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The Contourlet Transform: An Efficient Directional Multiresolution Image Representation
 IEEE TRANSACTIONS ON IMAGE PROCESSING
"... The limitations of commonly used separable extensions of onedimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a “true” twodimensional transform that can capture the intrinsic geometrical structure t ..."
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Cited by 510 (20 self)
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The limitations of commonly used separable extensions of onedimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a “true” twodimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discretedomain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discretedomain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and thus it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for Npixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuousdomain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.
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 137 (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
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 115 (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.
NONSUBSAMPLED CONTOURLET TRANSFORM: FILTER DESIGN AND APPLICATIONS IN DENOISING
"... In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields ..."
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Cited by 105 (4 self)
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In this paper we study the nonsubsampled contourlet transform. We address the corresponding filter design problem using the McClellan transformation. We show how zeroes can be imposed in the filters so that the iterated structure produces regular basis functions. The proposed design framework yields filters that can be implemented efficiently through a lifting factorization. We apply the constructed transform in image noise removal where the results obtained are comparable to the stateofthe art, being superior in some cases.
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 103 (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 85 (44 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.
Optimal Representation of Piecewise Hölder Smooth Bivariate Functions by the Easy Path Wavelet Transform
"... The Easy Path Wavelet Transform (EPWT) [22] 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 68 (3 self)
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The Easy Path Wavelet Transform (EPWT) [22] 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. Using polyharmonic spline interpolation, we show in this paper that the EPWT leads, for a suitable choice of the pathways, to optimal Nterm approximations for piecewise Hölder smooth functions with singularities along curves. Key words. sparse data representation, wavelet transform along pathways, Nterm approximation AMS Subject classifications. 41A25, 42C40, 68U10, 94A08 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 58 (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.
Weighted Averaging for Denoising with Overcomplete Dictionaries
"... We consider the scenario where additive, independent and identically distributed (i.i.d) noise in an image is removed using an overcomplete set of linear transforms and thresholding. Rather than the standard approach where one obtains the denoised signal by ad hoc averaging of the denoised estimates ..."
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Cited by 18 (3 self)
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We consider the scenario where additive, independent and identically distributed (i.i.d) noise in an image is removed using an overcomplete set of linear transforms and thresholding. Rather than the standard approach where one obtains the denoised signal by ad hoc averaging of the denoised estimates provided by denoising with each of the transforms, we formulate the optimal combination as a conditional linear estimation problem and solve it for optimal estimates. Our approach is independent of the utilized transforms and the thresholding scheme, and as we illustrate using oracle based denoisers, it extends established work by exploiting a separate degree of freedom that is in general not reachable using previous techniques. Our derivation of the optimal estimates specifically relies on the assumption that the utilized transforms provide sparse decompositions. At the same time, our work is robust as it does not require any assumptions about image statistics beyond sparsity. Unlike existing work which tries to devise ever more sophisticated transforms and thresholding algorithms to deal with the myriad types of image singularities, our work uses basic tools to obtain very high performance on singularities by taking better advantage of the sparsity that surrounds them. With wellestablished transforms we obtain results that are competitive with stateoftheart methods. EDICS: RSTDNOI, FLTLFLT