Ramesh A. Gopinath

Theory of regular M-band wavelet bases

Orthonormal M-band wavelet bases have been constructed and applied by several authors. This paper makes three main contributions. First, it generalizes the minimal length K-regular 2-band wavelets of Daubechies (1988) to the M-band case by deriving explicit formulas for K-regular M-band scaling filters. Several equivalent characterizations of K-regularity are given and their significance explained. Second, two approaches to the construction of the (M-1) wavelet filters and associated wavelet bases are described; one relies on a state-space characterization with a novel technique to obtain the unitary wavelet filters; the other uses a factorization approach. Third, this paper gives a set of necessary and sufficient condition on the M-band scaling filter for it to generate an orthonormal wavelet basis. The conditions are very similar to those obtained by Cohen (1990) and Lawton (1990) for 2-band wavelets. >

Maximum likelihood modeling with Gaussian distributions for classification

Maximum likelihood (ML) modeling of multiclass data for classification often suffers from the following problems: (a) data insufficiency implying overtrained or unreliable models, (b) large storage requirement, (c) large computational requirement and/or (d) the ML is not discriminating between classes. Sharing parameters across classes (or constraining the parameters) clearly tends to alleviate the first three problems. We show that in some cases it can also lead to better discrimination (as evidenced by reduced misclassification error). The parameters considered are the means and variances of the Gaussians and linear transformations of the feature space (or equivalently the Gaussian means). Some constraints on the parameters are shown to lead to linear discrimination analysis (a well-known result) while others are shown to lead to optimal feature spaces (a relatively new result). Applications of some of these ideas to the speech recognition problem are also given.

Wavelet based speckle reduction with application to SAR based ATD/R

The paper introduces a novel speckle reduction method based on thresholding the wavelet coefficients of the logarithmically transformed image. The method is computational efficient and can significantly reduce the speckle while preserving the resolution of the original image. Both soft and hard thresholding schemes are studied and the results are compared. When fully polarimetric SAR images are available, the authors propose several approaches to combine the data from different polarizations to achieve even better performance. Wavelet processed imagery is shown to provide better detection performance for the synthetic-aperture radar (SAR) based automatic target detection/recognition (ATD/R) problem.<<ETX>>

Short-time Gaussianization for robust speaker verification

In this paper, a novel approach for robust speaker verification, namely short-time Gaussianization, is proposed. Short-time Gaussianization is initiated by a global linear transformation of the features, followed by a short-time windowed cumulative distribution function (CDF) matching. First, the linear transformation in the feature space leads to local independence or decorrelation. Then the CDF matching is applied to segments of speech localized in time and tries to warp a given feature so that its CDF matches normal distribution. It is shown that one of the recent techniques used for speaker recognition, feature warping [l] can be formulated within the framework of Gaussianization. Compared to the baseline system with cepstral mean subtraction (CMS), around 20% relative improvement in both equal error rate(EER) and minimum detection cost function (DCF) is obtained on NIST 2001 cellular phone data evaluation.

A hybrid GMM/SVM approach to speaker identification

Proposes a classification scheme that incorporates statistical models and support vector machines. A hybrid system which appropriately combines the advantages of both the generative and discriminant model paradigms is described and experimentally evaluated on a text-independent speaker recognition task in matched and mismatched training and test conditions. Our results prove that the combination is beneficial in terms of performance and practical in terms of computation. We report relative improvements of up to 25% reduction in identification error rate compared to the baseline statistical model.

Optimal wavelet representation of signals and the wavelet sampling theorem

The wavelet representation using orthonormal wavelet bases has received widespread attention. Recently M-band orthonormal wavelet bases have been constructed and compactly supported M-band wavelets have been parameterized. This paper gives the theory and algorithms for obtaining the optimal wavelet multiresolution analysis for the representation of a given signal at a predetermined scale in a variety of error norms. Moreover, for classes of signals, this paper gives the theory and algorithms for designing the robust wavelet multiresolution analysis that minimizes the worst case approximation error among all signals in the class. All results are derived for the general M-band multiresolution analysis. An efficient numerical scheme is also described for the design of the optimal wavelet multiresolution analysis when the least-squared error criterion is used. Wavelet theory introduces the concept of scale which is analogous to the concept of frequency in Fourier analysis. This paper introduces essentially scale limited signals and shows that band limited signals are essentially scale limited, and gives the wavelet sampling theorem, which states that the scaling function expansion coefficients of a function with respect to an M-band wavelet basis, at a certain scale (and above) completely specify a band limited signal (i.e., behave like Nyquist (or higher) rate samples). >

Least squared error FIR filter design with transition bands

The authors propose the use of transition bands and transition functions in the ideal amplitude frequency response to allow the analytical design of optimal least-squared-error FIR digital filters with an explicit control of the transition band edges. Design formulas are derived for approximations to ideal frequency responses which use pth-order spline transition functions. A mixed analytical and numerical method for zero-error weight in the transition bands and passband and stopband error weighting functions with an integral squared error approximation are derived. A variable-order spline transition function is developed, and a method for choosing the optimal order to minimize the integral squared approximation error is given. >

The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform

This paper introduces an approximately shift invariant redundant dyadic wavelet transform - the phaselet transform - that includes the popular dual-tree complex wavelet transform of Kingsbury (see Phil. R. Soc. London A, Sept. 1999) as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way - and hence called phaselets - to achieve approximate shift-redundancy; the bigger the set, the better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented - building on the work of Selesnick (see IEEE Trans. Signal Processing) for the design of wavelet pairs for Kingsburys dual-tree complex wavelet transform. Construction of two-dimensional (2-D) directional bases from tensor products of one-dimensional (1-D) phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader, independent of the approximate shift invariance property that this paper argues they possess.

On upsampling, downsampling, and rational sampling rate filter banks

Solutions to the problem of design of rational sampling rate filter banks in one dimension has previosly been proposed. The ability to interchange the operations of upsampling, downsampling, and filtering plays an important role in these solutions. The present paper develops a complete theory for the analysis of arbitrary combinations of upsamplers, downsamplers and filters in multiple dimensions. Although some of the simpler results are well known, the more difficult results concerning swapping upsamplers and downsamplers and variations thereof are new. As an application of this theory, the authors obtain algebraic reductions of the general multidimensional rational sampling rate problem to a multidimensional uniform filter bank problem. However, issues concerning the design of the filters themselves are not addressed. In multiple dimensions, upsampling and downsampling operators are determined by integer matrices (as opposed to scalars in one dimension), and the noncommutativity of matrices makes the problem considerably more difficult. Cascades of upsamplers and downsamplers in one dimension are easy to analyze. The new results for the analysis of multidimensional upsampling and downsampling operators are derived using the Aryabhatta/Bezout identity over integer matrices as a fundamental tool. A number of new results in the theory of integer matrices that a relevant to the filter bank problem are also developed. Special cases of some of the results pertaining to the commutativity of upsamplers/downsamplers have been obtained in parallel by several authors. >

Subspace constrained Gaussian mixture models for speech recognition

A standard approach to automatic speech recognition uses hidden Markov models whose state dependent distributions are Gaussian mixture models. Each Gaussian can be viewed as an exponential model whose features are linear and quadratic monomials in the acoustic vector. We consider here models in which the weight vectors of these exponential models are constrained to lie in an affine subspace shared by all the Gaussians. This class of models includes Gaussian models with linear constraints placed on the precision (inverse covariance) matrices (such as diagonal covariance, maximum likelihood linear transformation, or extended maximum likelihood linear transformation), as well as the LDA/HLDA models used for feature selection which tie the part of the Gaussians in the directions not used for discrimination. In this paper, we present algorithms for training these models using a maximum likelihood criterion. We present experiments on both small vocabulary, resource constrained, grammar-based tasks, as well as large vocabulary, unconstrained resource tasks to explore the rather large parameter space of models that fit within our framework. In particular, we demonstrate significant improvements can be obtained in both word error rate and computational complexity.