Frame Operator Research Articles

Necessary and sufficient conditions to perform Spectral Tetris

Spectral Tetris is a powerful tool for constructing sparse equal norm Hilbert space frames. We introduce a new form of Spectral Tetris which works for non-equal-norm frames. It is known that this method cannot construct all frames—even in the new case introduced here. Until now, it has been a mystery why Spectral Tetris sometimes works and sometimes fails. We provide a complete answer by giving necessary and sufficient conditions for Spectral Tetris to construct frames with prescribed vector norms and prescribed frame operator. We present a variety of examples and include cases where Spectral Tetris always works.

Convergence of sequences of Calderón‐Zygmund operators with application to wavelet expansions

AbstractWe present an extrapolation theorem for the convergence of sequences of Calderón‐Zygmund operators on various function spaces. We show that if a sequence of Calderón‐Zygmund operators with the same smoothness and decay parameters is convergent in operator norm on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2(\mathbb {R}^n)$\end{document}, then it is also convergent as operators on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^p(\mathbb {R}^n)$\end{document}, 1 < p < ∞, and on a large class of Triebel‐Lizorkin, Besov and Hardy spaces. As an application, we show that as the sampling density tends to the infinity, the wavelet frame operator tends to the identity in operator norm on Triebel‐Lizorkin, Besov and Hardy spaces.

A characterisation of Weyl–Heisenberg frame operators

We give a simple characterisation of Gabor frame operators in L2(R).

Approximating the inverse frame operator from localized frames

This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that sampling with well-localized frames improves both the accuracy of the numerical frame approximation as well as the robustness and efficiency of the (finite) frame operator inversion. Moreover, in applications such as magnetic resonance imaging, where the given data often may not constitute a well-localized frame, a technique is devised to project the corresponding frame data onto a more suitable frame. As a result, the target function may be approximated as a finite expansion with its asymptotic convergence solely dependent on its smoothness. Numerical examples are provided.

Constructing finite frames of a given spectrum and set of lengths

When constructing finite frames for a given application, the most important consideration is the spectrum of the frame operator. Indeed, the minimum and maximum eigenvalues of the frame operator are the optimal frame bounds, and the frame is tight precisely when this spectrum is constant. Often, the second-most important design consideration is the lengths of frame vectors: Gabor, wavelet, equiangular and Grassmannian frames are all special cases of equal norm frames, and unit norm tight frame-based encoding is known to be optimally robust against additive noise and erasures. We consider the problem of constructing frames whose frame operator has a given spectrum and whose vectors have prescribed lengths. For a given spectrum and set of lengths, the existence of such frames is characterized by the Schur–Horn Theorem—they exist if and only if the spectrum majorizes the squared lengths—the classical proof of which is nonconstructive. Certain construction methods, such as harmonic frames and spectral tetris, are known in the special case of unit norm tight frames, but even these provide but a few examples from the manifold of all such frames, the dimension of which is known and nontrivial. In this paper, we provide a new method for explicitly constructing any and all frames whose frame operator has a prescribed spectrum and whose vectors have prescribed lengths. The method itself has two parts. In the first part, one chooses eigensteps—a sequence of interlacing spectra—that transform the trivial spectrum into the desired one. The second part is to explicitly compute the frame vectors in terms of these eigensteps; though nontrivial, this process is nevertheless straightforward enough to be implemented by hand, involving only arithmetic, square roots and matrix multiplication.

A Novel Approach of Digital Video Encryption

the recent years with the development of internet technologies, video technologies have been broadly used in TV, communication and multimedia, So security is required on video data. Although much video encryption technique has been develop but not give so much efficiency in terms of encryption and decryption process. However, they are more complex to implement as a system and are difficult to be applied in a widespread manner. Here we propose a new novel scheme for digital video encryption. In this paper we give a method to generate an encrypted video by encrypted Video-frame. Based on novel secure video scheme, an effective and generalized scheme of video encryption. It is a matrix computation scheme which uses a concept of Video- frame and xor() operation. This paper proves that proposed scheme is able to fully encrypt the video frame and have a better performance that can be measured by different Parameters. Further we can extend our approach into a digital video stenography.

Geometry of the Welch bounds

A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular, a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames of symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.

Frames, Semi-Frames, and Hilbert Scales

Given a total sequence in a Hilbert space, we speak of an upper (resp. lower) semi-frame if only the upper (resp. lower) frame bound is valid. Equivalently, for an upper semi-frame, the frame operator is bounded but has an unbounded inverse, whereas a lower semi-frame has an unbounded frame operator with bounded inverse. For upper semi-frames, in the discrete and the continuous case, we build two natural Hilbert scales which may yield a novel characterization of certain function spaces of interest in signal processing. We present some examples and, in addition, some results concerning the duality between lower and upper semi-frames, as well as some generalizations, including fusion semi-frames and Banach semi-frames.

Effect of feed frame design and operating parameters on powder attrition, particle breakage, and powder properties

Abstract Feed frame is a device used in rotary tablet presses to drive the powders into compression dies. It applies shear forces to the powders as they are stirred around the feed frame chambers. This study focused on understanding the particle attrition, powder properties and the flow property changes of the material processed. The results demonstrated that the impact of the feed frame and die disk on the particle size distribution (PSD) outlet depended on the initial mean particle size, the die size and the powder outlet position. It also impacted the flow properties. The scale-up effect using a feed frame for a high production tablet press shows a significant increment in powder attrition.

Alternating Direction Method for Balanced Image Restoration

This paper presents an efficient algorithm for solving a balanced regularization problem in the frame-based image restoration. The balanced regularization is usually formulated as a minimization problem, involving an l(2) data-fidelity term, an l(1) regularizer on sparsity of frame coefficients, and a penalty on distance of sparse frame coefficients to the range of the frame operator. In image restoration, the balanced regularization approach bridges the synthesis-based and analysis-based approaches, and balances the fidelity, sparsity, and smoothness of the solution. Our proposed algorithm for solving the balanced optimal problem is based on a variable splitting strategy and the classical alternating direction method. This paper shows that the proposed algorithm is fast and efficient in solving the standard image restoration with balanced regularization. More precisely, a regularized version of the Hessian matrix of the l(2) data-fidelity term is involved, and by exploiting the related fast tight Parseval frame and the special structures of the observation matrices, the regularized Hessian matrix can perform quite efficiently for the frame-based standard image restoration applications, such as circular deconvolution in image deblurring and missing samples in image inpainting. Numerical simulations illustrate the efficiency of our proposed algorithm in the frame-based image restoration with balanced regularization.

Spectral Tetris Fusion Frame Constructions

Spectral tetris is a flexible and elementary method to construct unit norm frames with a given frame operator, having all of its eigenvalues greater than or equal to two. One important application of spectral tetris is the construction of fusion frames. We first show how the assumption on the spectrum of the frame operator can be dropped and extend the spectral tetris algorithm to construct unit norm frames with any given spectrum of the frame operator. We then provide a sufficient condition for using this generalization of spectral tetris to construct fusion frames with prescribed spectrum for the fusion frame operator and with prescribed dimensions for the subspaces. This condition is shown to be necessary in the tight case of redundancy greater than or equal to two.

Subfusion Frames

Fusion frames are generalizations of frames in Hilbert spaces which were introduced by Casazza et al. (2008). In the present paper, we study the relations between fusion frames and subfusion frame operators. Specially, we introduce new construction of subfusion frames and derive new results.

Towards Frame-based Source Code Generation for Heterogeneous Real-time Environments

This work focusses on the source code generation of task and communication patterns for different real-time operating systems. On the basis of reusable frames, an initial approach for implementing distributed applications will be presented. Thereby, distributed real-time systems can be mapped to the operating system framework Hyper Real-time Operating System (HyROS). In this respect, it is of great importance to cleanly separate functional from non-functional properties. Mapping different systems to HyROS can be achieved by using HyFrameLang, a language for frame operations and HyAppLang, an embedded domain specific language. Programming examples for HyFrameLang and HyAppLang are presented for task patterns of PXROS-HR and OSEK/VDX, representative real-time operating systems. Moreover, a short introduction on framing communication aspects is given.

Non-orthogonal Fusion Frames and the Sparsity of Fusion Frame Operators

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator $S_{\mathcal{W}}$ , which in turn is heavily dependent on the sparsity of $S_{\mathcal{W}}$ . We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of non-orthogonal fusion frames. We show that for a fusion frame {W i ,v i } i∈I , if dim(W i )=k i , then the matrix of the non-orthogonal fusion frame operator ${\mathcal{S}}_{{\mathcal{W}}}$ has in its corresponding location at most a k i ×k i block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator ${\mathcal{S}}_{{\mathcal{W}}}$ is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of multiple fusion frames whose corresponding fusion frame operator becomes an diagonal operator is also examined.

GENERATING NEW FRAMES IN L 2 (R) BY CONVOLUTIONS

Let c = {c n n} n∈z ∈l 1 (Z) and {f n } n∈z be a frame (Riesz basis, respectively) of L 2 (R). We obtdn necessary and sufficient conditions on c under which {c* λ f n } n∈z becomes a frame (Riesz basis, respectively) of L 2 (R)，where λ > 0 and (c * λ f)(t) := Σ n∈z c n f (t-nλ). When {c* λ f n } n∈z becomes a frame of L 2 (R)，we present its frame operator and the canonical dual frame in a simple form. Some interesting examples are included.

The Intersection Between Deep Moral Frames and Rhetorical Style in the Struggle over U.S. Immigration Reform

This essay highlights and explores a point of tension between theoretical writings on style and moral frames. Past political communication scholarship points to the importance of the feminine style in today's televisual era of politics. In this same political era, the conservative strict parent moral frame has dominated most policy debates. Surprisingly, this highly successful moral frame appears squarely at odds with the feminine style so closely connected with political success. This essay attempts to unravel this tension between styles and frames by examining discourse drawn from the 2007 debate over comprehensive immigration reform. To account for the success of conservative messages within this debate, this essay both (a) calls into question the nature of the relationship between the television medium and the feminine style and (b) expands our understanding of the discursive operation of deep moral frames by drawing a distinction between intra-familial and extra-familial policy discourse.

Optimally Sparse Frames

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed from this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames.

Reproducing kernel bounds for an advanced wavelet frame via the theta function

This work presents an analysis of the reproducing kernel K(p) associated to a class of wavelets K(p) derived from the theory of multiplicatively advanced differential equations. These kernels K(p) are expressible in terms of a family of new wavelets fk(t), which are generated by two fundamental wavelets Cosq(t) and Sinq(t), all of which satisfy multiplicatively q-advanced perturbations of the second-order equation of the harmonic oscillator f″(t)=−f(t). As the parameter q→1+, Cosq(t) and Sinq(t) approach cos(t) and sin(t) uniformly, respectively, on compact subsets of R. Decay rates for K(p) as q→∞ are given, refining the understanding of the associated frame operator S, and providing for efficiency in inversion of S.

Orthogonal projection decomposition of matrices and construction of fusion frames

We investigate decompositions of positive matrices as weighted sums of orthogonal projections, and apply them to the construction of fusion frames when fusion frame operators are prescribed. Examples are provided to demonstrate the simplicity and flexibility in this practical construction of fusion frames. As an application, we provide an method constructing Parseval fusion frames that are optimal for the one packet erasure problem.

Optimization over finite frame varieties and structured dictionary design

A finite (μ,S)-frame variety consists of real or complex matrices F=[f1⋯fN] satisfying FF⁎=S and ‖fn‖=μn for all n=1,…,N. This paper introduces an approximate geometric gradient descent procedure over these varieties, which is powered by minimization algorithms for the frame operator distance and recent characterizations of these varietiesʼ tangent spaces. For almost all compatible pairings (μ,S), we demonstrate that minimization of the frame operator distance converges linearly under a threshold, we derive a process for constructing the orthogonal projection onto these varietiesʼ tangent spaces, and finally demonstrate that the approximate gradient descent procedure converges. Finally, we apply this procedure to numerically construct Grassmannian frames and Welch bound equality sequences with low mutual coherence.