New fast smoothers for multiscale systems

We consider the smoothing problem for multiscale stochastic systems based on the wavelet transform. These models involve processes indexed by the nodes of a dyadic tree. Each level of the dyadic tree represents one scale or resolution of the process; therefore, moving upward on the tree divides the resolution by 2, whereas moving downward multiplies it by 2. The processes are built according to a recursion in scale from coarse to fine to which random details are added. To operate the change in scale, one must perform an interpolation. This is achieved using the QMF pair of operators attached to a wavelet transform. These models have proved to be of great value to capture textures or fractal-like processes as well as to perform multiresolution sensor fusion (an example of which is given here). Up to now however, only subclasses of multiscale systems were amenable to fast algorithms and through different formalisms: those relying on Haar's wavelet and those involving only one of the two wavelet interpolators. We provide here a unifying framework that handles any system based on orthogonal wavelets. A smoothing theory is presented to define the field of fast algorithms for Markov random fields and give intuition on how to design them. This theory reveals the difficulties arising with general multiscale systems. We then prove that orthogonality properties of wavelets are the gate to fastness     

Asymptotic decorrelation of between-Scale Wavelet coefficients

In recent years there has been much interest in the analysis of time series using a discrete wavelet transform (DWT) based upon a Daubechies wavelet filter. Part of this interest has been sparked by the fact that the DWT approximately decorrelates certain stochastic processes, including stationary fractionally differenced (FD) processes with long memory characteristics and certain nonstationary processes such as fractional Brownian motion. It is shown that, as the width of the wavelet filter used to form the DWT increases, the covariance between wavelet coefficients associated with different scales decreases to zero for a wide class of stochastic processes. These processes are Gaussian with a spectral density function (SDF) that is the product of the SDF for a (not necessarily stationary) FD process multiplied by any bounded function that can serve as an SDF on its own. We demonstrate that this asymptotic theory provides a reasonable approximation to the between-scale covariance properties of wavelet coefficients based upon filter widths in common use. Our main result is one important piece of an overall strategy for establishing asymptotic results for certain wavelet-based statistics.     

Fusion algorithm for multisensor images based on discrete multiwavelet transform

The authors review the notion of multiwavelets and describe the use of the discrete multiwavelet transform (DMWT) in image fusion processing. Multiwavelets are extensions from scalar wavelets, and have several advantages in comparison with scalar wavelets. Multiwavelet analysis can offer more precise image analysis than wavelet multiresolution analysis. A novel fusion algorithm is presented for multisensor images based on the discrete multiwavelet transform that can be performed at pixel level. After the registering of source images, a pyramid for each source image can be obtained by applying decomposition with multiwavelets in each level. The multiwavelet decomposition coefficients of the input images are appropriately merged and a new fused image is obtained by reconstructing the fused multiwavelet coefficients. This image fusion algorithm may be used to combine images from multisensors to obtain a single composite with extended information content. The results of experiments indicate that this image fusion algorithm can provide a more satisfactory fusion outcome.     

Multiscale autoregressive models and wavelets

The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads. While the MAR framework was originally motivated by wavelets, the link between these two worlds has been previously established only in the simple case of the Haar wavelet. The first contribution of this paper is to provide a unification of the MAR framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem. The second contribution of this paper is to develop wavelet-based approximate internal MAR models for stochastic processes. This will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients. Taking advantage of the statistical machinery provided by the MAR framework, we will illustrate the application of our models to sample-path generation and estimation from noisy, irregular, and sparse measurements     

Multifocus and multispectral image fusion based on pixel significance using discrete cosine harmonic wavelet transform

The energy compaction and multiresolution properties of wavelets have made the image fusion successful in combining important features such as edges and textures from source images without introducing any artifacts for context enhancement and situational awareness. The wavelet transform is visualized as a convolution of wavelet filter coefficients with the image under consideration and is computationally intensive. The advent of lifting-based wavelets has reduced the computations but at the cost of visual quality and performance of the fused image. To retain the visual quality and performance of the fused image with reduced computations, a discrete cosine harmonic wavelet (DCHWT)-based image fusion is proposed. The performance of DCHWT is compared with both convolution and lifting-based image fusion approaches. It is found that the performance of DCHWT is similar to convolution-based wavelets and superior/similar to lifting-based wavelets. Also, the computational complexity (in terms of additions and multiplications) of the proposed method scores over convolution-based wavelets and is competitive to lifting-based wavelets.     

Orthogonal and Biorthogonal $sqrt 3$-Refinement Wavelets for Hexagonal Data Processing

The hexagonal lattice was proposed as an alternative method for image sampling. The hexagonal sampling has certain advantages over the conventionally used square sampling. Hence, the hexagonal lattice has been used in many areas. A hexagonal lattice allows radic3, dyadic and radic7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The radic3-refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale, and hence, it provides more resolution levels from which one can choose. This fact is the main motivation for the study of radic3-refinement surface subdivision, and it is also the main reason for the recommendation to use the radic3-refinement for discrete global grid systems. However, there is little work on compactly supported radic3 -refinement wavelets. In this paper, we study the construction of compactly supported orthogonal and biorthogonal radic3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with twofold symmetry and construct the associated orthogonal radic3-refinement wavelets. We study the sixfold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of sixfold symmetric radic3-refinement filter banks and construct the associated biorthogonal wavelets.     

Vector-valued wavelets and vector filter banks

In this paper, we introduce vector-valued multiresolution analysis and vector-valued wavelets for vector-valued signal spaces. We construct vector-valued wavelets by using paraunitary vector filter bank theory. In particular, we construct vector-valued Meyer wavelets that are band-limited. We classify and construct vector-valued wavelets with sampling property. As an application of vector-valued wavelets, multiwavelets can be constructed from vector-valued wavelets. We show that certain linear combinations of known scalar-valued wavelets may yield multiwavelets. We then present discrete vector wavelet transforms for discrete-time vector-valued (or blocked) signals, which can be thought of as a family of unitary vector transforms     

Nonseparable wavelet-based cone-beam reconstruction in 3-D rotational angiography

In this paper, we propose a new wavelet-based reconstruction method suited to three-dimensional (3-D) cone-beam (CB) tomography. It is derived from the Feldkamp algorithm and is valid for the same geometrical conditions. The demonstration is done in the framework of nonseparable wavelets and requires ideally radial wavelets. The proposed inversion formula yields to a filtered backprojection algorithm but the filtering step is implemented using quincunx wavelet filters. The proposed algorithm reconstructs slice by slice both the wavelet and approximation coefficients of the 3-D image directly from the CB projection data. The validity of this multiresolution approach is demonstrated on simulations from both mathematical phantoms and 3-D rotational angiography clinical data. The same quality is achieved compared with the standard Feldkamp algorithm, but in addition, the multiresolution decomposition allows to apply directly image processing techniques in the wavelet domain during the inversion process. As an example, a fast low-resolution reconstruction of the 3-D arterial vessels with the progressive addition of details in a region of interest is demonstrated. Other promising applications are the improvement of image quality by denoising techniques and also the reduction of computing time using the space localization of wavelets.     

时间尺度的多分辨率综合

本文根据小波分析的基本原理,对原子钟信号进行多分辨分解,将分解后的小波变换系数进行加权平均,得到不同小波尺度综合原子时的加权平均小波变换系数,然后由小波变换的重构条件,反演综合时间尺度,由于对原子钟信号进行小波分解,利用不同尺度的小波变换系数的小波方差进行加权平均,这样既要考虑不同原子钟在稳定性方面的差异,又顾及同一台原子在钟在不同小波尺度的变化特性,最后根据陕西天文台国家授时中心的实测数据对这种     

Algorithms for designing wavelets to match a specified signal

Algorithms for designing a mother wavelet /spl psi/(x) such that it matches a signal of interest and such that the family of wavelets {2/sup -(j/2)//spl psi/(2/sup -j/x-k)} forms an orthonormal Riesz basis of L/sup 2/(/spl Rscr/) are developed. The algorithms are based on a closed form solution for finding the scaling function spectrum from the wavelet spectrum. Many applications require wavelets that are matched to a signal of interest. Most current design techniques, however, do not design the wavelet directly. They either build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing multiresolution analysis or design a scaling function that generates a multiresolution analysis with some desired properties. In this paper, two sets of equations are developed that allow us to design the wavelet directly from the signal of interest. Both sets impose bandlimitedness, resulting in closed form solutions. The first set derives expressions for continuous matched wavelet spectrum amplitudes. The second set of equations provides a direct discrete algorithm for calculating close approximations to the optimal complex wavelet spectrum. The discrete solution for the matched wavelet spectrum amplitude is identical to that of the continuous solution at the sampled frequencies. An interesting byproduct of this work is the result that Meyer's spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary. Specific examples are given which demonstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals.     

Multiresolution FIR neural-network-based learning algorithm applied to network traffic prediction

In this paper, a multiresolution finite-impulse-response (FIR) neural-network-based learning algorithm using the maximal overlap discrete wavelet transform (MODWT) is proposed. The multiresolution learning algorithm employs the analysis framework of wavelet theory, which decomposes a signal into wavelet coefficients and scaling coefficients. The translation-invariant property of the MODWT allows alignment of events in a multiresolution analysis with respect to the original time series and, therefore, preserving the integrity of some transient events. A learning algorithm is also derived for adapting the gain of the activation functions at each level of resolution. The proposed multiresolution FIR neural-network-based learning algorithm is applied to network traffic prediction (real-world aggregate Ethernet traffic data) with comparable results. These results indicate that the generalization ability of the FIR neural network is improved by the proposed multiresolution learning algorithm.     

Multifractality Tests Using Bootstrapped Wavelet Leaders

Multifractal analysis, which mostly consists of measuring scaling exponents, is becoming a standard technique available in most empirical data analysis toolboxes. Making use of the most recent theoretical results, it is based here on the estimation of the cumulants of the log of the wavelet leaders, an elaboration on the wavelet coefficients. These log-cumulants theoretically enable discrimination between mono- and multifractal processes, as well as between simple log-normal multifractal models and more advanced ones. The goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data. Bootstrap issues together with six declinations of test designs are analyzed. Their statistical performance (significances, powers, and p-values) are assessed and compared by means of Monte Carlo simulations performed on synthetic stochastic processes whose multifractal properties (and log-cumulants) are known theoretically a priori. We demonstrate that the joint use of wavelet Leaders, log-cumulants, and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data. This tool is practically effective and can be applied to a single observation of data with finite length.     

Multiple scale correlation of signals by shift-invariant discrete wavelet transform

Correlation of signals at multiple scales of observation is useful for multiresolution interpretation of image, data and target signature analysis. Multiresolution analysis is inherent in the discrete wavelet transform (DWT), but shift-variance of the coefficients of the transform in dyadic orthogonal and biorthogonal basis spaces is the problem associated with it. Shift-variance of the transform and absence of a direct transform domain relationship make correlation of signals by the DWT inconvenient at multiple scales. The circulant shift property of the DWT coefficients is used in a novel way to produce correlation of signals at multiple scales with the critically sampled DWT only. The algorithm is derived in both discrete time and z-domain for signal vectors of finite duration. The algorithm is independent of signal waveform and wavelet kernel and is applied particularly for multiple scale correlation of radar signals, namely linear frequency modulated (LFM) chirp signals.     

Shift-orthogonal wavelet bases

Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level but not with respect to dilations across scales. We characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and synthesis function spaces and obtain the corresponding shift-orthogonal basis functions by suitable orthogonalization. In the second approach, we take the complementary view and start from the digital filterbank. We present several illustrative examples, including a hybrid version of the Battle-Lemarie (1987, 1988) spline wavelets. We also provide filterbank formulas for the fast wavelet algorithm. A shift-orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function; both transforms have essentially the same approximation properties. One experimentally confirmed benefit of relaxing the interscale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart     

Multiresolution wavelet analysis of evoked potentials

Neurological injury, such as from cerebral hypoxia, appears to cause complex changes in the shape of evoked potential (EP) signals. To characterize such changes we analyze EP signals with the aid of scaling functions called wavelets. In particular, we consider multiresolution wavelets that are a family of orthonormal functions. In the time domain, the multiresolution wavelets analyze EP signals at coarse or successively greater levels of temporal detail. In the frequency domain, the multiresolution wavelets resolve the EP signal into independent spectral bands. In an experimental demonstration of the method, somatosensory EP signals recorded during cerebral hypoxia in anesthetized cats are analyzed. Results obtained by multiresolution wavelet analysis are compared with conventional time-domain analysis and Fourier series expansions of the same signals. Multiresolution wavelet analysis appears to be a different, sensitive way to analyze EP signal features and to follow the EP signal trends in neurologic injury. Two characteristics appear to be of diagnostic value: the detail component of the MRW displays an early and a more rapid decline in response to hypoxic injury while the coarse component displays an earlier recovery upon reoxygenation     

小波变换实现图像导向滤波

导向滤波器是一类能够实现图像方向滤波的滤波器组,它比方向滤波器计算量小并且可以实现各个方向的滤波处理。该文把小波函数和多分辨关系引入到导向滤波处理过程中,实现了图像的方向滤波。实验验证了小波可以从图像中提取更多方向的信息,表明新导向滤波的有效性。     

基于小波分析的分形参数估计新方法

该文研究目的是估计1/f类分形随机过程参数矢量 (, 2,2w)。作者基于小波分析,对1/f过程观测值的小波系数方差进行一系列代数运算,并给出详尽的证明过程,最终求取了噪声中分数布朗运动(fBm)参数矢量的估计量。实验结果表明,与传统的极大似然估计(ML)相比,算法简洁,效果良好,估计参数范围广泛,同时对噪声也不再局限于高斯分布。     

Multiresolution reconstruction in fan-beam tomography

In this paper, a new multiresolution reconstruction approach for fan-beam tomography is established. The theoretical development assumes radial wavelets. An approximate reconstruction formula based on a near-radial quincunx multiresolution scheme is proposed. This multiresolution algorithm allows to compute both the quincunx approximation and detail coefficients of an image from its fan-beam projections. Simulations on mathematical phantoms show that wavelet decomposition is acceptable for small beam angles but deteriorates at high angles. The main applications of the method are denoising and wavelet-based image analysis.     

A matrix-valued wavelet KL-like expansion for wide-sense stationary random processes

Matrix-valued wavelet series expansions for wide-sense stationary processes are studied in this paper. The expansion coefficients a are uncorrelated matrix random process, which is a property similar to that of a matrix Karhunen-Loe/spl grave/ve (MKL) expansion. Unlike the MKL expansion, however, the matrix wavelet expansion does not require the solution of the eigen equation. This expansion also has advantages over the Fourier series, which is often used as an approximation to the MKL expansion in that it completely eliminates correlation. The basis functions of this expansion can be obtained easily from wavelets of the Matrix-valued Lemarie/spl acute/-Meyer type and the power-spectral density of the process.     

The contourlet transform: an efficient directional multiresolution image representation.

The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications. Index Terms-Contourlets, contours, filter banks, geometric image processing, multidirection, multiresolution, sparse representation, wavelets.