Dilation Matrix Research Articles

Microlocal analysis of edge flatness through directional multiscale representations

Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using `higher-than-parabolic' scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points.

How Much Do Ultrathin Polymers with Intrinsic Microporosity Swell in Liquids?

As synthetic membrane materials, polymers with intrinsic microporosity (PIMs) have demonstrated unprecedented permeation and molecular-separation properties. Here, we report the swelling characteristics of submicron-thick supported films of spirobisindane-based PIMs, PIM-1 and PIM-6FDA-OH, for six organic solvents and water using in situ spectroscopic ellipsometry. Surprisingly, PIMs swell significantly in most organic solvents, with swelling factors (SF = hswollen/hdry) as high as 2.5. This leads to the loss of the ultrarigid character of the polymer and produces equilibrated liquid-like swollen films. Filling of the excess frozen-in fractional free volume with liquid was discovered next to swelling-induced polymer matrix dilation. Water hardly swells the polymer matrix, but it penetrates into the intrinsic microporous structure. This study is the first to provide fundamental swelling data for PIMs, leading to better comprehension of their permeation properties. Such an understanding is indispensable for applications such as solvent filtration, natural-gas separation, and ion retention in flow batteries.

Multivariate symmetric refinable functions and function vectors

For any symmetry group [Formula: see text], any appropriate matrix dilation (compatible with [Formula: see text]) and any appropriate symmetry center [Formula: see text] we give an explicit method for the construction of [Formula: see text]-symmetric with respect to the center [Formula: see text] refinable masks which have sum rule of an arbitrary order [Formula: see text]. Moreover, we give a description of all these masks. For any symmetry group [Formula: see text], any appropriate matrix dilation (compatible with [Formula: see text]) and any appropriate row of symmetry centers [Formula: see text] we give two explicit methods for the construction of [Formula: see text]-symmetric with respect to the row of centers [Formula: see text] refinable matrix masks which have sum rule of an arbitrary order [Formula: see text]. A description of all such matrix masks is also presented.

Multivariate sampling-type approximation

Approximation properties of the expansions [Formula: see text], where [Formula: see text] is a linear differential operator and [Formula: see text] is a matrix dilation, are studied. The sampling expansions are a special case of such differential expansions. Error estimations in [Formula: see text]-norm, [Formula: see text], are given in terms of the Fourier transform of [Formula: see text]. The approximation order depends on the smoothness of [Formula: see text], the order of [Formula: see text], the order of Strang–Fix condition for [Formula: see text] and [Formula: see text]. A wide class of [Formula: see text] including both band-limited and compactly supported functions is considered, but a special condition of compatibility [Formula: see text] with [Formula: see text] is required. Such differential expansions may be useful for engineers.

Regularity of non-stationary subdivision: a matrix approach

In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case M = mI, m ge 2). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples.

On a trilinear singular integral form with determinantal kernel

We study a trilinear singular integral form acting on two- dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calderón commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the L p \mathrm {L}^p estimates hold for the considered form.

A Computation with the Connes–Thom Isomorphism

AbstractLet A ∊ Mn(ℝ) be an invertible matrix. Consider the semi-direct product ℝn ⋊ ℤ where the action of ℤ on ℝn is induced by the left multiplication by A. Let (α, τ) be a strongly continuous action of ℝn ⋊ ℤ on a C*-algebra B where α is a strongly continuous action of ℝn and τ is an automorphism. The map τ induces a map . We show that, at the K-theory level, τ commutes with the Connes–Thom map if det(A) > 0 and anticommutes if det(A) < 0. As an application, we recompute the K-groups of the Cuntz–Li algebra associated with an integer dilation matrix.

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$| \det A|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stockler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.

Polynomial spaces reproduced by elliptic scaling functions

In this paper, we consider the so-called elliptic scaling functions [V. G. Zakharov, Elliptic scaling functions as compactly supported multivariate analogs of the B-splines, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014) 1450018]. Any elliptic scaling function satisfies the refinement relation with a real isotropic dilation matrix; and, in the paper, we prove that any real isotropic matrix is similar to an orthogonal matrix and the similarity transformation matrix determines a positive-definite quadratic form. We prove that the polynomial space reproduced by integer shifts of a compactly supported function can be usually considered as a polynomial solution to a system of constant coefficient PDE’s. We show that the algebraic polynomials reproduced by a compactly supported elliptic scaling function belong to the kernel of a homogeneous elliptic differential operator that the differential operator corresponds to the quadratic form; and thus any elliptic scaling function reproduces only affinely-invariant polynomial spaces. However, in the paper, we present nonstationary elliptic scaling functions such that the scaling functions can reproduce no scale-invariant (only shift-invariant) polynomial spaces.

A characterization of multiwavelet packets on general lattices

The objective of this paper is to establish a complete characterization of multiwavelet packets associated with matrix dilation on general lattices $Gamma$ in $mathbb R^d$ by virtue of time-frequency analysis, matrix theory and operator theory.

Construction of Multivariate Tight Framelet Packets Associated with Dilation Matrix

Construction of Multivariate Tight Framelet Packets Associated with Dilation Matrix

Coorbit spaces and wavelet coefficient decay over general dilation groups

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on L 2 ( R d ) \textrm {L}^2(\mathbb {R}^d) . It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question of how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted mixed L p \textrm {L}^{p} -estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.

Subdivision schemes induced by approximating schemes

Construction of new interpolatory subdivision schemes from approximating subdivision schemes is a hot topic that recently emerges in the eld of subdivision method. However, rather less attention has been paid to the uni ed formula that can express the interpolatory subdivision generating function analytically in terms of the approximating subdivision symbol. This paper presents such a uni ed subdivision generating function formula that consists the generating function of the approximating subdivision scheme and one subsymbol. This uni ed subdivision generating function formula can be a generating function of an interpolatory scheme or that of an approximating scheme depending on the speci c choice of the subsymbol. This formula plays an important role in theories and applications. First, it can be applied to higher-dimensional subdivisions with arbitrary dilation matrix, and it can also induce approximating schemes. Second, it does not need to solve a system of linear equations. Third, this formula has signi cant geometric meanings. At last, the speci c structure of the formula also implies that zero condition properties of new subdivision scheme is closely related with the properties of the primal approximating subdivision. We illustrate the uni ed formula with three examples.

Extensions of the Heisenberg group by one‐parameter groups of dilations which are subgroups of the affine and the symplectic groups

Extensions of the multidimensional Heisenberg group by one‐parameter groups of matrix dilations are introduced and classified up to isomorphism. Each group is isomorphic to both, a subgroup of the symplectic group and a subgroup of the affine group, and its metaplectic representation splits into two irreducible subrepresentations, each of which is equivalent to a subrepresentation of the wavelet representation.

Constrained doubly coprime factorization for all stabilizing ℋ∞ controllers

The conventional Youla parameterization (equivalently Q-Parameterization) approach to solve ℋ∞ optimal control problems requires solving the well-known matrix dilation optimization as a method for satisfying the ℋ∞-norm constraint of the closed-loop transfer matrix. As an alternative, this paper presents a constrained doubly coprime factorization so that the ℋ∞-norm constraint of the closed-loop transfer matrix can be satisfied without the need for matrix dilation optimization. For a given ℋ∞-norm constraint, a constrained plant is suggested from a state estimator that includes the worst-case disturbance and estimation effects. Then, the constrained doubly coprime factorization is derived from the constrained plant model. All the stabilizing ℋ∞ controllers are expressed by using the constrained coprime factors. Finally, an application example is provided to demonstrate the effectiveness of the proposed method.

Pure- and mixed-gas CO2/CH4 separation properties of PIM-1 and an amidoxime-functionalized PIM-1

The prototypical solution-processable polymer of intrinsic microporosity, PIM-1, and derivatives thereof offer combinations of permeability and selectivity that make them potential candidate materials for membrane-based gas separations. Paramount to the design and evaluation of PIMs for economical natural gas sweetening is a high and stable CO2/CH4 selectivity under realistic, mixed-gas conditions. Here, amidoxime-functionalized PIM-1 (AO-PIM-1) was prepared and examined for fundamental structure/property relationships. Qualitative NLDFT pore-size distribution analyses of physisorption isotherms (N2 at -196 oC; CO2 at 0 oC) reveal a tightened microstructure indicating size-sieving ultra-microporosity (<7Å). AO-PIM-1 demonstrated a three-fold increase in αD(CO2/CH4) over PIM-1, surpassing the 2008 upper bound with P(CO2)=1153Barrer and ideal α(CO2/CH4)=34. Under a 50:50 CO2:CH4 mixed-gas feed, AO-PIM-1 showed less selectivity loss than PIM-1, maintaining a mixed-gas α(CO2/CH4) ~21 across a 20bar pressure range. Conversely, PIM-1 endured up to 60% increases in mixed-gas CH4 permeability over pure-gas values concurrent with a selectivity of only ~8 at 20bar. A pervasive intermolecular hydrogen bonding network in AO-PIM-1 predominantly yields a rigidified microstructure that mitigates CO2-induced matrix dilations, reducing detrimental mixed-gas CH4 copermeation.

Semi-orthogonal multiwavelet frames in &amp;lt;italic&amp;gt;L&amp;lt;/italic&amp;gt;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;(R&amp;lt;sup&amp;gt;&amp;lt;italic&amp;gt;n&amp;lt;/italic&amp;gt;&amp;lt;/sup&amp;gt;)

We investigate semi-orthogonal multiwavelet frames with matrix dilation that |det A |>1 in L 2 (R n ) and obtain some characterizations of strictly semi-orthogonal multiwavelet frames. We nd the equivalence of semi-orthogonal Parseval multiwavelet frames and Generalized Multiresolution Analysis (GMRA) multiwavelet Parseval frames. In particular, we discuss the semi-orthogonality of minimally support frequency (MSF) multi- wavelet frames, and construct some examples of semi-orthogonal multiwavelet frames by wavelet sets.

A Study of a Two-Directional Vector Multivariate Wavelet Wraps and Applications in Engineering Materials

The existence of compactly supported orthogonal two-directional vector-valued wavelets associated with a pair of orthogonal shortly supported vector-valued scaling functions is researched. We introduce a class of two-directional vector-valued four-dimensional wavelet wraps according to a dilation matrix, which are generalizations of univariate wavelet wraps. Three orthogonality formulas regarding the wavelet wraps are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet wraps. The sufficient condition for the existence of four-dimensional wavelet wraps is established based on the multiresolution analysis method.

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125: 68–186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.

Positive temperature coefficient and high Seebeck coefficient in ZnO–P2O5/Co composites

This article reports a study of electrical properties of new Zinc Phosphate glass/Cobalt composites (45mol.% ZnO–55mol.%P2O5) (ZP/Co). The measurements of electrical conductivity at room temperature as a function of cobalt's concentration showed a non-conducting to conducting phase transition at percolation threshold of 27vol.%. The Seebeck coefficient obtained under the same conditions, accompanies a sign, with high positive and negative values below and above the percolation threshold respectively, depicting a p- to n-type conducting phase transition, confirming the conductivity measurements. Then, the measurements of electrical conductivity and Seebeck coefficient above the percolation threshold as a function of temperature showed an original conducting to insulating phase transition, called Positive Temperature Coefficient (PTC) at T=420K, associated to a high negative value of S≤−8000μV/K, with the highest power factor PF=σS2≈8×10−3W m−1 K−2. The thermal measurements of volume expansion confirm this transition, indicating matrix dilation around this temperature. However, the thermal behavior of the electrical conductivity and Seebeck coefficient data obtained below the percolation threshold showed different mechanisms i.e.; Small Polaron Hopping (SPH) mechanism at high temperatures and Mott's Variable Range Hopping (VRH) at low temperatures.