Wavelet Frames Research Articles

Nonhomogeneous Wavelet Dual Frames and Extension Principles in Reducing Subspaces

It can be seen from the literature that nonhomogeneous wavelet frames are much simpler to characterize and construct than homogeneous ones. In this work, we address such problems in reducing subspaces of L2ℝd. A characterization of nonhomogeneous wavelet dual frames is obtained, and by using the characterization, an MOEP and an MEP are derived under general assumptions for such wavelet dual frames.

Generalized matrix spectral factorization and quasi-tight framelets with a minimum number of generators

As a generalization of orthonormal wavelets in L 2 ( R ) L_2({\mathbb {R}}) , tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large family of refinable functions cannot be used to construct tight framelets. This motivates us to introduce the notion of a quasi-tight framelet, which is a dual framelet but behaves almost like a tight framelet. It turns out that the study of quasi-tight framelets is intrinsically linked to the problem of the generalized matrix spectral factorization for matrices of Laurent polynomials. In this paper, we provide a systematic investigation on the generalized matrix spectral factorization problem and compactly supported quasi-tight framelets. As an application of our results on generalized matrix spectral factorization for matrices of Laurent polynomials, we prove in this paper that from any arbitrary compactly supported refinable function in L 2 ( R ) L_2({\mathbb {R}}) , we can always construct a compactly supported one-dimensional quasi-tight framelet having the minimum number of generators and the highest possible order of vanishing moments. Our proofs are constructive and supplemented by step-by-step algorithms. Several examples of quasi-tight framelets will be provided to illustrate the theoretical results and algorithms developed in this paper.

Orbits of bounded bijective operators and Gabor frames

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $$L^2(\mathbb {R})$$ , which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $$\mathbb {Z}$$ , which are orbits of bounded operators on $$L^2(\mathbb {R})$$ . Two classes of overcomplete Gabor frames which cannot be ordered over $$\mathbb {Z}$$ and represented by orbits of operators in $$GL(L^2(\mathbb {R}))$$ are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames.

Bayesian Deconvolution and Quantification of Metabolites from J-Resolved NMR Spectroscopy

Two-dimensional (2D) nuclear magnetic resonance (nmr) methods have become increasingly popular in metabolomics, since they have considerable potential to accurately identify and quantify metabolites within complex biological samples. 2D 1H J-resolved (jres) nmr spectroscopy is a widely used method that expands overlapping resonances into a second dimension. However, existing analytical processing methods do not fully exploit the information in the jres spectrum and, more importantly, do not provide measures of uncertainty associated with the estimates of quantities of interest, such as metabolite concentration. Combining the data-generating mechanisms and the extensive prior knowledge available in online databases, we develop a Bayesian method to analyse 2D jres data, which allows for automatic deconvolution, identification and quantification of metabolites. The model extends and improves previous work on one-dimensional nmr spectral data. Our approach is based on a combination of B-spline tight wavelet frames and theoretical templates, and thus enables the automatic incorporation of expert knowledge within the inferential framework. Posterior inference is performed through specially devised Markov chain Monte Carlo methods. We demonstrate the performance of our approach via analyses of datasets from serum and urine, showing the advantages of our proposed approach in terms of identification and quantification of metabolites.

Nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces in $$L^2({\mathbb {K}})$$

In this paper, we introduce the structure of nonuniform nonhomogeneous dual wavelet frames over non-Archimedean fields. A characterization of nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces over non-Archimedean fields is obtained.

A nonconvex [formula omitted] model for image restoration with impulse noise

In this paper, we propose a model for image restoration with blur and impulse noise, it is composed of l1 data fitting term and a nonconvex regularization term, which is the weighted difference of l1-norm and l2-norm based on wavelet frame. The combined model is difficult to solve by the corresponding Euler Lagrangian equations, here we solve it by alternating direction method of multipliers (ADMM). We describe the detailed process of the algorithm, and establish the convergence of the algorithm. The experimental outcomes on different blurs and different impulse noises demonstrate that the proposed approach is better than those existing methods in terms of standard signal-to-noise ratio (PSNR), relative error (ReErr), and visual quality.

Wavelet frames in $L^2(\mathbb{R}^d)$

We obtain a necessary and sufficient condition for the existence of wavelet frames. We define and study the synthesis and analysis operators associated with wavelet frames. We discuss some applications of operator value (OPV) frames in the theory of wavelet frames. Also, we discuss the minimal property of wavelet frame coefficients and study the property of over completeness of wavelet frames. Various characterizations of wavelet frame, Riesz wavelet basis and orthonormal wavelet basis are given. Further, dual wavelet frames are discussed and a characterization of dual wavelet frames is given. Finally, we give a characterization of a pair of biorthogonal Riesz bases.

The equivalence of F_{a}-frames

Structured frames such as wavelet and Gabor frames in L^{2}(mathbb {R}) have been extensively studied. But L^{2}(mathbb{ R}_{+}) cannot admit wavelet and Gabor systems due to mathbb{R}_{+} being not a group under addition. In practice, L^{2}(mathbb{R}_{+}) models the causal signal space. The function-valued inner product-based F_{a}-frame for L^{2}(mathbb{R}_{+}) was first introduced by Hasankhani Fard and Dehghan, where an F_{a}-frame was called a function-valued frame. In this paper, we introduce the notions of F_{a}-equivalence and unitary F_{a}-equivalence between F_{a}-frames, and present a characterization of the F_{a}-equivalence and unitary F_{a}-equivalence. This characterization looks like that of equivalence and unitary equivalence between frames, but the proof is nontrivial due to the particularity of F_{a}-frames.

Residual-Sparse Fuzzy C-Means Clustering Incorporating Morphological Reconstruction and Wavelet Frame

Instead of directly utilizing an observed image including some outliers, noise or intensity inhomogeneity, the use of its ideal value (e.g. noise-free image) has a favorable impact on clustering. Hence, the accurate estimation of the residual (e.g. unknown noise) between the observed image and its ideal value is an important task. To do so, we propose an $\ell_0$ regularization-based Fuzzy $C$-Means (FCM) algorithm incorporating a morphological reconstruction operation and a tight wavelet frame transform. To achieve a sound trade-off between detail preservation and noise suppression, morphological reconstruction is used to filter an observed image. By combining the observed and filtered images, a weighted sum image is generated. Since a tight wavelet frame system has sparse representations of an image, it is employed to decompose the weighted sum image, thus forming its corresponding feature set. Taking it as data for clustering, we present an improved FCM algorithm by imposing an $\ell_0$ regularization term on the residual between the feature set and its ideal value, which implies that the favorable estimation of the residual is obtained and the ideal value participates in clustering. Spatial information is also introduced into clustering since it is naturally encountered in image segmentation. Furthermore, it makes the estimation of the residual more reliable. To further enhance the segmentation effects of the improved FCM algorithm, we also employ the morphological reconstruction to smoothen the labels generated by clustering. Finally, based on the prototypes and smoothed labels, the segmented image is reconstructed by using a tight wavelet frame reconstruction operation. Experimental results reported for synthetic, medical, and color images show that the proposed algorithm is effective and efficient, and outperforms other algorithms.

On the use of the SOS metaheuristic algorithm in hybrid image fusion methods to achieve optimum spectral fidelity

ABSTRACTImage fusion aims to spatially enhance a low-resolution multispectral (MS) image by utilizing a high-resolution panchromatic (Pan) band. Various image fusion methodologies have been proposed with the aim to improve the spatial detail quality without deteriorating the colour content of the input MS image. Previous studies revealed the fact that there is no such thing as ‘the best image fusion method’ since all fusion methods cause either spectral distortion or spatial detail loss to some extent, which motivates the researchers to develop more advanced methods to keep the colour content while increasing the spatial detail quality. This study proposed to use the Symbiotic Organisms Search (SOS) metaheuristic algorithm in hybrid image fusion methods to achieve the optimum colour quality in the fused images. The SOS algorithm was used in two hybrid fusion approaches, one including the Intensity-Hue-Saturation (IHS) and Discrete Wavelet Transform (DWT) methods and the other one including the IHS and Discrete Wavelet Frame Transform (DWFT) methods. The results of the proposed methods were qualitatively and quantitatively compared in three test sites against those of eighteen widely-used image fusion methods. It was concluded that the proposed methods led to superior colour quality with both singlesensor and multisensor input images, regardless of the spatial resolution difference between the input images. The proposed methods were also found to be very successful in sharpening the images, despite the fact that their main purpose was to keep the colour content as much as possible.

Multiscaling frame multiresolution analysis and associated wavelet frames

Frame multiresolution analysis (FMRA) in [Formula: see text] is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in [Formula: see text], which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity [Formula: see text]. We characterize wavelet frames and Parseval wavelet frames for [Formula: see text] under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions [Formula: see text] in [Formula: see text] to be multiframe generators of [Formula: see text]. Especially, the second condition depends on the multiscaling FMRA and [Formula: see text] only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.

A Characterization of Multi-Wavelet Dual Frames in Sobolev Spaces

For the past few years, wavelet and multi-wavelet frames have attracted interest from researchers. In this paper, we address some of these problems in the setting of the Sobolev space, and characterize of multi-wavelet dual frames in these spaces by using a pair of equations.

Sparse View CT Image Reconstruction Based on Total Variation and Wavelet Frame Regularization

The sparse view problem of image reconstruction encountered in computed tomography (CT) is an important research issue due to its considerable potential in lowering radiation dose. Among the researches, the total variation (TV) method is especially effective in sparse view CT reconstruction for its good ability to preserve sharp edges and suppress noise. However, TV-based methods often produce undesired staircase artifacts in smooth regions of the reconstructed images since the reconstructed problem is usually ill-posed and TV regularization favors piecewise constant functions. Moreover, the image can be accurately approximated by sparse coefficients under a proper wavelet tight frame, which has good capability of sparsely estimating the piecewise smooth functions and the quality of reconstructed image can be improved by the sparse prior information. To deal with sparse view CT reconstruction problem, a minimization hybrid reconstruction model that incorporates TV with the wavelet frame has been proposed, which is to use the TV-norm of the low-frequency wavelet frame coefficients and the $\ell _{0} $ -norm of the high-frequency wavelet frame coefficients to eliminate staircase effect while maintaining sharp edges, simultaneously provide enough regularization in smooth regions. In addition, considering that the two regularization terms produce more parameters, an alternating direction method of multipliers (ADMM) algorithm has been applied to solve the minimization problem by iteratively minimization separately. Finally, compared with several iterative reconstruction methods, the experimental results demonstrate the competitiveness of the proposed method in terms of preserving edges, suppressing staircase artifacts and denoising.

Dualwavelet frames in Sobolev spaces on local fields of positive characteristic

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. In this article, we obtain the characterization of nonhomogeneous wavelet frames and nonhomogeneous dual wavelet frames in a Sobolev spaces on a local field of positive characteristic by means of a pair of equations.

Comparative Study of Compressed Sensing for Heart Sound Acquisition in Wireless Body Sensor Networks

With the application of wireless body sensing network to real-time monitoring of biological signals, compressed sensing (CS) has become a promising signal acquisition technology that can extend monitoring time, reduce equipment cost and decrease power consumption. This paper makes a comparative study on the acquisition of heart sound (HS) signals by CS, compares and analyzes the performance of wavelet basis, reconstruction algorithms and frames. Among a large number of experimental data records, the reconstruction performance under different compression rates is comparatively studied, and the suitable wavelet basis, reconstruction algorithm and signal frame size for HS acquisition are obtained. The presented performance records and comparative analyses can give benchmarks for future academic research, and are also believed to be beneficial for practical applications.

A Weighted Fidelity and Regularization-Based Method for Mixed or Unknown Noise Removal from Images on Graphs.

Image denoising technologies in a Euclidean domain have achieved good results and are becoming mature. However, in recent years, many real-world applications encountered in computer vision and geometric modeling involve image data defined in irregular domains modeled by huge graphs, which results in the problem on how to solve image denoising problems defined on graphs. In this paper, we propose a novel model for removing mixed or unknown noise in images on graphs. The objective is to minimize the sum of a weighted fidelity term and a sparse regularization term that additionally utilizes wavelet frame transform on graphs to retain feature details of images defined on graphs. Specifically, the weighted fidelity term with ℓ1-norm and ℓ2-norm is designed based on a analysis of the distribution of mixed noise. The augmented Lagrangian and accelerated proximal gradient methods are employed to achieve the optimal solution to the problem. Finally, some supporting numerical results and comparative analyses with other denoising algorithms are provided. It is noted that we investigate image denoising with unknown noise or a wide range of mixed noise, especially the mixture of Poisson, Gaussian, and impulse noise. Experimental results reported for synthetic and real images on graphs demonstrate that the proposed method is effective and efficient, and exhibits better performance for the removal of mixed or unknown noise in images on graphs than other denoising algorithms in the literature. The method can effectively remove mixed or unknown noise and retain feature details of images on graphs. It delivers a new avenue for denoising images in irregular domains.

Whale Optimization for Wavelet-Based Unsupervised Medical Image Segmentation: Application to CT and MR Images

Image segmentation plays crucial role in medical image analysis and forms the basis for clinical diagnosis and patient’s treatment planning. But the large variation in organ shapes, inhomogeneous intensities, poor contrast, organic nature of textures and complex boundaries in medical images makes segmentation process adverse and challenging. Further, the absence of annotated ground-truth dataset in medical field limits the advantages of the trending deep learning techniques causing several setbacks. Though numerous unsupervised methods are reported in literature to combat the challenges in medical domain, achieving better segmentation quality still remains as an open issue. This work aim to address this issue integrating the strength of multiresolution analysis and the meta-heuristic optimization techniques for unsupervised medical image segmentation. The proposed approach employs undecimated wavelet frames to extract translation invariant texture and gray information at different orientations to effectively characterize the textures in medical images. Next, the approach introduces the latest meta-heuristic whale optimization algorithm (WOA), the global optimizer to enhance the performance of unsupervised clustering algorithm with optimized cluster centers to cluster the extracted wavelet texture features. Moreover, the study contributes to fill the gap in literature investigating for the first time different intelligence algorithms such as fuzzy, genetic algorithm (GA) and particle swarm optimization (PSO) for unsupervised medical image segmentation to demonstrate the efficacy of the proposed approach. Evaluation was performed on medical CT and MR images based on feature similarity (FSIM), dice (DC) and feature of merits (FOMs). Experimental results demonstrates the supremacy of the proposed approach over other intelligence algorithms. Finally, statistical study with ANOVA analysis was carried out to confirm the significance of the proposed approach in determining the optimal solution and displaying promising segmentation results toward diagnostic support for radiologist.

Multiplicative noise removal via using nonconvex regularizers based on total variation and wavelet frame

Total variation based regularization and wavelet frame based regularization are two different methods for image restoration. Moreover, nonconvex regularization has advantages over convex regularization for preserving edges in images. In this paper, a variational model consisting of two nonconvex regularizers based respectively on total variation and wavelet frame is proposed for multiplicative noise removal. The proposed model combines the advantages of the nonconvex regularization and the hybrid regularization. So it can more effectively preserve edges and details in images than any single regularization can. The alternating minimization method is proposed to solve the novel model. Although the corresponding minimization problem is nonconvex, the convexity of all subproblems is guaranteed under some mild conditions. Therefore, we study the convergence property of the algorithm. Numerical experiments have proved that the proposed model with the corresponding algorithm outperforms some state-of-the-art models in terms of the peak signal-to-noise ratio and the relative error of the restored images.

On Parseval Wavelet Frames via Multiresolution Analyses in

AbstractWe give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

Embedding Distortion Analysis in Wavelet-domain Watermarking

Imperceptibility and robustness are two complementary fundamental requirements of any watermarking algorithm. Low-strength watermarking yields high imperceptibility, but exhibits poor robustness. High-strength watermarking schemes achieve good robustness but often infuse distortions resulting in poor visual quality in host images. This article analyses the embedding distortion for wavelet-based watermarking schemes. We derive the relationship between distortion, measured in mean square error (MSE), and the watermark embedding modification and propose the linear proportionality between MSE and the sum of energy of the selected wavelet coefficients for watermark embedding modification. The initial proposition assumes the orthonormality of discrete wavelet transform. It is further extended for non-orthonormal wavelet kernels using a weighting parameter that follows the energy conservation theorems in wavelet frames. The proposed analysis is verified by experimental results for both non-blind and blind watermarking schemes. Such a model is useful to find the optimum input parameters, including the wavelet kernel, coefficient selection, and subband choices for wavelet domain image watermarking.