Tight Wavelet Frames Research Articles

On Some Results Related to Non-Uniform Tight Wavelet Frames

On Some Results Related to Non-Uniform Tight Wavelet Frames

An Efficient Inexact Gauss–Seidel-Based Algorithm for Image Restoration with Mixed Noise

A challenge in image restoration is to recover a clear image from the blurry observation in the presence of different types of noise. There are few works addressing image deblurring under mixed noise. To handle this issue, we propose a general model based on classical wavelet tight frame regularization. We utilize a convexity-preserving term to obtain a component-wise convex model under a mild condition. Indeed, to reduce the cost of solving subproblems, the inexact Gauss–Seidel-based majorized semi-proximal alternating direction method of multipliers (sGS-imsPADMM) with relative error control is developed. Besides, the global convergence of sGS-imsPADMM is demonstrated. Numerical results for the image restoration problems show that the proposed model and solving approach are superior to some state-of-the-art methods both in numerical analysis and visual quality.

Tight wavelet frames on the space of M-positive vectors

Wavelets on the sets of [Formula: see text]-positive vectors in the Euclidean space are studied. These sets are multidimensional analogs of the half-line in the Walsh analysis. Following the ideas of the Walsh analysis, the space of [Formula: see text]-positive vectors is equipped with a coordinate-wise addition. Harmonic analysis on this space is also similar to the Walsh harmonic analysis, and the Fourier transform is such that there exists a class of so-called test functions (with a compact support of the function itself and of its Fourier transform). Tight wavelet frames consisting of the test functions are studied. A complete description of the masks generating such frames is given, and an algorithmic method for constructing them is developed. These frames may be very useful for applications to signal processing because some examples of such systems on the half-line were already investigated in this aspect, and it appeared that they have an advantage over classical wavelet systems when used for processing fractal signals and images.

On Approximation by Tight Wavelet Frames on the Field of $$p$$-Adic Numbers

On Approximation by Tight Wavelet Frames on the Field of $$p$$-Adic Numbers

Joint Non-Local Statistical and Wavelet Tight Frame Information-Based ℓ₀ Regularization Model for Image Deblurring

Joint Non-Local Statistical and Wavelet Tight Frame Information-Based ℓ₀ Regularization Model for Image Deblurring

Fast deep learning with tight frame wavelets

Fast deep learning with tight frame wavelets

Tight wavelet frames on zero-dimensional groups. Construction and approximation

We present a method for constructing tight wavelet frames on locally compact zero-dimensional groups. When constructing frames, we do not use the principle of unitary extension. We also consider the approximate properties of the resulting frames for functions from the Sobolev space with logarithmic weight.

On p-adic tight wavelet frames

We propose a simple method to constructing step mask and corresponding step wavelet functions that generate tight wavelet frames on the field Qp of p-adic numbers. To construct this mask we use three new ideas. First, we consider only an additive group of the field Qp. Second, to construct the mask we use a special tree. Thirdly, we do not use the principle of unitary extension, we use Pontryagin's principle of duality instead. Using this method, we obtain all step refinable functions.

Characterization of tight wavelet frames with composite dilations in L2(Rn)

Tight wavelet frames are different from the orthonormal wavelets because of redundancy. By sacrificing orthonormality and allowing redundancy, the tight wavelet frames become much easier to construct than the orthonormal wavelets. Guo, Labate, Lim, Weiss, and Wilson [Electron. Res. Announc. Am. Math. Soc. 10 (2004), 78-87] introduced the theory of wavelets with composite dilations in order to provide a framework for the construction of waveforms defined not only at various scales and locations but also at various orientations. In this paper, we provide the characterization of composite wavelet system to be tight frame for L2(Rn).

Kullback-Leibler Divergence-Based Fuzzy C-Means Clustering Incorporating Morphological Reconstruction and Wavelet Frames for Image Segmentation.

In this article, we elaborate on a Kullback-Leibler (KL) divergence-based Fuzzy C -Means (FCM) algorithm by incorporating a tight wavelet frame transform and morphological reconstruction (MR). To make membership degrees of each image pixel closer to those of its neighbors, a KL divergence term on the partition matrix is introduced as a part of FCM, thus resulting in KL divergence-based FCM. To make the proposed FCM robust, a filtered term is augmented in its objective function, where MR is used for image filtering. Since tight wavelet frames provide redundant representations of images, the proposed FCM is performed in a feature space constructed by tight wavelet frame decomposition. To further improve its segmentation accuracy (SA), a segmented feature set is reconstructed by minimizing the inverse process of its objective function. Each reconstructed feature is reassigned to the closest prototype, thus modifying abnormal features produced in the reconstruction process. Moreover, a segmented image is reconstructed by using tight wavelet frame reconstruction. Finally, supporting experiments coping with synthetic, medical, and real-world images are reported. The experimental results exhibit that the proposed algorithm works well and comes with better segmentation performance than other peers. In a quantitative fashion, its average SA improvements over its peers are 4.06%, 3.94%, and 4.41%, respectively, when segmenting synthetic, medical, and real-world images. Moreover, the proposed algorithm requires less time than most of the FCM-related algorithms.

Multivariate tight wavelet frames with few generators and high vanishing moments

Tight wavelet frames (TWFs) are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with TWF construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to TWFs with high numbers of vanishing moments for all of the wavelet masks, while still having few highpass masks: in fact, we match the best known upper bound on the number of highpass masks for general box spline TWF constructions, while typically achieving much better vanishing moments for all of the wavelet masks, proving a nontrivial lower bound on this quantity.

A Brief Survey of the Graph Wavelet Frame

In recent years, the research of wavelet frames on the graph has become a hot topic in harmonic analysis. In this paper, we mainly introduce the relevant knowledge of the wavelet frames on the graph, including relevant concepts, construction methods, and related theory. Meanwhile, because the construction of graph tight framelets is closely related to the classical wavelet framelets on ℝ, we give a new construction of tight frames on ℝ. Based on the pseudosplines of type II, we derive an MRA tight wavelet frame with three generators ψ1, ψ2, and ψ3 using the oblique extension principle (OEP), which generate a tight wavelet frame in L2(ℝ). We analyze that three wavelet functions have the highest possible order of vanishing moments, which matches the order of the approximation order of the framelet system provided by the refinable function. Moreover, we introduce the construction of the Haar basis for a chain and analyze the global orthogonal bases on a graph . Based on the sequence of framelet generators in L2(ℝ) and the Haar basis for a coarse‐grained chain, the decimated tight framelets on graphs can be constructed. Finally, we analyze the detailed construction process of the wavelet frame on a graph.

Image restoration: Structured low rank matrix framework for piecewise smooth functions and beyond

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

Large Graph Signal Denoising With Application to Differential Privacy

Over the last decade, signal processing on graphs has become a very active area of research. Specifically, the number of applications, for instance in statistical or deep learning, using frames built from graphs, such as wavelets on graphs, has increased significantly. We consider in particular the case of signal denoising on graphs via a data-driven wavelet tight frame methodology. This adaptive approach is based on a threshold calibrated using Stein's unbiased risk estimate adapted to a tight-frame representation. We make it scalable to large graphs using Chebyshev-Jackson polynomial approximations, which allow fast computation of the wavelet coefficients, without the need to compute the Laplacian eigendecomposition. However, the overcomplete nature of the tight-frame, transforms a white noise into a correlated one. As a result, the covariance of the transformed noise appears in the divergence term of the SURE, thus requiring the computation and storage of the frame, which leads to an impractical calculation for large graphs. To estimate such covariance, we develop and analyze a Monte-Carlo strategy, based on the fast transformation of zero mean and unit variance random variables. This new data-driven denoising methodology finds a natural application in differential privacy. A comprehensive performance analysis is carried out on graphs of varying size, from real and simulated data.

A fractional-order regularization with sparsity constraint for blind restoration of images

Blind image deconvolution is one of the most challenging problems in image restoration. Inspired by the work on sparsity constraint and deblurring of blind motion, we propose a model with fractional-order regularization and sparsity constraint for blind restoration and construct split Bregman combining an iterative thresholding algorithm. Fractional-order penalty term in Besov space is expanded by wavelet basis and computed using iterative thresholding algorithm. The regularized terms of blur kernel under tight wavelet frame systems are solved by the split Bregman method. Numerical experiments show that our algorithm can effectively remove different kinds of blur without requiring any prior information of the blur kernels and obtain higher signal-to-noise ratios and lower relative errors. In addition, fractional-order derivative in Besov space can preserve both edges and smoothness better than the integer-order derivative.

Density of Frame Wavelets and Tight Frame Wavelets in Local Fields

Density of Frame Wavelets and Tight Frame Wavelets in Local Fields

An Efficient Nonconvex Regularization Method for Wavelet Frame Based Compressed Sensing Recovery

Abstract In this paper, we propose a variation model which takes advantage of the wavelet tight frame and nonconvex shrinkage penalties for compressed sensing recovery. We address the proposed optimization problem by introducing a adjustable parameter and a firm thresholding operations. Numerical experiment results show that the proposed method outperforms some existing methods in terms of the convergence speed and reconstruction errors. JEL classification numbers: 68U10, 65K10, 90C25, 62H35. Keywords: Compressed Sensing, Nonconvex, Firm thresholding, Wavelet tight frame.

Frame multiresolution analysis of continuous piecewise linear functions

The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.

Sparse Regularization-Based Fuzzy C-Means Clustering Incorporating Morphological Grayscale Reconstruction and Wavelet Frames

The conventional fuzzy C-means (FCM) algorithm is not robust to noise and its rate of convergence is generally impacted by data distribution. Consequently, it is challenging to develop FCM-related algorithms that have good performance and require less computing time. In this article, we elaborate on a comprehensive FCM-related algorithm for image segmentation. To make FCM robust, we first utilize a morphological grayscale reconstruction (MGR) operation to filter observed images before clustering, which guarantees noise-immunity and image detail-preservation. Since real images can generally be approximated by sparse coefficients in a tight wavelet frame system, feature spaces of observed and filtered images can be obtained. Taking such features to be clustered, we investigate an improved FCM model in which a sparse regularization term is introduced into the objective function of FCM. We design a three-step iterative algorithm to solve the sparse regularization-based FCM model, which is constructed by the Lagrangian multiplier method, hard-threshold operator, and normalization operator, respectively. Such an algorithm can not only perform well for image segmentation, but also come with high computational efficiency. To further enhance the segmentation accuracy, we use MGR to filter the label set generated by clustering. Finally, a large number of supporting experiments and comparative studies with other FCM-related algorithms available in the literature are provided. The obtained results for synthetic, medical and color images indicate that the proposed algorithm has good ability for multiphase image segmentation, and performs better than other alternative FCM-related algorithms. Moreover, the proposed algorithm requires less time than most of the existing algorithms.

Univariate tight wavelet frames of minimal support

This work characterizes (dyadic homogeneous) wavelet frames for $$L^2({{\mathbb {R}}})$$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated with the dilation operator. The approach is closely related to usual Fourier domain fiberization techniques, dual Gramian analysis, and extension principles. Spectral formulas are used to determine all the tight wavelet frames for $$L^2({{\mathbb {R}}})$$ with a fixed finite number of generators of minimal support. The method associates wavelet frames of this type with certain inner operator-valued functions in Hardy spaces. The cases with one and two generators are completely solved.