have the advantage of being computationally relatively fast, while having a reconstruction performance that is similar to state-of-the-art techniques. Because the demosaicing rules are linear, it is fairly simple to integrate denoising into the demosaicing. In this paper, we present a method that performs joint denoising and demosaicing, using a Gaussian Scale Mixture (GSM) prior model, thereby modeling the local edge direction as a hidden variable. The results indicate that this technique offers a better reconstruction performance (in PSNR sense and visually) than sequential demosaicing and denoising. On a recent GPU, our algorithm takes 3.5 s for reconstructing a 12 megapixel RAW digital camera image.

Compressed Sensing (CS) theory has gained so much attention recently in the areas of signal processing. The sparsity of the transform coefficients has been widely employed in the early CS recovery techniques. However, except for the sparsity, there are other priors about transform coefficients such as the tree structure and the statistical dependencies that could be employed in CS reconstruction. In this paper, we propose to introduce the Gaussian Scale Mixtures (GSM) model into the tree structure based Orthogonal Matching Pursuit (TSOMP) reconstruction algorithm. This GSM model can efficiently denote the statistical dependencies between wavelet coefficients. And these statistical dependencies will improve the accuracy of the searching of the tree structure subspace in TSOMP algorithm. When both the inter-scale dependences (such as GSM model) of the coefficients and the intra-scale dependences (such as tree structure) of the coefficients are combined into the Orthogonal Matching Pursuit reconstruction algorithm, the noise and instability in TSOMP reconstruction are well reduced. Some state-of-the-art methods are compared with the proposed method. Experimental results show that the proposed method improves reconstruction accuracy for a given number of measurements and more image details are recovered.

In this article, we examine and provide insights into the frequency domain ICA approach. We develop the concept of a dynamic random process to model the frequency domain source signals. It formalizes the concept of signals that are stationary in a frame but exhibit dynamics at the frame level. Frame dynamics is an important characteristics of these signals and this work demonstrates its significant role in the success of the ICA methods in each frequency bin. We show that the independence between the marginal distributions of the source signals in each frequency bin is related to the independence of the frame dynamics of the time domain source signals. The frame dynamics also naturally leads to the marginal distribution of the source signal in each frequency bin being modeled by a Gaussian scale mixture (GSM). Concentrating on the bin-wise ICA methods, a significant contribution of the paper is to show that signals modeled using variance dependent GSM density can be separated using ICA even though they might be dependent on each other as long as the the frame dynamics of the source signals are different almost surely.

Attack to security of watermarking is to estimate secret keys used in watermarking communications. The security level of watermarking algorithms can be evaluated by Cramer-Rao Bound (CRB) for the estimation of secret keys. As a widely used method, Add-Spread-Spectrum (Add-SS) watermarking has drawn a lot of attention. Previous work on watermarking security is mainly based on the assumption that the host is Gaussian distributed and ignores the impact of the non-Gaussian characteristics of natural images. With the incorporation of Gaussian Scale Mixture (GSM) model for host signals, this paper presents a theoretical analysis on the security of the Add-SS watermarking algorithms. By giving the CRB and Modified Cramer-Rao Bound (MCRB) for the estimation of secret carriers under Known Message Attack (KMA) and Watermarked Only Attack (WOA), this paper also reveals the factors that may influence the security of Add-SS based watermarking algorithms. Results presented in this paper are very helpful for designing the new generation of secure and robust watermarking system.

Image restoration (deconvolution) is a basic step for image processing, analysis and computer vision. We addressed blurred image deconvolution problem using Expectation maximization (EM) based approach in the wavelet domain. The sparsity property of wavelet coefficients is modelled using the class of Gaussian Scale Mixture (GSM), which represents the heavy-tailed statistical distribution. The maximum a posterior (MAP) estimate is computed using EM, where scale factors of GSM plays the role of hidden variables. The estimated hidden scaling variables are then used to restore the original image. Although similar formulations have been proposed before but the resulting optimization problems have been computationally demanding and sometimes depends heavily on the initial values of parameters. We proposed an optimized Bayesian approach in wavelet domain to restore an image degraded by linear distortion (e.g., blur) and additive Gaussian noise. Simulation results are presented to demonstrate the quality of our method, over a wide range of blur and noise level, both visually and in terms of signal to noise ratio.

We describe here two ways to improve on recent results in image restoration using Bayes least squares estimation with local Gaussian scale mixtures (BLS-GSM) in overcomplete oriented pyramids. First one consists of allowing for a spatial adaptation of the covariance matrix defining the GSM model at each pyramid subband. This can be implemented in practice by dividing the subbands into spatial blocks. The other, more powerful, method is to generalize the GSM model to include more than one covariance matrices for each subband. The advantage of the latter method is its flexibility, as it allows for mixing Gaussian densities with different covariance matrices at every spatial location in every subband. It also allows for non-local selective processing, taking advantage of the repetition in the scene of image features that are not necessarily spatially grouped. We also describe an empirical method to adapt denoising algorithms for doing image restoration, with the only constraint on the denoising method of being applicable to non-white noise sources. Here we present mature results of the spatially adaptive method applied to denoising and deblurring, plus some estimation techniques and encouraging preliminary results of the multi-GSM concept.

In this paper, we propose a new class of iteratively re-weighted least squares (IRLS) for sparse recovery problems. The proposed methods are inspired by constrained maximum-likelihood estimation under a Gaussian scale mixture (GSM) distribution assumption. In the noise-free setting, we provide sufficient conditions ensuring the convergence of the sequences generated by these algorithms to the set of fixed points of the maps that rule their dynamics and derive conditions verifiable a posteriori for the convergence to a sparse solution. We further prove that these algorithms are quadratically fast in a neighborhood of a sparse solution. We show through numerical experiments that the proposed methods outperform classical IRLS for l_p-minimization with p\in(0,1] in terms of speed and of sparsity-undersampling tradeoff and are robust even in presence of noise. The simplicity and the theoretical guarantees provided in this paper make this class of algorithms an attractive solution for sparse recovery problems

A statistical model for images decomposed in an overcomplete wavelet pyramid is described. Each neighborhood of pyramid coefficients is modeled as the product of a Gaussian vector of known covariance, and an independent hidden positive scalar random variable. We propose an efficient Bayesian estimator for the pyramid coefficients of an image degraded by linear distortion (e.g., blur) and additive Gaussian noise. We demonstrate the quality of our results in simulations over a wide range of blur and noise levels.

A Bayesian method for texture model choice from blurred and noisy (i.e., indirect) observations is presented. The textures are modeled by stationary Random Fields, with various distribution laws, either Gaussian or Scale Mixtures of Gaussians. The power spectral densities of the fields are modeled by parametric functions and the aim is to select the most appropriate model among a set of candidates. This is achieved by computing the a posteriori model probabilities through parameter marginalization. The marginalization is done by sampling and harmonic mean approach, considering separately each model, in a within-model sampling strategy. The highly nonlinear dependency with respect to the parameters imposes the use of the Metropolis-Hastings sampler. Moreover, to achieve efficient sampling, the paper proposes a new fast algorithm based on the Fisher information matrix, the Fisher Metropolis-Hastings.

In this paper we present a new approach towards the extraction of affine image regions based on detecting shape-stable boundaries from a multi-scale image representation. We construct an affine morphological scale space (AMSS) representation [1], which performs anisotropic diffusion while preserving boundaries and being invariant to affine transformations. We extract the transition boundaries of the diffusivity velocity map and track their evolution at each level of the scale-space. We then determine the stability of the boundary shape through a minimization process over different scales. Unlike most state of the art detectors which use the Gaussian scale space for multi-scale image representation, our approach is intrinsically affine invariant. We evaluate our detector by measuring repeatability of regions in transformed images of the same scene and comparing it to the state-of-the-art region detectors [2].

It is proposed to apply the statistical analysis of the increments of fluctuating particle fluxes to analyzing the probability characteristics of turbulent transport processes in plasma. Such an approach makes it possible to analyze the dynamic probability characteristics of the process under study. It is shown that, in the plasmas of the L-2M stellarator and the TAU-1 linear device, the increments of local fluctuating particle fluxes are of stochastic character and that their distributions are scale mixtures of Gaussians. In particular, for TAU-1, the increments have the Laplacian distribution (which is a scale mixture of Gaussians with an exponential mixing distribution). This implies that the rate of flux variations is a diffusion process with random time. It is shown that the characteristic growth (damping) time of fluctuations is one order of magnitude shorter than their characteristic correlation time. Physical mechanisms that may be responsible for the random character of the growth (damping) of fluctuations are discussed.