Faculty Bibliography

In this correspondence, we introduce a dual-tree rational-dilation complex wavelet transform for oscillatory signal processing. Like the short-time Fourier transform and the dyadic dual-tree complex wavelet transform, the introduced transform employs quadrature pairs of time-frequency atoms which allow to work with the analytic signal. The introduced wavelet transform is a constant-transform, a property lacked by the short-time Fourier transform, which in turn makes the introduced transform more suitable for models that depend on scale. Also, the frequency resolution can be as high as desired, a property lacked by the dyadic dual-tree complex wavelet transform, which makes the introduced transform more suitable for processing oscillatory signals like speech, audio and various biomedical signals.

This paper describes a discrete-time wavelet transform for which the Q-factor is easily specified. Hence, the transform can be tuned according to the oscillatory behavior of the signal to which it is applied. The transform is based on a real-valued scaling factor (dilation-factor) and is implemented using a perfect reconstruction over-sampled filter bank with real-valued sampling factors. Two forms of the transform are presented. The first form is defined for discrete-time signals defined on all of. The second form is defined for discrete-time signals of finite-length and can be implemented efficiently with FFTs. The transform is parameterized by its Q-factor and its oversampling rate (redundancy), with modest oversampling rates (e.g., three to four times overcomplete) being sufficient for the analysis/synthesis functions to be well localized.

This paper describes the construction of chirp-like constant-modulus transmit waveforms designed so as to possess multiple notches in their frequency spectra at user-specified frequencies. We propose an iterative projection algorithm with low computational complexity. In wide-band radar systems, such frequency-notched transmit waveforms are needed so as to avoid transmitting into frequency bands that are used by other systems such as for communication, navigation, etc.

Due to the prevalence of edges in image content, various directional transforms have been proposed for the efficient representation of images. Such transforms are useful for coding, denoising, and image restoration using sparse signal representation techniques. This paper describes a new non-separable 2D DCT-like orthonormal block transform that is optimized for a specified orientation angle. The approach taken in this paper is to extend to two-dimensions one approach (of several) for constructing the standard 1D DCT. The proposed transform is obtained as the eigenvectors of particular matrices, as is the standard 1D DCT.

The tunable Q-factor wavelet transform (TQWT) is a fully-discrete wavelet transform for which the Q-factor, Q, of the underlying wavelet and the asymptotic redundancy (over-sampling rate), r, of the transform are easily and independently specified. In particular, the specified parameters Q and r can be real-valued. Therefore, by tuning Q, the oscillatory behavior of the wavelet can be chosen to match the oscillatory behavior of the signal of interest, so as to enhance the sparsity of a sparse signal representation. The TQWT is well suited to fast algorithms for sparsity-based inverse problems because it is a Parseval frame, easily invertible, and can be efficiently implemented using radix-2 FFTs. The TQWT can also be used as an easily-invertible discrete approximation of the continuous wavelet transform.

We study an alternative to the Bessel K form (BKF) distribution that results when instead of gamma, the scale random variable in the Gaussian scale mixture (GSM) parameterization follows a generalized exponential distribution. The new distribution is expressed as a sum of simpler Laplace densities. Some properties of this distribution are provided. We investigate parameter estimation under the additive Gaussian noise model using simple cascades of estimators. Exact and approximate Bayesian estimators are derived for the generalized K form (GKF) in the spherically contoured case that make use of the generalized incomplete Gamma function.

We examine one way to extend recently proposed wavelet-based maximum a posteriori estimation rules for image denoising to video. The proposed approach takes into account both spatial and temporal dependencies between wavelet coefficients, and is general enough to incorporate different spherically contoured prior distributions on noiseless coefficients, as well as different spatiotemporal coefficient neighborhoods. Presented extensions of the algorithm have reasonable complexity and are suited to vectorized, convolution-based implementations. (C) 2010 SPIE and IS&T. [DOI: 10.1117/1.3514739]