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 folk-belief — some would say folk-theorem — 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 near-optimal 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 m-term 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 m-term approximation, the construction of efficient adaptive representation, the construction of the curvelet frame, and a crude analysis of the performance of curvelet schemes.

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.

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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

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.

We construct a new orthonormal basis for L 2 (R 2), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The basis elements are smooth and of rapid decay in the spatial domain, and in the frequency domain are localized near angular wedges which, at radius r =2 j,haveradialextent∆r≈2 j and angular extent ∆θ ≈ 2 −j. Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approxi-mation: they give very 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 objects are sparse: they belong to every ℓp, p>0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme for such objects. Orthonormal ridgelets may be viewed as L2 substitutes for approximation by sums of ridge functions, and so can perform many of the same tasks as the ridgelets systems constructed by Candès (1997, 1998). Orthonormal ridgelets make available the machinery of orthogonal decompositions, which is not availble for ridge functions as they are not in L2 (R2). The ridgelet orthobasis is constructed as the isometric image of a special wavelet basis for

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 curvelet-based biorthogonal decomposition

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...

Let C(t):I↦ → R² be a simple closed unit-speed 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 m-term 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 highest-quality m-term approximations. Here performance is measured by considering worst-case 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 recently-introduced tight frames of scalar curvelets, we construct a vector frame of curvelets for this problem. Invoking results on the near-optimality of scalar curvelets in representing objects with edges, we argue that vector curvelets provide near-optimal quality m-term approximations. We show that they dramatically outperform both wavelet and Fourier-based representations in terms of m-term approximation error. The m-term 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.

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 direction-specific 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 multi-orientation and on the other hand because of the multi-scale 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