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The Gaussian Transform of Distributions: Definition, Computation and Application

The Gaussian Transform of Distributions: Definition, Computation and Application,10.1109/TSP.2006.877657,IEEE Transactions on Signal Processing,Teodor

The Gaussian Transform of Distributions: Definition, Computation and Application   (Citations: 8)
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This paper introduces the general purpose Gaussian Transform of distributions, which aims at representing a generic symmetric distribution as an infinite mixture of Gaussian distributions. We start by the mathematical formulation of the problem and continue with the investigation of the conditions of existence of such a transform. Our analysis leads to the derivation of analytical and numerical tools for the computation of the Gaussian Transform, mainly based on the Laplace and Fourier transforms, as well as of the afferent properties set (e.g. the transform of sums of independent variables). The Gaussian Transform of distributions is then analytically derived for the Gaussian and Laplacian distributions, and obtained numerically for the Generalized Gaussian and the Generalized Cauchy distribution families. In order to illustrate the usage of the proposed transform we further show how an infinite mixture of Gaussians model can be used to estimate/denoise non-Gaussian data with linear estimators based on the Wiener filter. The decomposition of the data into Gaussian components is straightforwardly computed with the Gaussian Transform, previously derived. The estimation is then based on a two-step procedure, the first step consisting in variance estimation, and the second step in data estimation through Wiener filtering. To this purpose we propose new generic variance estimators based on the Infinite Mixture of Gaussians prior. It is shown that the proposed estimators compare favorably in terms of distortion with the shrinkage denoising technique, and that the distortion lower bound under this framework is lower than the classical MMSE bound.
Journal: IEEE Transactions on Signal Processing - TSP , vol. 54, no. 8, pp. 2976-2985, 2006
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    • ...LLIPTICALLY symmetric distributions appear recurrently as a subject of interest in the signal-processing literature; recent contributions such as [1]–[4] are examples...
    • ...Reference [1] focuses on the problem of denoising of a univariate Gaussian scale mixture random variable corrupted by Gaussian noise; various statistical estimators are proposed and their performance are compared...
    • ...We extend as far as possible the results of Alecu et al. [1] and Selesnick et al. [2], [3]...
    • ...This fact is sometimes omitted in recent papers [1], [4]...
    • ... In the univariate case, this class of distributions is studied by Alecu et al. [1] and generally known as generalized -Gaussian distributions or gpG; this denomination can also be found for [14]...
    • ...We deal with the same context as in [1] but generalized to an arbitrary dimension and to the general elliptical framework (or at least to the wide-sense GSM context), and with the same context as in [3]...
    • ...1) Under condition , (19)–(21) have at least one solution, and all the solutions are necessarily in [0; 1] and in [0; 1], respectively...
    • ...1) Under condition , (19)–(21) have at least one solution, and all the solutions are necessarily in [0; 1] and in [0; 1], respectively...
    • ...As a first illustration, let us consider a vector following an exponential power distribution with density generator and generating function given in Table I. For , this class of distributions, known as generalized Gaussian (gpG), is that studied by Alecu et al. [1]; the case was previously studied in [51] considering only the MMSE estimator...
    • ...However, in the present example, the MAP can be analytically solved (see also [1]–[3]) and leads to...
    • ...(c) and (d) Illustration of function and of (arbitrarily rescaled) versus [0; 1] (c) for (solid line), 6.2 (dashed line), 6.5 dash-dotted line), and SNR dB and (d) for (solid line), (dashed line), (dash-dotted line), and SNR dB...
    • ...proposed in [1] in the univariate context , which appears to be worse than that of the Wiener filter (and obviously worse than that of the MMSE)...
    • ...We have no explanation yet on why the Wiener estimator is so accurate in this example, nor why it is so much better than the estimator of [1]...
    • ...For a finer comparison, we refer the reader to [1]...
    • ...Possible fixed point(s) of is (are) not necessarily in [0; 1] and the fixed-point algorithm does not guarantee the convergence to the fixed point(s) of . However, in this very special example, one can check that is a trivial fixed point of and that the other fixed points are the solutions of the quartic equation . The roots of the corresponding quartic polynomial can be sought using Ferrari’s method [56, Sec...
    • ...This is illustrated in Fig. 5, which depicts and , showing that the maxima (one or two) are not necessarily in [0; 1]...
    • ...Moreover, since is negative, one concentrates on a fixed point of in the interval [0; 1]. But since and are log-concave here, the fixed point approach is not guaranteed to converge to the unique solution of . The behaviors of and are also depicted in Fig. 5...
    • ...In this paper, we have revisited the denoising, or estimation problem of a random vector embedded in noise, extending the approach of [1], which deals only with scalar Gaussian scale mixtures, to the more general class of -dimensional elliptically distributed vectors...
    • ...Moreover, since and , by the continuity hypothesis on , function must have at least one zero in [0; 1]. 2) Differentiating against , one easily achieves...

    Steeve Zozoret al. Some results on the denoising problem in the elliptically distributed ...

    • ...In this study, the maximum a posteriori (MAP) approach is presented, extending recent works by Alecu et al. [1] and Selesnick [2, 3]: (i) the estimation is performed in a multivar iate context, (ii) the corrupting noise is not limited to be Ga ussian...
    • ...Investigations on elliptically symmetric distributions a ppear recurrently in the signal processing literature; very rece nt contributions such as [1, 2, 3] are examples...
    • ...The first study is that of Alecu et al. which dealt with what they call the Gaussian transform of a symmetrically distributed scalar random variable [1]...
    • ...As in [1, 2, 3], this work can find applications in image denoising, or even in radar data processing since radar clutter is often modeled b y GSM processes [8]...
    • ...As in [1, 3], we concentrate here on the maximum a posteriori estimator (MAP)...
    • ...We consider the generalization of the problem of [1, 3] of estimating a d-dimensional random vector X elliptically distributed and with covariance matrix RX, from a noisy observation...
    • ...where the noise Z is assumed independent of X and elliptically distributed with covariance matrix RZ. As in [1, 3], we assume that both covariance matrices RX and RZ and pdfs pX and pZ are known...
    • ...This problem generalizes [1] in the sense that dimension d is arbitrary and that Z can be non-Gaussian...
    • ...In such situations, the maximum a posteriori (MAP) approach is a good alternative, as shown in [1, 2, 3]. The MAP consists in seeking the vector b...
    • ...Xw 2 [0 ; 1], and that in the strict sense GSM, the MMSE estimator magnitude is in [0 ; 1] as well...
    • ...Xw 2 [0 ; 1], and that in the strict sense GSM, the MMSE estimator magnitude is in [0 ; 1] as well...
    • ...1. If both dX and dZ are continuously differentiable and monotonic (necessarily decreasing), then equation (8) has at least one solution, which, from (8), is necessarily in [0 ; 1]. Indeed, writing x = G(x) where G(x) is the right hand-side of (8), it is immediate that G(0) � 0 and G(1) � 1, hence, from the continuity assumption, there exists at least one x so that x = G(x)...
    • ...the fixed-point method xk+1 = G(xk) with x0 2 [0 ; 1] always converges to a solution of (8), which is a maximum of the posterior distribution (except if it is initialized a t a minimum)...
    • ...Under the monotony assumption, one has Xmap 2 [0 ; 1]d and under the log-convexity assumption, the fixed point method converges to a local maximum...
    • ...Furthermore, in the case p = 1, one can easily show that the MAP has an explicit form [1, 2, 3]: Xmap = � 1...
    • ...where A denotes the indicator function of set A. For p = .7, the MAP was determined by a exhaustive search of the maximum of dX(x2kyk2/σ 2) dZ((1 x)2kyk2) for x 2 [0 ; 1]...
    • ...This study extends: (i) the approach of [1], that deals only with scalar GSM (and with a Gaussian corrupting noise); (ii) the study of [2, 3] where th e dimension is not restricted to 1, but where the noise is still considered Gaussian and the vectors to estimate restricted to Laplacian or radially exponential distributed...

    S. Zozoret al. ON THE MAP ESTIMATION IN THE CONTEXT OF ELLIPTICAL DISTRIBUT IONS

    • ...In this paper, we revisit the denoising problem in the context of elliptical distributions; this approach extends a work by Alecu et al. [1] in two directions: (i) we address the multivariate case and (ii) our study is not restricted to the scale mixtures of Gaussians, but extends to the whole family of elliptical distributions...
    • ...Recently, Alecu et al. have introduced the notion of “Gaussian transform” of a random variable with symmetric probability density function (pdf) [1]...
    • ...The so-called “Gaussian transform” of the pdf of X as defined in [1] is in fact nothing but the pdf of the square A2 of the mixing variable A...
    • ...Some conclusions are drawn from these examples, and comments on Alecu’s results are provided [1]...
    • ...We deal with the same context as in [1], but generalized to any dimension d and to the general elliptical framework...
    • ...where 0 < p < 2. For d = 1, this class of distributions is that studied by Alecu et al. and generally known as pgeneralized Gaussian distributions [1]...
    • ...The performances of the estimators proposed in [1] are not depicted here, but they appear worse than the Wiener filter in general (and obviously worse than the MMSE)...
    • ...In this paper we have revisited the denoising problem, extending the approach of [1], that deals only with scalar scale mixture of Gaussians, to the more general class of ddimensional elliptically distributed vectors...
    • ...We note, however, that the suboptimal estimation technique proposed in [1] also required numerical integration...
    • ...In [1], the maximum a posteriori (MAP) estimation is discussed, although as presented it is equivalent to the MMSE (i.e...

    S. Zozoret al. REVISITING THE DENOISING PROBLEM IN THE CONTEXT OF ELLIPTICAL DISTRIBU...

    • ...The model of the noise statistics is derived following the ideas of inverse Gaussian transform of the PDF of the noise variance [8], [9]...
    • ...From a mathematical point of view, (27) is derived following the ideas of inverse Gaussian transform of the PDF of the noise variance [8]...

    Xinning Weiet al. MMSE Detection Based on Noise Statistics with Random Noise Variance

    • ...The algorithm is based on the representation of non-Gaussian distributions as an Infinite Mixture of Gaussians [10] and relies on an iterative procedure consisting of alternated variance estimation and linear inversion operations...

    Thierry Punet al. Brain-computer interaction research at the computer vision and multime...

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