Raymond O. Wells

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 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.

Biorthogonal Wavelet Space: Parametrization and Factorization

In this paper we study the algebraic and geometric structure of the space of compactly supported biorthogonal wavelets. We prove that any biorthogonal wavelet matrix pair (which consists of the scaling filters and wavelet filters) can be factored as the product of primitive para-unitary matrices, a pseudo identity matrix pair, an invertible matrix, and the canonical Haar matrix. Compared with the factorization results of orthogonal wavelets, it now becomes apparent that the difference between orthogonal and biorthogonal wavelets lies in the pseudo identity matrix pair and the invertible matrix, which in the orthogonal setting will be the identity matrix and a unitary matrix. Thus by setting the pseudo identity matrix pair to be the identity matrix and using the Schmidt orthogonalization method on the invertible matrix, it is very straightforward to convert a biorthogonal wavelet pair into an orthogonal wavelet.

Fast implementation of wavelet transform for m-band filter banks

An orthogonal m-band discrete wavelet transform has an O(m2) complexity. In this paper, we present a fast implementation of such a discrete wavelet transform. In an orthonormal m-band wavelet system, the vanishing moments and orthogonality conditions are imposed on the scaling filter only. Given a scaling filter, one can design the other m-1 wavelet filters. It is well-known that there are infinitely many solutions in such designing procedure. Here we choose one specific type of solutions and implement the corresponding wavelet transform in a scheme which has complexity O(m). Thus for any scaling filter, one can always construct a full orthogonal m-band wavelet matrix with an O(m) discrete wavelet transform.

Representing the geometry of domains by wavelets with applications to partial differential equations

The recently introduced compactly supported wavelets of Daubechies have proven to be very useful in various aspects of signal processing, notably in image compression. The fundamental idea in the application to image compression was to take a digitized image and to use wavelets to provide a multiscale representation of it, and to then discard some information at some scales, and leave information more intact at other scales. The Daubechies wavelets have differentiability properties in addition to their compact support and orthogonality properties. A number of authors have used wavelet system for solving various problems in differential and integral equations. An additional point of view for approaching boundary value problems in partial differential equations were introduced, in which the boundary, the boundary data, and the unknown solution (of a boundary value problem) on the interior of a domain are all uniformly represented in terms of compactly supported wavelet functions in an extrinsic ambient Euclidean space. In this paper we describe multiscale representations of domains and their boundaries and obtain multiscale representations of some of the basic elements of geometric calculus (line integrals, surface measures, etc.) which are then in turn useful for specific numerical calculations in problems in approximate solutions of differential equations. This is similar to the spectral method in solving differential equations (essentially using an orthonormal expansion), but here we use the localization property of wavelets to extend this orthonormal representation to boundary data and geometric boundaries. In this paper we want to indicate how to carry this out.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

On the conditioning of numerical boundary measures in wavelet Galerkin methods

The paper investigates the accuracy and numerical stability of a class of wavelet Galerkin formulations on irregular domains. The method of numerical boundary measures is based upon a domain embedding strategy in which the irregular domain of interest is embedded in a larger domain having regular geometry. One advantage of the domain embedding method is that the boundary conditions on the larger, regular domain can be enforced in a straightforward manner, and the solution procedure can exploit the highly structured form of the resulting governing equations. The defining characteristic of this method is that the calculation of integrals along the irregular boundary are carried out using recently derived numerical boundary measures. In addition, the coercive bilinear forms characterizing the boundary value problem of interest must be calculated when restricted to the actual domain. In the case of wavelet Galerkin formulations, this calculation is accomplished with the three term connection coefficients that characterize the numerical boundary measure. The numerical stability and accuracy of the domain embedding procedure is compared to a newly developed wavelet-based finite element formulation.

Optimally smooth symmetric quadrature mirror filters for image coding

Symmetric quadrature mirror filters (QMFs) offer several advantages for wavelet-based image coding. Symmetry and odd-length contribute to efficient boundary handling and preservation of edge detail. Symmetric QMFs can be obtained by mildly relaxing the filter bank orthogonality conditions. We describe a computational algorithm for these filter banks which is also symmetric in the sense that the analysis and synthesis operations have identical implementations, up to a delay. The essence of a wavelet transform is its multiresolution decomposition, obtained by iterating the lowpass filter. This allows one to introduce a new design criterion, smoothness (good behavior) of the lowpass filter under iteration. This design constraint can be expressed solely in terms of the lowpass filter tap values (via the eigenvalue decomposition of a certain finite-dimensional matrix). Our innovation is to design near- orthogonal QMFs with linear-phase symmetry which are optimized for smoothness under iteration, not for stopband rejection. The new class of optimally smooth QMF filter banks yields high performance in a practical image compression system.

Wavelets and wave propagation modeling

This paper discusses several current attempts to use acoustic and electromagnetic wave propagation for modeling physical phenomena and the role that wavelet analysis is playing in these efforts. The first problem involves geophysical modeling of the ocean floor using acoustic waves, and wavelets have recently been shown to play an important role. The second problem involves modeling of SAR radar images in the context of automatic target recognition efforts. The third problem is global illumination in computer graphics, i.e., simulation of reflected and absorbed light for everyday environments. The role of wavelets is more embryonic in these latter two areas, but there are some common principles in all of these modeling efforts, and the methodology of wavelets seems well suited to certain aspects of these problems.

Scale-band-dependent thresholding for signal denoising using undecimated discrete wavelet packet transforms

The purpose of this paper is to study signal denoising by thresholding coefficients of undecimated discrete wavelet packet transforms (UDWPT). The undecimated filterbank implementation of UDWPT is first considered, and the best basis selection algorithm that prunes the complete undecimated discrete wavelet packet binary tree is studied for the purpose of signal denoising. Distinct from the usual approach which selects the best subtree based on the original (unthresholded) transform coefficients, our selection is based on the thresholded coefficients, since we believe discarding the small coefficients permits to choose the best basis from the set of coefficients that will really contribute to the reconstructed signal. Another feature of the algorithm is the thresholding scheme. To threshold coefficients which are correlated differently from scale to scale and from band to band, a uniform threshold is not appropriate. Alternatively, two scale-band-dependent thresholding schemes are designed: a correlation-dependent model and a Monte Carlo simulation-based model. The cost function for the pruning algorithm is specifically designed for the purpose of signal denoising. We consider it profitable to split a band if more noise can be discarded by thresholding while signal components are preserved. So, higher SNR is desirable in the process of selection. Experiments conducted for 1D and 2D signals shows that the algorithm achieves good SNR performance while preserving high frequency details of signals.