Faculty Bibliography

This paper uses the UEP approach for the construction of wavelet tight frames with two (anti-) symmetric wavelets, and provides some results and examples that complement recent results by Q. Jiang. A description of a family of solutions when the lowpass scaling filter is of even-length is provided. When one wavelet is symmetric and the other is antisymmetric, the wavelet filters can be obtained by a simple procedure based on matching the roots of associated polynomials. The design examples in this paper begin with the construction of a lowpass filter h(0)(n) that is designed so as to ensure that both wavelets have at least a specified number of vanishing moments. (C) 2004 Elsevier Inc. All rights reserved.

This paper introduces the double-density dual-tree discrete wavelet transform (DWT), which is a DWT that combines the double-density DWT and the dual-tree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractional-delay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant.

In this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Grobner algorithms offer an effective way to obtain solutions in some of these cases. (C) 2003 Elsevier Ltd. All rights reserved.

This paper uses probability models on expansive wavelet transform coefficients with interpolation constraints to estimate missing blocks in images. We use simple probability models on wavelet coefficients to formulate the estimation process as a linear programming problem and solve it to recover the missing pixels. Our formulation is general and can be augmented with more sophisticated probability models to obtain even better estimates on a variety of image regions. The presented approach has many parallels to recently introduced dictionary based signal representations with which it shares certain optimality properties. We provide simulation examples over edge regions using both critically-sampled and expansive (over-complete) wavelet transforms.

This paper examines the properties of a recently introduced 3-D dual-tree discrete wavelet transform (DDWT) for video coding. The 3-D DDWT is an attractive video representation because it isolates motion along different directions in separate subbands. However, it is an overcomplete transform with 8:1 redundancy. We examine the effectiveness of the iterative projection-based noise shaping scheme proposed by Kingsbury [3] on reducing the number of coefficients. We also investigate the correlation between subbands at the same spatial/temporal location, both in the significance map and in actual coefficient values.

Video processing techniques using true 3D transforms are largely unexploited, partly because of the drawbacks of traditional separable 3D transforms. In this paper we use a new type of non-separable 3D wavelet transform for video denoising and overcome the motion-mixture problem by using oriented complex wavelets. This wavelet transform is a 3D version of Kingsbury's 1D and 2D dual-tree wavelet transforms. We also investigate video denoising techniques using a combination of both 2D and 3D oriented wavelet transforms. The results are compared with those obtained by separable wavelet transforms.

Complex discrete wavelet transforms (DWT) have significant advantages over real wavelet transforms for certain signal processing problems. Two approaches to the implementation of complex wavelet transforms have been proposed earlier. Both approaches require discrete-time allpass systems having approximately linear-phase and (fractional) delay. This paper compares the results when different allpass systems are used. In the earlier work, maximally flat delay allpass systems were used. In this paper, it is shown that an allpass system designed according to the minimax criterion yields improvements for the complex DWT. (C) 2003 Elsevier Science B.V. All rights reserved.