Showing papers on "Harmonic wavelet transform published in 2003"

An EM algorithm for wavelet-based image restoration

Abstract: This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in the wavelet coefficients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. Thus, it is a general-purpose approach to wavelet-based image restoration with computational complexity comparable to that of standard wavelet denoising schemes or of frequency domain deconvolution methods. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring O(NlogN) operations per iteration. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach performs competitively with, in some cases better than, the best existing methods in benchmark tests.

Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method

Abstract: SUMMARY In this paper wavelet transforms and a logarithmic barrier method are applied to the inversion of large-scale magnetic data to recover a 3-D distribution of magnetic susceptibility. The fast wavelet transform is used, along with thresholding the small wavelet coefficients, to form a sparse representation of the sensitivity matrix. The reduced size of the resultant matrix allows the solution of large problems that are otherwise intractable. The compressed matrix is used to carry out fast forward modelling by performing matrix-vector multiplications in the wavelet domain. The reduction in CPU time is directly proportional to the compression ratio of the matrix. A second important feature of the algorithm used here is the use of an interior-point method of optimization to enforce positivity constraints. In this approach, the positivity is incorporated into the inversion by a sequence of non-linear optimizations approximated by truncated Newton steps. At the heart of the algorithm, a linear system of equations is solved. The conjugate gradient technique has been used as the basic solver to take the advantage of the efficient forward modelling offered by the sparse matrix representation. Overall, the combination of wavelet transforms, interior point optimization and conjugate gradient solutions readily allows us to solve magnetic inverse problems that have a few hundred thousand parameters and tens of thousands of data.

Non-Equispaced Fast Fourier Transforms with Applications to Tomography

Abstract: In this article we describe a non-equispaced fast Fourier transform. It is similar to the algorithms of Dutt and Rokhlin and Beylkin but is based on an exact Fourier series representation. This results in a greatly simplified analysis and increased flexibility. The latter can be used to achieve more efficiency. Accuracy and efficiency of the resulting algorithm are illustrated by numerical examples. In the second part of the article the non-equispaced FFT is applied to the reconstruction problem in Computerized Tomography. This results in a different view of the gridding method of O’Sullivan and in a new ultra fast reconstruction algorithm. The new reconstruction algorithm outperforms the filtered backprojection by a speedup factor of up to 100 on standard hardware while still producing excellent reconstruction quality.

Video denoising using 2D and 3D dual-tree complex wavelet transforms

Abstract: The denoising of video data should take into account both temporal and spatial dimensions, however, true 3D transforms are rarely used for video denoising. Separable 3-D transforms have artifacts that degrade their performance in applications. This paper describes the design and application of the non-separable oriented 3-D dual-tree wavelet transform for video denoising. This transform gives a motion-based multi-scale decomposition for video - it isolates in its subbands motion along different directions. In addition, we investigate the denoising of video using the 2-D and 3-D dual-tree oriented wavelet transforms, where the 2-D transform is applied to each frame individually.

A new framework for complex wavelet transforms

Abstract: Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce two-stage mapping-based complex wavelet transforms that consist of a mapping onto a complex function space followed by a DWT of the complex mapping. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, non-redundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we can exploit this flexibility to create the complex double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of (3/sup m/-1/2/sup m/-1) in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy.

Wavelet Analysis and its Application

Abstract: The wavelet analysis is a new subject in the signal processing. It has many good characteristic and applied perspective. The basic principle of the wavelet transform is briefly introduced in this paper. The application of the wavelet transform in the estimation of the signal virtual value is also put forward.

An efficient texture classification algorithm using Gabor wavelet

Abstract: In this paper we have investigated the application of nonseparable Gabor wavelet transform for texture classification We have compared the effect of applying the dyadic wavelet transform as a traditional method with Gabor wavelet for texture extraction It is well known that Gabor wavelets attain maximum joint space-frequency resolution which is highly significant in the process of texture extraction in which the conflicting objectives of accuracy in texture representation and texture spatial localization are both important This fact has been explored in our results as they show that the classification rate obtained for Gabor wavelet is higher that those obtained using dyadic wavelets Based on our experiments, the Gabor wavelet is more appropriate than dyadic wavelets for texture classification as it leads to a better discrimination of textures

Skeletonization of Ribbon-like shapes based on a new wavelet function

Abstract: A wavelet-based scheme to extract skeleton of Ribbon-like shape is proposed in this paper, where a novel wavelet function plays a key role in this scheme, which possesses three significant characteristics, namely, 1) the position of the local maximum moduli of the wavelet transform with respect to the Ribbon-like shape is independent of the gray-levels of the image. 2) When the appropriate scale of the wavelet transform is selected, the local maximum moduli of the wavelet transform of the Ribbon-like shape produce two new parallel contours, which are located symmetrically at two sides of the original one and have the same topological and geometric properties as that of the original shape. 3) The distance between these two parallel contours equals to the scale of the wavelet transform, which is independent of the width of the shape. This new scheme consists of two phases: 1) Generation of wavelet skeleton-based on the desirable properties of the new wavelet function, symmetry analyses of the maximum moduli of the wavelet transform is described. Midpoints of all pairs of contour elements can be connected to generate a skeleton of the shape, which is defined as wavelet skeleton. 2) Modification of the wavelet skeleton. Thereafter, a set of techniques are utilized for modifying the artifacts of the primary wavelet skeleton. The corresponding algorithm is also developed in this paper. Experimental results show that the proposed scheme is capable of extracting exactly the skeleton of the Ribbon-like shape with different width as well as different gray-levels. The skeleton representation is robust against noise and affine transformation.

Prediction Based on a Multiscale Decomposition

Abstract: A wavelet-based forecasting method for time series is introduced. It is based on a multiple resolution decomposition of the signal, using the redundant "a trous" wavelet transform which has the advantage of being shift-invariant. The result is a decomposition of the signal into a range of frequency scales. The prediction is based on a small number of coefficients on each of these scales. In its simplest form it is a linear prediction based on a wavelet transform of the signal. This method uses sparse modelling, but can be based on coefficients that are summaries or characteristics of large parts of the signal. The lower level of the decomposition can capture the long-range dependencies with only a few coefficients, while the higher levels capture the usual short-term dependencies. We show the convergence of the method towards the optimal prediction in the autoregressive case. The method works well, as shown in simulation studies, and studies involving financial data.

Method and apparatus for three-dimensional imaging in the fourier domain

Abstract: A method is described for acquiring two or more two-dimensional Fourier transforms from different perspectives of a three-dimensional object region (9). A three-dimensional Fourier transform can then be constructed using tomographic methods, permitting the application of image analysis algorithms (52) analogous to those used for two-dimensional images.

Wavelet transform with spectral post-processing for enhanced feature extraction [machine condition monitoring]

Abstract: The quality of machine condition monitoring techniques and their applicability in the industry are determined by the effectiveness and efficiency, with which characteristic signal features are extracted and identified. Because of the weak amplitude and short duration of structural defect signals at the incipient stage, it is generally difficult to extract hidden features from the data measured using conventional spectral techniques. A new approach, based on a combined wavelet and Fourier transformation, is presented in this paper. Experimental studies on a rolling bearing with a localized point defect of 0.25 mm diameter have shown that this new technique provides significantly improved feature extraction capability over the spectral technique.

Application of the S transform to prestack noise attenuation filtering

Abstract: [1] The S transform is a time-frequency localization technique that bridges the gap between the short-time Fourier transform and wavelet transforms. We propose a new method, designed for application to multiple time series that all have similar time dependence in their spectra, that exploits the properties of the S transform to estimate signal frequency characteristics as a function of time. The method gives an average time-frequency distribution that forms the basis for an adaptive filter for noise attenuation. Using prestack (multichannel) seismic data from a crustal seismic reflection profile in Canada, we show that this technique is particularly effective for determining residual statics corrections for data with a low signal-to-noise ratio.

Matching-pursuit for simulations of quantum processes

Abstract: The matching-pursuit algorithm is implemented to develop an extension of the split-operator Fourier transform method to a nonorthogonal, nonuniform and dynamically adaptive coherent-state representation. The accuracy and efficiency of the computational approach are demonstrated in simulations of deep tunneling and long time dynamics by comparing our simulation results with the corresponding benchmark calculations.

Generalized S Transform and Seismic Response Analysis of Thin Interbedss Surrounding Regions by Gps

Abstract: S transform (ST) proposed by Stockwell et al. is the unique transform that provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. This feature is very important for applications. However, the ST can't work well for seismic data analysis since its basic wavelet is not appropriate. In this paper, the ST is generalized with two steps, and two kinds of new transforms are obtained, which are called generalized S transform (GST). First, the basic wavelet in ST is replaced by a modulated harmonic wave with four undetermined coefficients, and then a new transform and its inverse are given, called GST1. Second, taking a linear combination of the basic wavelets in step 1 as a new basic wavelet, called GST2, and its inverse is constructed. To compare ST with GST, the ST and GST method are used to analyze several typical models of thin beds, respectively. The results show that the resolution of GST is better than that of ST. The GST method can determine accurately the location of interfaces of acoustic impedance in thin interbeds of thickness being only an eighth wavelength, while ST method can't. In this study, the effectiveness of GST method is also verified by processing results of real data.

Wavelet Analysis and Application

Abstract: With the development of Fourier analysis,wavelet analysis become the concerned question of many subjects.On the basis of conception of wavelet transform,the Mallat algorithm has been introduced and applied on signal analysis and disposal,so a failure signal can been decomposed and restructured.Through analysis of experiment,the high frequency part can reflect the turning point,and the position of fault-point can be gained by the important information,so drew a conclusion that application of wavelet analysis is effective on the aspect of singularity measurement.

The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform

Abstract: This paper introduces an approximately shift invariant redundant dyadic wavelet transform - the phaselet transform - that includes the popular dual-tree complex wavelet transform of Kingsbury (see Phil. R. Soc. London A, Sept. 1999) as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way - and hence called phaselets - to achieve approximate shift-redundancy; the bigger the set, the better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented - building on the work of Selesnick (see IEEE Trans. Signal Processing) for the design of wavelet pairs for Kingsbury's dual-tree complex wavelet transform. Construction of two-dimensional (2-D) directional bases from tensor products of one-dimensional (1-D) phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader, independent of the approximate shift invariance property that this paper argues they possess.

Digital Ridgelet Transform Based on True Ridge Functions

Abstract: We study a notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator. There are fast algorithms for analysis, for synthesis, and for partial reconstruction. Associated with this is a transform which is a digital analog of the orthonormal ridgelet transform (but not orthonormal for finite n). In either approach, we get an overcomplete frame; the result of ridgelet transforming an n × n array is a 2n × 2n array. The analysis operator is invertible on its range; the appropriately preconditioned operator has a tightly controlled spread of singular values. There is a near-parseval relationship. Our construction exploits the recent development by Averbuch et al. (2001) of the Fast Slant Stack, a Radon transform for digital image data; it may be viewed as following a Fast Slant Stack with fast 2-d wavelet transform. A consequence of this construction is that it offers discrete objects (discrete ridgelets, discrete Radon transform, discrete Pseudopolar Fourier domain) which obey inter-relationships paralleling those in the continuum ridgelet theory (between ridgelets, Radon transform, and polar Fourier domain). We make comparisons with other notions of ridgelet transform, and we investigate what we view as the key issue: the summability of the kernel underlying the constructed frame. The sparsity observed in our current implementation is not nearly as good as the sparsity of the underlying continuum theory, so there is room for substantial progress in future implementations.

A new local multiscale Fourier analysis for medical imaging

Abstract: The Stockwell transform (ST), recently developed for geophysics, combines features of the Fourier, Gabor and wavelet transforms; it reveals frequency variation over time or space. This valuable information is obtained by Fourier analysis of a small segment of a signal at a time. Localization of the Fourier spectrum is achieved by filtering the signal with frequency-dependent Gaussian scaling windows. This multi-scale time–frequency analysis provides information about which frequencies occur and more importantly when they occur. Furthermore, the Stockwell domain can be directly inferred from the Fourier domain and vice versa. These features make the ST a potentially effective tool to visualize,analyze, and process medical imaging data. The ST has proven useful in noise reduction and tissue texture analysis. Herein, we focus on the theory and effectiveness of the ST for medical imaging. Its effectiveness and comparison with other linear time–frequency transforms, such as the Gabor and wavelet transforms, are discussed and demonstrated using functional magnetic resonance imaging data.

A Method for Fast Computation of the Fourier Transform over a Finite Field

Abstract: We consider the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of algorithms for the Fourier transform with complexity less than that of the best known analogs are given.

Speech enhancement using 2-D Fourier transform

Abstract: This paper presents an innovative way of using the two-dimensional (2-D) Fourier transform for speech enhancement. The blocking and windowing of the speech data for the 2-D Fourier transform are explained in detail. Several techniques of filtering in the 2-D Fourier transform domain are also proposed. They include magnitude spectral subtraction, 2-D Wiener filtering as well as a hybrid filter which effectively combines the one-dimensional (1-D) Wiener filter with the 2-D Wiener filter. The proposed hybrid filter compares favorably against other techniques using an objective test.

Fractional Fourier transform of the Gaussian and fractional domain signal support

Abstract: The fractional Fourier transform (FrFT) provides an important extension to conventional Fourier theory for the analysis and synthesis of linear chirp signals. It is a parameterised transform which can be used to provide extremely compact representations. The representation is maximally compressed when the transform parameter, /spl alpha/, is matched to the chirp rate of the input signal. Existing proofs are extended to demonstrate that the fractional Fourier transform of the Gaussian function also has Gaussian support. Furthermore, expressions are developed which allow calculation of the spread of the signal representation for a Gaussian windowed linear chirp signal in any fractional domain. Both continuous and discrete cases are considered. The fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT. This is equated with a restatement of the uncertainty principle for linear chirp signals and the fractional Fourier domains. The calculated values for the fractional domain support are tested empirically through comparison with the discrete transform output for a synthetic signal with known parameters. It is shown that the same expressions are appropriate for predicting the support of the ordinary Fourier transform of a Gaussian windowed linear chirp signal.

Method and apparatus for noise reduction using discrete wavelet transform

Abstract: An improved noise reduction process by wavelet thresholding utilizes a discrete wavelet transform to decompose the image into different resolution levels. A thresholding function is then applied in different resolution levels with different threshold values to eliminate insignificant wavelet coefficients which mainly correspond to the noise in the original image. Finally, an inverse discrete wavelet transform is applied to generate the noise-reduced video image. The threshold values are based on the relationships between the noise standard deviations of different decomposition levels in the wavelet domain and the noise standard deviation of the original image.