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A complete framework for the linear time-invariant (LTI) filtering theory of bivariate signals is proposed based on tailored quaternion Fourier transform. This framework features a direct description of LTI filters in terms of their eigenproperties enabling compact calculus and physically interpretable filtering relations in the frequency domain. The design of filters exhibiting fundamental properties of polarization optics (birefringence and diattenuation) is straightforward. It yields an efficient spectral synthesis method and new insights on Wiener filtering for bivariate signals with prescribed frequency-dependent polarization properties. This generic framework facilitates original descriptions of bivariate signals in two components with specific geometric or statistical properties. Numerical experiments support our theoretical analysis and illustrate the relevance of the approach on synthetic data. [ABSTRACT FROM AUTHOR]

A novel approach toward the spectral analysis of stationary random bivariate signals is proposed. Unlike existing approaches, the proposed framework exhibits a natural link between well-defined statistical objects and physical parameters for bivariate signals. Using the quaternion Fourier transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as complex-valued sequences. This makes possible the definition of a scalar quaternion-valued spectral density and the corresponding autocovariance for bivariate signals. This spectral density can be meaningfully interpreted in terms of frequency-dependent polarization attributes. A natural decomposition of the spectral density of any random bivariate signal in terms of unpolarized and polarized components is introduced. Nonparametric spectral density estimation is investigated, and we introduce the polarization periodogram of a random bivariate signal. Numerical experiments support our theoretical analysis, illustrating the relevance of the approach on synthetic data. [ABSTRACT FROM PUBLISHER]

Sparse representations have proven their efficiency in solving a wide class of inverse problems encountered in signal and image processing. Conversely, enforcing the information to be spread uniformly over representation coefficients exhibits relevant properties in various applications such as robust encoding in digital communications. Antisparse regularization can be naturally expressed through an \ell \infty -norm penalty. This paper derives a probabilistic formulation of such representations. A new probability distribution, referred to as the democratic prior, is first introduced. Its main properties as well as three random variate generators for this distribution are derived. Then this probability distribution is used as a prior to promote antisparsity in a Gaussian linear model, yielding a fully Bayesian formulation of antisparse coding. Two Markov chain Monte Carlo algorithms are proposed to generate samples according to the posterior distribution. The first one is a standard Gibbs sampler. The second one uses Metropolis–Hastings moves that exploit the proximity mapping of the log-posterior distribution. These samples are used to approximate maximum a posteriori and minimum mean square error estimators of both parameters and hyperparameters. Simulations on synthetic data illustrate the performances of the two proposed samplers, for both complete and over-complete dictionaries. All results are compared to the recent deterministic variational FITRA algorithm. [ABSTRACT FROM PUBLISHER]

Multifractal random Walks are defined as integrals of infinitely divisible stationary multifractal cascades with respect to fractional Brownian motion. Their key properties are studied, such as finiteness of moments and scaling, with respect to the chosen values of the self-similarity and infinite divisibility parameters. The range of these parameters is larger than that considered previously in the literature, and the cases of both exact and nonexact scale invariance are considered. Special attention is paid to various types of definitions of multifractal random walks. The resulting random walks are of interest in modeling multifractal processes whose marginals exhibit stationarity and symmetry. [ABSTRACT FROM AUTHOR]

We propose to model the statistics of natural images, thanks to the large class of stochastic processes called Infinitely Divisible Cascades (IDCs). IDCs were first introduced in one dimension to provide multifractal time series to model the so-called intermittency phenomenon in hydrodynamical turbulence. We have extended the definition of scalar IDCs from one to N dimensions and commented on the relevance of such a model in fully developed turbulence in [1]. In this paper, we focus on the particular 2D case. IDCs appear as good candidates to model the statistics of natural images. They share most of their usual properties and appear to be consistent with several independent theoretical and experimental approaches of the literature. We point out the interest of IDCs for applications to procedural texture synthesis. [ABSTRACT FROM AUTHOR]

Multiplicative processes, multifractals, and more recently also infinitely divisible cascades have seen increased popularity in a host of applications requiring versatile multiscale models, ranging from hydrodynamic turbulence to computer network traffic, from image processing to economics. The methodologies prevalent as of today rely to a large extent on iterative schemes used to produce infinite detail and repetitive structure across scales. While appealing, due to their simplicity, these constructions have limited applicability as they lead by default to power law progression of moments through scales, to non- stationary increments and often to inherent log-periodic scaling which favors an exponential set of scales. This paper studies and develops a wide class of infinitely divisible cascades (IDC), thereby establishing the first reported cases of controllable scaling of moments in non-power-law font Embedded in the framework of IDC, these processes exhibit stationary increments and scaling over a continuous range of scales. Criteria for convergence, further statistical properties, as well as MATLAB routines are provided. [ABSTRACT FROM AUTHOR]