Showing papers by "David L. Donoho published in 2000"

Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges

Abstract: : 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(sup 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.

Digital curvelet transform: strategy, implementation, and experiments

Abstract: Recently, Candes and Donoho 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 R2. 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, Curvelet 256, implementing this strategy in the case of 256 X 256 images, and we describe some experiments we have conducted using it. Examples are available for viewing by web browser.

Orthonormal ridgelets and linear singularities

Abstract: We construct a new orthonormal basis for $L^2({\Bbb R}^2)$, whose elements are angularly integrated ridge functions---{\it 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 $r = 2^j$, have radial extent $\Delta r \approx 2^j$ and angular extent $\Delta \theta \approx 2\pi/2^{j}$.Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approximation: they give very efficient approximations to objects such as $1_{\{ x_1\cos\theta+ x_2\sin\theta > a\}} \ e^{-x^2_1-x^2_2}$ which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such objects are sparse: they belong to every $\ell^p$, 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 ...

Curvelets, multiresolution representation, and scaling laws

Abstract: Curvelets provide a new multiresolution representation with several features that set them apart from existing representations such as wavelets, multiwavelets, steerable pyramids, and so on They are based on an anisotropic notion of scaling The frame elements exhibit very high direction sensitivity and are highly anisotropic In this paper we describe these properties and indicate why they may be important for both theory and applications

Curvelets and curvilinear integrals

Abstract: Let C(t):I@?R^2 be a simple closed unit-speed C^2 curve in R^2 with normal [formula](t). The curve C generates a distribution @C which acts on vector fields [formula](x"1, x"2):R^2@?R^2 by line integration according to[formula] We consider the problem of efficiently approximating such functionals. Suppose we have a vector basis or frame @F=[formula] with dual @F*=[formula]; then an m-term approximation to @C can be formed by selecting m terms (@m"i:1=

Beamlet pyramids: a new form of multiresolution analysis suited for extracting lines, curves, and objects from very noisy image data

Abstract: We describe a multiscale pyramid of line segments and develop algorithms which exploit that pyramid to recover image features - lines, curves, and blobs - from very noisy data.

Nonlinear pyramid transforms based on median-interpolation

Abstract: We introduce a nonlinear refinement subdivision scheme based on median-inter- polation. The scheme constructs a polynomial interpolating adjacent block medians of an underlying object. The interpolating polynomial is then used to impute block medians at the next finer triadic scale. Perhaps surprisingly, expressions for the refinement operator can be obtained in closed-form for the scheme interpolating by polynomials of degree D = 2. Despite the nonlinearity of this scheme, convergence and regularity can be established using techniques reminiscent of those developed in analysis of linear refinement schemes. The refinement scheme can be deployed in multiresolution fashion to construct a nonlinear pyra- mid and an associated forward and inverse transform. In this paper we discuss the basic properties of these transforms and their possible use in removing badly non-Gaussian noise. Analytic and computational results are presented to show that in the presence of highly non-Gaussian noise, the coefficients of the nonlinear transform have much better properties than traditional wavelet coeffi- cients.

Curvelets and reconstruction of images from noisy radon data

Abstract: The problem of recovering an input signal form noisy and linearly distorted data arises in many different areas of scientific investigation; e.g., noisy Radon inversion is a problem of special interest and considerable practical relevance in medical imaging. We will argue that traditional methods for solving inverse problems - damping of the singular value decomposition or cognate methods - behave poorly when the object to recover has edges. We apply a new system of representation, namely the curvelets in this setting. Curvelets provide near-optimal representations of otherwise smooth objects with discontinuities along smooth C2 edges. Inspired by some recent work on nonlinear estimation, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build a reconstruction based on the shrinkage of the noisy curvelet coefficients. This novel approach is shown to give a new theoretical understanding of the problem of edges in the Radon inversion problem.© (2000) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Digital ridgelet transform via digital polar coordinate transform

Abstract: A data processing technique called a Ridgelet transform is disclosed for more efficiently representing information. Original data samples (e.g., in the time domain) are received and transformed into frequency domain values provided in Cartesian coordinates. The frequency domain values provided in Cartesian coordinates are then transformed to digital polar coordinates (provided in a digital polar grid). Because the polar grid is non-uniform, the polar coordinate values can be weighted or normalized. A Wavelet transform is performed on data derived the frequency domain values provided in digital polar coordinates to generate Wavelet coefficients (or Ridgelet coefficients). Next a thresholding or filtering process can be performed on the Ridgelet coefficients to select a group of larger Wavelet coefficients and to discard the remaining Wavelet coefficients.

Karhunen-Loeve multispectral and multiscale image restoration

Abstract: We introduce in this paper the notion of WT-KLT and apply it to the problem of noise removal. Decorrelating first the data in the spatial domain using the WT and afterwards using the KLT in spectral domain allows us to derive a roust noise modeling in the WT-KLT space, and hence to filter the transformed data in an efficient way. Experiments are performed in order to derive (i) the best way to calculate the covariance matrix in the case of noisy data, (ii) the best method to correct the noisy WT-KLT coefficients. Finally we investigate if the curvelet transform could be an alternative to the wavelet transform for color image filtering.