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The easy path wavelet transform: A new adaptive wavelet transform for sparse representation of twodimensional data
 Multiscale Model. Simul
"... Dedicated to Manfred Tasche on the occasion of his 65th birthday We introduce a new locally adaptive wavelet transform, called Easy Path Wavelet Transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate ma ..."
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Cited by 136 (9 self)
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Dedicated to Manfred Tasche on the occasion of his 65th birthday We introduce a new locally adaptive wavelet transform, called Easy Path Wavelet Transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of twodimensional data. Key words. wavelet transform along pathways, data compression, adaptive wavelet bases, directed wavelets AMS Subject classifications. 65T60, 42C40, 68U10, 94A08 1
A review of curvelets and recent applications
 IEEE Signal Processing Magazine
, 2009
"... Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing ..."
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Cited by 126 (10 self)
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Multiresolution methods are deeply related to image processing, biological and computer vision, scientific computing, etc. The curvelet transform is a multiscale directional transform which allows an almost optimal nonadaptive sparse representation of objects with edges. It has generated increasing interest in the community of applied mathematics and signal processing over the past years. In this paper, we present a review on the curvelet transform, including its history beginning from wavelets, its logical relationship to other multiresolution multidirectional methods like contourlets and shearlets, its basic theory and discrete algorithm. Further, we consider recent applications in image/video processing, seismic exploration, fluid mechanics, simulation of partial different equations, and compressed sensing.
Optimally Sparse Image Representation by the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of f ..."
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Cited by 114 (8 self)
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The Easy Path Wavelet Transform (EPWT) [19] has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and it exploits the local correlations of the given data in a simple appropriate manner. In this paper, we show that the EPWT leads, for a suitable choice of the pathways, to optimal Nterm approximations for piecewise Hölder continuous functions with singularities along curves.
A New Hybrid Method for Image Approximation using the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of functi ..."
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Cited by 109 (4 self)
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The Easy Path Wavelet Transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image compression that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure. Key words. sparse data representation, tensor product wavelet transform, easy path wavelet transform, linear diffusion, smoothing filters, adaptive wavelet bases, Nterm approximation AMS Subject classifications. 41A25, 42C40, 68U10, 94A08 1
Shearlet Coorbit Spaces: Compactly Supported Analyzing Shearlets
 Traces and Embeddings, J. Fourier Anal. Appl
"... Abstract. We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Bes ..."
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Cited by 86 (14 self)
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Abstract. We show that compactly supported functions with sufficient smoothness and enough vanishing moments can serve as analyzing vectors for shearlet coorbit spaces. We use this approach to prove embedding theorems for subspaces of shearlet coorbit spaces resembling shearlets on the cone into Besov spaces. Furthermore, we show embedding relations of traces of these subspaces with respect to the real axes. 1.
OPTIMALLY SPARSE MULTIDIMENSIONAL REPRESENTATION USING SHEARLETS
"... Abstract. In this paper we show that the shearlets, an affinelike system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2–dimensional functions f that are C2 except for discontinuities along C2 curves. More specifically, if f S N is ..."
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Cited by 71 (29 self)
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Abstract. In this paper we show that the shearlets, an affinelike system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2–dimensional functions f that are C2 except for discontinuities along C2 curves. More specifically, if f S N is the N–term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as ‖f − f S N ‖2 2 ≃ N −2 (log N) 3, N → ∞, which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N −1 associated with wavelet approximations. Unlike the curvelets, that have similar sparsity properties, the shearlets form an affinelike system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations and translations to a single welllocalized window function.
Sparse Directional Image Representations using the Discrete Shearlet Transform
 Appl. Comput. Harmon. Anal
"... It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a n ..."
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Cited by 69 (41 self)
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It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. This paper introduces a new discrete multiscale directional representation called the Discrete Shearlet Transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the Discrete Shearlet Transform is very competitive in denoising applications both in terms of performance and computational efficiency.
Resolution of the Wavefront Set using Continuous Shearlets
, 2008
"... Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unabl ..."
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Cited by 65 (39 self)
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Abstract. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SHψf(a, s, t) = 〈f, ψast〉, where the analyzing elements ψast are dilated and translated copies of a single generating function ψ. The dilation matrices form a twoparameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements {ψast} form a system of smooth functions at continuous scales a> 0, locations t ∈ R 2, and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. 1.
A Shearlet Approach to Edge Analysis and Detection”, IEEE Trans. Image Proc. 18(5) 929–941, 2009. 7. Results of ball spiraling surface detected for 2D and 3D shearlet routines without noise added to data. Figure 8. Results of ball spiraling surface detect
"... Abstract—It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities suc ..."
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Cited by 49 (27 self)
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Abstract—It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions. Index Terms—Curvelets, edge detection, feature extraction, shearlets, singularities, wavelets. I.
The uncertainty principle associated with the continuous shearlet transform
 International Journal of Wavelets, Multiresolution and Information Processing
"... Abstract. Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, se ..."
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Cited by 37 (15 self)
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Abstract. Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the socalled Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure squareintegrability of the derived minimizers by considering weighted L2spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.