Digital Implementation of Ridgelet Packets

Flesia, Ana Georgina; Hel‐Or, Hagit; Averbuch, Amir; Candès, Emmanuel J.; Coifman, Ronald R.; Donoho, David L.

doi:10.1016/s1570-579x(03)80030-7

Cited by 26 publications

(21 citation statements)

References 26 publications

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“…In [24], the authors followed this organization principle and constructed a family of tight frames they call ridgelet packets. One problem with this work is that the family of tight frames or orthobases they exhibit is missing a key ingredient, a translation parameter, which may limit their applicability.…”

Section: Geometry and Tilings: Ridgelet Packetsmentioning

confidence: 99%

New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

Candès

Donoho

2003

Comm Pure Appl Math

1,410

1,108

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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 obeysThis 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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Section: Geometry and Tilings: Ridgelet Packetsmentioning

confidence: 99%

New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

Candès

Donoho

2003

Comm Pure Appl Math

1,410

1,108

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“…Curvelets [8][9][10]20,33] correspond to α = 1, β = 1/2, wavelets [13,27,30,33] are α = β = 1, ridgelets [6,21] are α = 1, β = 0, the Gabor transform is α = β = 0, and the new "wave atoms" are the point α = β = 1/2. A continuous Gaussian transform with geometry α = β = 1/2 was introduced by Cordóba and Fefferman [14].…”

Section: Wave Atoms Curvelets and Other Phase-space Tilingsmentioning

confidence: 99%

Wave atoms and time upscaling of wave equations

Demanet

Ying

2009

Numer. Math.

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We present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying, two-dimensional time-independent periodic media. The method consists in representing the time-dependent Green's function in wave atoms, a tight frame of multiscale, directional wave packets obeying a precise parabolic balance between oscillations and support size, namely wavelength ∼(diameter). 2 Wave atoms offer a uniquely structured representation of the Green's function in the sense that• the resulting matrix is universally sparse over the class of C ∞ coefficients, even for "large" times; • the matrix has a natural low-rank block-structure after separation of the spatial indices.The parabolic scaling is essential for these properties to hold. As a result, it becomes realistic to accurately build the full matrix exponential in the wave atom frame, using repeated squaring up to some time typically of the form t ∼ √ x, which is bigger than the standard CFL timestep. Once the "expensive" precomputation of the Green's function has been carried out, it can be used to perform unusually large, upscaled, "cheap" time steps. The algorithm is relatively simple in that it does not require an underlying geometric optics solver. We prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks. On a N -by-N grid, the "expensive" precomputation takes somewhere between O(N 3 log N ) and O(N 4 log N ) steps L. Demanet (B) 123 2 L. Demanet, L. Ying depending on the separability of the acoustic medium. The complexity of upscaled timestepping, however, beats the O(N 3 log N ) bottleneck of pseudospectral methods on an N -by-N grid, for a wide range of physically relevant situations. In particular, we show that a naive version of the wave atom algorithm provably runs in O(N 2+δ ) operations for arbitrarily small δ-but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant. As a result, we get estimates between O(N 2.5 log N ) and O(N 3 log N ) for upscaled timestepping. We also show several numerical examples. In practice, the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions, as in reflection seismology. In academic examples of accurate propagation of bandlimited wavefronts, if the precomputation step is factored out, then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3-5 at N = 512, and a factor 10-20 at N = 1,024, for the same accuracy. Very similar gains are obtained in comparison versus a finite difference method.

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“…This is a toolbox for multiscale geometric analysis, originating from Xiaoming Huo's and Ofer Levi's theses [6,7], Georgina Flesia's postdoctoral work [8,9], and other related journal papers [10,11,12,13,14,15].…”

Section: What Is Reproducible Computational Science?mentioning

confidence: 99%

RunMyCode.Org: A Novel Dissemination and Collaboration Platform for Executing Published Computational Results

2012

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Digital Implementation of Ridgelet Packets

Cited by 26 publications

References 26 publications

New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

Wave atoms and time upscaling of wave equations

RunMyCode.Org: A Novel Dissemination and Collaboration Platform for Executing Published Computational Results

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Digital Implementation of Ridgelet Packets

Cited by 26 publications

References 26 publications

New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

Wave atoms and time upscaling of wave equations

RunMyCode.Org: A Novel Dissemination and Collaboration Platform for Executing Published Computational Results

Contact Info

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Resources

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New tight frames of curvelets and optimal representations of objects with piecewise C² singularities

New tight frames of curvelets and optimal representations of objects with piecewise C² singularities