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Finite tight frames and some applications
, 2009
"... A finitedimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler desc ..."
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Cited by 8 (2 self)
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A finitedimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler
Existence And Construction Of Finite Tight Frames
, 2002
"... The space of finite tight frames of M vectors in l N with prescribed norms {bj}__x and frame constant A corresponds to the first N columns of matrices in O(M) ( the orthogonal group ) with the property that the norms of the first N elements of their rows equal the values j=M 2<land 2=N. aj = bj/v ..."
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Cited by 19 (6 self)
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The space of finite tight frames of M vectors in l N with prescribed norms {bj}__x and frame constant A corresponds to the first N columns of matrices in O(M) ( the orthogonal group ) with the property that the norms of the first N elements of their rows equal the values j=M 2<land 2=N. aj = bj
PHYSICAL LAWS GOVERNING FINITE TIGHT FRAMES
"... We give a physical interpretation for finite tight frames along the lines of Columb’s Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an Ndimensional Hilbert space which are the ..."
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We give a physical interpretation for finite tight frames along the lines of Columb’s Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an Ndimensional Hilbert space which
GENERATION OF FINITE TIGHT FRAMES BY HOUSEHOLDER THRANSFORMATIONS
"... Abstract. Finite tight frames are used widely for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such purpose. Our algorithm employs the Householder transformation ..."
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Cited by 14 (1 self)
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Abstract. Finite tight frames are used widely for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such purpose. Our algorithm employs the Householder
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
“A physical interpretation for finite tight frames, ” preprint, 2003.
"... i=1 which automatically holds if 8 is symmetric. By Corollary 3.2, 8 is a 2design, and such a 0, 1, 2design is a 3design if 8 is symmetric. This result can be found in the literature (cf. [5] and [10]). In Eldar and Forney [6], the relationship between tight frames and rankone quantum measurem ..."
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i=1 which automatically holds if 8 is symmetric. By Corollary 3.2, 8 is a 2design, and such a 0, 1, 2design is a 3design if 8 is symmetric. This result can be found in the literature (cf. [5] and [10]). In Eldar and Forney [6], the relationship between tight frames and rankone quantum
New tight frames of curvelets and optimal representations of objects with piecewise C² singularities
 COMM. ON PURE AND APPL. MATH
, 2002
"... This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshap ..."
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Cited by 428 (21 self)
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This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle
Prime tight frames
 Adv. Comput. Math
"... Abstract: We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames constr ..."
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Cited by 5 (1 self)
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Abstract: We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. We show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames as well as all prime frames
A physical interpretation of tight frames
 In: Harmonic Analysis and Applications, Applied and Numerical Harmonic Analysis
, 2006
"... Summary. We find finite tight frames when the lengths of the frame elements are predetermined. In particular, we derive a “fundamental inequality ” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical physic ..."
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Cited by 18 (6 self)
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Summary. We find finite tight frames when the lengths of the frame elements are predetermined. In particular, we derive a “fundamental inequality ” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical
Curvelets: a surprisingly effective nonadaptive representation of objects with edges
 IN CURVE AND SURFACE FITTING: SAINTMALO
, 2000
"... It is widely believed that to efficiently represent an otherwise smooth object with discontinuities along edges, one must use an adaptive representation that in some sense ‘tracks ’ the shape of the discontinuity set. This folkbelief — some would say folktheorem — is incorrect. At the very least ..."
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Cited by 395 (21 self)
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least, the possible quantitative advantage of such adaptation is vastly smaller than commonly believed. We have recently constructed a tight frame of curvelets which provides stable, efficient, and nearoptimal representation of otherwise smooth objects having discontinuities along smooth curves
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