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Basis Pursuit
, 1994
"... The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decompos ..."
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Cited by 119 (15 self)
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The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including superresolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries  signal representation schemes  of which the Wavelets dictionary is only the most wellknown. Wavelet dictionaries, Gabor dictionaries, Multiscale...
Stable Numerical Algorithms for Equilibrium Systems
 SIAM J. Matrix Anal. Appl
, 1992
"... An equilibrium system (also known as a KKT system, a saddlepoint system, or a sparse tableau) is a square linear system with a certain structure. G. Strang has observed that equilibrium systems arise in optimization, finite elements, structural analysis, and electrical networks. Recently, G. W. Stew ..."
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Cited by 28 (3 self)
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An equilibrium system (also known as a KKT system, a saddlepoint system, or a sparse tableau) is a square linear system with a certain structure. G. Strang has observed that equilibrium systems arise in optimization, finite elements, structural analysis, and electrical networks. Recently, G. W. Stewart established a norm bound for a type of equilibrium system in the case that the "stiffness" portion of the system is very illconditioned. In this paper we investigate the algorithmic implications of Stewart's result. We show that all standard textbook algorithms for equilibrium systems are unstable. Then we show that a certain hybrid method has the right stability property. 1 Equilibrium systems Recently, Strang [1986] has observed that the problem of solving the structured linear system / D \GammaA A T 0 !/ x y ! = / b c ! (1) This work supported by an NSF Presidential Young Investigator grant, with matching funds received from Xerox Corp. y Department of Computer Scie...
An Affine Scaling Algorithm For Minimizing Total Variation In Image Enhancement
, 1994
"... . A computational algorithm is proposed for image enhancement based on total variation minimization with constraints. This constrained minimization problem is introduced by Rudin et al [13, 14, 15] to enhance blurred and noisy images. Our computational algorithm solves the constrained minimization p ..."
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Cited by 16 (1 self)
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. A computational algorithm is proposed for image enhancement based on total variation minimization with constraints. This constrained minimization problem is introduced by Rudin et al [13, 14, 15] to enhance blurred and noisy images. Our computational algorithm solves the constrained minimization problem directly by adapting the affine scaling method for the unconstrained l 1 problem [3]. The resulting computational scheme, when viewed as an image enhancement process, has the feature that it can be used in an interactive manner in situations where knowledge of the noise level is either unavailable or unreliable. This computational algorithm can be implemented with a conjugate gradient method. It is further demonstrated that the iterative enhancement process is efficient. Key Words. image enhancement, image reconstruction, deconvolution, minimal total variation, affine scaling algorithm, projected gradient method Department of Computer Science and Advanced Computing Research Institut...
In vivo Impedance Imaging with Total Variation Regularization
, 2009
"... We show that electrical impedance tomography (EIT) image reconstruction algorithms with regularization based on the Total Variation (TV) functional are suitable for in vivo imaging of physiological data. This reconstruction approach helps to preserve discontinuities in reconstructed profiles, such ..."
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Cited by 2 (1 self)
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We show that electrical impedance tomography (EIT) image reconstruction algorithms with regularization based on the Total Variation (TV) functional are suitable for in vivo imaging of physiological data. This reconstruction approach helps to preserve discontinuities in reconstructed profiles, such as step changes in electrical properties at interorgan boundaries, which are typically smoothed by traditional reconstruction algorithms. The use of the TV functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation. This cannot be efficiently solved with traditional optimization techniques such as the Newton Method. We explore two implementations methods for regularization with the TV functional: the Lagged Diffusivity method and the Primal Dual – Interior Point Method (PD–IPM). First we clarify the implementation details of these algorithms for EIT reconstruction. Next, we analyze the performance of these algorithms on noisy simulated data. Finally, we show reconstructed EIT images of in–vivo data for ventilation and gastric emptying studies. In comparison to traditional quadratic regularization, TV regularization shows improved ability to reconstruct sharp contrasts.
Segmentation of Pulmonary Nodule Images Using Total Variation Minimization
, 1998
"... Total variation minimization has edge preserving and enhancing properties which make it suitable for image segmentation. We present Image Simplification, a new formulation and algorithm for image segmentation. We illustrate the edge enhancing properties of total variation minimization in a discrete ..."
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Cited by 1 (0 self)
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Total variation minimization has edge preserving and enhancing properties which make it suitable for image segmentation. We present Image Simplification, a new formulation and algorithm for image segmentation. We illustrate the edge enhancing properties of total variation minimization in a discrete setting by giving exact solutions to the problem for piecewise constant functions in the presence of noise. In this case, edges can be exactly recovered if the noise is sufficiently small. After optimization, segmentation is completed using edge detection. We find that our image segmentation approach yields good results when applied to the segmentation of pulmonary nodules. 1 Introduction Image segmentation is the partitioning of an image into regions so that each region corresponds to one object in the image. Due to the importance of the segmentation problem in computer vision, numerous strategies have been proposed. See the surveys [7, 9, 14] for some examples. We propose Image Simplifica...
Total Variation Regularization in Electrical Impedance Tomography
, 2007
"... This paper presents an evaluation of the use of Primal Dual Methods for efficiently regularizing the electric impedance tomography (EIT) problem with the Total Variation (TV) functional. The Total Variation functional is assuming an important role in the regularization of inverse problems thanks to ..."
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Cited by 1 (1 self)
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This paper presents an evaluation of the use of Primal Dual Methods for efficiently regularizing the electric impedance tomography (EIT) problem with the Total Variation (TV) functional. The Total Variation functional is assuming an important role in the regularization of inverse problems thanks to its ability to preserve discontinuities in reconstructed profiles. This property is desirable in many fields of application of EIT imaging, such as the medical and the industrial, where interorgan boundaries, in the first case, and interphase boundaries, in the latter case, present step changes in electrical properties which are difficult to be reconstructed with traditional regularization methods, as they tend to smooth the reconstructed image. Though desirable, the TV functional leads to the formulation of the inverse problem as a minimization of a nondifferentiable function which cannot be efficiently solved with traditional optimization techniques
Recovery Of Blocky Images
 SIAM J. Appl. Math
, 1996
"... The purpose of this investigation is to understand situations under which an enhancement method succeeds in recovering an image from data which are noisy and blurred. The method in question is due to Rudin and Osher. The method selects, from a class of feasible images, one that has the least total v ..."
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The purpose of this investigation is to understand situations under which an enhancement method succeeds in recovering an image from data which are noisy and blurred. The method in question is due to Rudin and Osher. The method selects, from a class of feasible images, one that has the least total variation.
ISSN 17499097Total Variation Regularization in Electrical Impedance Tomography
, 2007
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