Two-dimensional directional wavelets and the scale-angle representation

Antoine, Jean-Pierre; Murenzi, Romain

doi:10.1016/0165-1684(96)00065-5

Cited by 142 publications

(110 citation statements)

References 20 publications

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“…Here, W, V are smooth windows compactly supported near the intervals [1,2] and [−1/2, 1/2] respectively. Whereas in the spatial domain curvelets live near an oriented rectangle R of length 2 −j/2 and width 2 −j , in the frequency domain, they are located in a parabolic wedge of length 2 j and width 2 j/2 and whose orientation is orthogonal to that of R. The joint localization in both space allows us to think about curvelets as occupying a 'Heisenberg cell' in phase-space with parabolic scaling in both domains.…”

Section: A New Form Of Multiscale Analysismentioning

confidence: 99%

The curvelet representation of wave propagators is optimally sparse

Candès

Demanet

2005

Comm Pure Appl Math

255

248

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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 [9,5] 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 wellapproximated by simply translating the center of the curvelet along the Hamiltonian flow-hence the diagonal shift in the curvelet representation. 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.

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Section: A New Form Of Multiscale Analysismentioning

confidence: 99%

The curvelet representation of wave propagators is optimally sparse

Candès

Demanet

2005

Comm Pure Appl Math

255

248

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“…where x = (x, y) represents 2-D spatial coordinates, and k0 = (k0, k1) is the wave-vector of the mother wavelet, which determines scale-resolving power (SRP) and angularresolving power (ARP) of the wavelet [22]. The frequency domain representation, ψ M (k), of a Morlet wavelet iŝ…”

Section: End-stopped Waveletsmentioning

confidence: 99%

“…Morlet wavelets can be used to detect linear structures having a specific orientation. In spatial domain, the two dimensional (2-D) Morlet wavelet is given by [22] ψM (x) = (e jk 0 .…”

Section: End-stopped Waveletsmentioning

confidence: 99%

Image Authentication Under Geometric Attacks Via Structure Matching

Monga

Vats

Evans

2005 IEEE International Conference on Multimedia and Expo

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Surviving geometric attacks in image authentication is considered to be of great importance. This is because of the vulnerability of classical watermarking and digital signature based schemes to geometric image manipulations, particularly local geometric attacks. In this paper, we present a general framework for image content authentication using salient feature points. We first develop an iterative feature detector based on an explicit modeling of the human visual system. Then, we compare features from two images by developing a generalized Hausdorff distance measure. The use of such a distance measure is crucial to the robustness of the scheme, and accounts for feature detector failure or occlusion, which previously proposed methods do not address. The proposed algorithm withstands standard benchmark (e.g. Stirmark) attacks including compression, common signal processing operations, global as well as local geometric transformations, and even hard to model distortions such as print and scan. Content changing (malicious) manipulations of image data are also accurately detected.

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“…In the following, to briefly introduce the 2D-CWT, we follow the formalism of [31,32], to which the interested reader is addressed.…”

Section: The 2d-cwtmentioning

confidence: 99%

Wind Direction Extraction from SAR in Coastal Areas

Zecchetto

2018

Remote Sensing

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This paper aims to illustrate and test a method, based on the Two-Dimensional Continuous Wavelet Transform (2D-CWT), developed to extract the wind directions from the Synthetic Aperture Radar (SAR) images. The knowledge of the wind direction is essential to retrieve the wind speed by using the radar-backscatter versus wind speed algorithms. The method has been applied to 61 SAR images from different satellites (Envisat, COSMO-SkyMed, Radarsat-2 and Sentinel-1A,B), and the results have been compared with the analysis wind fields from the European Centre for Medium-range Weather Forecasts (ECMWF) model, with in situ reports and with scatterometer data when available. The 2D-CWT method provides satisfactory results, both in areas a few kilometres from the coast and offshore. It is reliable as it produces good direction estimates, no matter what the characteristics of the SAR are. Statistics reports a success in the SAR wind direction estimates in 95% of cases (in 83% of cases the SAR-ECMWF wind direction difference is < ±20 • , in 92% < ±30 • ) with a mean directional bias B θ < 7 • . The SAR derived wind directions cannot be said to be validated, as the data available at present cannot be really representative of the wind field in the coastal area. However, the figures given by SAR winds are highly valuable even not properly validated, providing an independent and unique view of the spatial variability of the wind over the sea, which is possible by using the 2D-CWT method to derive the wind directions.

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Two-dimensional directional wavelets and the scale-angle representation

Cited by 142 publications

References 20 publications

The curvelet representation of wave propagators is optimally sparse

The curvelet representation of wave propagators is optimally sparse

Image Authentication Under Geometric Attacks Via Structure Matching

Wind Direction Extraction from SAR in Coastal Areas

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