In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial on
In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.
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Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L-2(R-d). We obtain an MOEP for (...
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Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L-2(R-d). We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames. (C) 2017 Elsevier Inc. All rights reserved.
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Let Λ=K×L be a full rank time-frequency lattice in a, ~d ×a, ~d. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ~2(...
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Let Λ=K×L be a full rank time-frequency lattice in a, ~d ×a, ~d. In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ~2(a, ~d) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419-433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel-Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ) (_(j=1)~N G(g _j,Λ)) for L ~2(a, ~d). We show that this is true whenever v(Λ) N. In particular, when v(Λ) 1, any Bessel-Gabor system is a subset of a tight Gabor frame G(g,Λ) G(h,Λ) for L ~2(a, ~d). Related results for affine systems are also discussed.
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A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf R})$ is said to be a Bessel system if the associated analysis operator $$ \txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X} $$ is...
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A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf R})$ is said to be a Bessel system if the associated analysis operator $$ \txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X} $$ is well-defined and bounded. A Bessel system is a fundamental frame if $\txs{X}$ is injective and its range is closed. This paper considers the above two properties for a generalized shift-invariant system $X$. By definition, such a system has the form $$ X=\bigcup_{j\in J} Y_j, $$ where each $Y_j$ is a shift-invariant system (i.e., is comprised of lattice translates of some function(s)) and $J$ is a countable (or finite) index set. The definition is general enough to include wavelet systems, shift-invariant systems, Gabor systems, and many variations of wavelet systems such as quasi-affine ones and nonstationary ones. The main theme of this paper is the “fiberization” of $\txs{X}$, which allows one to study the frame and Bessel properties of $X$ via the spectral properties of a collection of finite-order Hermitian nonnegative matrices.
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摘要 :Let $\Lambda=\mathcal{K}\times\mathcal{L}$ be a full rank time-frequency lattice in ? d ×? d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pa...
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Let $\Lambda=\mathcal{K}\times\mathcal{L}$ be a full rank time-frequency lattice in ? d ×? d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L 2(? d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ j=1 N G(g j ,Λ)) for L 2(? d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L 2(? d ). Related results for affine systems are also discussed.
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In this paper, we characterize the space of almost periodic (AP) functions in one variable using either a Weyl-Heisenberg (WH) system or an affine system. Our observation is that the sought-for characterization of the AP space is ...
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In this paper, we characterize the space of almost periodic (AP) functions in one variable using either a Weyl-Heisenberg (WH) system or an affine system. Our observation is that the sought-for characterization of the AP space is valid if and only if the given WH (respectively, affine) system is an L-2(R)-frame. Moreover, the frame bounds of the system are also the sharpest bounds in our characterization. This draws an intriguing and quite unexpected connection between L-2(R) representations and AP-representations.
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When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry...
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When bivariate filter banks and wavelets are used for surface multiresolution processing, it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry. This symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. Recently lifting-scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets have certain smoothness, then the analysis or synthesis scaling function or both have big supports in general. In particular, when the synthesis low-pass filter is a commonly used scheme such as Loop's scheme or CatmullClark's scheme, the corresponding analysis low-pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness. Big supports of scaling functions, or in other words big templates of multiresolution algorithms, are undesirable for surface processing. On the other hand, a frame provides flexibility for the construction of "basis" systems. This paper concerns the construction of wavelet (or affine) bi-frames with high symmetry. In this paper we study the construction of wavelet bi-frames with 4-fold symmetry for quadrilateral surface multiresolution processing, with both the dyadic and √2 refinements considered. The constructed bi-frames have 4 framelets (or frame generators) for the dyadic refinement, and 2 framelets for the √2 refinement. Namely, with either the dyadic or √2 refinement, a frame system constructed in this paper has only one more generator than a wavelet system. The constructed bi-frames have better smoothness and smaller supports than biorthogonal wavelets. Furthermore, all the frame algorithms considered in this paper are given by templates so that one can easily implement them.
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This paper is about the construction of univariate wavelet bi-frames with each framelet being symmetric. As bivariate filter banks are used for surface multiresolution processing, it is required that the corresponding decompositio...
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This paper is about the construction of univariate wavelet bi-frames with each framelet being symmetric. As bivariate filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices. For open surfaces, special multiresolution algorithms are designed to process boundary vertices. When the multiresolution algorithms derived from univariate wavelet bi-frames are used as the boundary algorithms, it is desired that not only the scaling functions but also all framelets be symmetric. In addition, the algorithms for curve/surface multiresolution processing should be given by templates so that they can be easily implemented. In this paper, first, by appropriately associating the lowpass and highpass outputs to the nodes of Z, we show that both biorthogonal wavelet multiresolution algorithms and bi-frame multiresolution algorithms can be represented by templates. Then, using the idea of the lifting scheme, we provide frame algorithms given by several iterative steps with each step represented by a symmetric template. Finally, with the given templates of algorithms, we obtain the corresponding filter banks and construct bi-frames based on their smoothness and vanishing moments. Two types of symmetric bi-frames are studied in this paper. In order to provide a clearer picture on the template-based procedure for bi-frame construction, in this paper we also consider the template-based construction of biorthogonal wavelets. The approach of the template-based bi-frame construction introduced in this paper can be extended easily to the construction of bivariate bi-frames with high symmetry for surface multiresolution processing.
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We give a simple and explicit construction of compactly supported affine tight frames with small number of generators, associated to multivariate box splines (with respect to the dilation matrix 2I). Moreover, the same technique a...
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We give a simple and explicit construction of compactly supported affine tight frames with small number of generators, associated to multivariate box splines (with respect to the dilation matrix 2I). Moreover, the same technique applied to the case of bivariate box splines on the four-directions mesh with dilation matrix ((11)(1-1)) gives tight frames with at most five generators. (C) 2004 Elsevier Inc. All rights reserved.
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Using the range function approach to shift invariant spaces in L-2(R ") we give a simple characterization of frames and Riesz families generated by shifts of it countable set of generators in terms of their behavior on subspaces o...
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Using the range function approach to shift invariant spaces in L-2(R ") we give a simple characterization of frames and Riesz families generated by shifts of it countable set of generators in terms of their behavior on subspaces of l(2)(Z(n)). This in turn gives a simplified approach to the analysis of frames and Riesz families done by Gramians and dual Gramians. We prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator. As an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function. (C) 2000 Academic Press. [References: 11]
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