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62
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
, 2008
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Uncertainty principles and ideal atomic decomposition
 IEEE Transactions on Information Theory
, 2001
"... Suppose a discretetime signal S(t), 0 t<N, is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1ft = g and sinusoids expf2 iwt=N) = p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every d ..."
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Cited by 583 (20 self)
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Suppose a discretetime signal S(t), 0 t<N, is a superposition of atoms taken from a combined time/frequency dictionary made of spike sequences 1ft = g and sinusoids expf2 iwt=N) = p N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discretetime signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time/frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the `1 norm of the coe cients among all decompositions. Here \highly sparse " means that Nt + Nw < p N=2 where Nt is the number of time atoms, Nw is the number of frequency atoms, and N is the length of the discretetime signal.
The curvelet transform for image denoising
 IEEE TRANS. IMAGE PROCESS
, 2002
"... We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform [2] and the curvelet transform [6], [5]. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A cen ..."
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Cited by 404 (40 self)
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We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform [2] and the curvelet transform [6], [5]. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourierdomain computation of an approximate digital Radon transform. We introduce a very simple interpolation in Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudopolar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of à trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with “state of the art ” techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including treebased Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than waveletbased reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.
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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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, 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. By applying naive thresholding to the curvelet transform of such an object, one can form mterm approximations with rate of L 2 approximation rivaling the rate obtainable by complex adaptive schemes which attempt to ‘track ’ the discontinuity set. In this article we explain the basic issues of efficient mterm approximation, the construction of efficient adaptive representation, the construction of the curvelet frame, and a crude analysis of the performance of curvelet schemes.
Ridgelets: A key to higherdimensional intermittency?
, 1999
"... In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and ..."
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Cited by 170 (11 self)
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In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and other nonpointlike structures, for which wavelets are poorly adapted. We discuss in this paper a new subject, ridgelet analysis, which can effectively deal with linelike phenomena in dimension 2, planelike phenomena in dimension 3 and so on. It encompasses a collection of tools which all begin from the idea of analysis by ridge functions ψ(u1x1+...+unxn) whose ridge profiles ψ are wavelets, or alternatively from performing a wavelet analysis in the Radon domain. The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.
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 102 (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.’
Recovering Edges in IllPosed Inverse Problems: Optimality of Curvelet Frames
, 2000
"... We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such in ..."
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Cited by 80 (14 self)
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We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model Mean Squared Errors (MSEs) that tend to zero with noise level ɛ only as O(ɛ1/2)asɛ → 0. A recent innovation – nonlinear shrinkage in the wavelet domain – visually improves edge sharpness and improves MSE convergence to O(ɛ2/3). However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recentlyintroduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curveletbased biorthogonal decomposition
Digital Curvelet Transform: Strategy, Implementation and Experiments
 in Proc. Aerosense 2000, Wavelet Applications VII
, 1999
"... Recently, Candes and Donoho (1999) introduced the curvelet transform, a new multiscale representation suited for objects which are smooth away from discontinuities across curves. Their proposal was intended for functions f defined on the continuum plane R 2 . In this paper, we consider the pro ..."
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Cited by 55 (8 self)
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Recently, Candes and Donoho (1999) introduced the curvelet transform, a new multiscale representation suited for objects which are smooth away from discontinuities across curves. Their proposal was intended for functions f defined on the continuum plane R 2 . In this paper, we consider the problem of realizing this transform for digital data. We describe a strategy for computing a digital curvelet transform, we describe a software environment, Curvelet256, implementing this strategy in the case of 256 256 images, and we describe some experiments we have conducted using it. Examples are available for viewing by web browser. Dedication. To the Memory of Dr. Revira Singer, z"l. Keywords. Wavelets, Curvelets, Ridgelets, Digital Ridgelet Transform. Acknowledgements. We would like to thank Emmanuel Candes and Xiaoming Huo for many constructive suggestions, for editorial comments, and lengthy discussions. A reproduction of the Picasso engraving was kindly provided by Ruth Kozodoy...
Multilayered Image Representation: Application to Image Compression
 IEEE Trans. Image Process
"... Abstract The main contribution of this work is a new paradigm for image representation and image compression. We describe a new multilayered representation technique for images. An image is parsed into a superposition of coherent layers: smoothregions layer, textures layer, etc. The multilayere ..."
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Cited by 39 (2 self)
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Abstract The main contribution of this work is a new paradigm for image representation and image compression. We describe a new multilayered representation technique for images. An image is parsed into a superposition of coherent layers: smoothregions layer, textures layer, etc. The multilayered decomposition algorithm consists in a cascade of compressions applied successively to the image itself and to the residuals that resulted from the previous compressions. During each iteration of the algorithm, we code the residual part in a lossy way: we only retain the most significant structures of the residual part, which results in a sparse representation. Each layer is encoded independently with a different transform, or basis, at a different bitrate; and the combination of the compressed layers can always be reconstructed in a meaningful way. The strength of the multilayer approach comes from the fact that different sets of basis functions complement each others: some of the basis functions will give reasonable account of the large trend of the data, while others will catch the local transients, or the oscillatory patterns. This multilayered representation has a lot of beautiful applications in image understanding, and image and video coding. We have implemented the algorithm and we have studied its capabilities.
Curvelets and Curvilinear Integrals
, 1999
"... Let C(t):I↦ → R² be a simple closed unitspeed C² curve in R 2 with normal ⃗n(t). The curve C generates a distribution Γ which acts on vector fields ⃗v(x1,x2):R² ↦ → R² by line integration according to Γ(⃗v) = ⃗v(C(t)) · ⃗n(t)dt. We consider the problem of efficiently approximating such functional ..."
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Cited by 39 (2 self)
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Let C(t):I↦ → R² be a simple closed unitspeed C² curve in R 2 with normal ⃗n(t). The curve C generates a distribution Γ which acts on vector fields ⃗v(x1,x2):R² ↦ → R² by line integration according to Γ(⃗v) = ⃗v(C(t)) · ⃗n(t)dt. We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame Φ = ( ⃗ φµ); then an mterm approximation to Γ can be formed by selecting m terms (µi:1≤i≤m) and taking ˜Γm(⃗v) = m∑ i=1 Γ ( ⃗ φµ i)[⃗v, ⃗ φµ i Here the µi can be chosen adaptively based on the curve C. We are interested in finding a vector basis or frame for which the above scheme yields the highestquality mterm approximations. Here performance is measured by considering worstcase error on vector fields which are smooth in an L 2 Sobolev sense: Err(Γ, ˜ Γm) =sup{Γ(⃗v) − ˜ Γm(⃗v)  : ‖Div(⃗v)‖2 ≤ 1}. We establish an isometry between this problem and the problem of approximating objects with edges in L 2 norm. Starting from the recentlyintroduced tight frames of scalar curvelets, we construct a vector frame of curvelets for this problem. Invoking results on the nearoptimality of scalar curvelets in representing objects with edges, we argue that vector curvelets provide nearoptimal quality mterm approximations. We show that they dramatically outperform both wavelet and Fourierbased representations in terms of mterm approximation error. The mterm approximations to Γ are built from terms with support approaching more and more closely the curve C with increasing m; the terms have support obeying the scaling law width ≈ length 2. Comparable results can be developed, with additional work, for scalar curvelet approximation in the case of scalar integrands I(f) = f(C(t))dt.