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Xiaosheng Zhuang

Publications and source records attributed to Xiaosheng Zhuang.

32 records · Page 2Linked to original sources
arXiv2017

Linear multiscale transforms based on even-reversible subdivision operators

Multiscale transforms for real-valued data, based on interpolatory subdivision operators have been studied in recent year. They are easy to define, and can be extended to other types of data, for example to manifold-valued data. In this paper we define linear multiscale transforms, based on certain linear, non-interpolatory subdivision operators, termed "even-reversible". For such operators, we prove, using Wiener's lemma, the existence of an inverse to the linear operator defined by the even part of the subdivision mask, and termed it "even-inverse". We show that the non-interpolatory subdivision operators, with spline or with pseudo-spline masks, are even-reversible, and derive explicitly, for the quadratic and cubic spline subdivision operators, the symbols of the corresponding even-inverse operators. We also analyze properties of the multiscale transforms based on even-reversible subdivision operators, in particular, their stability and the rate of decay of the details.

math.NA↗
arXiv2017

Directional Compactly supported Box Spline Tight Framelets with Simple Structure

To effectively capture singularities in high-dimensional data and functions, multivariate compactly supported tight framelets, having directionality and derived from refinable box splines, are of particular interest in both theory and applications. The $d$-dimensional Haar refinable function $χ_{[0,1]^d}$ is a simple example of refinable box splines. For every dimension $d\in \N$, in this paper we construct a directional compactly supported $d$-dimensional Haar tight framelet such that all its high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs and they exhibit totally $(3^d-1)/2$ directions in dimension $d$. Furthermore, applying the projection method to such directional Haar tight framelets, from every refinable box spline in every dimension, we construct a directional compactly supported box spline tight framelet with simple structure such that all the high-pass filters in its underlying tight framelet filter bank have only two nonzero coefficients with opposite signs. Moreover, such compactly supported box spline tight framelets can achieve arbitrarily high numbers of directions by using refinable box splines with increasing supports.

cs.IT↗
arXiv2017

Explicit Bernstein type inequalities for wavelet coefficients in $L_p(R^n)$

In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p=\{f: \|(i ω)^k\hat{f}(ω)\|_p\leq 1\}, k\in N, p\in(1,\infty)$ using an important quantity $C_{k,p}(ψ)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(ψ)$ with $ψ$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity $C_{k,p}(ψ)$.

math.FA↗
arXiv2016

Representation of functions on big data associated with directed graphs

This paper is an extension of the previous work of Chui, Filbir, and Mhaskar (Appl. Comput. Harm. Anal. 38 (3) 2015:489-509), not only from numeric data to include non-numeric data as in that paper, but also from undirected graphs to directed graphs (called digraphs, for simplicity). Besides theoretical development, this paper introduces effective mathematical tools in terms of certain data-dependent orthogonal systems for function representation and analysis directly on the digraphs. In addition, this paper also includes algorithmic development and discussion of various experimental results on such data-sets as CORA, Proposition, and Wiki-votes.

math.CA↗
arXiv2015

Symmetric Canonical Quincunx Tight Framelets with High Vanishing Moments and Smoothness

We propose an approach to construct a family of two-dimensional compactly supported real-valued symmetric quincunx tight framelets $\{ϕ; ψ_1,ψ_2,ψ_3\}$ in $L_2(R^2)$ with arbitrarily high orders of vanishing moments. Such symmetric quincunx tight framelets are associated with quincunx tight framelet filter banks $\{a;b_1,b_2,b_3\}$ having increasing orders of vanishing moments and enjoying the additional double canonical properties: \[ b_1(k_1,k_2)=(-1)^{1+k_1+k_2} a(1-k_1,-k_2), b_3(k_1,k_2)=(-1)^{1+k_1+k_2} b_2(1-k_1,-k_2). \] For a low-pass filter $a$ which is not a quincunx orthonormal wavelet filter, we show that a quincunx tight framelet filter bank $\{a;b_1,\ldots,b_L\}$ with $b_1$ taking the above canonical form must have $L\ge 3$ high-pass filters. Thus, our family of symmetric double canonical quincunx tight framelets has the minimum number of generators. Numerical calculation indicates that this family of symmetric double canonical quincunx tight framelets can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry. In particular, the second approach yields a family of $6$-multiple canonical real-valued quincunx tight framelets in $L_2(R^2)$ and a family of double canonical complex-valued quincunx tight framelets in $L_2(R^2)$ such that both of them have symmetry and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in $L_2(R^2)$ with symmetry, high vanishing moments, and smoothness. Symmetric quincunx tight framelets constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing.

cs.IT↗
arXiv2014

Directional Tensor Product Complex Tight Framelets with Low Redundancy

Though high redundancy rate of a tight frame can improve performance in applications, as the dimension increases, it also makes the computational cost skyrocket and the storage of frame coefficients increase exponentially. This seriously restricts the usefulness of such tight frames for problems in moderately high dimensions such as video processing in dimension three. Inspired by the directional tensor product complex tight framelets ${TP-CTF}_m$ with $m\ge 3$ in [14,18] and their impressive performance for image processing in [18,30] in this paper we introduce a directional tensor product complex tight framelet ${TP-CTF}^!_6$ (called reduced ${TP-CTF}_6$) with low redundancy. Such ${TP-CTF}_6^!$ is a particular example of tight framelet filter banks with mixed sampling factors. The ${TP-CTF}^!_6$ in $d$ dimensions not only offers good directionality but also has the low redundancy rate $\frac{3^d-1}{2^d-1}$ (e.g., the redundancy rates are $2, 2\mathord{\frac{2}{3}}, 3\mathord{\frac{5}{7}}, 5\mathord{\frac{1}{3}}$ and $7\mathord{\frac{25}{31}}$ for dimension $d=1,..., 5$, respectively). Moreover, our numerical experiments on image/video denoising and inpainting show that the performance using our proposed ${TP-CTF}^!_6$ is often comparable or sometimes better than several state-of-the-art frame-based methods which have much higher redundancy rates than that of ${TPCTF}^!_6$.

cs.IT↗
arXiv2013

Smooth affine shear tight frames with MRA structure

Finding efficient representations is one of the most challenging and heavily sought problems in mathematics. Representation using shearlets recently receives a lot of attention due to their desirable properties in both theory and applications. Using the framework of frequency-based affine systems, in this paper we introduce and systematically study affine shear tight frames which include all known shearlet tight frames as special cases. Our results in this paper will resolve several key questions on shearlets. We provide a complete characterization for an affine shear tight frame and then use it to obtain smooth affine shear tight frames with all their generators in the Schwarz class. Though multiresolution analysis (MRA) is the foundation and key feature of wavelet analysis for fast numerical implementation of a wavelet transform, all the known shearlets so far do not possess any MRA structure and filter banks. In order to study affine shear tight frames with MRA structure, we introduce the notion of a sequence of affine shear tight frames and then we provide a complete characterization for it. Based on our characterizations, we present two different approaches, i.e., non-stationary and quasi-stationary, for the construction of sequences of affine shear tight frames with MRA structure such that all their generators are smooth (in the Schwarz class) and they have underlying filter banks. Consequently, their associated transforms can be efficiently implemented using filter banks similarly as a fast wavelet transform does.

math.NA↗
arXiv2013

Gabor Shearlets

In this paper, we introduce Gabor shearlets, a variant of shearlet systems, which are based on a different group representation than previous shearlet constructions: they combine elements from Gabor and wavelet frames in their construction. As a consequence, they can be implemented with standard filters from wavelet theory in combination with standard Gabor windows. Unlike the usual shearlets, the new construction can achieve a redundancy as close to one as desired. Our construction follows the general strategy for shearlets. First we define group-based Gabor shearlets and then modify them to a cone-adapted version. In combination with Meyer filters, the cone-adapted Gabor shearlets constitute a tight frame and provide low-redundancy sparse approximations of the common model class of anisotropic features which are cartoon-like functions.

math.FA↗
arXiv2012

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is completely missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely l_1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that -- provided the size of the missing part is asymptotically to the size of the analyzing functions -- asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.

math.FA↗
arXiv2011

Digital Shearlet Transform

Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.

math.NA↗
arXiv2011

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.

math.NA↗
arXiv2010

Matrix Extension with Symmetry and Construction of Biorthogonal Multiwavelets

Let $(\pP,\wt\pP)$ be a pair of $r \times s$ matrices of Laurent polynomials with symmetry such that $\pP(z) \wt\pP^*(z)=I_\mrow$ for all $z\in \CC \bs \{0\}$ and both $\pP$ and $\wt\pP$ have the same symmetry pattern that is compatible. The biorthogonal matrix extension problem with symmetry is to find a pair of $s \times s$ square matrices $(\pP_e,\wt\pP_e)$ of Laurent polynomials with symmetry such that $[I_r, \mathbf{0}] \pP_e =\pP$ and $[I_r,\mathbf{0}]\wt\pP_e=\wt\pP$ (that is, the submatrix of the first $r$ rows of $\pP_e,\wt\pP_e$ is the given matrix $\pP,\wt\pP$, respectively), $\pP_e$ and $\wt\pP_e$ are biorthogonal satisfying $\pP_e(z)\wt\pP_e^*(z)=I_\mcol$ for all $z\in \CC \bs \{0\}$, and $\pP_e,\wt\pP_e$ have the same compatible symmetry. In this paper, we satisfactorily solve this matrix extension problem with symmetry by constructing the desired pair of extension matrices $(\pP_e,\wt\pP_e)$ from the given pair of matrices $(\pP,\wt\pP)$. Matrix extension plays an important role in many areas such as wavelet analysis, electronic engineering, system sciences, and so on. As an application of our general results on matrix extension with symmetry, we obtain a satisfactory algorithm for constructing symmetric biorthogonal multiwavelets by deriving high-pass filters with symmetry from any given pair of biorthgonal low-pass filters with symmetry. Several examples of symmetric biorthogonal multiwavelets are provided to illustrate the results in this paper.

math.FA↗
arXiv2010

Matrix Extension with Symmetry and Its Application to Filter Banks

In this paper, we completely solve the matrix extension problem with symmetry and provide a step-by-step algorithm to construct such a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Furthermore, using a cascade structure, we obtain a complete representation of any $r\times s$ paraunitary matrix $\mathsf{P}$ having compatible symmetry, which in turn leads to an algorithm for deriving a desired matrix $\mathsf{P}_e$ from a given matrix $\mathsf{P}$. Matrix extension plays an important role in many areas such as electronic engineering, system sciences, applied mathematics, and pure mathematics. As an application of our general results on matrix extension with symmetry, we obtain a satisfactory algorithm for constructing symmetric paraunitary filter banks and symmetric orthonormal multiwavelets by deriving high-pass filters with symmetry from any given low-pass filters with symmetry. Several examples are provided to illustrate the proposed algorithms and results in this paper.

cs.IT↗