Showing papers in "International Journal of Wavelets, Multiresolution and Information Processing in 2008"

The uncertainty principle associated with the continuous shearlet transform

Abstract: Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.

FUSION FRAMES AND g-FRAMES IN HILBERT C*-MODULES

Abstract: The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

Hilbert–schmidt operators and frames — classification, best approximation by multipliers and algorithms

Abstract: In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

Bandwidth empirical mode decomposition and its application

Abstract: There are some methods to decompose a signal into different components such as: Fourier decomposition and wavelet decomposition But they have limitations in some aspects Recently, there is a new signal decomposition algorithm called the Empirical Mode Decomposition (EMD) Algorithm which provides a powerful tool for adaptive multiscale analysis of nonstationary signals Recent works have demonstrated that EMD has remarkable effect in time series decomposition, but EMD also has several problems such as scale mixture and convergence property This paper proposes two key points to design Bandwidth EMD to improve on the empirical mode decomposition algorithm By analyzing the simulated and actual signals, it is confirmed that the Intrinsic Mode Functions (IMFs) obtained by the bandwidth criterion can approach the real components and reflect the intrinsic information of the analyzed signal In this paper, we use Bandwidth EMD to decompose electricity consumption data into cycles and trend which help us recognize the structure rule of the electricity consumption series

Compactly supported multivariate, pairs of dual wavelet frames obtained by convolution

Abstract: In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e. the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function.

Sampling expansion in shift invariant spaces

Abstract: For any ϕ(t) in L2(ℝ), let V(ϕ) be the closed shift invariant subspace of L2(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L2(ℝ) that belongs to a sampling subspace of L2(ℝ). Several illustrating examples are also provided.

Multisensor remote sensing image fusion using stationary wavelet transform: effects of basis and decomposition level

Abstract: Stationary wavelet transform is an efficient algorithm for remote sensing image fusion. In this paper, we investigate the effects of orthogonal/biorthogonal filters and decomposition depth on using...

MONO-COMPONENTS VS IMFs IN SIGNAL DECOMPOSITION

Abstract: The concepts of intrinsic mode functions and mono-components are investigated in relation to the empirical mode decomposition Mono-components are defined to be the functions for which non-negative analytic instantaneous frequency is well defined We show that a great variety of functions are mono-components based on which adaptive decomposition of signals are theoretically possible We justify the role of empirical mode decomposition in signal decomposition in relation to mono-components

The dyadic lifting schemes and the denoising of digital images

Abstract: The dyadic lifting schemes, which generalize Sweldens lifting schemes, have been proposed for custom-design of dyadic and bi-orthogonal wavelets and their duals. Starting with dyadic wavelets and exploiting the control provided in the form of free parameters, one can custom-design dyadic as well as bi-orthogonal wavelets adapted to a particular application. To validate the usefulness of the schemes, two construction methods have been proposed for designing dyadic wavelet filters with higher number of vanishing moments; using these design techniques, spline dyadic wavelet filters have been custom-designed for denoising of digital images, which exhibit enhanced denoising effects.

A wavelet-based image quality assessment method

Abstract: Image quality is a key characteristic in image processing,10,11 image retrieval,12,13 and biometrics.14 In this paper, a novel reduced-reference image quality assessment method is proposed based on wavelet transform. By simulating the human visual system, we take the variance of the visual sensitive coefficients into account to measure a distorted image. The computational complexity of the proposed method is much lower compared with some existing methods. Experimental results demonstrate its advantages in terms of correlation coefficient, outlier ratio, transmitted information, and CPU cost. Moreover, it is also illustrated that the proposed method has a good accordance with human subjective perception.

The continuous wavelet transform on conic sections

Abstract: We review the coherent state ( or group-theoretical) construction of the continuous wavelet transform (CWT) on the two-sphere. Next, we describe the construction of a CWT on the upper sheet of a two-sheeted hyperboloid, emphasizing the similarities between the two cases. Finally, we give some indications on the CWT on a paraboloid and we introduce a unified approach to the CWT on conic sections.

Reversible watermarking algorithm based on wavelet lifting scheme

Abstract: Nowadays, digital watermarking algorithms are widely applied to ownership protection and tampering detection of digital images. In this paper, a new reversible watermarking algorithm based on wavelet lifting scheme is proposed. In the algorithm, the image is firstly divided into some no-overlapping blocks, and then the wavelet lifting scheme where on every block, is performed watermarking data is embedded into the image according to the attribute of the subband of every block. In order to guarantee the security of algorithm, chaotic system is used to shuffle the position of blocks. The interesting point and usefulness of the algorithm lies in the fact that the watermarked image can be exactly restored into the original image, and the watermarking is robust to cropping. The experimental results show the effectiveness of this scheme.

Parametrization for balanced multifilter banks

Abstract: The parametrization for two kinds of multifilter banks generating balanced multiwavelets is presented in this paper. In case (I), both lowpass and highpass filters are flipping filters. Filters in ...

A wavelet support vector machine coupled method for time series prediction

Abstract: In this paper, a hybrid scheme for time series prediction is developed based on wavelet decomposition combined with Bayesian Least Squares Support Vector Machine regression. As a filtering step, us...

Subband variance computation of homoscedastic additive noise in discrete dyadic wavelet transform

Abstract: The paper deals with noise power variation that occurs when Discrete Dyadic Wavelet Transform (DDWT) is applied to signals affected by Wide Sense Stationary (WSS) additive white noise owing to the use of a non orthonormal expansion. An exact relationship between the noise variance in the original signal and the noise variance in the wavelet coefficients at a generic level is derived. This relationship is crucial in the application of wavelet thresholding for signal denoising to properly select the threshold in each subband.

Extraction of noise tolerant, gray-scale transform and rotation invariant features for texture segmentation using wavelet frames

Abstract: In this paper, we propose a texture feature extraction scheme at multiple scales and discuss the issues of rotation and gray-scale transform invariance as well as noise tolerance of a texture analysis system. The nonseparable discrete wavelet frame analysis is employed which gives an overcomplete wavelet decomposition of the image. The texture is decomposed into a set of frequency channels by a circularly symmetric wavelet filter, which in essence gives a measure of edge magnitudes of the texture at different scales. The texture is characterized by local energies over small overlapping windows around each pixel at different scales. The features so extracted are used for the purpose of multi-texture segmentation. A simple clustering algorithm is applied to this signature to achieve the desired segmentation. The performance of the segmentation algorithm is evaluated through extensive testing over various types of test images.

Depth estimation using multiwavelet analysis based stereo vision approach

Abstract: The problem of dimensional defects in aluminum die-casting is widespread throughout the foundry industry and their detection is of paramount importance in maintaining product quality. Due to the unpredictable factory environment and metallic, with highly reflective, nature of aluminum die-castings, it is extremely hard to estimate true dimensionality of the die-casting, autonomously. In this work, we propose a novel robust 3D reconstruction algorithm capable of reconstructing dimensionally accurate 3D depth models of the aluminum die-castings. The developed system is very simple and cost effective as it consists of only a stereo camera pair and a simple fluorescent light. The developed system is capable of estimating surface depths within the tolerance of 1.5 mm. Moreover, the system is invariant to illuminative variations and orientation of the objects in the input image space, which makes the developed system highly robust. Due to its hardware simplicity and robustness, it can be implemented in different factory environments without a significant change in the setup.

Construction of symmetric tight wavelet frames from quasi-interpolatory subdivision masks and their applications

Abstract: This paper presents tight wavelet frames with two compactly supported symmetric generators of more than one vanishing moments in the Unitary Extension Principle. We determine all possible free tension parameters of the quasi-interpolatory subdivision masks whose corresponding refinable functions guarantee our wavelet frame. In order to reduce shift variance of the standard discrete wavelet transform, we use the three times oversampling filter bank and eventually obtain a ternary (low, middle, high) frequency scale. In applications to signal/image denoising and erasure recovery, the results demonstrate reduced shift variance and better performance of our wavelet frame than the usual wavelet systems such as Daubechies wavelets.

Intelligent optimal control of robotic manipulators using wavelets

Abstract: The paper is concerned with the application of wavelet-based neural networks for optimal control of robotic manipulators motion. The model of robotic manipulators with regard to frictions and disturbances is nonlinear and uncertain. Optimal control law is found by the optimization of the Hamilton–Jacobi–Bellman (H-J-B) equation and it shows how wavelet-based neural networks can overcome nonlinearities through optimization without preliminary off-line learning phase. The neural network is learned as on-line and an adaptive learning algorithm is derived from the Lyapunov theory. This is so that both tracking stability and error convergence of the estimation for the nonlinear function can be guaranteed in the closed-loop system. The Lyapunov function for the nonlinear analysis is derived from the user input in terms of a specified quadratic performance index. Simulation results on a three-link robot manipulator show the satisfactory performance of the proposed control schemes even in the presence of large modeling uncertainties and external disturbances. Furthermore, it is shown that the tracking error for wavelet neural networks is less than conventional neural networks.

Perturbation techniques in irregular spline-type spaces

Abstract: In this paper we study various perturbation techniques in the context of irregular spline-type spaces. We first present the sampling problem in this general setting and prove a general result on the possibility of perturbing sampling sets. This result can be regarded as a spline-type space analogue in the spirit of Kadec's Theorem for bandlimited functions (see Refs. 14 and 15). We further derive some quantitative estimates on the amount by which a sampling set can be perturbed, and finally prove a result on the existence of optimal perturbations (with the stability of reconstruction being the optimality criterion). Finally, the techniques developed in the earlier parts of the paper are used to study the problem of disturbing a basis for a spline-type space, in order to derive a sufficient criterion for a space generated by irregular translations to be a spline-type space.

Iterative regularization and nonlinear inverse scale space based on translation invariant wavelet shrinkage

Abstract: In this paper, we present a new class of iterative regularization methods in the setting of Besov spaces, which can be seen as generalizations of J. Xu's method. By incorporating translation invariant wavelet transform, minimizers of the new methods can be understood as the alternative to translation invariant wavelet shrinkage with weight that is dependent on the wavelet decomposition scale and the Besov smooth order. And we generalize the iterative regularization methods to a new class of nonlinear inverse scale spaces with scale and Besov smooth order dependent weight. The numerical results show an excellent denoising effect and improvement over J. Xu's method.

An optimization algorithm for biorthogonal wavelet filter banks design

Abstract: A new approach for designing the Biorthogonal Wavelet Filter Bank (BWFB) for the purpose of image compression is presented in this paper. The approach is broken into two steps. First, an optimal filter bank is designed in the theoretical sense, based on Vaidyanathan's coding gain criterion in the SubBand Coding (SBC) system. Then, the above filter bank is optimized based on the criterion of Peak Signal-to-Noise Ratio (PSNR) in the JPEG2000 image compression system, resulting in a BWFB in practical application sense. With the approach, a series of BWFBs for a specific class of applications related to image compression, such as gray-level images, can be quickly designed. Here, new 7/5 BWFBs are presented based on the above approach for image compression applications. Experiments show that the 7/5 BWFBs not only have excellent compression performance, but also easy computation and are more suitable for VLSI hardware implementations. They perform equally well with respect to 7/5 filters in the JPEG2000 standard.

A wavelet based significance test for periodicities in indian monsoon rainfall

Abstract: This paper is a sequel to a recent study of the authors' that uses a combination of multiresolution analysis (MRA) and classical Fourier spectral methods, to identify 17 peaks in the power spectral...

Irregular sampling in shift invariant spaces of higher dimensions

Abstract: We consider irregular sampling in shift invariant spaces V of higher dimensions. The problem that we address is: find e so that given perturbations (λk) satisfying sup|λk| < e, we can reconstruct an arbitrary function f of V as a Riesz basis expansions from its irregular sample values f(k + λk). A framework for dealing with this problem is outlined and in which one can explicitly calculate sufficient limits e for the reconstruction. We show how it works in two concrete situations.

The construction of 2d rotationally invariant wavelets and their application in image edge detection

Abstract: Construction of rotationally invariant 2D wavelets is important in image processing, but is difficult. In this paper, the discrete form of a 2D rotationally invariant wavelet is constructed by back-projection from a 1D symmetrical wavelet. Such rotationally invariant 2D wavelets allow effective edge detection in any direction. These wavelets are combined with the 2D directional wavelets for the use in non-maximum suppression edge detection. The resulting binary edges are characterized by finer contours, differential detection characteristics and noise robustness compared to other edge detectors in various test images. In particular, where fine binary edges in noisy images are required, this novel approach compares favorably to the classical methods of Canny and Mallat with detection of more edges thanks to the implicit denoising properties and the full rotational invariance of the method.

Locally adaptive multiscale bayesian method for image denoising based on bivariate normal inverse gaussian distributions

Abstract: Recently, the use of wavelet transform has led to significant advances in image denoising applications. Among wavelet-based denoising approaches, the Bayesian techniques give more accurate estimates. Considering interscale dependencies, these estimates become closer to the original image. In this context, the choice of an appropriate model for wavelet coefficients is an important issue. The performance can also be improved by estimating model parameters in a local neighborhood. In this paper, we propose the bivariate normal inverse Gaussian (NIG) distribution, which can model a wide range of heavy-tailed to less heavy-tailed processes, to model the local wavelet coefficients at adjacent scales. We will show that this new statistical model is superior to the conventional generalized Gaussian (GG) model. Then, a minimum mean square error-based (MMSE-based) Bayesian estimator is designed to effectively remove noise from wavelet coefficients. Exploiting this new statistical model in the dual-tree complex wavelet domain, we achieved state-of-the-art performance among related recent denoising approaches, both visually and in terms of peak signal-to-noise ratio (PSNR).

Structure and texture image inpainting using sparse representations and an iterative curvelet thresholding approach

Abstract: Representing the image to be inpainted in an appropriate sparse dictionary, we introduce a novel method for the filling-in of structure and texture in regions of missing image information. In the morphological component analysis (MCA) inpainting approach, a TV penalty is added to better reduce ringing artifacts. However, the incorporation of TV penalty terms leads to PDE schemes that are numerically intensive. Inspired by the works of Daubechies–Teschke and Borup–Nielsen, we replace the TV term by a term. It results in an iterative curvelet thresholding scheme for the structure image inpainting. In the whole inpainting process, an alternative approach is presented to layer inpainting. Experimental results show the performance of the algorithm.

Secure and progressive image transmission through shadows generated by multiwavelet transform

Abstract: In this paper, multiwavelet transform is used in the context of image sharing employing the SPIHT-generated bit stream with the modified Shamir's (r, m) threshold scheme over the Galois field. This provides multiwavelet-based embedded shadow images for progressive transmission and reconstruction. Both integer multiwavelet transform and balanced multiwavelet transform are considered in the method. It is secure and fault-tolerant, and gives small shadow size and excellent rate-distortion performance. Experimental results and sample applications are given to demonstrate the characteristics of this method.

On convolution for wavelet transform

Abstract: A basic function D(x,y,z) associated with the general wavelet transform is defined and its properties are investigated. Using D(x,y,z) associated with the wavelet transform, translation and convolution for this transform are defined and certain existence theorems are proved. An approximation theorem involving wavelet convolution is also proved.

Characterizations of a class of orthogonal multiple vector-valued wavelet packets

Abstract: The notion of multiple vector-valued wavelet packets is introduced. A procedure for constructing the multiple vector-valued wavelet packets is presented. Their characteristics are investigated by means of integral transformation and operator theory, and three orthogonality formulas concerning the multiple vector-valued wavelet packets. Finally, new orthogonal bases of L2(R, Cs × s) are constructed from these multiple vector-valued wavelet packets.