Jan E. Odegard

Noise reduction using an undecimated discrete wavelet transform

A new nonlinear noise reduction method is presented that uses the discrete wavelet transform. Similar to Donoho (1995) and Donohoe and Johnstone (1994, 1995), the authors employ thresholding in the wavelet transform domain but, following a suggestion by Coifman, they use an undecimated, shift-invariant, nonorthogonal wavelet transform instead of the usual orthogonal one. This new approach can be interpreted as a repeated application of the original Donoho and Johnstone method for different shifts. The main feature of the new algorithm is a significantly improved noise reduction compared to the original wavelet based approach. This holds for a large class of signals, both visually and in the l/sub 2/ sense, and is shown theoretically as well as by experimental results.

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>>

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). >

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.

Smooth biorthogonal wavelets for applications in image compression

We introduce a new family of smooth, symmetric biorthogonal wavelet basis. The new wavelets are a generalization of the Cohen, Daubechies and Feauveau (CDF) biorthogonal wavelet systems proposed in 1992. Smoothness is controlled independently in the analysis and synthesis bank and is achieved by optimization of the discrete finite variation (DFV) measure introduced for orthogonal wavelet design. The DFV measure dispenses with a measure of differentiability (for smoothness) which requires-a large number of vanishing wavelet moments (e.g., Holder and Sobolev exponents) in favor of a smoothness measure that uses the fact that only a finite depth of the filter bank tree is involved in most practical applications. Image compression examples applying the new filters using the embedded wavelet zerotree (EZW) compression algorithm due to Shapiro (1993) shows that the new basis functions performs better when compared to the classical CDF 7/9 wavelet basis.

Optimal wavelets for signal decomposition and the existence of scale-limited signals

Wavelet methods give a flexible alternative to Fourier methods in nonstationary signal analysis. The concept of band-limitedness plays a fundamental role in Fourier analysis. Since wavelet theory replaces frequency with scale, a natural question is whether there exists a useful concept of scale-limitedness. Obvious definitions of scale-limitedness are too restrictive, in that there would be few or no useful scale-limited signals. The authors introduce a viable definition for scale-limited signals, and show that the class is rich enough to include bandlimited signals, and impulse trains, among others. Moreover, for a wide choice of criteria, it is shown how to design the optimal wavelet for representing a given signal, and how to design robust wavelets that optimally represent certain classes of signals.<<ETX>>

Nearly symmetric orthogonal wavelets with non-integer DC group delay

This paper investigates the design of Coiflet-like nearly symmetric compactly supported orthogonal wavelets. The group delay is used as the main vehicle by which near symmetry is achieved. By requiring a specified degree of flatness of the group delay at /spl omega/=0 (equivalent to appropriate moment conditions), near symmetry is achieved. The way in which the group delay approximates a constant is a traditional measure of symmetry in filter design. Grobner bases are used to obtain the solutions to the defining nonlinear equations. It is found that the DC group delay that maximizes the group delay flatness at /spl omega/=0 is irrational, and for a length 10 orthogonal wavelet with three vanishing moments, the solution is presented.

Instantaneous Frequency Estimation using the Reassignment method

This paper explores the method of reassignment for extracting instantaneous frequency attributes from seismic data. The reassignment method was rst applied to the spectrogram by Kodera, Gendrin and de Villedary [5] and later generalized to any bilinear time-frequency or time-scale representation by Auger and Flandrin [1]. Key to the method is a nonlinear convolution where the value of the convolution is not placed at the center of the convolution kernel but rather reassigned to the center of mass of the function within the kernel. The resulting reassigned representation yields signi cantly improved component localization. In this paper we will study the impact of the reassigned time-frequency representation on our ability to reliably estimate instantaneous frequency for a given seismic signal.

Simultaneous noise reduction and SAR image data compression using best wavelet packet basis

We propose a novel method for simultaneous noise reduction and data compression based on shrinking, quantizing and coding the wavelet packet (WP) coefficients. A dynamic programming and fast pruning algorithm is used to efficiently choose the best basis from the entire library of admissible WP bases, and jointly optimize the bit allocation strategy and the quantization scheme in the rate-distortion framework. Soft-thresholding in the wavelet domain can significantly suppress noise, e.g., the speckles of the synthetic aperture radar 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 image.