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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 231 (17 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 needleshaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2−j, each element has an envelope which is aligned along a ‘ridge ’ of length 2−j/2 and width 2−j. We prove that curvelets provide an essentially optimal representation of typical objects f which are C2 except for discontinuities along C2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the nterm partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ‖f − f C n ‖ 2 L2 ≤ C · n−2 · (log n) 3, n → ∞. This rate of convergence holds uniformly over a class of functions which are C 2 except for discontinuities along C 2 curves and is essentially optimal. In comparison, the squared error of nterm wavelet approximations only converges as n −1 as n → ∞, which is considerably worst than the optimal behavior.
On Calderón’s conjecture
, 1999
"... This paper is a successor of [4]. In that paper we considered bilinear operators of the form (1) Hα(f1,f2)(x): = p.v. f1(x − t)f2(x + αt) dt ..."
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Cited by 82 (25 self)
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This paper is a successor of [4]. In that paper we considered bilinear operators of the form (1) Hα(f1,f2)(x): = p.v. f1(x − t)f2(x + αt) dt
New Multiscale Transforms, Minimum Total Variation Synthesis: Applications to EdgePreserving Image Reconstruction
, 2001
"... This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and acc ..."
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Cited by 77 (11 self)
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This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and accurate numerical implementations with computational complexities of at most N log N. In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘denoising.’
L p estimates on the bilinear Hilbert transform for 2
 Ann. of Math
, 1997
"... The purpose of this paper is to prove Theorem 1.1 below. Consider the Schwartz space S(IR) of smooth and rapidly decaying functions f on the real line. There is a tempered distribution de ned by ..."
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Cited by 55 (21 self)
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The purpose of this paper is to prove Theorem 1.1 below. Consider the Schwartz space S(IR) of smooth and rapidly decaying functions f on the real line. There is a tempered distribution de ned by
Continuous Curvelet Transform: I. Resolution of the Wavefront Set
, 2003
"... We discuss a Continuous Curvelet Transform (CCT), a transform f ↦ → Γf (a, b, θ) of functions f(x1,x2) onR 2,into atransform domain with continuous scale a>0, location b ∈ R 2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) =〈f,γabθ 〉 where the inner products project f onto an ..."
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Cited by 41 (4 self)
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We discuss a Continuous Curvelet Transform (CCT), a transform f ↦ → Γf (a, b, θ) of functions f(x1,x2) onR 2,into atransform domain with continuous scale a>0, location b ∈ R 2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) =〈f,γabθ 〉 where the inner products project f onto analyzing elements called curvelets γabθ which are smooth and of rapid decay away from an a by √ a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track ’ the behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0,θ0), Γf (a, x0,θ0) decays rapidly as a → 0iff is smooth near x0, orifthe singularity of f at x0 is oriented in a different direction than θ0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0,θ0) for fixed (x0,θ0), as a → 0 can precisely identify the wavefront set and the H m
Resolution of the Wavefront Set using Continuous Shearlets
, 2008
"... Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unabl ..."
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Cited by 36 (20 self)
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Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SHψf(a, s, t) = 〈f, ψast〉, where the analyzing elements ψast are dilated and translated copies of a single generating function ψ. The dilation matrices form a twoparameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {ψast} form a system of smooth functions at continuous scales a> 0, locations t ∈ R 2, and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. 1.
Uniqueness in the Inverse Conductivity Problem for Nonsmooth Conductivities in Two Dimensions
, 1997
"... this paper) imply, via Sobolev embedding, that the conductivity is continuous. It is interesting to note that the only uniqueness results available for conductivities which are discontinuous are due to Kohn and Vogelius [9] who study conductivities which are piecewise analytic and V. Isakov [8] who ..."
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Cited by 34 (9 self)
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this paper) imply, via Sobolev embedding, that the conductivity is continuous. It is interesting to note that the only uniqueness results available for conductivities which are discontinuous are due to Kohn and Vogelius [9] who study conductivities which are piecewise analytic and V. Isakov [8] who considers a class of conductivities which are piecewise C
The Bernstein problem in the Heisenberg group
, 2005
"... We establish the following theorem of Bernstein type for the first Heisenberg group H¹: Let S be a C² connected Hminimal surface which is a graph over some plane P, then S is either a noncharacteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition ..."
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Cited by 31 (10 self)
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We establish the following theorem of Bernstein type for the first Heisenberg group H¹: Let S be a C² connected Hminimal surface which is a graph over some plane P, then S is either a noncharacteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.
On Leray's SelfSimilar Solutions Of The NavierStokes Equations Satisfying Local Energy Estimates
 Arch. Rational Mech. Anal
, 1998
"... . This paper proves that Leray's selfsimilar solutions of the threedimensional NavierStokes equations must be trivial under very general assumptions, for example, if they satisfy local energy estimates. 1. Introduction In 1934 Leray [Le] raised the question of the existence of selfsimilar soluti ..."
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Cited by 25 (1 self)
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. This paper proves that Leray's selfsimilar solutions of the threedimensional NavierStokes equations must be trivial under very general assumptions, for example, if they satisfy local energy estimates. 1. Introduction In 1934 Leray [Le] raised the question of the existence of selfsimilar solutions of the NavierStokes equations. For a long time, the selfsimilar solutions had appeared to be a good candidate for constructing singular solutions of the NavierStokes equations. Leray's question was unanswered until 1995, when Necas, Ruzicka, and Sver'ak [NRS] showed, among other things, that the only selfsimilar solution satisfying the global energy estimates is zero. Although they answered Leray's original problem, some important questions were left open. For example, can a selfsimilar solution satisfying local energy estimates exist? The goal of this paper is to show that the selfsimilar solutions must be zero under very general assumptions, for example, if they satisfy the local...