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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 513 (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.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 427 (36 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
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 102 (32 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.
Directional Multiscale Modeling of Images using the Contourlet Transform
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2004
"... The contourlet transform is a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks. The contourlet expansion is composed of basis images oriented at varying directions in multiple scales, with flexible aspect ratios. With this rich set of basis ima ..."
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Cited by 90 (5 self)
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The contourlet transform is a new extension to the wavelet transform in two dimensions using nonseparable and directional filter banks. The contourlet expansion is composed of basis images oriented at varying directions in multiple scales, with flexible aspect ratios. With this rich set of basis images, the contourlet transform can effectively capture the smooth contours that are the dominant features in natural images with only a small number of coefficients. We begin with a detailed study on the statistics of the contourlet coefficients of natural images, using histogram estimates of the marginal and joint distributions, and mutual information measurements to characterize the dependencies between coefficients. The study reveals the nonGaussian marginal statistics and strong intrasubband, crossscale, and crossorientation dependencies of contourlet coefficients. It is also found that conditioned on the magnitudes of their generalized neighborhood coefficients, contourlet coefficients can approximately be modeled as Gaussian variables. Based on these statistics, we model contourlet coefficients using a hidden Markov tree (HMT) model that can capture all of their interscale, interorientation, and intrasubband dependencies. We experiment this model in the image denoising and texture retrieval applications where the results are very promising. In denoising, contourlet HMT outperforms wavelet HMT and other classical methods in terms of visual quality. In particular, it preserves edges and oriented features better than other existing methods. In texture retrieval, it shows improvements in performance over wavelet methods for various oriented textures.
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 79 (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.
Wave atoms and sparsity of oscillatory patterns
 Appl. Comput. Harmon. Anal
, 2006
"... We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, ..."
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Cited by 73 (10 self)
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We introduce “wave atoms ” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength ∼ (diameter) 2. We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping ” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shiftinvariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.
Wavelets with composite dilations and their MRA properties
 Appl. Comput. Harmon. Anal
"... Affine systems are reproducing systems of the form AC = {Dc Tk ψ ` : 1 ≤ ` ≤ L, k ∈ Zn, c ∈ C}, which arise by applying lattice translation operators Tk to one or more generators ψ ` in L2(Rn), followed by the application of dilation operators Dc, associated with a countable set C of invertible m ..."
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Cited by 65 (24 self)
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Affine systems are reproducing systems of the form AC = {Dc Tk ψ ` : 1 ≤ ` ≤ L, k ∈ Zn, c ∈ C}, which arise by applying lattice translation operators Tk to one or more generators ψ ` in L2(Rn), followed by the application of dilation operators Dc, associated with a countable set C of invertible matrices. In the wavelet literature, C is usually taken to be the group consisting of all integer powers of a fixed expanding matrix. In this paper, we develop the properties of much more general systems, for which C = {c = a b: a ∈ A, b ∈ B} where A and B are not necessarily commuting matrix sets. C need not contain a single expanding matrix. Nonetheless, for many choices of A and B, there are wavelet systems with multiresolution properties very similar Preprint submitted to Elsevier Science on February 4, 2005; revised: July 25, 2005 to those of classical dyadic wavelets. Typically, A expands or contracts only in certain directions, while B acts by volumepreserving maps in transverse directions. Then the resulting wavelets exhibit the geometric properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for multidimensional signal and image processing applications. Our method is a systematic approach to the theory of affinelike systems yielding these and more general features.
Fast adaptive wavelet packet image compression
 IEEE Transactions on Image Processing
, 2000
"... Abstract—Wavelets are illsuited to represent oscillatory patterns: rapid variations of intensity can only be described by the small scale wavelet coefficients, which are often quantized to zero, even at high bit rates. Our goal in this paper is to provide a fast numerical implementation of the best ..."
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Cited by 60 (18 self)
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Abstract—Wavelets are illsuited to represent oscillatory patterns: rapid variations of intensity can only be described by the small scale wavelet coefficients, which are often quantized to zero, even at high bit rates. Our goal in this paper is to provide a fast numerical implementation of the best wavelet packet algorithm [1] in order to demonstrate that an advantage can be gained by constructing a basis adapted to a target image. Emphasis in this paper has been placed on developing algorithms that are computationally efficient. We developed a new fast twodimensional (2D) convolutiondecimation algorithm with factorized nonseparable 2D filters. The algorithm is four times faster than a standard convolutiondecimation. An extensive evaluation of the algorithm was performed on a large class of textured images. Because of its ability to reproduce textures so well, the wavelet packet coder significantly out performs one of the best wavelet coder [2] on images such as Barbara and fingerprints, both visually and in term of PSNR. Index Terms—Adaptive transform, best basis, image compression, ladder structure, wavelet packet. I.
Directionlets: Anisotropic Multidirectional Representation with Separable Filtering
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2004
"... In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges a ..."
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Cited by 58 (6 self)
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In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. Onedimensional (1D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To capture efficiently these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (MDIR) and anisotropic transform is required. We present a new latticebased perfect reconstruction and critically sampled anisotropic MDIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard twodimensional (2D) WT. The corresponding anisotropic basis functions (directionlets) have directional vanishing moments (DVM) along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation (NLA) of images, achieving the approximation power O(N −1.55), which is competitive to the rates achieved by the other oversampled transform constructions.
Adaptive directional liftingbased wavelet transform for image coding
 IEEE TRANS. IMAGE PROCESS
, 2007
"... We present a novel 2D wavelet transform scheme of adaptive directional lifting (ADL) in image coding. Instead of alternately applying horizontal and vertical lifting, as in present practice, ADL performs liftingbased prediction in local windows in the direction of high pixel correlation. Hence, it ..."
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Cited by 44 (12 self)
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We present a novel 2D wavelet transform scheme of adaptive directional lifting (ADL) in image coding. Instead of alternately applying horizontal and vertical lifting, as in present practice, ADL performs liftingbased prediction in local windows in the direction of high pixel correlation. Hence, it adapts far better to the image orientation features in local windows. The ADL transform is achieved by existing 1D wavelets and is seamlessly integrated into the global wavelet transform. The predicting and updating signals of ADL can be derived even at the fractional pixel precision level to achieve high directional resolution, while still maintaining perfect reconstruction. To enhance the ADL performance, a ratedistortion optimized directional segmentation scheme is also proposed to form and code a hierarchical image partition adapting to local features. Experimental results show that the proposed ADLbased image coding technique outperforms JPEG 2000 in both PSNR and visual quality, with the improvement up to 2.0 dB on images with rich orientation features.