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196
An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems
, 2009
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Highly sparse representations from dictionaries are unique and independent of the sparseness measure
, 2003
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A frameletbased image inpainting algorithm
 Applied and Computational Harmonic Analysis
"... Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the c ..."
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Cited by 90 (40 self)
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Abstract. Image inpainting is a fundamental problem in image processing and has many applications. Motivated by the recent tight frame based methods on image restoration in either the image or the transform domain, we propose an iterative tight frame algorithm for image inpainting. We consider the convergence of this frameletbased algorithm by interpreting it as an iteration for minimizing a special functional. The proof of the convergence is under the framework of convex analysis and optimization theory. We also discuss the relationship of our method with other waveletbased methods. Numerical experiments are given to illustrate the performance of the proposed algorithm. Key words. Tight frame, inpainting, convex analysis 1. Introduction. The problem of inpainting [2] occurs when part of the pixel data in a picture is missing or overwritten by other means. This arises for example in restoring ancient drawings, where a portion of the picture is missing or damaged due to aging or scratch; or when an image is transmitted through a noisy channel. The task of inpainting is to recover the missing region from the incomplete data observed. Ideally, the restored image should possess shapes and patterns consistent
Framing Pyramids
 IEEE TRANSACTIONS ON SIGNAL PROCESSING
, 2003
"... In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal li ..."
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Cited by 65 (6 self)
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In 1983, Burt and Adelson introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous domain frames associated with the LP.
Pairs of Dual Wavelet Frames From Any Two Refinable Functions
, 2000
"... Starting from any two compactly supported refinable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual dwavelet frames in L 2 (R). Moreover, the number of vanishing moments of ..."
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Cited by 42 (21 self)
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Starting from any two compactly supported refinable functions in L 2 (R) with dilation factor d, we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual dwavelet frames in L 2 (R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L 2 (R), it is possible to construct explicitly and easily wavelets that are finite linear combinations of translates (d k), and that generate a wavelet frame with arbitrarily preassigned number of vanishing moments. We illustrate the general theory by examples of such pairs of dual wavelet frames derived from Bspline functions.
A survey on wavelet applications in data mining
 SIGKDD Explor. Newsl
"... Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework tha ..."
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Cited by 37 (4 self)
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Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework that reduces the overall process into smaller components. Then applications of wavelets for each component are reviewd. The paper concludes by discussing the impact of wavelets on data mining research and outlining potential future research directions and applications. 1.
Linearized Bregman Iterations for FrameBased Image Deblurring
, 2008
"... Abstract. Real images usually have sparse approximations under some tight frame systems derived from either framelets, over sampled discrete (window) cosine or Fourier transform. In this paper, we propose a method for image deblurring in tight frame domains. It is reduced to finding a sparse solutio ..."
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Cited by 35 (17 self)
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Abstract. Real images usually have sparse approximations under some tight frame systems derived from either framelets, over sampled discrete (window) cosine or Fourier transform. In this paper, we propose a method for image deblurring in tight frame domains. It is reduced to finding a sparse solution of a system of linear equations whose coefficients matrix is rectangular. Then, a modified version of the linearized Bregman iteration proposed and analyzed in [10,11,43,50] can be applied. Numerical examples show that the method is very simple to implement, robust to noise and effective for image deblurring. 1. Introduction. Image
Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix
 J. Comput. Appl. Math
"... Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in ..."
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Cited by 33 (21 self)
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Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in L2(R d) of exponential decay, which are derived from compactly supported Mrefinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several
Restoration of chopped and nodded images by framelets
 SIAM J. Sci. Comput
"... Abstract. In infrared astronomy, an observed image from a chop and nod process can be considered as the result of passing the original image through a highpass filter. Here we propose a restoration algorithm which builds up a tight framelet system that has the highpass filter as one of the framelet ..."
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Cited by 30 (20 self)
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Abstract. In infrared astronomy, an observed image from a chop and nod process can be considered as the result of passing the original image through a highpass filter. Here we propose a restoration algorithm which builds up a tight framelet system that has the highpass filter as one of the framelet filters. Our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition. The framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image. A proof of the convergence of the algorithm based on convex analysis is also provided. Simulated and real images are tested to illustrate the efficiency of our method over the projected Landweber method. Key words. analysis Tight frame, chopped and nodded image, projected Landweber method, convex AMS subject classifications. 42C40, 65T60, 68U10, 94A08 1. Introduction. We