Most simple nonlinear thresholding rules for wavelet-based denoising assume that the wavelet coefficients are independent. However, wavelet coefficients of natural images have significant dependencies. In this paper, we will only consider the dependencies between the coefficients and their parents in detail. For this purpose, new non-Gaussian bivariate distributions are proposed, and corresponding nonlinear threshold functions (shrinkage functions) are derived from the models using Bayesian estimation theory. The new shrinkage functions do not assume the independence of wavelet coefficients. We will show three image denoising examples in order to show the performance of these new bivariate shrinkage rules. In the second example, a simple subband-dependent data-driven image denoising system is described and compared with effective data-driven techniques in the literature, namely VisuShrink, SureShrink, BayesShrink, and hidden Markov models. In the third example, the same idea is applied to the dual-tree complex wavelet coefficients.

We consider the problem of efficiently encoding a signal by transforming it to a new representation whose components are statistically independent. A widely studied linear solution, known as independent component analysis (ICA), exists for the case when the signal is generated as a linear transformation of independent nongaussian sources. Here, we examine a complementary case, in which the source is nongaussian and elliptically symmetric. In this case, no invertible linear transform suffices to decompose the signal into independent components, but we show that a simple nonlinear transformation, which we call radial gaussianization (RG), is able to remove all dependencies. We then examine this methodology in the context of natural image statistics. We first show that distributions of spatially proximal bandpass filter responses are better described as elliptical than as linearly transformed independent sources. Consistent with this, we demonstrate that the reduction in dependency achieved by applying RG to either nearby pairs or blocks of bandpass filter responses is significantly greater than that achieved by ICA. Finally, we show that the RG transformation may be closely approximated by divisive normalization, which has been used to model the nonlinear response properties of visual neurons.

This article presents an overview of the theoretical and practical issues associated with the development, analysis, and application of detection algorithms to exploit hyperspectral imaging data. We focus on techniques that exploit spectral information exclusively to make decisions regarding the type of each pixel—target or nontarget—on a pixel-by-pixel basis in an image. First we describe the fundamental structure of the hyperspectral data and explain how these data influence the signal models used for the development and theoretical analysis of detection algorithms. Next we discuss the approach used to derive detection algorithms, the performance metrics necessary for the evaluation of these algorithms, and a taxonomy that presents the various algorithms in a systematic manner. We derive the basic algorithms in each family, explain how they work, and provide results for their theoretical performance. We conclude with empirical results that use hyperspectral imaging data from the HYDICE and Hyperion sensors to illustrate the operation and performance of various detectors.

Designing and operating a classification system becomes drastically more difficult as the data dimensionality increases. A feature extraction (FE) step is often used to reduce the data dimensionality to mitigate this complexity. Thus FE may be viewed as a form of data compression whos objective is to minimize the consequences reducing the dimensionality has on class separability. This differs from the normal objective of data compression which is to minimize distortion, typically measured in the mean squared sense. It is often unclear whether the resulting features from a FE method provide an optimum set for classification. Further, extracting discrimination features from finite data sets increases in difficulty as the dimensionality of the data increases. The need for features to reduce complexity, combined with the difficulties of extracting features, justifies the need for studying ways of ranking feature sets for classification, i.e. feature set evaluation (FSE) techniques. This ...

In this paper we discuss the use of Boundary Methods (BM) for distribution analysis. We view these methods as tools which can be used to extract useful information from sample distributions. We believe that the information thus extracted has utility for a number of applications, but in particular we discuss the use of BM as a new mechanism to Feature Set Evaluation (FSE) and as an aid to constructing robust and efficient Neural Networks (NN) to solve classification problems. In the first case, BM can stablish which feature set is most appropriate for classification. We demonstrate experimentally that the derived ranking is consistent with alternative ranking techniques based on Bayes error (ffl), showing the theoretical relationship between Overlap Sum (OS), the BM measure of class separability, and ffl. Next, we investigate complexity issues associated with using BMs for FSE and compare with other techniques used for FSE. Finally, BM are used as Sample Selecction (SS) mechanism to tra...

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The purpose of this paper is to study the estimation problem of a multivariate elliptically symmetric random variable corrupted by a multivariate elliptically symmetric noise. 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 multivariate context, (ii) the corrupting noise is not limited to be Gaussian. This paper also extends our previous work that dealt with the minimum mean square error (MMSE) approach [4]. The MMSE is briefly recalled and the MAP is derived. Then the practical use of the MAP in a general setting is discussed and compared to that of the MMSE and of the Wiener estimator. Several examples illustrate the behaviors of these estimators and exhibit their performances. 1.

This paper develops and compares the MAP and MMSE estimators for spherically-contoured multivariate Laplace random vectors in additive white Gaussian noise. The MMSE estimator is expressed in closed-form using the generalized incomplete gamma function. We also find a computationally efficient yet accurate approximation for the MMSE estimator. In addition, this paper develops an expression for the mean square error MSE for any estimator of spherically-contoured multivariate Laplace random vectors in AWGN, the development of which again depends on the generalized incomplete gamma function. The estimators are motivated and tested on the problem of wavelet-based image denoising.