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 functionφwith multiplicity greater than one,we prove that we can always derive fromφa 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 functionφ(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 fromφsatisfying 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.

Framelet变换高光谱图像光谱加权稀疏解混

空域中高光谱数据由于信息过于分散,冗余过多,且易受噪声的影响,其特征提取难度较大。为了提高高光谱图像解混的鲁棒性和稀疏性,提出了一种framelet变换高光谱图像光谱加权稀疏解混方法。介绍了高光谱稀疏解混和framelet变换方法的理论知识,接着利用framelet变换对高光谱图像解混建模,并且在该模型上加入变换域光谱加权稀疏正则项,提出framelet变换的高光谱图像光谱加权稀疏解混模型。最后,利用交替方向乘子法对模型进行求解。实验结果表明:信号与重建误差比(SRE)提高12.4%~1045%,丰度重构正确率(Ps)保持在16%的误差内。与其他相关稀疏解混方法相比,本文提出的算法具有良好的抗噪性和稀疏性能,获得了更好的解混结果。

Construction of Multivariate Tight Framelet Packets Associated with Dilation Matrix

In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for L2 （JRa） by replacing some mother framelets.

Framelet变换结合HSV的多模态图像融合方法

为了获得轮廓清晰、细节丰富的多模态医学图像融合结果,提出framelet变换结合人类视觉系统(HVS)的图像融合方法.首先,将所有输入图像分解为低频和高频图像;然后,根据不同频率图像的物理意义融合图像和人类视觉系统,提出两种基于HVS的图像融合规则,分别用于融合低频和高频图像,即使用可见性融合方法融合低频图像,使用HVS模型的纹理信息融合高频图像;最后,通过反变换将所有framelet变换图像重建为融合图像.实验采用CT/MRI两种模态的脑部图像,以及老年痴呆临床PET/MRI图像,与主成分分析法、对比度法、梯度金字塔法、小波变换法和轮廓变换法相比,提出的方法融合结果在熵、互信息等多个评估标准上均有较大提升,可视化信息更加丰富.

FRAMELET BASED DECONVOLUTION

In this paper, two framelet based deconvolution algorithms are proposed. The basic idea of framelet based approach is to convert the deconvolution problem to the problem of inpainting in a frame domain by constructing a framelet system with one of the masks being the given （discrete） convolution kernel via the unitary extension principle of [26], as introduced in [6-9] . The first algorithm unifies our previous works in high resolution image reconstruction and infra-red chopped and nodded image restoration, and the second one is a combination of our previous frame-based deconvolution algorithm and the iterative thresholding algorithm given by [14, 16]. The strong convergence of the algorithms in infinite dimensional settings is given by employing proximal forward-backward splitting （PFBS） method. Consequently, it unifies iterative algorithms of infinite and finite dimensional setting and simplifies the proof of the convergence of the aluorithms of [6].

ON ALGORITHMS FOR AUTOMATIC DEBLURRING FROM A SINGLE IMAGE

In this paper, we study two variational blind deblurring models for a single linage,The first model is to use the total variation prior in both image and blur, while the second model is to use the flame based prior in both image and blur. The main contribution of this paper is to show how to employ the generalized cross validation （GCV） method efficiently and automatically to estimate the two regularization parameters associated with the priors in these two blind motion deblurring models. Our experimental results show that the visual quality of restored images by the proposed method is very good, and they are competitive with the tested existing methods. We will also demonstrate the proposed method is also very efficient.

Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system

The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling theory to recover a function by its nonuniform sampling.In the present paper,based on dual framelet systems for the Sobolev space pair(H^(s)(R^(d)),H^(-s)(R^(d))),where d/2<s<■,we investigate the problem of constructing the approximations to all the functions in H^(■)(R^(d))by nonuniform sampling.We first establish the convergence rate of the framelet series in(H^(s)(R^(d)),H^(-s)(R^(d))),and then construct the framelet approximation operator that acts on the entire space H^(■)(R^(d)).We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters,and obtain an estimate bound for the perturbation error.Our result shows that under the condition d/2<s<■,the approximation operator is robust to shift perturbations.Motivated by Hamm(2015)’s work on nonuniform sampling and approximation in the Sobolev space,we do not require the perturbation sequence to be in■^(α)(Z^(d)).Our results allow us to establish the approximation for every function in H^(■)(R^(d))by nonuniform sampling.In particular,the approximation error is robust to the jittering of the samples.