Sparse fusion frames: existence and construction

Calderbank, Robert; Casazza, Peter G.; Heinecke, Andreas M.; Kutyniok, Gitta; Pezeshki, Ali

doi:10.1007/s10444-010-9162-3

Cited by 46 publications

(84 citation statements)

References 49 publications

(75 reference statements)

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“…Because of this, fusion frame theory in used in applications where twostage (local/global) analysis is required, with applications situated in areas which require distributed processing, such as: distributed sensing, parallel processing, packet encoding and optimal packings [13]. In finite frame theory, a fusion frame is a spanning collection of subspaces, which were first studied in [14], and have been further analyzed in [4,12,13,15].…”

Section: Fusion Framesmentioning

confidence: 99%

“…Moreover, seeing that the construction of UNTFs via Spectral Tetris is now completely characterized, one might then ask the question: Can Spectral Tetris be used to construct non-tight, unit norm frames? In [4], they answered this question positively and adapted Spectral Tetris to construct non-tight, unit norm frames with spectrum greater than or equal to two. They called this adaptation Sparse Unit Norm Frame Construction for Real Eigenvalues (SFR).…”

Section: Spectral Tetris For Non-tight Unit Norm Framesmentioning

confidence: 99%

“…The SFR construction method also utilizes the 2 × 2 building block A (x) at (5.1), to help build a unit norm frame with prescribed spectrum one or two vectors at a time. In [4], they provide sufficient conditions for when SFR can construct a unit norm frame. In particular, SFR can construct a unit norm frame…”

Section: Spectral Tetris For Non-tight Unit Norm Framesmentioning

confidence: 99%

See 2 more Smart Citations

The Fundamentals of Spectral Tetris Frame Constructions

Casazza

Woodland

2015

Sampling Theory, a Renaissance

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Abstract. In a landmark paper [9], Casazza, Fickus, Mixon, Wang and Zhou introduced a fundamental method for constructing unit norm tight frames, which they called Spectral Tetris. This was a significant advancement for finite frame theory -especially constructions of finite frames. This paper then generated a vast amount of literature as Spectral Tetris was steadily developed, refined, and generalized until today we have a complete picture of what are the broad applications as well as the limitations of Spectral Tetris. In this paper, we will put this vast body of literature into a coherent theory.

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Section: Fusion Framesmentioning

confidence: 99%

Section: Spectral Tetris For Non-tight Unit Norm Framesmentioning

confidence: 99%

Section: Spectral Tetris For Non-tight Unit Norm Framesmentioning

confidence: 99%

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The Fundamentals of Spectral Tetris Frame Constructions

Casazza

Woodland

2015

Sampling Theory, a Renaissance

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“…The detailed steps are provided in Algorithm 1. We remark that an extension to arbitrary spectra of the frame operator is described in Calderbank et al (2011).…”

Section: Spectral Tetris -An Algorithm To Construct Sparse Tight Framesmentioning

confidence: 99%

Sparse matrices in frame theory

2013

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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.

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“…We refer to [15] for an introduction to frame theory and to [8] for an overview of the current research in the field. Frames have traditionally played a significant role in the theory of signal processing, but today they have found application to packet based network communication [7,18], wireless sensor networks [9,10,11,12], distributed processing [7], quantum information theory, bio-medical engineering [2,25], compressed sensing [3,14], fingerprinting [26], spectral theory [6,19,20], and much more.…”

Section: Introductionmentioning

confidence: 99%

Frame scalings: A condition number approach

Casazza

2017

Linear Algebra and its Applications

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Abstract. Scaling frame vectors is a simple and noninvasive way to construct tight frames. However, not all frames can be modifed to tight frames in this fashion, so in this case we explore the problem of finding the best conditioned frame by scaling, which is crucial for applications like signal processing. We conclude that this problem is equivalent to solving a convex optimization problem involving the operator norm, which is unconventional since this problem was only studied in the perspective of Frobenius norm before. We also further study the Frobenius norm case in relation to the condition number of the frame operator, and the convexity of optimal scalings.

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Sparse fusion frames: existence and construction

Abstract: Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert

Cited by 46 publications

References 49 publications

The Fundamentals of Spectral Tetris Frame Constructions

The Fundamentals of Spectral Tetris Frame Constructions

Sparse matrices in frame theory

Frame scalings: A condition number approach

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