The Experts below are selected from a list of 462 Experts worldwide ranked by ideXlab platform
David L Donoho - One of the best experts on this subject based on the ideXlab platform.
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Ridge functions and orthonormal Ridgelets
2013Co-Authors: David L DonohoAbstract:Orthonormal Ridgelets are a specialized set of angularly-integrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal Ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t), and we study the properties of the linear approximation operator which ‘kills ’ coefficients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give efficient nonlinear approximations to pure ridge functions. In effect, the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation. Key Words and Phrases. Wavelets. Ridge function. Ridgelet. Radon transform. Best m-term approximation. Thresholding of wavelet coefficients
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Tight Frames of k-Plane Ridgelets and the Problem of Representing Objects Which Are Smooth Away from d-Dimensional Singularities in R^n
2010Co-Authors: David L DonohoAbstract:For each pair (n, k) with 1 ≤ k<n, we construct a tight frame (ρλ: λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane Ridgelets. The intent is to efficiently represent functions which are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether in fact k-plane Ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle Xn,k – the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to Xn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space R k(n−k) × R n−k. We then use the X-ray isometry [Solmon, 1976] to map this tight frame isometrically to a tight frame for L 2 (R n) – the k-plane Ridgelets. This construction makes analysis of a function f ∈ L2 (Rn)byk-plane Ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for Xn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1
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Tight Frames of k-Plane Ridgelets and the Problem of Representing Objects Which Are Smooth Away from d-Dimensional Singularities in R^n
2007Co-Authors: David L DonohoAbstract:For each pair (n, k) with 1 # k < n, we construct a tight frame (# # : # # #) for L 2 (R n ), which we call a frame of k-plane Ridgelets. The intent is to efficiently represent functions which are smooth away from singularities along k-planes in R n . We also develop tools to help decide whether in fact k-plane Ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle X n,k -- the fiber bundle having the Grassman manifold G n,k of k-planes in R n for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to X n,k , via the smooth local coordinates of G n,k , of an orthonormal basis of tensor Meyer wavelets on Euclidea..
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astronomical image representation by the curvelet transform
Astronomy and Astrophysics, 2003Co-Authors: J L Starck, David L Donoho, Emmanuel J CandesAbstract:We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We show that the curvelet transform allows us also to well enhance elongated features contained in the data. Finally, we describe the Morphological Component Analysis, which consists in separating features in an image which do not present the same morphological characteristics. A range of examples illustrates the results.
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Flesia, Digital ridgelet transform based on true ridge functions
Academic Press, 2003Co-Authors: David L Donoho, Ana Georgina FlesiaAbstract:Abstract. We study a notion of ridgelet transform for arrays of digital data in which the analysis operator uses true ridge functions, as does the synthesis operator. There are fast algorithms for analysis, for synthesis, and for partial reconstruction. Associated with this is a transform which is a digital analog of the orthonormal ridgelet transform (but not ortho-normal for finite n). In either approach, we get an overcomplete frame; the result of ridgelet transforming an n × n array is a 2n × 2n array. The analysis operator is invertible on its range; the appropriately precondi-tioned operator has a tightly controlled spread of singular values. There is a near-parseval relationship. Our construction exploits the recent development by Averbuch et al. (2001) of the Fast Slant Stack, a Radon transform for digital image data; it may be viewed as following a Fast Slant Stack with fast 2-d wavelet transform. A consequence of this construction is that it offers discrete objects (discrete Ridgelets, discrete Radon transform, discrete Pseudopolar Fourier domain) which obey inter-relationships paralleling those in the continuum ridgelet theory (between Ridgelets, Radon transform, and polar Fourier domain). We make comparisons with other notions of ridgelet transform, and we investigate what we view as the key issue: the summability of the kernel underlying the constructed frame. The sparsity observed in our current implementation is not nearly as good as the sparsity of the under-lying continuum theory, so there is room for substantial progress in future implementations
E Campmany - One of the best experts on this subject based on the ideXlab platform.
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automatic detection of ship tracks in atsr 2 satellite imagery
Atmospheric Chemistry and Physics, 2009Co-Authors: E Campmany, R G Grainger, S M Dean, A M SayerAbstract:Abstract. Ships modify cloud microphysics by adding cloud condensation nuclei (CCN) to a developing or existing cloud. These create lines of larger reflectance in cloud fields that are observed in satellite imagery. An algorithm has been developed to automate the detection of ship tracks in Along Track Scanning Radiometer 2 (ATSR-2) imagery. The scheme has been integrated into the Global Retrieval of ATSR Cloud Parameters and Evaluation (GRAPE) processing chain. The algorithm firstly identifies intensity Ridgelets in clouds which have the potential to be part of a ship track. This identification is done by comparing each pixel with its surrounding ones. If the intensity of three adjacent pixels is greater than the intensity of their neighbours, then it is classified as a ridgelet. These Ridgelets are then connected together, according to a set of connectivity rules, to form tracks which are classed as ship tracks if they are long enough. The algorithm has been applied to two years of ATSR-2 data. Ship tracks are most frequently seen off the west coast of California, and the Atlantic coast of both West Africa and South-Western Europe. The global distribution of ship tracks shows strong seasonality, little inter-annual variability and a similar spatial pattern to the distribution of ship emissions.
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Detection of ship tracks in ATSR2 satellite imagery
Copernicus Publications, 2008Co-Authors: E Campmany, R G Grainger, S M DeanAbstract:Ships modify cloud microphysics by adding cloud condensation nuclei (CCN) to a developing or existing cloud. These create lines of larger reflectance in cloud fields that are observed in satellite imagery. Ship tracks are most frequently seen off the west coast of California, and the Atlantic coast of both west Africa and south-western Europe. In order to automate their detection within the Along Track Scanning Radiometer 2 (ATSR2) data set an algorithm was developed and integrated with the Global Retrieval of ATSR Cloud Parameters and Evaluation (GRAPE) processing chain. The algorithm firstly identifies intensity Ridgelets in clouds which have the potential to be part of a ship track. This identification is done by comparing each pixel with its surrounding ones. If the intensity of three adjacent pixels is greater than the intensity of its neighbours, then it is classified as a ridgelet. These Ridgelets are then connected together, according to a set of connectivity rules, to form tracks which are classed as ship tracks if they are long enough. The algorithm has been applied to two years of ATSR2 data. A month of results have been compared with other satellite datasets to validate the algorithm. There is a high ratio of false detections. Nevertheless the global distribution of ship tracks shows a similar pattern to the ship emissions distribution
Martin Vetterli - One of the best experts on this subject based on the ideXlab platform.
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the finite ridgelet transform for image representation
IEEE Transactions on Image Processing, 2003Co-Authors: Martin VetterliAbstract:The ridgelet transform was introduced as a sparse expansion for functions on continuous spaces that are smooth away from discontinuities along lines. We propose an orthonormal version of the ridgelet transform for discrete and finite-size images. Our construction uses the finite Radon transform (FRAT) 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.
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image denoising using orthonormal finite ridgelet transform
SPIE Conference on Wavelet Applications in Signal and Image Processing VIII, 2000Co-Authors: Martin VetterliAbstract:This paper addresses the image denoising problem using a newly proposed digital image transform: the finite rigdelet transform (FRIT). The transform is invertible, non-redundant and achieved via fast algorithms. Furthermore this transform can be designed to be orthonormal thus indicating its potential in many other image processing applications. We then propose various improvements on the initial design of the FRIT in order to make it to have better energy compaction and to reduce the border effect. Experimental results show that the new transform outperforms wavelets in denoising images with linear discontinuities.
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orthonormal finite ridgelet transform for image compression
International Conference on Image Processing, 2000Co-Authors: Martin VetterliAbstract:A finite implementation of the ridgelet transform is presented. The transform is invertible, non-redundant and achieved via fast algorithms. Furthermore we show that this transform is orthogonal hence it allows one to use non-linear approximations for the representation of images. Numerical results on different test images are shown. Those results conform with the theory of the ridgelet transform in the continuous domain-the obtained representation can represent efficiently images with linear singularities. Thus it indicates the potential of the proposed system as a new transform for coding of images.
Emmanuel J Candes - One of the best experts on this subject based on the ideXlab platform.
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Ridgelets: estimating with ridge functions
2010Co-Authors: Emmanuel J CandesAbstract:Feedforward neural networks, projection pursuit regression, and more generally, estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences. To address problems inherent to these methods – ranging from the construction of neural networks to their efficiency and capability – Candes (1999d) developed a new system that allows the representation of arbitrary functions as superpositions of specific ridge functions, the Ridgelets. In a nonparametric regression setting, this article suggests expanding noisy data into a ridgelet series and applying a scalar nonlinearity to the coefficients (dumping); this is unlike existing approaches based on stepwise additions of elements. The procedure is simple, constructive, stable and spatially adaptive – and fast algorithms have been developed to implement it. The ridgelet estimator is nearly optimal for estimating functions with certain kinds of spatial inhomogeneities. In addition, Ridgelets help to identify new classes of estimands – corresponding to a new notion of smoothness – that are well suited for ridge functions estimation. While the results are stated in a decision theoretic framework, numerical experiments are also presented to illustrate the practical performance of the methodology. Key Words and Phrases. Nonparametric regression, Ridgelets, ridge functions, Projection Pursuit Regression, minimax decision theory, Radon transform, spatial inhomogeneities, edges, thresholding of ridgelet coefficients. Acknowledgments. I am especially grateful to David Donoho for serving as my adviser and for many fruitful discussions. It is a pleasure to acknowledge conversations with Iain Johnstone. This research was supported by National Science Foundation grants dms 95–05151 and dms 98–72890 (KDI) and by afosr muri 95–P49620–96–1–0028. 1
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astronomical image representation by the curvelet transform
Astronomy and Astrophysics, 2003Co-Authors: J L Starck, David L Donoho, Emmanuel J CandesAbstract:We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We show that the curvelet transform allows us also to well enhance elongated features contained in the data. Finally, we describe the Morphological Component Analysis, which consists in separating features in an image which do not present the same morphological characteristics. A range of examples illustrates the results.
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Ridgelets: ESTIMATING WITH RIDGE FUNCTIONS
2003Co-Authors: Emmanuel J CandesAbstract:Feedforward neural networks, projection pursuit regression, and more generally, estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences. To address problems inherent to these methods—ranging from the construction of neural networks to their efficiency and capability—Candès [Appl. Comput. Harmon. Anal. 6 (1999) 197–218] developed a new system that allows the representation of arbitrary functions as superpositions of specific ridge functions, the Ridgelets. In a nonparametric regression setting, this article suggests expanding noisy data into a ridgelet series and applying a scalar nonlinearity to the coefficients (damping); this is unlike existing approaches based on stepwise additions of elements. The procedure is simple, constructive, stable and spatially adaptive—and fast algorithms have been developed to implement it. The ridgelet estimator is nearly optimal for estimating functions with certain kinds of spatial inhomogeneities. In addition, Ridgelets help to identify new classes of estimands—corresponding to a new notion of smoothness— that are well suited for ridge functions estimation. While the results are stated in a decision theoretic framework, numerical experiments are also presented to illustrate the practical performance of the methodology
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the curvelet transform for image denoising
IEEE Transactions on Image Processing, 2002Co-Authors: J L Starck, Emmanuel J Candes, David L DonohoAbstract:We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a/spl grave/ trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement.
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very high quality image restoration by combining wavelets and curvelets
Wavelets : applications in signal and image processing. Conference, 2001Co-Authors: J L Starck, David L Donoho, Emmanuel J CandesAbstract:We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We develop a methodology to combine wavelets together these new systems to perform noise removal by exploiting all these systems simultaneously. The results of the combined reconstruction exhibits clear advantages over any individual system alone. For example, the residual error contains essentially no visually intelligible structure: no structure is lost in the reconstruction.
S M Dean - One of the best experts on this subject based on the ideXlab platform.
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automatic detection of ship tracks in atsr 2 satellite imagery
Atmospheric Chemistry and Physics, 2009Co-Authors: E Campmany, R G Grainger, S M Dean, A M SayerAbstract:Abstract. Ships modify cloud microphysics by adding cloud condensation nuclei (CCN) to a developing or existing cloud. These create lines of larger reflectance in cloud fields that are observed in satellite imagery. An algorithm has been developed to automate the detection of ship tracks in Along Track Scanning Radiometer 2 (ATSR-2) imagery. The scheme has been integrated into the Global Retrieval of ATSR Cloud Parameters and Evaluation (GRAPE) processing chain. The algorithm firstly identifies intensity Ridgelets in clouds which have the potential to be part of a ship track. This identification is done by comparing each pixel with its surrounding ones. If the intensity of three adjacent pixels is greater than the intensity of their neighbours, then it is classified as a ridgelet. These Ridgelets are then connected together, according to a set of connectivity rules, to form tracks which are classed as ship tracks if they are long enough. The algorithm has been applied to two years of ATSR-2 data. Ship tracks are most frequently seen off the west coast of California, and the Atlantic coast of both West Africa and South-Western Europe. The global distribution of ship tracks shows strong seasonality, little inter-annual variability and a similar spatial pattern to the distribution of ship emissions.
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Detection of ship tracks in ATSR2 satellite imagery
Copernicus Publications, 2008Co-Authors: E Campmany, R G Grainger, S M DeanAbstract:Ships modify cloud microphysics by adding cloud condensation nuclei (CCN) to a developing or existing cloud. These create lines of larger reflectance in cloud fields that are observed in satellite imagery. Ship tracks are most frequently seen off the west coast of California, and the Atlantic coast of both west Africa and south-western Europe. In order to automate their detection within the Along Track Scanning Radiometer 2 (ATSR2) data set an algorithm was developed and integrated with the Global Retrieval of ATSR Cloud Parameters and Evaluation (GRAPE) processing chain. The algorithm firstly identifies intensity Ridgelets in clouds which have the potential to be part of a ship track. This identification is done by comparing each pixel with its surrounding ones. If the intensity of three adjacent pixels is greater than the intensity of its neighbours, then it is classified as a ridgelet. These Ridgelets are then connected together, according to a set of connectivity rules, to form tracks which are classed as ship tracks if they are long enough. The algorithm has been applied to two years of ATSR2 data. A month of results have been compared with other satellite datasets to validate the algorithm. There is a high ratio of false detections. Nevertheless the global distribution of ship tracks shows a similar pattern to the ship emissions distribution
