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56
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 278 (23 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 112 (10 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.
Approximation of functions over redundant dictionaries using coherence
 Proc. of SODA
, 2003
"... ..."
The Finite Ridgelet Transform for Image Representation
 IEEE Transactions on Image Processing
, 2003
"... The ridgelet transform [6] was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we propose an orthonormal version of the ridgelet transform for discrete and finite size images. Our construction uses the finite ..."
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Cited by 71 (2 self)
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The ridgelet transform [6] was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we propose an orthonormal version of the ridgelet transform for discrete and finite size images. Our construction uses the finite Radon transform (FRAT) [11], [20] as a building block. To overcome the periodization effect of a finite transform, we introduce a novel ordering of the FRAT coefficients. We also analyze the FRAT as a frame operator and derive the exact frame bounds. The resulting finite ridgelet transform (FRIT) is invertible, nonredundant and computed via fast algorithms. Furthermore, this construction leads to a family of directional and orthonormal bases for images. Numerical results show that the FRIT is more effective than the wavelet transform in approximating and denoising images with straight edges.
Orthonormal Ridgelets and Linear Singularities
, 1998
"... We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge func ..."
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Cited by 56 (16 self)
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We construct a new orthonormal basis for L2 (R2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L2 (R2) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge functions are not in L2 (R2). Orthonormal ridgelet expansions have an interesting application in nonlinear approximation: the problem of efficient approximations to objects such as 1 {x1 cos θ+x2 sin θ>a} e−x2 1−x2 2 which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such an object are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a nearideal nonlinear approximation scheme. The ridgelet orthobasis is the isometric image of a special wavelet basis for Radon space; as a consequence, ridgelet analysis is equivalent to a special wavelet analysis in the Radon domain. This means that questions of ridgelet analysis of linear singularities can be answered by wavelet analysis of point singularities. At the heart of our nonlinear approximation result is the study of a certain tempered distribution on R2 defined formally by S(u, v) =v  −1/2σ(u/v) with σ a certain smooth bounded function; this is singular at (u, v) =(0,0) and C ∞ elsewhere. The key point is that the analysis of this point singularity by tensor Meyer wavelets yields sparse coefficients at high frequencies; this is reflected in the sparsity of the ridgelet coefficients and the good nonlinear approximation properties of the ridgelet basis.
Wavelets, Approximation, and Compression
, 2001
"... this article is to look at recent wavelet advances from a signal processing perspective. In particular, approximation results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed ..."
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Cited by 51 (6 self)
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this article is to look at recent wavelet advances from a signal processing perspective. In particular, approximation results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed
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 50 (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 36 (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...
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 29 (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.
Can recent innovations in harmonic analysis `explain' key findings in natural image statistics
 Network: Computation in Neural Systems
"... Recently, applied mathematicians have been pursuing the goal of sparse coding of certain mathematical models of images with edges. They have found by mathematical analysis that, instead of wavelets and Fourier methods, sparse coding leads towards new systems: ridgelets and curvelets. These new syste ..."
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Cited by 26 (1 self)
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Recently, applied mathematicians have been pursuing the goal of sparse coding of certain mathematical models of images with edges. They have found by mathematical analysis that, instead of wavelets and Fourier methods, sparse coding leads towards new systems: ridgelets and curvelets. These new systems have elements distributed across a range of scales and locations, but also orientations. In fact they have highly directionspecific elements and exhibit increasing numbers of distinct directions as we go to successively finer scales. Meanwhile, researchers in Natural Scene Statistics (NSS) have been attempting to find sparse codes for natural images. The new systems they have found by computational optimization have elements distributed across a range of scales and locations, but also orientations. The new systems are certainly unlike wavelet and gabor systems, on the one hand because of the multiorientation and on the other hand because of the multiscale nature. There is a certain degree of visual resemblance between the findings in the two fields, which suggests the hypothesis that certain important findings in the NSS literature