C. Sidney Burrus

Complex wavelet transforms with allpass filters

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.

Wavelet Based SAR Speckle Reduction and Image Compression

This paper evaluates the performance of the recently published wavelet-based algorithm for speckle reduction of SAR images. The original algorithm, based on the theory of wavelet thresholding due to Donoho and Johnstone, has been shown to improve speckle statistics. In this paper, we give more extensive results based on tests performed at Lincoln Laboratory (LL). The LL benchmarks show that the SAR imagery is significantly enhanced perceptually. Although the wavelet processed data results in an increase in the number of natural clutter false alarms, an appropriately modified CFAR detector (i.e., by clamping the estimated clutter standard deviation) eliminates the extra false alarms. The paper also gives preliminary results on the performance of the new and improved wavelet denoising algorithm based on the shift invariant wavelet transform. By thresholding the shift invariant discrete wavelet transform we can further reduce speckle to achieve a perceptually superior SAR image with ground truth information significantly enhanced. Preliminary results on the speckle statistics of this new algorithm is improved over the classical wavelet denoising algorithm. Finally, we show that the classical denoising algorithm as proposed by Donoho and Johnstone and applied to SAR has the added benefit of achieving about 3:1 compression with essentially no loss in image fidelity.

Nonlinear processing of a shift-invariant discrete wavelet transform (DWT) for noise reduction

A novel approach for noise reduction is presented. Similar to Donoho, we employ thresholding in some wavelet transform domain but use a nondecimated and consequently redundant wavelet transform instead of the usual orthogonal one. Another difference is the shift invariance as opposed to the traditional orthogonal wavelet transform. We show that this new approach can be interpreted as a repeated application of Donohos original method. The main feature is, however, a dramatically improved noise reduction compared to Donohos approach, both in terms of the l2 error and visually, for a large class of signals. This is shown by theoretical and experimental results, including synthetic aperture radar (SAR) images.

Directional Complex-Wavelet Processing

Poor directional selectivity, a major disadvantage of the separable 2D discrete wavelet transform (DWT), has previously been circumvented either by using highly redundant, nonseparable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees. In this paper, we demonstrate that superior directional selectivity may be obtained with no redundancy in any separable wavelet transform. We achieve this by projecting the wavelet coefficients to separate approximately the positive and negative frequencies. Subsequent decimation maintains non-redundancy. A novel reconstruction step guarantees perfect reconstruction within this critically- sampled framework. Although our transform generates complex- valued coefficients, it may be implemented with a fast algorithm that uses only real arithmetic. We also explain how redundancy may be judiciously introduced into our transform to benefit certain applications. To demonstrate the efficacy of our projection technique, we show that it achieves state-of-the-art performance in a seismic image- processing application.

Optimum FIR and IIR multistage multirate filter design

A theorem is introduced which is useful in deriving equivalent multirate filter structures. Frequency responses of multistage multirate filters are derived and defined by deriving their equivalent one-stage filters. A design principle is proposed to reduce filtering requirements at each stage and move the filter operations to low-sampling-rate stages and thus result in a lower arithmetic rate. Optimum FIR and IIR multistage multirate filter designs are developed based on this principle. The new design has a one-point passband specification for each decimator and/or interpolator stage resulting in a wider transition region and lower filter order. Examples are given to explain the design procedure, and comparisons are made to show the superiority of the new filters.

Simultaneous speckle reduction and data compression using best wavelet packet bases with application to synthetic aperture radar (SAR) based ATD/R

We propose a novel method for simultaneous speckle reduction and data compression based on shrinking, quantizing and coding the wavelet packet coefficients of the logarithmically transformed image. A fast algorithm is used to find the best wavelet packet basis in the rate- distortion sense from the entire library of admissible wavelet packet bases. Soft-thresholding in wavelet domain can significantly suppress the speckles of the synthetic aperture radar (SAR) images while maintaining bright reflections for subsequent detection and recognition. Optimal bit allocation, quantization and entropy coding achieve the goal of compression while maintaining the fidelity of the SAR image.

Fast approximate Fourier transform via wavelets transform

We propose an algorithm that uses the discrete wavelet transform as a tool to compute the discrete Fourier transform (DFT). The Cooley-Tukey FFT is shown to be a special case of the proposed algorithm when the wavelets in use are trivial. If no intermediate coefficients are dropped and no approximations are made, the proposed algorithm computes the exact results, and its computational complexity is on the same order of the FFT. The main advantage of the proposed algorithm is that the good time and frequency localization of wavelets can be exploited to approximate the Fourier transform for many classes of signals resulting in much less computation. Thus the new algorithm provides an efficient complexity vs accuracy tradeoff. When approximations are allowed, under certain sparsity conditions, the algorithm can achieve linear complexity. It has been shown that the thresholding of the wavelet coefficients has near optimal noise reduction property for many classes of signals. We show that for the same reason, the proposed algorithm also reduces the noise while doing the approximation. If we need to compute the DFT of noisy signals, the proposed algorithm not only can reduce the numerical complexity, but also can produce cleaner results. In summary, we propose a novel fast approximate Fourier transform algorithm using the wavelet transform. Since wavelets are the conditional basis of many classes of signals, the algorithm is very efficient and has built-in de-noising capacity.

Speckle reduction via wavelet shrinkage with application to synthetic-aperture-radar-based automatic target detection/reduction (ATD/R)

We propose a novel speckle reduction method based on shrinking 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. Wavelet processed imagery is shown to provide better detection performance for synthetic-aperture radar based automatic target detection/recognition problem.

Viewpoint Global warming toward open educational resources

Seeking to realize the potential for significantly improving and advancing the worlds standard of education.

Iterative Design of l p FIR and IIR Digital Filters

This paper presents a family of algorithms to design FIR and IIR digital filters using lp norms as optimality criteria. The algorithms presented are based on the Iterative Reweighted Least Squares (IRLS) method, and enjoy the same flexibility that traditional IRLS methods have. While other FIR methods use l2 or l¿ norms as design criteria, one can design filters that optimally compromise between these two criteria by using general lp norms. Several important design problems can be solved by posing them as lp problems (including the Constrained Least Squares and Magnitude lp problems). This paper also presents IRLS algorithms to design complex lp and magnitude lp IIR filters. It is worth noting that the methods presented are based on a common theory, with changes mostly on the design of associated weighting functions. The methods are flexible, robust and typically converge rapidly.