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591
A fast iterative shrinkagethresholding algorithm with application to . . .
, 2009
"... We consider the class of Iterative ShrinkageThresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterat ..."
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Cited by 984 (8 self)
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We consider the class of Iterative ShrinkageThresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterative ShrinkageThresholding Algorithm (FISTA) which preserves the computational simplicity of ISTA, but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for waveletbased image deblurring demonstrate the capabilities of FISTA.
Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition
, 1998
"... Abstract. This paper is a survey of the inverse scattering problem for timeharmonic acoustic and electromagnetic waves at fixed frequency. We begin by a discussion of “weak scattering ” and Newtontype methods for solving the inverse scattering problem for acoustic waves, including a brief discussi ..."
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Cited by 877 (45 self)
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Abstract. This paper is a survey of the inverse scattering problem for timeharmonic acoustic and electromagnetic waves at fixed frequency. We begin by a discussion of “weak scattering ” and Newtontype methods for solving the inverse scattering problem for acoustic waves, including a brief discussion of Tikhonov’s method for the numerical solution of illposed problems. We then proceed to prove a uniqueness theorem for the inverse obstacle problems for acoustic waves and the linear sampling method for reconstructing the shape of a scattering obstacle from far field data. Included in our discussion is a description of Kirsch’s factorization method for solving this problem. We then turn our attention to uniqueness and reconstruction algorithms for determining the support of an inhomogeneous, anisotropic media from acoustic far field data. Our survey is concluded by a brief discussion of the inverse scattering problem for timeharmonic electromagnetic waves. 1.
PrivacyPreserving Data Mining
, 2000
"... A fruitful direction for future data mining research will be the development of techniques that incorporate privacy concerns. Specifically, we address the following question. Since the primary task in data mining is the development of models about aggregated data, can we develop accurate models with ..."
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Cited by 774 (3 self)
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A fruitful direction for future data mining research will be the development of techniques that incorporate privacy concerns. Specifically, we address the following question. Since the primary task in data mining is the development of models about aggregated data, can we develop accurate models without access to precise information in individual data records? We consider the concrete case of building a decisiontree classifier from tredning data in which the values of individual records have been perturbed. The resulting data records look very different from the original records and the distribution of data values is also very different from the original distribution. While it is not possible to accurately estimate original values in individual data records, we propose anovel reconstruction procedure to accurately estimate the distribution of original data values. By using these reconstructed distributions, we are able to build classifiers whose accuracy is comparable to the accuracy of classifiers built with the original data.
An iterative regularization method for total variationbased image restoration, Multiscale Model
 Simul
"... Abstract. We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total varia ..."
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Cited by 154 (26 self)
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Abstract. We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.
A Compressive Landweber Iteration for Solving IllPosed Inverse Problems
, 2008
"... In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear illposed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for ..."
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Cited by 122 (4 self)
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In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear illposed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landweber’s method is numerically very intensive. Therefore we propose an adaptive variant of Landweber’s iteration that significantly may reduce the computational expense, i.e. leading to a compressed version of Landweber’s iteration. We lend the concept of adaptivity that was primarily developed for wellposed operator equations (in particular, for elliptic PDE’s) essentially exploiting the concept of wavelets (frames), Besov regularity, best Nterm approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landweber’s iteration. In combination with an adequate refinement/stopping rule (apriori as well as aposteriori principles) we prove that the proposed procedure is an regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging.
Adaptive Wavelet Methods and Sparsity Reconstruction for Inverse Heat Conduction Problems
, 2009
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Image Deblurring with Blurred/Noisy Image Pairs
"... Taking satisfactory photos under dim lighting conditions using a handheld camera is challenging. If the camera is set to a long exposure time, the image is blurred due to camera shake. On the other hand, the image is dark and noisy if it is taken with a short exposure time but with a high camera g ..."
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Cited by 119 (4 self)
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Taking satisfactory photos under dim lighting conditions using a handheld camera is challenging. If the camera is set to a long exposure time, the image is blurred due to camera shake. On the other hand, the image is dark and noisy if it is taken with a short exposure time but with a high camera gain. By combining information extracted from both blurred and noisy images, however, we show in this paper how to produce a high quality image that cannot be obtained by simply denoising the noisy image, or deblurring the blurred image alone. Our approach is image deblurring with the help of the noisy image. First, both images are used to estimate an accurate blur kernel, which otherwise is difficult to obtain from a single blurred image. Second, and again using both images, a residual deconvolution is proposed to significantly reduce ringing artifacts inherent to image deconvolution. Third, the remaining ringing artifacts in smooth image regions are further suppressed by a gaincontrolled deconvolution process. We demonstrate the effectiveness of our approach using a number of indoor and outdoor images taken by offtheshelf handheld cameras in poor lighting environments.
Recovery algorithms for vector valued data with joint sparsity constraints
, 2006
"... Vector valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take ..."
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Cited by 105 (22 self)
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Vector valued data appearing in concrete applications often possess sparse expansions with respect to a preassigned frame for each vector component individually. Additionally, different components may also exhibit common sparsity patterns. Recently, there were introduced sparsity measures that take into account such joint sparsity patterns, promoting coupling of nonvanishing components. These measures are typically constructed as weighted ℓ1 norms of componentwise ℓq norms of frame coefficients. We show how to compute solutions of linear inverse problems with such joint sparsity regularization constraints by fast thresholded Landweber algorithms. Next we discuss the adaptive choice of suitable weights appearing in the definition of sparsity measures. The weights are interpreted as indicators of the sparsity pattern and are iteratively updated after each new application of the thresholded Landweber algorithm. The resulting twostep algorithm is interpreted as a doubleminimization scheme for a suitable target functional. We show its ℓ2norm convergence. An implementable version of the algorithm is also formulated, and its norm convergence is proven. Numerical experiments in color image restoration are presented.
A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions
, 1999
"... Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for 1dimensional problems and blockToeplitz Toeplitzblock matrices for 2dim ..."
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Cited by 98 (23 self)
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Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for 1dimensional problems and blockToeplitz Toeplitzblock matrices for 2dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitzplus Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also s...