Faculty Bibliography

In this paper, we present a new image denoising algorithm. We assume a mixture of bivariate circular symmetric Laplacian probability density functions (pdfs) where for each wavelet coefficients may have different local parameter. This pdf characterizes simultaneously 1) the heavy-tailed nature, 2) the interscale dependencies of the wavelet coefficients and also 3) the empirically observed correlation between the coefficient amplitudes. We employ this local bivariate mixture model to derive a Bayesian image denoising technique. This proposed pdf, potentially can fits better the statistical properties of the wavelet coefficients than several other existing models. Our simulation results reveal that the proposed denoising method is among the best reported in the literature. This is justified since the accuracy of the employed distribution for noise-free data determines the denoising performance.

In this paper, we are interested in modeling groups of wavelet coefficients using a zero-mean, elliptically contoured multivariate Laplace probability distribution function (pdf). Specifically, we are interested in the problem of estimating a d-point Laplace vector, s. in additive white Gaussian noise (AWGN), n, from an observation, y = s + n. In the scalar case (d = 1), the MAP and MMSE estimators are already known; and in the vector case (d > 1), the MAP estimator can be obtained by an iterative successive substitution algorithm. For the special case where the contour of the Laplace pdf is spherical, the MMSE estimators for the vector case (d > 1) have been derived in our previous work; we have shown that the MMSE estimator can be expressed in terms of the generalized incomplete Gamma function. For the general elliptically-contoured case, the MMSE estimator can not be expressed as such. In this paper, we therefore investigate approximations to the MMSE estimator of a Laplace vector in AWGN.

In this paper, we present a new video denoising algorithm using bivariate Cauchy probability density function (pdf) with local scaling factor for distribution of wavelet coefficients in each subband. The bivariate pdf takes into account the statistical dependency among wavelet coefficients and the local scaling factor model the empirically observed correlation between the coefficient amplitudes. Using maximum a posteriori (MAP) estimator and minimum mean squared estimator (MMSE), we describe two methods for video denoising which rely on the bivariate Cauchy random variables with high local correlation. Because separate 3-D transforms, such as ordinary 3-D wavelet transforms (DWT), have artifacts that degrade their performance for denoising, we implement our algorithms in 3-D complex wavelet transform (DCWT) domain. In addition, we use our denoising algorithm in 2-D DCWT domain, where the 2-D transform is applied to each frame individually. The simulation results show that our denoising algorithms achieve better performance than several published methods both visually and in terms of peak signal-to-noise ratio (PSNR).

Although wavelet-based image denoising is a powerful tool for image processing applications, relatively few publications have addressed so far wavelet-based video denoising. The main reason is that the standard 3-D data transforms do not provide useful representations with good energy compaction property, for most video data. For example, the multi-dimensional standard separable discrete wavelet transform (M-D DWT) mixes orientations and motions in its Subbands, and produces the checkerboard artifacts. So, instead of M-D DWT, usually oriented transforms suchas multi-dimensional complex wavelet transform (M-D DCWT) are proposed for video processing. In this paper we use a Laplace distribution with local variance to model the statistical properties of noise-free wavelet coefficients. This distribution is able to simultaneously model the heavy-tailed and intrascale dependency properties of wavelets. Using this model, simple shrinkage functions are obtained employing maximum a posteriori (MAP) and minimum mean squared error (MMSE) estimators. These shrinkage functions are proposed for video denoising in DCWT domain. The Simulation results shows that this simple denoising method has impressive performance visually and quantitatively.

In this paper, we design a bivariate maximum a posteriori (MAP) estimator that supposes the prior of wavelet coefficients as a mixture of bivariate Cauchy distributions. This model not only is a mixture but is also bivariate. Since mixture models are able to capture the heavy-tailed property of wavelets and bivaraite distributions can model the intrascale dependences of wavelet coefficients, this bivariate mixture probability density function (pdf) can better capture statistical properties of wavelet coefficients. The simulation results show that our proposed technique achieves better performance than other methods employing non mixture pdfs such as bivariate Cauchy pdf and circular symmetric Laplacian pdf visually and in terms of peak signal-to-noise ratio (PSNR). We also compare our algorithm with several recently published denoising methods and see that it is among the best reported in the literature.

Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for time-scale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechies-type filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Grobner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.

This paper describes a new set of dyadic wavelet frames with two generators. The construction is simple, yet the wavelets cover the time-frequency plane in an arrangement that provides a higher sampling in both time and frequency. Specifically, the spectrum of the first wavelet is concentrated halfway between the spectrum of the second wavelet and the spectrum of its dilated version. In addition, the second wavelet is translated by half integers rather than whole integers in the frame construction. This arrangement leads to an expansive wavelet transform that is approximately shift invariant and has intermediate scales. The wavelet frames presented in this paper are compactly supported and have vanishing moments.

BACKGROUND: Power in the gamma band EEG increases during saccades in normal subjects. OBJECTIVE: To develop a potential method to quantify signs of cortical responsiveness in persistent vegetative state (PVS) we quantified gamma range EEG in association with conjugate slow ballistic eye movements (SBEM). METHODS: The EEG and the simultaneous electro-oculogram were recorded in 14 (8F/6M) PVS patients. Clinical scoring was based on the Glasgow Coma Scale (GCS) and Coma Rating Scale (CRS). The Wavelet Transform, followed by Hilbert transform was applied to the EEG and gamma power distribution was quantified relative to the timing of an eye movement. We correlated the clinical and the neurophysiological measures. RESULTS: Gamma activity was present in all PVS patients. Its power was modulated in association with eye movements only in less severely affected patients, with minimum power prior to, and maximum power during the eye movement. In severely affected patients there was no evidence of a temporal relationship between gamma power and the phase of the eye movement. CONCLUSIONS: Detecting changes in the time course of gamma power in relation to conjugate ballistic eye movements provides a quantitative neurophysiological method for prospective longitudinal studies to explore if the preservation of this CNS function relates to the potential for recovery in PVS patients.