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Compressed sensing
 IEEE Trans. Inform. Theory
"... Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measureme ..."
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Cited by 1746 (18 self)
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Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1
Ridge functions and orthonormal ridgelets
 J. Approx. Theory
"... Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coeff ..."
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Cited by 14 (2 self)
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Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1D wavelet coefficients of the ridge profile r(t), and we study the properties of the linear approximation operator which ‘kills ’ coefficients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give efficient nonlinear approximations to pure ridge functions. In effect, the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1D wavelet coefficients of the 1D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation. Key Words and Phrases. Wavelets. Ridge function. Ridgelet. Radon transform. Best mterm approximation. Thresholding of wavelet coefficients.
Normed groups: dichotomy and duality
"... The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a ri ..."
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Cited by 9 (8 self)
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The key vehicle of the recent development of a topological theory of regular variation based on topological dynamics [BOst13], and embracing its classical univariate counterpart (cf. [BGT]) as well as fragmentary multivariate (mostly Euclidean) theories (eg [MeSh], [Res], [Ya]), are groups with a rightinvariant metric carrying ‡ows. Following the vector paradigm, they are best seen as normed groups. That concept only occasionally appears explicitly in the literature despite its frequent disguised presence, and despite a respectable lineage traceable back to the Pettis closedgraph theorem, to the Birkho¤Kakutani metrization theorem and further back still to Banach’s Théorie des opérations linéaires. We collect together known salient features and develop their theory including Steinhaus theory uni…ed by the Category Embedding Theorem [BOst11], the associated themes of subadditivity and convexity, and a topological duality inherent to topological
Nonlinear Wavelet Approximation In Anisotropic Besov Spaces
 Indiana Univ. Math. J
"... We introduce new anisotropic wavelet decompositions associated with the smoothness #, # =(#1 ,...,# d ), #1 ,...,# d > 0 of multivariate functions as measured in anisotropic Besov spaces B # . We give the rate of nonlinear approximation of functions f # B # by these wavelets. Finally, we prove ..."
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Cited by 2 (0 self)
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We introduce new anisotropic wavelet decompositions associated with the smoothness #, # =(#1 ,...,# d ), #1 ,...,# d > 0 of multivariate functions as measured in anisotropic Besov spaces B # . We give the rate of nonlinear approximation of functions f # B # by these wavelets. Finally, we prove that, among a general class of anisotropic wavelet decompositions of a function f # B # , the anisotropic wavelet decomposition associated with # gives the optimal rate of compression of the wavelet decomposition of f . Chapter 1
Absolute and Relative CutOff in Adaptive Approximation By Wavelets
, 1996
"... Given the wavelet expansion of a function v, a nonlinear adaptive approximation of v is obtained by neglecting those coefficients whose size drops below a certain threshold. We propose several ways to define the threshold: all are based on the characterization of the local regularity of v (in a Sob ..."
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Cited by 1 (1 self)
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Given the wavelet expansion of a function v, a nonlinear adaptive approximation of v is obtained by neglecting those coefficients whose size drops below a certain threshold. We propose several ways to define the threshold: all are based on the characterization of the local regularity of v (in a Sobolev or Besov scale) in terms of summability of properly defined subsets of its coefficients. Apriori estimates of the approximation error are derived. For the Haar system, the asymptotic behavior of both the approximation error and the number of survived coefficients is thoroughly investigated for a class of functions having Holdertype singularities. Corresponding author: Claudio Canuto Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi, 24 I10129 TORINO Italy phone: +39 11 564 7543 fax: +39 11 564 7599 email: ccanuto@polito.it 1 Introduction Wavelets have proven a powerful tool in signal processing and related topics (see [11], [16], [30]). For instance, the i...
INTERPOLATION SPACES AND INTERPOLATION GROUPS
"... This is a temporary version of the notes which were used for the second part of my lecture of 30/9/2010. Hopefully a somewhat more definitive version will be available later. Some parts of the lecture do not appear in these notes, and of course we did not get to the latter part of these notes in the ..."
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This is a temporary version of the notes which were used for the second part of my lecture of 30/9/2010. Hopefully a somewhat more definitive version will be available later. Some parts of the lecture do not appear in these notes, and of course we did not get to the latter part of these notes in the lecture. 1. Compatible couples of quasinormed abelian groups. Definition 1. Let A be an abelian group where the group operation is denoted by + and the identity element by 0. A functional ‖· ‖ on A is said to be a quasinorm or a cquasinorm on A if it satisfies (i) ‖a ‖ ≥ 0 for all a ∈ A. (ii) ‖a ‖ = 0 if and only if a = 0. (iii) ‖−a ‖ = ‖a ‖ for all a ∈ A. (iv) There exists a positive constant c such that ‖a + b ‖ ≤ c (‖a ‖ + ‖b‖) for all a and b in A. A quasinormed abelian group is thus an abelian group equipped with a quasinorm. If c = 1 then the 1quasinorm ‖· ‖ is also referred to as a norm, and we have a normed abelian group. Theorem 2. Let A be a cquasinormed abelian group with cquasinorm denoted by ‖·‖A. Let ρ be the positive on A such that number which satisfies (2c) ρ = 2. Then there exists a 1quasinorm ‖· ‖ ∗ A (1.1) ‖a ‖ ∗