Optimal dual frames for erasures

For any given frame (for the purpose of encoding) in a finite dimensional Hilbert space, we investigate its dual frames that are optimal for erasures (for the purpose of decoding). We show that in general the canonical dual is not necessarily optimal. Moreover, optimal dual frames are not necessarily unique. We present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem. As an application, we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight.     

等距丢失模型下的框架张量积重构方法

框架理论常应用于信号重构.当编码系数在传输过程中发生等距丢失时,基于框架张量积的一些性质,我们可以利用框架张量积对信号进行编码从而降低数据丢失对重构信号的影响.本文由此提出了一种等距丢失模型,并在此模型下,研究了数据等距丢失下的最优对偶框架张量积,得出了对偶框架和正则对偶框架的张量积是最优对偶框架张量积的两个充分必要条件.最后数值实验也说明了：在等距丢失模型下,最优对偶框架张量积比一般对偶框架张量积的信号重构结果更优.     

Linearly connected sequences and spectrally optimal dual frames for erasures

In the case that a frame is prescribed for applications and erasures occur in the process of data transmissions, we examine optimal dual frames for the recovery from single erasures. In contrast to earlier papers, we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality. This notion of optimality is natural when the Neumann series is used to recover the original data in an iterative manner. We obtain a complete characterization of spectrally one-erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame. Our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame. We prove that the linear connectivity property is equivalent to the intersection dependent property, and is also closely related to the well-known concept of a k-independent set. Additionally, we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work.     

Complex equiangular tight frames and erasures

In this paper we demonstrate that there are distinct differences between real and complex equiangular tight frames (ETFs) with regards to erasures. For example, we prove that there exist arbitrarily large non-trivial complex equiangular tight frames which are optimal against three erasures, and that such frames come from a unique class of complex ETFs. In addition, we extend certain results in Bodmann and Paulsen (2005) [2] to complex vector spaces as well as show that other results regarding real ETFs are not valid for complex ETFs.     

Approximate duality of g-frames in Hilbert spaces

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

APPROXIMATE DUALITY OF g-FRAMES IN HILBERT SPACES

In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.     

Robust dimension reduction,fusion frames,and Grassmannian packings

We consider estimating a random vector from its measurements in a fusion frame, in presence of noise and subspace erasures. A fusion frame is a collection of subspaces, for which the sum of the projection operators onto the subspaces is bounded below and above by constant multiples of the identity operator. We first consider the linear minimum mean-squared error (LMMSE) estimation of the random vector of interest from its fusion frame measurements in the presence of additive white noise. Each fusion frame measurement is a vector whose elements are inner products of an orthogonal basis for a fusion frame subspace and the random vector of interest. We derive bounds on the mean-squared error (MSE) and show that the MSE will achieve its lower bound if the fusion frame is tight. We then analyze the robustness of the constructed LMMSE estimator to erasures of the fusion frame subspaces. We limit our erasure analysis to the class of tight fusion frames and assume that all erasures are equally important. Under these assumptions, we prove that tight fusion frames consisting of equi-dimensional subspaces have maximum robustness (in the MSE sense) with respect to erasures of one subspace among all tight fusion frames, and that the optimal subspace dimension depends on signal-to-noise ratio (SNR). We also prove that tight fusion frames consisting of equi-dimensional subspaces with equal pairwise chordal distances are most robust with respect to two and more subspace erasures, among the class of equi-dimensional tight fusion frames. We call such fusion frames equi-distance tight fusion frames. We prove that the squared chordal distance between the subspaces in such fusion frames meets the so-called simplex bound, and thereby establish connections between equi-distance tight fusion frames and optimal Grassmannian packings. Finally, we present several examples for the construction of equi-distance tight fusion frames.     

Robust Dual Reconstruction Systems and Fusion Frames

We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.     

The symmetry group of a finite frame

We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of Vale and Waldron (2005) [12] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames.     

Stable recovery of signals from frame coefficients with erasures at unknown locations

In an earlier work, we proposed a frame-based kernel analysis approach to the problem of recovering erasures from unknown locations. The new approach led to the stability question on recovering a signal from noisy partial frame coefficients with erasures occurring at unknown locations. In this continuing work, we settle this problem by obtaining a complete characterization of frames that provide stable reconstructions. We show that an encoding frame provides a stable signal recovery from noisy partial frame coefficients at unknown locations if and only if it is totally robust with respect to erasures. We present several characterizations for either totally robust frames or almost robust frames. Based on these characterizations several explicit construction algorithms for totally robust and almost robust frames are proposed. As a consequence of the construction methods, we obtain that the probability for a randomly generated frame to be totally robust with respect to a fixed number of erasures is one.     

Sparse matrices in frame theory

Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.     

Equal-Norm Tight Frames with Erasures

Equal-norm tight frames have been shown to be useful for robust data transmission. The losses in the network are modeled as erasures of transmitted frame coefficients. We give the first systematic study of the general class of equal-norm tight frames and their properties. We search for efficient constructions of such frames. We show that the only equal-norm tight frames with the group structure and one or two generators are the generalized harmonic frames. Finally, we give a complete classification of frames in terms of their robustness to erasures.     

Time-domain characterization of multiwindow Gabor systems on discrete periodic sets

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.     

SOME RESULTS ON CONTROLLED FRAMES IN HILBERT SPACES

We use two appropriate bounded invertible operators to define a controlled frame with optimal frame bounds.We characterize those operators that produces Parseval controlled frames also we state a way to construct nearly Parseval controlled frames.We introduce a new perturbation of controlled frames to obtain new frames from a given one.Also we reduce the distance of frames by appropriate operators and produce nearly dual frames from two given frames which are not dual frames for each other.     

Quantized Frame Expansions with Erasures

Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly.     

Dual fusion frames

The definition of dual fusion frame presents technical problems related to the domain of the synthesis operator. The notion commonly used is the analogue to the canonical dual frame. Here a new concept of dual is studied in infinite-dimensional separable Hilbert spaces. It extends the commonly used notion and overcomes these technical difficulties. We show that with this definition in many cases dual fusion frames behave similar to dual frames. We present examples of non-canonical dual fusion frames.     

Affine systems inL 2 (ℝ d ) II: Dual systems

The fiberization of affine systems via dual Gramian techniques, which was developed in previous papers of the authors, is applied here for the study of affine frames that have an affine dual system. Gramian techniques are also used to verify whether a dual pair of affine frames is also a pair of bi-orthogonal Riesz bases. A general method for a painless derivation of a dual pair of affine frames from an arbitrary MRA is obtained via the mixed extension principle. This work was partially sponsored by the National Science Foundation under Grants DMS-9102857, DMS-9224748, and DMS-9626319, by the United States Army Research Office under Contracts DAAL03-G-90-0090, DAAH04-95-1-0089, and by the Strategic Wavelet Program Grant from the National University of Singapore.     

Sobolev Duals in Frame Theory and Sigma-Delta Quantization

A new class of alternative dual frames is introduced in the setting of finite frames for ? ^d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (ΣΔ) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order $\mathcal{O}(N^{-r})$ for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.     

Erratum to: Sobolev Duals in Frame Theory and Sigma-Delta Quantization

A new class of alternative dual frames is introduced in the setting of finite frames for ℝ ^d . These dual frames, called Sobolev duals, provide a high precision linear reconstruction procedure for Sigma-Delta (ΣΔ) quantization of finite frames. The main result is summarized as follows: reconstruction with Sobolev duals enables stable rth order Sigma-Delta schemes to achieve deterministic approximation error of order O(N^-r)\mathcal{O}(N^{-r}) for a wide class of finite frames of size N. This asymptotic order is generally not achievable with canonical dual frames. Moreover, Sobolev dual reconstruction leads to minimal mean squared error under the classical white noise assumption.     

Robustness and surgery of frames

We characterize frames in Rⁿ that are robust to k erasures. The characterization is given in terms of the support of the null space of the synthesis operator of the frame. A necessary and sufficient condition is given for when an (r, k)-surgery on unit-norm tight frames in R² are possible. Also a generalization of a known characterization of the existence of tight frames with prescribed norms is given.