Ercan E. Kuruoglu

Image denoising using bivariate α-stable distributions in the complex wavelet domain

Recently, the dual-tree complex wavelet transform has been proposed as an analysis tool featuring near shift-invariance and improved directional selectivity compared to the standard wavelet transform. Within this framework, we describe a novel technique for removing noise from digital images. We design a bivariate maximum a posteriori estimator, which relies on the family of isotropic α-stable distributions. Using this relatively new statistical model we are able to better capture the heavy-tailed nature of the data as well as the interscale dependencies of wavelet coefficients. We test our algorithm for the Cauchy case, in comparison with several recently published methods. The simulation results show that our proposed technique achieves state-of-the-art performance in terms of root mean squared (RMS) error.

SAR image filtering based on the heavy-tailed Rayleigh model

Synthetic aperture radar (SAR) images are inherently affected by a signal dependent noise known as speckle, which is due to the radar wave coherence. In this paper, we propose a novel adaptive despeckling filter and derive a maximum a posteriori (MAP) estimator for the radar cross section (RCS). We first employ a logarithmic transformation to change the multiplicative speckle into additive noise. We model the RCS using the recently introduced heavy-tailed Rayleigh density function, which was derived based on the assumption that the real and imaginary parts of the received complex signal are best described using the alpha-stable family of distribution. We estimate model parameters from noisy observations by means of second-kind statistics theory, which relies on the Mellin transform. Finally, we compare the proposed algorithm with several classical speckle filters applied on actual SAR images. Experimental results show that the homomorphic MAP filter based on the heavy-tailed Rayleigh prior for the RCS is among the best for speckle removal

Bayesian Separation of Images Modeled With MRFs Using MCMC

We investigate the source separation problem of random fields within a Bayesian framework. The Bayesian formulation enables the incorporation of prior image models in the estimation of sources. Due to the intractability of the analytical solution, we resort to numerical methods for the joint maximization of the a posteriori distribution of the unknown variables and parameters. We construct the prior densities of pixels using Markov random fields based on a statistical model of the gradient image, and we use a fully Bayesian method with modified-Gibbs sampling. We contrast our work to approximate Bayesian solutions such as iterated conditional modes (ICM) and to non-Bayesian solutions of ICA variety. The performance of the method is tested on synthetic mixtures of texture images and astrophysical images under various noise scenarios. The proposed method is shown to outperform significantly both its approximate Bayesian and non-Bayesian competitors.

Source separation in astrophysical maps using independent factor analysis

A microwave sky map results from a combination of signals from various astrophysical sources, such as cosmic microwave background radiation, synchrotron radiation and galactic dust radiation. To derive information about these sources, one needs to separate them from the measured maps on different frequency channels. Our insufficient knowledge of the weights to be given to the individual signals at different frequencies makes this a difficult task. Recent work on the problem led to only limited success due to ignoring the noise and to the lack of a suitable statistical model for the sources. In this paper, we derive the statistical distribution of some source realizations, and check the appropriateness of a Gaussian mixture model for them. A source separation technique, namely, independent factor analysis, has been suggested recently in the literature for Gaussian mixture sources in the presence of noise. This technique employs a three layered neural network architecture which allows a simple, hierarchical treatment of the problem. We modify the algorithm proposed in the literature to accommodate for space-varying noise and test its performance on simulated astrophysical maps. We also compare the performances of an expectation-maximization and a simulated annealing learning algorithm in estimating the mixture matrix and the source model parameters. The problem with expectation-maximization is that it does not ensure global optimization, and thus the choice of the starting point is a critical task. Indeed, we did not succeed to reach good solutions for random initializations of the algorithm. Conversely, our experiments with simulated annealing yielded initialization-independent results. The mixing matrix and the means and coefficients in the source model were estimated with a good accuracy while some of the variances of the components in the mixture model were not estimated satisfactorily.

Finite mixture of α-stable distributions

Over the last decades, the @a-stable distribution has proved to be a very efficient model for impulsive data. In this paper, we propose an extension of stable distributions, namely mixture of @a-stable distributions to model multimodal, skewed and impulsive data. A fully Bayesian framework is presented for the estimation of the stable density parameters and the mixture parameters. As opposed to most previous work on mixture models, the model order is assumed unknown and is estimated using reversible jump Markov chain Monte Carlo. It is important to note that the Gaussian mixture model is a special case of the presented model which provides additional flexibility to model skewed and impulsive phenomena. The algorithm is tested using synthetic and real data, accurately estimating @a-stable parameters, mixture coefficients and the number of components in the mixture.

Skewed α-stable distributions for modelling textures

In this letter, we introduce a novel family of texture models which provide alternatives to texture models which are based on Gaussian distributions. In particular, we introduce linear textures generated with a member of the α-stable distribution family, which is a generalisation of the Gaussian distribution. The new family of texture models is capable of representing both impulsive and unsymmetric (skewed) image data which cannot be accommodated by the Gaussian model. We present new techniques for texture model estimation and we demonstrate the success of the techniques on synthetic data.

Modelling with mixture of symmetric stable distributions using Gibbs sampling

The stable distribution is a very useful tool to model impulsive data. In this work, a fully Bayesian mixture of symmetric stable distribution model is presented. Despite the non-existence of closed form for @a-stable distributions, the use of the product property make it possible to infer on parameters using a straightforward Gibbs sampling. This model is compared to the mixture of Gaussians model. Our proposed methodology is proved to be more robust to outliers than the mixture of Gaussians. Therefore, it is suitable to model mixture of impulsive data. Moreover, as Gaussian is a particular case of the @a-stable distribution, the proposed model is a generalization of mixture of Gaussians. Mixture of symmetric @a-stable is intensively tested in both, simulated and real data.

Image separation using particle filters

In this work, we will analyze the problem of source separation in the case of superpositions of different source images, which need to be extracted from a set of noisy observations. This problem occurs, for example, in the field of astrophysics, where the contributions of various Galactic and extra-Galactic components need to be separated from a set of observed noisy mixtures. Most of the previous work on the problem performed blind source separation, assuming noiseless models, and in the few cases when noise is taken into account, it is assumed that it is Gaussian and space-invariant. In this paper we review the theoretical fundamentals of particle filtering, an advanced Bayesian estimation method which can deal with non-Gaussian non-linear models and additive space-varying noise, and we introduce a hierarchical model and a fusion of multiple particle filters for the solution of the image separation problem. Our simulations on realistic astrophysical data show that the particle filter approach provides significantly better results in comparison with one of the most widespread algorithms for source separation (FastICA), especially in the case of low SNR.

Source separation in noisy astrophysical images modelled by Markov random fields

Astrophysical radiation maps provide images which are superpositions of various cosmological components such as the cosmic microwave background (CMB) radiation, galactic dust, synchrotron, free-free emission and extragalactic radio sources. All these components are of great interest to cosmologists and in particular CMB, in addition to being the picture of the early universe, carries important information that would help us to choose between existing evolution theories of the universe. In this work we present a technique for the separation of these components in the presence of receiver noise. In contrast with most work in the literature, we make use of the spatial information in the images in the form of correlation between pixels which we model using Markov Random Fields. The spatial information is included in the MRF model through a Bayesian estimation framework. We provide comparisons with the results obtained by FastICA.

Modeling of non-stationary autoregressive alpha-stable processes by particle filters

In the literature, impulsive signals are mostly modeled by symmetric alpha-stable processes. To represent their temporal dependencies, usually autoregressive models with time-invariant coefficients are utilized. We propose a general sequential Bayesian modeling methodology where both unknown autoregressive coefficients and distribution parameters can be estimated successfully, even when they are time-varying. In contrast to most work in the literature on signal processing with alpha-stable distributions, our work is general and models also skewed alpha-stable processes. Successful performance of our method is demonstrated by computer simulations. We support our empirical results by providing posterior Cramer-Rao lower bounds. The proposed method is also tested on a practical application where seismic data events are modeled.