Spectral tetris fusion frame constructions

Casazza, Peter G.; Fickus, Matthew; Heinecke, Andreas M.; Yang, Wang; Zhou, Zhengfang

doi:10.1117/12.892708

Cited by 14 publications

(45 citation statements)

References 11 publications

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“…We can find in the frame literature work on sparse tight frames created by spectral tetris [37] with less than 3N nonzero entries, sparse Steiner equiangular tight frames [38] with less than √ 2mN nonzero entries, and an approach based on discrete Gabor expansions [39]. The goal of these constructions is to reduce the computational burden of using the frame as an analysis/synthesis operator.…”

Section: Sparse Incoherent Framesmentioning

confidence: 99%

Algorithms for the construction of incoherent frames under various design constraints

2018

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Unit norm finite frames are generalizations of orthonormal bases with many applications in signal processing. An important property of a frame is its coherence, a measure of how close any two vectors of the frame are to each other. Low coherence frames are useful in compressed sensing applications. When used as measurement matrices, they successfully recover highly sparse solutions to linear inverse problems. This paper describes algorithms for the design of various low coherence frame types: real, complex, unital (constant magnitude) complex, sparse real and complex, nonnegative real and complex, and harmonic (selection of rows from Fourier matrices). The proposed methods are based on solving a sequence of convex optimization problems that update each vector of the frame. This update reduces the coherence with the other frame vectors, while other constraints on its entries are also imposed. Numerical experiments show the effectiveness of the methods compared to the Welch bound, as well as other competing algorithms, in compressed sensing applications.

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Section: Sparse Incoherent Framesmentioning

confidence: 99%

Algorithms for the construction of incoherent frames under various design constraints

2018

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“…Proof. Note that the system of equations (5) can be reduced to a system of rank(G) + 1 equations. When the b-rule algorithm is applied to k rank(G) + 1 systems of equations to find the minimal scalings, it follows that…”

Section: From (3) the Polytopementioning

confidence: 99%

“…One of the active areas of research is the construction of tight frames. Various methods of constructing tight frames have been developed for specific types of frames, including unit-norm tight frames, equiangular tight frames, tight frames of vectors having a given sequence of norms, tight fusion frames, sparse equal norm tight frames using spectral tetris, etc [3,17,6,5,13]. In the last couple of years the theme of scalable frames have been developed as a method of constructing tight frames from general frames by manipulating the length of frame vectors.…”

Section: Introductionmentioning

confidence: 99%

Minimal scalings and structural properties of scalable frames

Chan

Domagalski

Kim³

et al. 2017

Operators and Matrices

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Abstract. For a unit-norm frame F = { f i } k i=1 in R n , a scaling is a vector c = (c(1),... ,c(kis a Parseval frame in R n . If such a scaling exists, F is said to be scalable. A scaling c is a minimal scaling if { f i : c(i) > 0} has no proper scalable subframe. In this paper, we provide an algorithm to find all possible contact points for the John's decomposition of the identity by applying the b-rule algorithm to a linear system which is associated with a scalable frame. We also give an estimate of the number of minimal scalings of a scalable frame. We provide a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings c = (c(1),... ,c(k)) ∈ R k >0 of F have the same structural property. That is, the collections of all tight subframes of strictly scaled frames are the same up to a permutation of the frame elements. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame. Mathematics subject classification (2010): Primary 42C15, 05B20, 15A03.

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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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Spectral tetris fusion frame constructions

Cited by 14 publications

References 11 publications

Algorithms for the construction of incoherent frames under various design constraints

Algorithms for the construction of incoherent frames under various design constraints

Minimal scalings and structural properties of scalable frames

Frame scalings: A condition number approach

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