The Experts below are selected from a list of 8943 Experts worldwide ranked by ideXlab platform
Felix J. Herrmann - One of the best experts on this subject based on the ideXlab platform.
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True Amplitude depth migration using Curvelets
GEOPHYSICS, 2021Co-Authors: Hamideh Sanavi, Peyman P. Moghaddam, Felix J. HerrmannAbstract:We propose a true amplitude solution to the seismic imaging problem. We derive a diagonal scaling approach for the normal operator approximation in the curvelet domain. This is based on the theorem which states that Curvelets remain approximately invariant under the action of the normal operator. We use Curvelets as essential tools for both approximation and inversion. We also exploit the theorem which states that curvelet-domain approximation should be smooth in phase space by enforcing smoothness of curvelet coefficients in angle and space domain.We analyze our method using a reverse time migration-demigration code, simulating the acoustic wave equation on different synthetic models. Our method produces a good resolution with reflecting dips and reproduces true amplitude reflectors and compensates for incomplete illumination in seismic images.
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true amplitude seismic imaging in presence of noise using Curvelets match filtering
81st EAGE Conference and Exhibition 2019, 2019Co-Authors: Hamideh Sanavi, P Moghaddam, Felix J. HerrmannAbstract:True amplitude seismic migration in presence of noise is covered in this paper. Existence of noise has a deteriorating effect on seismic migration which could severely corrupt the seismic amplitudes and creates artifacts in the seismic image. We address this problem in this article and provide a solution. we propose a diagonal scaling approximation for inverse of normal operator using Curvelets. We exploit invariance of Curvelet under action of normal operator for this approximation. We also exploit the fact that the coefficients of Curvelets are smooth in both physical and phase space domain in context of Curvelet match filtering.
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non parametric seismic data recovery with curvelet frames
Geophysical Journal International, 2008Co-Authors: Felix J. Herrmann, Gilles HennenfentAbstract:SUMMARY Seismic data recovery from data with missing traces on otherwise regular acquisition grids forms a crucial step in the seismic processing flow. For instance, unsuccessful recovery leads to imaging artefacts and to erroneous predictions for the multiples, adversely affecting the performance of multiple elimination. A non-parametric transform-based recovery method is presented that exploits the compression of seismic data volumes by recently developed curvelet frames. The elements of this transform are multidimensional and directional and locally resemble wave fronts present in the data, which leads to a compressible representation for seismic data. This compression enables us to formulate a new curvelet-based seismic data recovery algorithm through sparsity-promoting inversion. The concept of sparsity-promoting inversion is in itself not new to geophysics. However, the recent insights from the field of ‘compressed sensing’ are new since they clearly identify the three main ingredients that go into a successful formulation of a recovery problem, namely a sparsifying transform, a sampling strategy that subdues coherent aliases and a sparsity-promoting program that recovers the largest entries of the curvelet-domain vector while explaining the measurements. These concepts are illustrated with a stylized experiment that stresses the importance of the degree of compression by the sparsifying transform. With these findings, a curvelet-based recovery algorithms is developed, which recovers seismic wavefields from seismic data volumes with large percentages of traces missing. During this construction, we benefit from the main three ingredients of compressive sampling, namely the curvelet compression of seismic data, the existence of a favourable sampling scheme and the formulation of a large-scale sparsity-promoting solver based on a cooling method. The recovery performs well on synthetic as well as real data by virtue of the sparsifying property of Curvelets. Our results are applicable to other areas such as global seismology.
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seismic imaging and processing with Curvelets
69th EAGE Conference and Exhibition incorporating SPE EUROPEC 2007, 2007Co-Authors: Felix J. Herrmann, Gilles Hennenfent, Peyman P. MoghaddamAbstract:B030 Seismic Imaging and Processing with Curvelets F.J. Herrmann* (University of British Columbia) G. Hennenfent (EOS-UBC) & P.P. Moghaddam (EOS-UBC) SUMMARY In this paper we present a nonlinear curvelet-based sparsity-promoting formulation for three problems in seismic processing and imaging namely seismic data regularization from data with large percentages of traces missing; seismic amplitude recovery for subsalt images obtained by reverse-time migration and primary-multiple separation given an inaccurate multiple prediction. We argue why these nonlinear formulations are beneficial. EAGE 69 th Conference & Exhibition — London UK 11 - 14 June 2007 In this paper we report recent developments on the
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Recent developments in curvelet-based seismic processing
69th EAGE Conference and Exhibition - Workshop Package, 2007Co-Authors: Felix J. HerrmannAbstract:Recent developments in curvelet-based seismic processing Felix J. Herrmann (Seismic Laboratory for Imaging and Modeling Department of Earth and Ocean Sciences The University of British Columbia) SUMMARY____________________________________________________________ In this paper we present recent developments in nonlinear curvelet-based sparsity-promoting formulations of problems in the seismic data processing flow. We present our latest work on a parallel curvelet transform and recent work on a curvelet-regularized formulation for the focal transform the prediction of multiples and the computation of the inverse data space. We show that the curvelet’s wavefront detection capability and invariance under wave propagation lead to a formulation of these problems
Hervé Chauris - One of the best experts on this subject based on the ideXlab platform.
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ensemble based data assimilation with Curvelets regularization
Journal of Petroleum Science and Engineering, 2015Co-Authors: Yanhui Zhang, Hervé Chauris, Dean S Oliver, Daniela DonnoAbstract:Abstract Geologic facies modeling plays an important role in reservoir characterization as a way to reproduce important patterns of heterogeneity in petroleum reservoirs and to facilitate the modeling of petrophysical properties of reservoir rocks. Because the data and general geological knowledge (well log, core, outcrop analog, etc.) are usually not sufficient to determine the distribution of geologic facies uniquely, it is advantageous to assimilate production data to reduce the uncertainty. Geologic facies are commonly represented with categorical indicators, which are intrinsically non-Gaussian. When the ensemble Kalman filter (EnKF) or related ensemble-based methods are used to assimilate data in a straightforward way, the updated model variables would lose the discrete nature and be contaminated with noise. This paper focuses on the use of the curvelet transform to reduce the noise from the posterior realizations after the assimilation of production data with ensemble-based methods. Curvelets provide an almost optimal sparse representation of objects with edges, making them well-suited for denoising estimated geologic facies distributions. The denoising of the updated model variables is implemented in the curvelet domain by minimizing an objective function which promotes the sparsity of curvelet coefficients. Because preservation of the data match is an important measure of the performance of the denoising method, the role of the approximation of the inverse posterior covariance is examined in the minimization. We demonstrate the application of Curvelets to denoising with two examples. The results show that Curvelets are useful for denoising in the problem concerned in this paper but lose data match unless the covariance is included. In that case, the data match remains relatively good, but not as good as achieved at the end of history matching.
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Solving the Wave Equation Using Curvelets
2010Co-Authors: Bingbing Sun, Hervé Chauris, Huizhu YangAbstract:Seismic imaging is a key step in seismic exploration to retrieve the Earth properties from seismic measurements at the surface. One needs to properly model the response of the Earth by solving the wave equation. We present how Curvelets can be used in that respect. Curvelets can be seen from the geophysical point of view as the representation of local plane waves. The unknown pressure, solution of the wave equation, is decomposed in the curvelet domain. We derive the new associated equation for the curvelet coefficients and show how to solve it. In this paper, we focus on a simple homogeneous model to illustrate the feasibility of the curvelet-based method. This is a first step towards the modeling in more complex models. In particular, we express the derivative of the wave field in the curvelet domain. The simulation results show that our algorithm can give a multi-scale and multi-directional view of the wave propagation. A potential application is to model the wave motion in some specific directions.
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2D Multiple Prediction in the Curvelet Domain
72nd EAGE Conference and Exhibition incorporating SPE EUROPEC 2010, 2010Co-Authors: Daniela Donno, Hervé Chauris, Mark NobleAbstract:The suppression of multiples is a crucial task when processing seismic reflection data. We investigate how Curvelets could be used for surface-related multiple prediction. From a geophysical point of view, a curvelet can be seen as the representation of a local plane wave, and is particularly well suited for seismic data decomposition. For the prediction of multiples in the curvelet domain, we propose to first decompose the input data into curvelet coefficients. These coefficients are then convolved together to predict the coefficients associated to multiples, and the final result is obtained by applying the inverse curvelet transform. The curvelet transform offers two advantages. The directional characteristic of Curvelets allows to exploit Snell’s law at the sea surface. Moreover, the possible aliasing in the predicted multiples can be better managed by using the curvelet multi-scale property to weight the prediction according to the low-frequency part of the data. 2D synthetic and field data examples show that some artifacts and aliasing effects can be indeed reduced in the multiple prediction with the use of Curvelets.
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Curvelet-based multiple prediction
GEOPHYSICS, 2010Co-Authors: Daniela Donno, Hervé Chauris, Mark NobleAbstract:The suppression of multiples is a crucial task when processing seismic reflection data. Using the curvelet transform for surface-related multiple prediction is investigated. From a geophysical point of view, a curvelet can be seen as the representation of a local plane wave and is particularly well suited for seismic data decomposition. For the prediction of multiples in the curvelet domain, first it is proposed to decompose the input data into curvelet coefficients. These coefficients are then convolved together to predict the coefficients associated with multiples, and the final result is obtained by applying the inverse curvelet transform. The curvelet transform offers two advantages. The directional characteristic of Curvelets allows for exploitation of Snell's law at the sea surface. Moreover, the possible aliasing in the predicted multiple is better managed by using the curvelet multiscale property to weight the prediction according to the low-frequency part of the data. 2D synthetic and field data examples show that some artifacts and aliasing effects are indeed reduced in the multiple prediction with the use of Curvelets, thus allowing for an improved multiple subtraction result.
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time demigration migration using Curvelets
71st EAGE Conference and Technical Exhibition Eur. Ass. of Geoscientists and Engineers, 2009Co-Authors: Hervé Chauris, Truong Nguyen, Gilles LambaréAbstract:From a geophysical point of view, Curvelets can be seen as the representation of local plane waves. They can be useful for a series of seismic processing tasks. Here we propose to use the decomposition in the curvelet domain in the context of velocity mod
Ghassan Alregib - One of the best experts on this subject based on the ideXlab platform.
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Curvelet transform with learning-based tiling
Signal Processing: Image Communication, 2017Co-Authors: Hasan Al-marzouqi, Ghassan AlregibAbstract:Compact signal and image representations are of crucial importance in a variety of application areas. Wavelet and wavelet-like transforms typically divide the frequency plane in a systematic non-adaptive approach. In this paper, we propose a learning-based method for adapting frequency domain tiling using the curvelet transform as the basis algorithm. The optimal tiling that better represents a single image or a given class of images is computed using denoising performance as the cost function. Simulated additive white Gaussian noise is removed from a given image using a thresholding algorithm. The curvelet tiling that generates maximal denoising performance as measured by PSNR or the logarithm of mean squared error (MSE) is considered optimal. The major adaptations considered are the number of scale decompositions, angular decompositions per scale/quadrant, and scale locations. A global optimization algorithm combining the three adaptations is proposed. Signal representations by adaptive Curvelets are shown to outperform default Curvelets in partial reconstruction error. Furthermore, adaptive Curvelets are used in compressed sensing recovery of incomplete seismic datasets and face images. Visual and numerical improvements across a variety of images and different subsampling ratios are reported. Finally, adaptive Curvelets denoising performance is tested on seismic datasets. Our results establish clear numerical and visual performance advantages over the default curvelet transform and the non-local means algorithm (NLM). HighlightsWavelet and wavelet-like transforms typically divide the frequency plane in a systematic non-adaptive approach.A learning-based method for adapting frequency domain tilings using Curvelets as the basis algorithm is presented.Our results establish numerical and visual performance advantages over the default curvelet transform.
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ICASSP - Searching for the optimal curvelet tiling
2013 IEEE International Conference on Acoustics Speech and Signal Processing, 2013Co-Authors: Hasan Al-marzouqi, Ghassan AlregibAbstract:Curvelets were recently introduced as a popular extension of wavelets. In the curvelet domain the input image is represented by sets of coefficients representing signal energy in different scales and angular directions. In this paper an algorithm that searches for optimal tilings for use with the curvelet transform is introduced. We consider two adaptations: scale locations, and the number of angular divisions per scale. A search algorithm that searches for the optimal tiling with respect to denoising performance is introduced. Results show significant improvement over original curvelet tilings. Tiling results were also tested with a seismic compressed sensing recovery problem. A similar performance advantage is reported.
David L. Donoho - One of the best experts on this subject based on the ideXlab platform.
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Fast Discrete Curvelet Transforms
Multiscale Modeling & Simulation, 2006Co-Authors: Emmanuel J. Candès, David L. Donoho, Laurent Demanet, Lexing YingAbstract:This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate Curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of Curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.
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Continuous curvelet transform: II. Discretization and frames
Applied and Computational Harmonic Analysis, 2005Co-Authors: Emmanuel J. Candès, David L. DonohoAbstract:We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length ≈ width^2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f → Γ_f (a, b, θ) of functions f(x_1, x_2) on R^2, with parameter space indexed by scale a > 0, location b ∈ R^2, and orientation θ. The transform projects f onto a curvelet γ_(abθ), yielding coefficient Γ_f (a, b, θ) = f, _(γabθ); the corresponding curvelet γ_(abθ) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these Curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on true affine parabolic scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith’s transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smith’s transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales. We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a_j = 2^−j, at equispaced intervals in direction, θ_(jl), = 2π2^(−j/2)l, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the ‘Curvelets 2002’ frame constructed by Emmanuel Candes et al. [1, 2, 3]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but at fine scales is based on sampling Smith’s transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smith’s continuous transform was intended for use in forming molecular decompositions of Fourier Integral Operators (FIO’s). Our results showing close approximation of the curvelet frame by a composite frame using true affine paraboblic scaling at fine scales allow us to cross-apply Smith’s results, proving that the discrete curvelet transform gives sparse representations of FIO’s of order zero. This yields an alternate proof of a recent result of Candes and Demanet about the sparsity of FIO representations in discrete curvelet frames.
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Continuous curvelet transform
Applied and Computational Harmonic Analysis, 2005Co-Authors: Emmanuel J. Candès, David L. DonohoAbstract:AbstractWe discuss a Continuous Curvelet Transform (CCT), a transform f↦Γf(a,b,θ) of functions f(x1,x2) on R2 into a transform domain with continuous scale a>0, location b∈R2, and orientation θ∈[0,2π). Here Γf(a,b,θ)=〈f,γabθ〉 projects 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 [E.J. Candès, F. Guo, New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Process. 82 (2002) 1519–1543; E.J. Candès, L. Demanet, Curvelets and Fourier integral operators, C. R. Acad. Sci. Paris, Sér. I (2003) 395–398; E.J. Candès, D.L. Donoho, Curvelets: a surprisingly effective nonadaptive representation of objects with edges, in: A. Cohen, C. Rabut, L.L. Schumaker (Eds.), Curve and Surface Fitting: Saint-Malo 1999, Vanderbilt Univ. Press, Nashville, TN, 2000]. 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→0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ0. Generalizing these examples, we show that decay properties of Γf(a,x0,θ0) for fixed (x0,θ0), as a→0 can precisely identify the wavefront set and the Hm-wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0,θ0) near which Γf(a,x,θ) is not of rapid decay as a→0; the Hm-wavefront set is the closure of those points (x0,θ0) where the ‘directional parabolic square function’ Sm(x,θ)=(∫|Γf(a,x,θ)|2daa3+2m)1/2 is not locally integrable. The CCT is closely related to a continuous transform pioneered by Hart Smith in his study of Fourier Integral Operators. Smith's transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier–Bros–Iagolnitzer) and Wave Packets (Cordoba–Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set
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new tight frames of Curvelets and optimal representations of objects with piecewise c2 singularities
Communications on Pure and Applied Mathematics, 2004Co-Authors: Emmanuel J. Candès, David L. DonohoAbstract:This paper introduces new tight frames of Curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped 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 that 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 that are C 2 except for discontinuities along piecewise C 2 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 n-term 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 that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior.
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Continuous Curvelet Transform: II. Discretization and Frames
2003Co-Authors: Emmanuel J. Candès, David L. DonohoAbstract:We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length ≈ width 2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f ↦ → Γf (a, b, θ) offunctions f(x1,x2) on R 2, with parameter space indexed by scale a>0, location b ∈ R 2, and orientation θ. The transform projects f onto a curvelet γabθ, yielding coefficient Γf (a, b, θ) =〈f,γabθ〉; the corresponding curvelet γabθ is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these Curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on true affine parabolic scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith'
Emmanuel J. Candès - One of the best experts on this subject based on the ideXlab platform.
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Fast Discrete Curvelet Transforms
Multiscale Modeling & Simulation, 2006Co-Authors: Emmanuel J. Candès, David L. Donoho, Laurent Demanet, Lexing YingAbstract:This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate Curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of Curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.
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3D discrete curvelet transform
Wavelets XI, 2005Co-Authors: Lexing Ying, Laurent Demanet, Emmanuel J. CandèsAbstract:In this paper, we present the first 3D discrete curvelet transform. This transform is an extension to the 2D transform described in Candes et al..1 The resulting curvelet frame preserves the important properties, such as parabolic scaling, tightness and sparse representation for singularities of codimension one. We describe three different implementations: in-core, out-of-core and MPI-based parallel implementations. Numerical results verify the desired properties of the 3D Curvelets and demonstrate the efficiency of our implementations.
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Continuous curvelet transform: II. Discretization and frames
Applied and Computational Harmonic Analysis, 2005Co-Authors: Emmanuel J. Candès, David L. DonohoAbstract:We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length ≈ width^2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f → Γ_f (a, b, θ) of functions f(x_1, x_2) on R^2, with parameter space indexed by scale a > 0, location b ∈ R^2, and orientation θ. The transform projects f onto a curvelet γ_(abθ), yielding coefficient Γ_f (a, b, θ) = f, _(γabθ); the corresponding curvelet γ_(abθ) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these Curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on true affine parabolic scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith’s transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smith’s transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales. We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a_j = 2^−j, at equispaced intervals in direction, θ_(jl), = 2π2^(−j/2)l, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the ‘Curvelets 2002’ frame constructed by Emmanuel Candes et al. [1, 2, 3]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but at fine scales is based on sampling Smith’s transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smith’s continuous transform was intended for use in forming molecular decompositions of Fourier Integral Operators (FIO’s). Our results showing close approximation of the curvelet frame by a composite frame using true affine paraboblic scaling at fine scales allow us to cross-apply Smith’s results, proving that the discrete curvelet transform gives sparse representations of FIO’s of order zero. This yields an alternate proof of a recent result of Candes and Demanet about the sparsity of FIO representations in discrete curvelet frames.
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Continuous curvelet transform
Applied and Computational Harmonic Analysis, 2005Co-Authors: Emmanuel J. Candès, David L. DonohoAbstract:AbstractWe discuss a Continuous Curvelet Transform (CCT), a transform f↦Γf(a,b,θ) of functions f(x1,x2) on R2 into a transform domain with continuous scale a>0, location b∈R2, and orientation θ∈[0,2π). Here Γf(a,b,θ)=〈f,γabθ〉 projects 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 [E.J. Candès, F. Guo, New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Process. 82 (2002) 1519–1543; E.J. Candès, L. Demanet, Curvelets and Fourier integral operators, C. R. Acad. Sci. Paris, Sér. I (2003) 395–398; E.J. Candès, D.L. Donoho, Curvelets: a surprisingly effective nonadaptive representation of objects with edges, in: A. Cohen, C. Rabut, L.L. Schumaker (Eds.), Curve and Surface Fitting: Saint-Malo 1999, Vanderbilt Univ. Press, Nashville, TN, 2000]. 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→0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ0. Generalizing these examples, we show that decay properties of Γf(a,x0,θ0) for fixed (x0,θ0), as a→0 can precisely identify the wavefront set and the Hm-wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0,θ0) near which Γf(a,x,θ) is not of rapid decay as a→0; the Hm-wavefront set is the closure of those points (x0,θ0) where the ‘directional parabolic square function’ Sm(x,θ)=(∫|Γf(a,x,θ)|2daa3+2m)1/2 is not locally integrable. The CCT is closely related to a continuous transform pioneered by Hart Smith in his study of Fourier Integral Operators. Smith's transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier–Bros–Iagolnitzer) and Wave Packets (Cordoba–Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set
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The Curvelet Representation of Wave Propagators is Optimally Sparse
arXiv: Analysis of PDEs, 2004Co-Authors: Emmanuel J. Candès, Laurent DemanetAbstract:This paper argues that Curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ~ length^2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized. It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e. faster than any negative polynomial), and well-organized in the sense that the very few nonnegligible entries occur near a shifted diagonal. Indeed, we actually show that the action of the wave-group on a curvelet is well-approximated by simply translating the center of the curvelet along the Hamiltonian flow--hence the diagonal shift in the curvelet epresentation. A physical interpretation of this result is that Curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.
