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Life Beyond Bases: The Advent of Frames (Part I)
, 2007
"... Redundancy is a common tool in our daily lives. Before we leave the house, we double and triplecheck that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just onc ..."
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Cited by 72 (8 self)
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Redundancy is a common tool in our daily lives. Before we leave the house, we double and triplecheck that we turned off gas and lights, took our keys, and have money (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming “just once more” they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in another system, typically a basis, where its characteristics are more readily apparent in the transform coefficients. However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be serious. In comes redundancy; we build a safety net into our representation so that we can avoid those disasters. The redundant counterpart of a basis is called a frame [no one seems to know why they are called frames, perhaps because of the bounds in (25)?]. It is generally acknowledged (at least in the signal processing and harmonic analysis communities) that frames were born in 1952 in the paper by Duffin and Schaeffer [32]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneers—Daubechies, Grossman, and Meyer [29]. Framelike ideas, that is, building redundancy into a signal expansion, can be found in pyramid
Painless Reconstruction from Magnitudes of Frame Coefficients
 J FOURIER ANAL APPL (2009) 15: 488–501
, 2009
"... The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present line ..."
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Cited by 49 (10 self)
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The goal of this paper is to develop fast algorithms for signal reconstruction from magnitudes of frame coefficients. This problem is important to several areas of research in signal processing, especially speech recognition technology, as well as state tomography in quantum theory. We present linear reconstruction algorithms for tight frames associated with projective 2designs in finitedimensional real or complex Hilbert spaces. Examples of such frames are twouniform frames and mutually unbiased bases, which include discrete chirps. The number of operations required for reconstruction with these frames grows at most as the cubic power of the dimension of the Hilbert space. Moreover, we present a very efficient algorithm which gives reconstruction on the order of d operations for a ddimensional Hilbert space.
Life Beyond Bases: The Advent of Frames
"... Redundancy is a common tool in our daily lives. We doubleand triplecheck that we turned off gas and lights, took our keys, money, etc. (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming âjust once more â they are on top of i ..."
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Cited by 15 (1 self)
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Redundancy is a common tool in our daily lives. We doubleand triplecheck that we turned off gas and lights, took our keys, money, etc. (at least those worrywarts among us do). When an important date is coming up, we drive our loved ones crazy by confirming âjust once more â they are on top of it. Of course, the reason we are doing that is to avoid a disaster by missing or forgetting something, not to drive our loved ones crazy. The same idea of removing doubt is present in signal representations. Given a signal, we represent it in another system, typically a basis, where its characteristics are more readily apparent in the transform coefficients (for example, waveletbased compression). However, these representations are typically nonredundant, and thus corruption or loss of transform coefficients can be fatal. In comes redundancy; we build a safety net into our representation so that we can avoid those fatal disasters. The redundant counterpart of a basis is called a frame (no one seems to know why they are called frames, perhaps because they are bounded from two sides as in (15)?). It is generally acknowledged 1 that frames were born in 1952 in the paper by Duffin and Schaeffer [57]. Despite being over half a century old, frames gained popularity only in the last decade, due mostly to the work of the three wavelet pioneersâ
Reconstruction of signals from magnitudes of redundant representations
"... Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm ..."
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Cited by 12 (6 self)
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Abstract. This paper is concerned with the question of reconstructing a vector in a finitedimensional real or complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well an iterative algorithm that finds the leastsquare solution and is robust in the presence of noise. We analyze its numerical performance by comparing it to two versions of the CramerRao lower bound. 1.
Tight pfusion frames
"... Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce pfusion frames, a sharpening of the notion of fusion frames. Tight pfusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann s ..."
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Cited by 8 (4 self)
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Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce pfusion frames, a sharpening of the notion of fusion frames. Tight pfusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the pfusion frame potential, derive bounds for its value, and discuss the connections to tight pfusion frames.
K.: Frame potential and finite abelian groups
 Contemp. Math
, 2008
"... Abstract. This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of generators by elem ..."
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Cited by 8 (3 self)
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Abstract. This article continues a prior investigation of the authors with the goal of extending characterization results of convolutional tight frames from the context of cyclic groups to general finite abelian groups. The collections studied are formed by translating a number of generators by elements of a fixed subgroup and it is shown, under certain norm conditions, that tight frames with this structure are characterized as local minimizers of the frame potential. Natural analogs to the downsampling and upsampling operators of finite cyclic groups are studied for arbitrary subgroups of finite abelian groups. Directions of further study are also proposed. 1.
Frames for Linear Reconstruction without Phase
"... Abstract — The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communication ..."
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Cited by 7 (1 self)
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Abstract — The objective of this paper is the linear reconstruction of a vector, up to a unimodular constant, when all phase information is lost, meaning only the magnitudes of frame coefficients are known. Reconstruction algorithms of this type are relevant for several areas of signal communications, including wireless and fiberoptical transmissions. The algorithms discussed here rely on suitable rankone operator valued frames defined on finitedimensional real or complex Hilbert spaces. Examples of such frames are the rankone Hermitian operators associated with vectors from maximal sets of equiangular lines and maximal sets of mutually unbiased bases. We also study erasures and show that in addition to loss of phase, a maximal set of mutually unbiased bases can correct up to one lost frame coefficient occurring in each basis except for one without loss. I.
Tight frames generated by finite nonabelian groups
, 2007
"... Let H be a Hilbert space of finite dimension d, such as the finite signals ℓ2(d) or a space of multivariate orthogonal polynomials, and n ≥ d. There is a finite number of tight frames of n vectors for H which can be obtained as the orbit of a single vector under the unitary action of an abelian grou ..."
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Cited by 7 (0 self)
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Let H be a Hilbert space of finite dimension d, such as the finite signals ℓ2(d) or a space of multivariate orthogonal polynomials, and n ≥ d. There is a finite number of tight frames of n vectors for H which can be obtained as the orbit of a single vector under the unitary action of an abelian group G (of symmetries of the frame). Each of these so called harmonic frames or geometrically uniform frames can be obtained from the character table of G in a simple way. These frames are used in signal processing and information theory. For a nonabelian group G there are in general uncountably many inequivalent tight frames of n vectors for H which can be obtained as such a G–orbit. However, by adding an additional natural symmetry condition (which automatically holds if G is abelian), we obtain a finite class of such frames which can be constructed from the character table of G in a similar fashion to the harmonic frames. This is done by identifying each G–orbit with an element of the group algebra CG (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.
A Physical Interpretation for Finite Tight Frames
, 2003
"... Though finite tight frames arise in many applications, they have often proved difficult to understand and construct. We investigate the nonlinear problem of finding a tight frame for which the lengths of the frame elements have been prescribed in advance. Borrowing several ideas from Classical Mecha ..."
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Cited by 6 (0 self)
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Though finite tight frames arise in many applications, they have often proved difficult to understand and construct. We investigate the nonlinear problem of finding a tight frame for which the lengths of the frame elements have been prescribed in advance. Borrowing several ideas from Classical Mechanics, we show that this problem has a natural, intuitive interpretation. In particular, we show that such frames may be characterized as the minimizers of a potential energy function, and justify their interpretation as “maximally orthogonal” sequences. By exploiting this idea, we are able toshowthat such frames always exist, provided the requisite lengths satisfy a “fundamental inequality.” In so doing, we characterize those sequences of nonnegative numbers which arise as the lengths of a tight frame’s elements.