Tight Framelets Research Articles

Exploring Structural Sparsity of Coil Images from 3-Dimensional Directional Tight Framelets for SENSE Reconstruction

Exploring Structural Sparsity of Coil Images from 3-Dimensional Directional Tight Framelets for SENSE Reconstruction

Convolutional Neural Networks for Spherical Signal Processing via Area-Regular Spherical Haar Tight Framelets.

In this article, we develop a general theoretical framework for constructing Haar-type tight framelets on any compact set with a hierarchical partition. In particular, we construct a novel area-regular hierarchical partition on the two spheres and establish its corresponding spherical Haar tight framelets with directionality. We conclude by evaluating and illustrate the effectiveness of our area-regular spherical Haar tight framelets in several denoising experiments. Furthermore, we propose a convolutional neural network (CNN) model for spherical signal denoising, which employs fast framelet decomposition and reconstruction algorithms. Experiment results show that our proposed CNN model outperforms threshold methods and processes strong generalization and robustness.

Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2×2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results.

A new time–frequency representation based on the tight framelet packet for telephone-band speech coding

To improve the quality and intelligibility of telephone-band speech coding, a new time–frequency representation based on a tight framelet packet transform is proposed in this paper. In the context of speech coding, the effectiveness of this representation stems from its resilience to quantization noise, and reconstruction stability. Moreover, it offers a sub-band decomposition and good time–frequency localization according to the critical bands of the human ear. The coded signal is obtained using dynamic bit allocation and optimal quantization of normalized framelet coefficients. The performances of the corresponding method are compared to the critically sampled wavelet packet transform. Extensive simulation revealed that the proposed speech coding scheme, which incorporates the tight framelet packet transform performs better than that based on the critically sampled wavelet packet transform. Furthermore, it ensures a high bit-rate reduction with negligible degradation in speech quality. The proposed coder is found to outperform the standard telephone-band speech coders in term of objective measures and subjective evaluations including a formal listening test. The subjective quality of our codec at 4 kbps is almost identical to the reference G.711 codec operating at 64 kbps.

High quality low-dose SPECT reconstruction using CGAN-based transformer network with geometric tight framelet

Single-photon emission computed tomography (SPECT) is a widely used diagnostic tool, but radioactive radiation during imaging poses potential health risks to subjects. Accurate low-dose single-photon emission computed tomography reconstruction is crucial in clinical applications of single-photon emission computed tomography. However, it remains a challenging problem due to the high noise and low spatial resolution of the low-dose reconstructed single-photon emission computed tomography images. The aim of the study is to develop a deep learning based framework for high quality low-dose single-photon emission computed tomography reconstruction. In the proposed framework, the conditional generative adversarial network (CGAN) was used as backbone structure and a Residual Attention CSwin Transformer (RACT) block was introduced as the basic building block for the generator of conditional generative adversarial network. The proposed residual attention CSwin transformer block has a dual-branch structure, which integrates the local modeling capability of CNN and the global dependency modeling capability of Transformer to improve the quality of single-photon emission computed tomography reconstructed images. More importantly, a novel loss term based on the geometric tight framelet (GTF) was designed to better suppress noise for the single-photon emission computed tomography reconstructed image while preserving the details of image to the greatest extent. Monte Carlo simulation software SIMIND was used to produce low-dose single-photon emission computed tomography images dataset to evaluate the performance of the proposed method. The simulation results showed that the proposed method can reduce more noise and preserve more details of various situations compared to several recent methods. To further validate the performance of the proposed method, we also verified the generalization ability of the proposed method, which is more adaptable to different noise level scenarios than other methods. Our results indicated that the proposed framework has the potential to reduce the tracer dose required for single-photon emission computed tomography imaging without compromising the quality of the reconstructed images.

Piecewise Weighted Smoothing Regularization in Tight Framelet Domain for Hyperspectral Image Restoration

Hyperspectral images captured by remote-sensing satellites are easily corrupted by various types of noise. Generally, hyperspectral signatures appear to be scattered in spatial-spectral domain, as well as noise. In transform domain, however, the principal components of a image are often centralized in the low-frequency band, while noise and some details are mainly contained in high-frequency components. The traditional transformation domain based smoothing methods take no account of the respective information distribution carried by different frequency bands. Hyperspectral image restoration aims to remove mixed noise for clean data, which usually amounts to an ill-posed inverse problem. Moreover, the matrix-decomposition-based model needs to reshape hyperspectral datacube into matrix form, which will lead to certain loss of spatial structure information. In order to address these issues, this paper incorporates piecewise weighted smoothing regularization in tight framelet domain to reformulate a novel convex model for hyperspectal image restoration, which maintains image signature to a great extent. Based on the noise-perturbed degradation model, the smoothing regularizer in tight framelet domain is imposed on abundance signature, in which the transformation matrix, as well as the weighted coefficient, is well designed for different frequency bands. As a surrogate of sparsity, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{q}$ </tex-math></inline-formula> -norm is employed to denoise for the enhancement of hyperspectral signatures. To the end, an efficient solver is carefully designed to derive the closed-form solutions by proximal alternating optimization. The experimental results on several synthetic and real datasets, not only demonstrate that the performance of proposed method is better than that of the current state-of-the-art approaches, but also verify the validation of regularization terms for hyperspectral image restoration.

A tailor-made 3-dimensional directional Haar semi-tight framelet for pMRI reconstruction

In this paper, we propose a model for parallel magnetic resonance imaging (pMRI) reconstruction, regularized by a carefully designed tight framelet system, that can lead to reconstructed images with much less artifacts in comparison to those from existing models. Our model is motivated from the observations that each receiver coil in a pMRI system is more sensitive to the specific object nearest to the coil, and all coil images are correlated. To exploit these observations, we first stack all coil images together as a 3-dimensional (3D) data matrix, and then design a 3D directional Haar tight framelet (3DHTF) to represent it. After analyzing sparse information of the coil images provided by the high-pass filters of the 3DHTF, we separate the high-pass filters into effective ones and ineffective ones, and we then devise a 3D directional Haar semi-tight framelet (3DHSTF) from the 3DHTF by replacing its ineffective filters with only one filter. This 3DHSTF is tailor-made for coil images, meanwhile, giving a significant saving in computation comparing to the 3DHTF. With the 3DHSTF, we propose an ℓ 1 -3DHSTF model for pMRI reconstruction. Numerical experiments for MRI phantom and in-vivo data sets are provided to demonstrate the superiority of our ℓ 1 -3DHSTF model in terms of the efficiency of reducing aliasing artifacts in the reconstructed images.

Inverting Incomplete Fourier Transforms by a Sparse Regularization Model and Applications in Seismic Wavefield Modeling

We propose a sparse regularization model for inversion of incomplete Fourier transforms and apply it to seismic wavefield modeling. The objective function of the proposed model employs the Moreau envelope of the \(\ell _0\) norm under a tight framelet system as a regularization to promote sparsity. This model leads to a non-smooth, non-convex optimization problem for which traditional iteration schemes are inefficient or even divergent. By exploiting special structures of the \(\ell _0\) norm, we identify a local minimizer of the proposed non-convex optimization problem with a global minimizer of a convex optimization problem, which provides us insights for the development of efficient and convergence guaranteed algorithms to solve it. We characterize the solution of the regularization model in terms of a fixed-point of a map defined by the proximity operator of the \(\ell _0\) norm and develop a fixed-point iteration algorithm to solve it. By connecting the map with an \(\alpha \)-averaged nonexpansive operator, we prove that the sequence generated by the proposed fixed-point proximity algorithm converges to a local minimizer of the proposed model. Our numerical examples confirm that the proposed model outperforms significantly the existing model based on the \(\ell _1\)-norm. The seismic wavefield modeling in the frequency domain requires solving a series of the Helmholtz equation with large wave numbers, which is a computationally intensive task. Applying the proposed sparse regularization model to the seismic wavefield modeling requires data of only a few low frequencies, avoiding solving the Helmholtz equation with large wave numbers. This makes the proposed model particularly suitable for the seismic wavefield (SW) modeling. Numerical results show that the proposed method performs better than the existing method based on the \(\ell _1\) norm in terms of the SNR values and visual quality of the restored synthetic seismograms.

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.

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.

Regularization with multilevel non-stationary tight framelets for image restoration

Variational regularization models are one of the popular and efficient approaches for image restoration. The regularization functional in the model carries prior knowledge about the image to be restored. The prior knowledge, in particular for natural images, are the first-order (i.e. variance in luminance) and second-order (i.e. contrast and texture) information. In this paper, we propose a model for image restoration, using a multilevel non-stationary tight framelet system that can capture the image's first-order and second-order information. We develop an algorithm to solve the proposed model and the numerical experiments show that the model is effective and efficient as compared to other higher-order models.

Adaptive Directional Haar Tight Framelets on Bounded Domains for Digraph Signal Representations

Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set $$K\subseteq \mathbb {R}^d$$ . In particular, on the unit block $$[0,1]^d$$ , such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.

On the Approximation Solutions of linear and nonlinear Volterra Integral Equation of First and Second kinds by Using B-spline Tight Framelets Generated by Unitary Extension Principle and Oblique Extension Principle

On the Approximation Solutions of linear and nonlinear Volterra Integral Equation of First and Second kinds by Using B-spline Tight Framelets Generated by Unitary Extension Principle and Oblique Extension Principle

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.

Bi-orthogonal wavelets for investigating Gibbs effects via oblique extension principle

Gibbs effect is generally known for Fourier and Wavelets expansions of a function in the neighborhood of its discontinuities points which deals with the nonuniform convergence of its truncated sums of these expansions. We study this phenomenon using bi-orthogonal wavelets (or simply, framelets) using pseudo-spline tight framelets generated using the oblique extension principle. We present some examples to illustrate the results.

A collocation method via the quasi-affine biorthogonal systems for solving weakly singular type of Volterra-Fredholm integral equations

Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples.

Biorthogonal-Wavelet-Based Method for Numerical Solution of Volterra Integral Equations

Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work is devoted to shedding some light on the benefits of using such framelets in the area of numerical computations of integral equations. We introduce a new numerical method for solving Volterra integral equations. It is based on pseudo-spline quasi-affine tight framelet systems generated via the oblique extension principles. The resulting system is converted into matrix equations via these generators. We present examples of the generated pseudo-splines quasi-affine tight framelet systems. Some numerical results to validate the proposed method are presented to illustrate the efficiency and accuracy of the method.

Causal FIR symmetric paraunitary matrix extension and construction of symmetric tight M-dilated framelets

We consider a causal FIR symmetric paraunitary (PU) matrix extension with parameters, and the constructions of symmetric orthogonal wavelets and symmetric tight framelets with an integer dilation factor M ⩾ 2 . Firstly, we propose an algorithm for factorizing a causal FIR symmetric Laurent polynomial vector into the product of order-one paraunitary matrices and constant vector. Secondly, based on the factorization algorithm, we propose a method for a causal symmetric PU extension with parameters of a Laurent polynomial vector. This method enables us to parameterize polyphase matrix whose first row is equal to the Laurent polynomial vector composed of polyphase components of given lowpass filter. And this symmetric PU extension provides a minimal factorization structure. Thirdly, we consider the constructions of symmetric orthogonal wavelets and symmetric tight framelets with integer dilation factor by the causal symmetric PU extension. Finally, several examples are provided to illustrate the construction methods proposed in this paper.

On the Gibbs Effect Based on the Quasi-Affine Dual Tight Framelets System Generated Using the Mixed Oblique Extension Principle

Gibbs effect represents the non-uniform convergence of the nth Fourier partial sums in approximating functions in the neighborhood of their non-removable discontinuities (jump discontinuities). The overshoots and undershoots cannot be removed by adding more terms in the series. This effect has been studied in the literature for wavelet and framelet expansions. Dual tight framelets have been proven useful in signal processing and many other applications where translation invariance, or the resulting redundancy, is very important. In this paper, we will study this effect using the dual tight framelets system. This system is generated by the mixed oblique extension principle. We investigate the existence of the Gibbs effect in the truncated expansion of a given function by using some dual tight framelets representation. We also give some examples to illustrate the results.

Directional tensor product complex tight framelets for compressed sensing MRI reconstruction

Compressed sensing magnetic resonance imaging (CS-MRI) is an effective way of reducing the sampling data in the k-space and shortening the scanning time. Motivated by the high performance of directional tensor product complex tight framelets (TPCTFs) for the image denoising problem, the authors proposed a novel framework that integrated TPCTF for sparse representation and projected fast iterative soft-thresholding algorithm (pFISTA) for CS-MRI reconstruction. Furthermore, to take advantage of the cross-scale relations in the wavelet tree of frame coefficients, the bivariate shrinkage (BS) function with local variance estimation is proposed to shrink thresholding. Such TPCTFs can provide sparse directional representations very well for MR image. When compared with other the state-of-the-art CS-MRI algorithms in numerical experiments, the proposed TPCTF-BS method achieves a higher reconstruction quality with respect to image edge preservation and the artefact suppression.