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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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The Split Bregman Method for L1-Regularized Problems SIAM J. Imaging Sci. 2, pp. 323-343 (21 pages) Online Publication Date: April 01, 2009
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The class of L1-regularized optimization problems has received much attention recently because of the introduction of “compressed sensing,” which allows images and signals to be reconstructed from small amounts of data. Despite this recent attention, many L1-regularized problems still remain difficult to solve, or require techniques that are very problem-specific. In this paper, we show that Bregman iteration can be used to solve a wide variety of constrained optimization problems. Using this technique, we propose a “split Bregman” method, which can solve a very broad class of L1-regularized problems. We apply this technique to the Rudin–Osher–Fatemi functional for image denoising and to a compressed sensing problem that arises in magnetic resonance imaging. |
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Multivalued Geodesic Ray-Tracing for Computing Brain Connections Using Diffusion Tensor Imaging SIAM J. Imaging Sci. 5, pp. 483-504 (22 pages) Online Publication Date: April 04, 2012
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Diffusion tensor imaging (DTI) is a magnetic resonance technique used to explore anatomical fibrous structures, like brain white matter. Fiber-tracking methods use the diffusion tensor (DT) field to reconstruct the corresponding fibrous structure. A group of fiber-tracking methods trace geodesics on a Riemannian manifold whose metric is defined as a function of the DT. These methods are more robust to noise than more commonly used methods where just the main eigenvector of the DT is considered. Until now, geodesic-based methods were not able to resolve all geodesics, since they solved the Eikonal equation, and therefore were not able to deal with multivalued solutions. Our algorithm computes multivalued solutions using an Euler–Lagrange form of the geodesic equations. The multivalued solutions become relevant in regions with sharp anisotropy and complex geometries, or when the first arrival time does not describe the geodesic close to the anatomical fibrous structure. In this paper, we compare our algorithm with the commonly used Hamilton–Jacobi (HJ) equation approach. We describe and analyze the characteristics of both methods. In the analysis we show that in cases where, e.g., U-shaped bundles appear, our algorithm can capture the underlying fiber structure, while other approaches will fail. A feasibility study with results for synthetic and real data is shown. |
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Image Denoising Using Mean Curvature of Image Surface SIAM J. Imaging Sci. 5, pp. 1-32 (32 pages) Online Publication Date: January 17, 2012
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We propose a new variational model for image denoising, which employs the $L^{1}$-norm of the mean curvature of the image surface $(x,f(x))$ of a given image $f:\Omega\rightarrow\mathbb{R}$. Besides eliminating noise and preserving edges of objects efficiently, our model can keep corners of objects and greyscale intensity contrasts of images and also remove the staircase effect. In this paper, we analytically study the proposed model and justify why our model can preserve object corners and image contrasts. We apply the proposed model to the denoising of curves and plane images, and also compare the results with those obtained by using the classical Rudin–Osher–Fatemi model [Phys. D, 60 (1992), pp. 259–268]. |
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The Natural Vectorial Total Variation Which Arises from Geometric Measure Theory SIAM J. Imaging Sci. 5, pp. 537-563 (27 pages) Online Publication Date: April 12, 2012
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Several ways to generalize scalar total variation to vector-valued functions have been proposed in the past. In this paper, we give a detailed analysis of a variant we denote by $\text{TV}_J$, which has not been previously explored as a regularizer. The contributions of the manuscript are twofold: on the theoretical side, we show that $\text{TV}_J$ can be derived from the generalized Jacobians from geometric measure theory. Thus, within the context of this theory, $\text{TV}_J$ is the most natural form of a vectorial total variation. As an important feature, we derive how $\text{TV}_J$ can be written as the support functional of a convex set in $\mathcal{L}^2$. This property allows us to employ fast and stable minimization algorithms to solve inverse problems. The analysis also shows that in contrast to other total variation regularizers for color images, the proposed one penalizes across a common edge direction for all channels, which is a major theoretical advantage. Our practical contribution consist of an extensive experimental section, where we compare the performance of a number of provable convergent algorithms for inverse problems with our proposed regularizer. In particular, we show in experiments for denoising, deblurring, superresolution, and inpainting that its use leads to a significantly better restoration of color images, both visually and quantitatively. Source code for all algorithms employed in the experiments is provided online. |
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A Semismooth Newton Method for Nonlinear Parameter Identification Problems with Impulsive Noise SIAM J. Imaging Sci. 5, pp. 505-536 (32 pages) Online Publication Date: April 12, 2012
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This work is concerned with nonlinear parameter identification in partial differential equations subject to impulsive noise. To cope with the non-Gaussian nature of the noise, we consider a model with $\mbox{L}^1$ fitting. However, the nonsmoothness of the problem makes its efficient numerical solution challenging. By approximating this problem using a family of smoothed functionals, a semismooth Newton method becomes applicable. In particular, its superlinear convergence is proved under a second-order condition. The convergence of the solution to the approximating problem as the smoothing parameter goes to zero is shown. A strategy for adaptively selecting the regularization parameter based on a balancing principle is suggested. The efficiency of the method is illustrated on several benchmark inverse problems of recovering coefficients in elliptic differential equations, for which one- and two-dimensional numerical examples are presented. |
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A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems SIAM J. Imaging Sci. 2, pp. 183-202 (20 pages) Online Publication Date: March 04, 2009
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We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding 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 wavelet-based image deblurring demonstrate the capabilities of FISTA which is shown to be faster than ISTA by several orders of magnitude. |
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One Shot Inverse Scattering via Rational Approximation SIAM J. Imaging Sci. 5, pp. 465-482 (18 pages) Online Publication Date: March 29, 2012
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We consider the two-dimensional inverse obstacle problem for the Helmholtz equation and aim for localizing several scatterers from the far field of the scattered wave for one fixed incident field and fixed frequency. Our method is independent of the physical properties of the scatterers and is based on a careful investigation of the decay of the tail of the Fourier coefficients of the given far field. Using Prony's method or, equivalently, certain rational Padé approximants, we determine a discrete set of point sources that produces a far field with approximately the same tail of Fourier coefficients. We further show how a repetition of this procedure for different virtual points of origin can be turned into a means for imaging the scatterers. Although this method suffers from a certain lack of stability in the presence of noise, it may provide a useful alternative imaging technique when the scatterers are small inhomogeneities and the number of measurements is as limited as described above. |
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Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing SIAM J. Img. Sci. 1, pp. 143-168 (26 pages) Online Publication Date: March 20, 2008
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We propose simple and extremely efficient methods for solving the basis pursuit problem $\min\{\|u\|_1 : Au = f, u\in\mathbb{R}^n\},$ which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem $\min_{u\in\mathbb{R}^n} \mu\|u\|_1+\frac{1}{2}\|Au-f^k\|_2^2$ for given matrix $A$ and vector $f^k$. We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving $A$ and $A^\top$ can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC. |
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A Butterfly Algorithm for Synthetic Aperture Radar Imaging SIAM J. Imaging Sci. 5, pp. 203-243 (41 pages) Online Publication Date: February 28, 2012
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In spite of an extensive literature on fast algorithms for synthetic aperture radar (SAR) imaging, it is not currently known if it is possible to accurately form an image from $N$ data points in provable near-linear time complexity. This paper seeks to close this gap by proposing an algorithm which runs in complexity $O(N \log N \log(1/\epsilon))$ without making the far-field approximation or imposing the beam pattern approximation required by time-domain backprojection, with $\epsilon$ the desired pixelwise accuracy. It is based on the butterfly scheme, which unlike the FFT works for vastly more general oscillatory integrals than the discrete Fourier transform. A complete error analysis is provided: the rigorous complexity bound has additional powers of $\log N$ and $\log(1/\epsilon)$ that are not observed in practice. |
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A New Alternating Minimization Algorithm for Total Variation Image Reconstruction SIAM J. Img. Sci. 1, pp. 248-272 (25 pages) Online Publication Date: August 20, 2008
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We propose, analyze, and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also the isotropic forms of TV discretizations. The per-iteration computational complexity of the algorithm is three fast Fourier transforms. We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or $q$-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the lagged diffusivity algorithm for TV-based deblurring. Some extensions of our algorithm are also discussed. |
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A Total Variation Model for Retinex SIAM J. Imaging Sci. 4, pp. 345-365 (21 pages) Online Publication Date: March 22, 2011
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Human vision has the ability to recognize color under varying illumination conditions. Retinex theory is introduced to explain how the human visual system perceives color. The main aim of this paper is to present a total variation model for Retinex. Different from the existing methods, we consider and study two important elements which include illumination and reflection. We assume spatial smoothness of the illumination and piecewise continuity of the reflection, where the total variation term is employed in the model. The existence of the solution of the model is shown in the paper. We employ a fast computation method to solve the proposed minimization problem. Numerical examples are presented to illustrate the effectiveness of the proposed model. |
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Dictionary Learning for Noisy and Incomplete Hyperspectral Images SIAM J. Imaging Sci. 5, pp. 33-56 (24 pages) Online Publication Date: January 17, 2012
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We consider analysis of noisy and incomplete hyperspectral imagery, with the objective of removing the noise and inferring the missing data. The noise statistics may be wavelength dependent, and the fraction of data missing (at random) may be substantial, including potentially entire bands, offering the potential to significantly reduce the quantity of data that need be measured. To achieve this objective, the imagery is divided into contiguous three-dimensional (3D) spatio-spectral blocks of spatial dimension much less than the image dimension. It is assumed that each such 3D block may be represented as a linear combination of dictionary elements of the same dimension, plus noise, and the dictionary elements are learned in situ based on the observed data (no a priori training). The number of dictionary elements needed for representation of any particular block is typically small relative to the block dimensions, and all the image blocks are processed jointly (“collaboratively") to infer the underlying dictionary. We address dictionary learning from a Bayesian perspective, considering two distinct means of imposing sparse dictionary usage. These models allow inference of the number of dictionary elements needed as well as the underlying wavelength-dependent noise statistics. It is demonstrated that drawing the dictionary elements from a Gaussian process prior, imposing structure on the wavelength dependence of the dictionary elements, yields significant advantages, relative to the more conventional approach of using an independent and identically distributed Gaussian prior for the dictionary elements; this advantage is particularly evident in the presence of noise. The framework is demonstrated by processing hyperspectral imagery with a significant number of voxels missing uniformly at random, with imagery at specific wavelengths missing entirely, and in the presence of substantial additive noise. |
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Multistatic Imaging of Extended Targets SIAM J. Imaging Sci. 5, pp. 564-600 (37 pages) Online Publication Date: April 12, 2012
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In this paper we develop iterative approaches for imaging extended inclusions from multistatic response measurements at single or multiple frequencies. Assuming measurement noise, we perform a detailed stability and resolution analysis of the proposed algorithms in two different asymptotic regimes. We consider both the Born approximation in the nonmagnetic case and a high-frequency regime in the general case. Based on a high-frequency asymptotic analysis of the measurements, an algorithm for finding a good initial guess for the illuminated part of the inclusion is provided and its optimality is shown. The initial guess, obtained through standard statistical arguments, turns out to be Kirchhoff migration. We illustrate the efficiency and the limitations of the proposed algorithms with a variety of numerical examples. |
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Parametric Level Set Methods for Inverse Problems SIAM J. Imaging Sci. 4, pp. 618-650 (33 pages) Online Publication Date: June 21, 2011
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In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results in a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, reinitialization, and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the way for easier use of Newton and quasi-Newton methods. Specifically for the purposes of this paper, we parameterize the level set function in terms of adaptive compactly supported radial basis functions, which, used in the proposed manner, provide flexibility in presenting a larger class of shapes with fewer terms. Also they provide a “narrow-banding” advantage which can further reduce the number of active unknowns at each step of the evolution. The performance of the proposed approach is examined in three examples of inverse problems, i.e., electrical resistance tomography, X-ray computed tomography, and diffuse optical tomography. |
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SIAM J. Imaging Sci. 3, pp. 492-526 (35 pages) Online Publication Date: September 09, 2010
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The novel concept of total generalized variation of a function $u$ is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of $u$. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect. |
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Edge Detection Filter based on Mumford–Shah Green Function SIAM J. Imaging Sci. 5, pp. 343-365 (23 pages) Online Publication Date: March 08, 2012
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In this paper, we propose an edge detection algorithm based on the Green function associated with the Mumford–Shah segmentation model. This Green function has a singularity at its center. A regularization method is therefore proposed here to obtain an edge detection filter known here as the Bessel filter. This filter is robust in the presence of noise, and its implementation is simple. It is demonstrated here that this filter is scale invariant. A mathematical argument is also provided to prove that the gradient magnitude of the convolved image with this filter has local maxima in discontinuities of the original image. The Bessel filter enjoys better overall performance (the product of the detection performance and localization indices) in Canny-like criteria than the state-of-the-art filters in the literature. Quantitative and qualitative evaluations of the edge detection algorithms investigated in this paper on synthetic and real world benchmark images confirm the theoretical results presented here, indicating the scale invariant property of the Bessel filter. The numerical complexity of the algorithm proposed here is as low as any convolution-based edge detection algorithm. |
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SIAM J. Imaging Sci. 5, pp. 119-149 (31 pages) Online Publication Date: January 24, 2012
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Recently, some primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation image restoration. This paper focuses on the convergence analysis of these primal-dual algorithms and shows that their involved parameters (including step sizes) can be significantly enlarged if some simple correction steps are supplemented. Some new primal-dual–based methods are thus proposed for solving the saddle-point problem. We show that these new methods are of the contraction type: the iterative sequences generated by these new methods are contractive with respect to the solution set of the saddle-point problem. The global convergence of these new methods thus can be obtained within the analytic framework of contraction-type methods. The novel study on these primal-dual algorithms from the perspective of contraction methods substantially simplifies existing convergence analysis. Finally, we show the efficiency of the new methods numerically. |
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Approximating Symmetric Positive Semidefinite Tensors of Even Order SIAM J. Imaging Sci. 5, pp. 434-464 (31 pages) Online Publication Date: March 20, 2012
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Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space $\mathcal{P}_0^{2m}$ of $2m$th-order symmetric positive semidefinite tensors is known to be a convex cone, enforcing positivity constraint directly on $\mathcal{P}_0^{2m}$ is usually not straightforward computationally because there is no known analytic description of $\mathcal{P}_0^{2m}$ for $m>1$. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone $\mathcal{P}_0^{2m}$ for the cases $0<m<3$, and presenting an explicit characterization of the approximation $\Sigma_{2m}\subset\Omega_{2m}$ for $m\geq1$, using the subset $\Omega_{2m}\subset\mathcal{P}_0^{2m}$ of semidefinite tensors that can be written as a sum of squares of tensors of order $m$. Furthermore, we show that this approximation leads to a nonnegative linear least-squares optimization problem with the complexity that equals the number of generators in $\Sigma_{2m}$. Finally, we experimentally validate the proposed approach and present an application for computing $2m$th-order diffusion tensors from diffusion weighted magnetic resonance images. |
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Adaptive Compressed Image Sensing Using Dictionaries SIAM J. Imaging Sci. 5, pp. 57-89 (33 pages) Online Publication Date: January 24, 2012
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In recent years, the theory of compressed sensing has emerged as an alternative for the Shannon sampling theorem, suggesting that compressible signals can be reconstructed from far fewer samples than required by the Shannon sampling theorem. In fact the theory advocates that nonadaptive, “random” functionals are in some sense optimal for this task. However, in practice, compressed sensing is very difficult to implement for large data sets, particularly because the recovery algorithms require significant computational resources. In this work, we present a new alternative method for simultaneous image acquisition and compression called adaptive compressed sampling. We exploit wavelet tree structures found in natural images to replace the “universal” acquisition of incoherent measurements with a direct and fast method for adaptive wavelet tree acquisition. The main advantages of this direct approach are that no complex recovery algorithm is in fact needed and that it allows more control over the compressed image quality, in particular, the sharpness of edges. Our experimental results show, by way of software simulations, that our adaptive algorithms perform better than existing nonadaptive methods in terms of image quality and speed. |
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SIAM J. Imaging Sci. 4, pp. 543-572 (30 pages) Online Publication Date: June 07, 2011
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The cryo-electron microscopy reconstruction problem is to find the three-dimensional (3D) structure of a macromolecule given noisy samples of its two-dimensional projection images at unknown random directions. Present algorithms for finding an initial 3D structure model are based on the “angular reconstitution” method in which a coordinate system is established from three projections, and the orientation of the particle giving rise to each image is deduced from common lines among the images. However, a reliable detection of common lines is difficult due to the low signal-to-noise ratio of the images. In this paper we describe two algorithms for finding the unknown imaging directions of all projections by minimizing global self-consistency errors. In the first algorithm, the minimizer is obtained by computing the three largest eigenvectors of a specially designed symmetric matrix derived from the common lines, while the second algorithm is based on semidefinite programming (SDP). Compared with existing algorithms, the advantages of our algorithms are five-fold: first, they accurately estimate all orientations at very low common-line detection rates; second, they are extremely fast, as they involve only the computation of a few top eigenvectors or a sparse SDP; third, they are nonsequential and use the information in all common lines at once; fourth, they are amenable to a rigorous mathematical analysis using spectral analysis and random matrix theory; and finally, the algorithms are optimal in the sense that they reach the information theoretic Shannon bound up to a constant for an idealized probabilistic model. |
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