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2010

Volume 3, Issue 3, pp. 253-701


Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction

Xiaoqun Zhang, Martin Burger, Xavier Bresson, and Stanley Osher

SIAM J. Imaging Sci. 3, pp. 253-276 (24 pages) | Cited 2 times

Online Publication Date: July 01, 2010

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Bregman methods introduced in [S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, Multiscale Model. Simul., 4 (2005), pp. 460–489] to image processing are demonstrated to be an efficient optimization method for solving sparse reconstruction with convex functionals, such as the $\ell^1$ norm and total variation [W. Yin, S. Osher, D. Goldfarb, and J. Darbon, SIAM J. Imaging Sci., 1 (2008), pp. 143–168; T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323–343]. In particular, the efficiency of this method relies on the performance of inner solvers for the resulting subproblems. In this paper, we propose a general algorithm framework for inverse problem regularization with a single forward-backward operator splitting step [P. L. Combettes and V. R. Wajs, Multiscale Model. Simul., 4 (2005), pp. 1168–1200], which is used to solve the subproblems of the Bregman iteration. We prove that the proposed algorithm, namely, Bregmanized operator splitting (BOS), converges without fully solving the subproblems. Furthermore, we apply the BOS algorithm and a preconditioned one for solving inverse problems with nonlocal functionals. Our numerical results on deconvolution and compressive sensing illustrate the performance of nonlocal total variation regularization under the proposed algorithm framework, compared to other regularization techniques such as the standard total variation method and the wavelet-based regularization method.

A Multiphase Image Segmentation Method Based on Fuzzy Region Competition

Fang Li, Michael K. Ng, Tie Yong Zeng, and Chunli Shen

SIAM J. Imaging Sci. 3, pp. 277-299 (23 pages)

Online Publication Date: July 13, 2010

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The goal of this paper is to develop a multiphase image segmentation method based on fuzzy region competition. A new variational functional with constraints is proposed by introducing fuzzy membership functions which represent several different regions in an image. The existence of a minimizer of this functional is established. We propose three methods for handling the constraints of membership functions in the minimization. We also add auxiliary variables to approximate the membership functions in the functional such that Chambolle's fast dual projection method can be used. An alternate minimization method can be employed to find the solution, in which the region parameters and the membership functions have closed form solutions. Numerical examples using grayscale and color images are given to demonstrate the effectiveness of the proposed methods.

Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models

Chunlin Wu and Xue-Cheng Tai

SIAM J. Imaging Sci. 3, pp. 300-339 (40 pages)

Online Publication Date: July 22, 2010

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In image processing, the Rudin–Osher–Fatemi (ROF) model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259–268] based on total variation (TV) minimization has proven to be very useful. So far many researchers have contributed to designing fast numerical schemes and overcoming the nondifferentiability of the model. Methods considered to be particularly efficient for the ROF model include the Chan–Golub–Mulet (CGM) primal-dual method [T.F. Chan, G.H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20 (1999), pp. 1964–1977], Chambolle's dual method [A. Chambolle, J. Math. Imaging Vis., 20 (2004), pp. 89–97], the splitting and quadratic penalty-based method [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248–272], and the split Bregman iteration [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323–343], as well as the augmented Lagrangian method [X.C. Tai and C. Wu, Lecture Notes in Comput. Sci. 5567, Springer-Verlag, Berlin, 2009, pp. 502–513]. In this paper, we first review the augmented Lagrangian method for the ROF model and then provide some convergence analysis and extensions to vectorial TV and high order models. All the algorithms and analysis will be presented in the discrete setting, which is much clearer for practical implementation than the continuous setting as in Tai and Wu, above. We also present, in the discrete setting, the connections between the augmented Lagrangian method, the dual methods, and the split Bregman iteration. Using our extensions and observations, we can easily figure out CGM and the split Bregman iteration for vectorial TV and high order models, which, to the best of our knowledge, have not been presented in the literature. Numerical examples demonstrate the efficiency and accuracy of our method, especially in the image deblurring case.

Reconstruction of Thin Tubular Inclusions in Three-Dimensional Domains Using Electrical Impedance Tomography

Roland Griesmaier

SIAM J. Imaging Sci. 3, pp. 340-362 (23 pages)

Online Publication Date: July 22, 2010

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We consider the inverse problem of reconstructing thin tubular inclusions inside a three-dimensional body from measurements of electrostatic currents and potentials on its boundary. By inclusions we mean objects with an electrical conductivity differing from that of the background material of the body. We apply an asymptotic expansion of the electrostatic potential on the boundary of the body as the thickness of the inclusions tends to zero to establish an asymptotic characterization of these inclusions in terms of the measurement data. This characterization is implemented in a noniterative reconstruction method similar to the factorization method for crack detection problems in two-dimensional domains. We present several numerical examples to illustrate our theoretical findings and to highlight the potentials and limitations of our method.

Multigrid Algorithm for High Order Denoising

Carlos Brito-Loeza and Ke Chen

SIAM J. Imaging Sci. 3, pp. 363-389 (27 pages)

Online Publication Date: August 19, 2010

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Image denoising has been a research topic deeply investigated within the last two decades. Excellent results have been obtained by using such models as the total variation (TV) minimization by Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259–268], which involves solving a second order PDE. In more recent years some effort has been made [Y.-L. You and M. Kaveh, IEEE Trans. Image Process., 9 (2000), pp. 1723–1730; M. Lysaker, S. Osher, and X.-C. Tai, IEEE Trans. Image Process., 13 (2004), pp. 1345–1357; M. Lysaker, A. Lundervold, and X.-C. Tai, IEEE Trans. Image Process., 12 (2003), pp. 1579–1590; Y. Chen, S. Levine, and M. Rao, SIAM J. Appl. Math., 66 (2006), pp. 1383–1406] in improving these results by using higher order models, particularly to avoid the staircase effect inherent to the solution of the TV model. However, the construction of stable numerical schemes for the resulting PDEs arising from the minimization of such high order models has proved to be very difficult due to high nonlinearity and stiffness. In this paper, we study a curvature-based energy minimizing model [W. Zhu and T. F. Chan, Image Denoising Using Mean Curvature, preprint, http://www.math.nyu.edu/ wzhu/], for which one has to solve a fourth order PDE. For this model we develop two new algorithms: a stabilized fixed point method and, based upon this, an efficient nonlinear multigrid (MG) algorithm. We will show numerical experiments to demonstrate the very good performance of our MG algorithm.

Reciprocity Theorems for One-Way Wave Fields in Curvilinear Coordinate Systems

Martijn Frijlink and Kees Wapenaar

SIAM J. Imaging Sci. 3, pp. 390-415 (26 pages) | Cited 1 time

Online Publication Date: August 19, 2010

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One-way wave equations conveniently describe wave propagation in media with discontinuous and/or rapid variations in one direction, but with smooth and slow variations in the complementary transverse directions. In the past, reciprocity theorems have been developed in terms of one-way wave fields. The boundaries of the integration volumes and the variations of the medium parameters must adhere to strict conditions. The variations must have the smoothness required by pseudodifferential operators, while the boundaries have to be flat. To extend the applicability to nonflat boundaries, this paper formulates one-way wave equations and corresponding reciprocity theorems in terms of curvilinear coordinates of the semiorthogonal (SO) type. In SO coordinate systems, one of the covariant basis vectors is orthogonal to the others, which can be nonorthogonal among each other. The same applies to the contravariant basis vectors. Furthermore, the orthogonal directions coincide; that is, the orthogonal co- and contravariant basis vectors coincide. SO coordinates are characterized by a local property of the basis vectors. An extra specification is necessary to make them conform in any way to nonflat boundaries. This can be done in terms of so-called lateral Cartesian (LC) coordinates. Cartesian coordinates are mapped to LC coordinates by applying an invertible transformation to one coordinate while keeping the others the same. LC coordinates are a straightforward means to describe or conform to nonflat boundaries. Applications of the extended reciprocity theorems include removal of multiple reflections, removal of complex propagation effects, wave field extrapolation, and synthesis of unrecorded data.

Higher Order Positive Semidefinite Diffusion Tensor Imaging

Liqun Qi, Gaohang Yu, and Ed X. Wu

SIAM J. Imaging Sci. 3, pp. 416-433 (18 pages)

Online Publication Date: August 19, 2010

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Due to the well-known limitations of diffusion tensor imaging, high angular resolution diffusion imaging (HARDI) is used to characterize non-Gaussian diffusion processes. One approach to analyzing HARDI data is to model the apparent diffusion coefficient (ADC) with higher order diffusion tensors. The diffusivity function is positive semidefinite. In the literature, some methods have been proposed to preserve positive semidefiniteness of second order and fourth order diffusion tensors. None of them can work for arbitrarily high order diffusion tensors. In this paper, we propose a comprehensive model to approximate the ADC profile by a positive semidefinite diffusion tensor of either second or higher order. We call this the positive semidefinite diffusion tensor (PSDT) model. PSDT is a convex optimization problem with a convex quadratic objective function constrained by the nonnegativity requirement on the smallest Z-eigenvalue of the diffusivity function. The smallest Z-eigenvalue is a computable measure of the extent of positive definiteness of the diffusivity function. We also propose some other invariants for the ADC profile analysis. Experiment results show that higher order tensors could improve the estimation of anisotropic diffusion and that the PSDT model can depict the characterization of diffusion anisotropy which is consistent with known neuroanatomy.

Reinterpretation and Enhancement of Signal-Subspace-Based Imaging Methods for Extended Scatterers

Fred K. Gruber and Edwin A. Marengo

SIAM J. Imaging Sci. 3, pp. 434-461 (28 pages) | Cited 1 time

Online Publication Date: August 26, 2010

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Interior sampling and exterior sampling (or enclosure) signal-subspace-based imaging methodologies for extended scatterers derived in previous work are reformulated and reinterpreted in terms of the concepts of angles and distances between subspaces. The insight gained from this reformulation renders a broader, more encompassing inversion methodology based on a (pseudo) cross-coherence matrix associated to the singular vectors of the scattering or response matrix and the singular vectors intrinsic to a given, hypothesized support region for the scatterers (under a known background Green's function associated to a known embedding medium where the scatterers reside). A number of new imaging functionals based on that cross-coherence matrix are proposed and numerically shown to perform well in both imaging and shape reconstruction problems. The proposed approaches do not require for their implementation the estimation of a cutoff in the singular value spectrum separating signal from noise subspaces, which is a common computational difficulty in signal subspace methods. In the shape reconstruction context it is also shown how to combine the signal subspace approach with the level set method.

Sparse Signal Reconstruction via Iterative Support Detection

Yilun Wang and Wotao Yin

SIAM J. Imaging Sci. 3, pp. 462-491 (30 pages)

Online Publication Date: August 26, 2010

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We present a novel sparse signal reconstruction method, iterative support detection (ISD), aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical $\ell_1$ minimization approach. ISD addresses failed reconstructions of $\ell_1$ minimization due to insufficient measurements. It estimates a support set $I$ from a current reconstruction and obtains a new reconstruction by solving the minimization problem $\min\{\sum_{i\notin I}|x_i|:Ax=b\}$, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit method, as well as its variants, because (i) the index set $I$ in ISD is not necessarily nested or increasing, and (ii) the minimization problem above updates all the components of $x$ at the same time. We generalize the null space property to the truncated null space property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called threshold-ISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that threshold-ISD has significant advantages over the classical $\ell_1$ minimization approach, as well as two state-of-the-art algorithms: the iterative reweighted $\ell_1$ minimization algorithm (IRL1) and the iterative reweighted least-squares algorithm (IRLS). MATLAB code is available for download from http://www.caam.rice.edu/ optimization/L1/ISD/.

Total Generalized Variation

Kristian Bredies, Karl Kunisch, and Thomas Pock

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.

Nonparametric Regression between General Riemannian Manifolds

Florian Steinke, Matthias Hein, and Bernhard Schölkopf

SIAM J. Imaging Sci. 3, pp. 527-563 (37 pages)

Online Publication Date: September 09, 2010

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We study nonparametric regression between Riemannian manifolds based on regularized empirical risk minimization. Regularization functionals for mappings between manifolds should respect the geometry of input and output manifold and be independent of the chosen parametrization of the manifolds. We define and analyze the three most simple regularization functionals with these properties and present a rather general scheme for solving the resulting optimization problem. As application examples we discuss interpolation on the sphere, fingerprint processing, and correspondence computations between three-dimensional surfaces. We conclude with characterizing interesting and sometimes counterintuitive implications and new open problems that are specific to learning between Riemannian manifolds and are not encountered in multivariate regression in Euclidean space.

Automatic Grid Resolution and Efficient Triangulation of Implicit Vector Field

Marc Fournier

SIAM J. Imaging Sci. 3, pp. 564-577 (14 pages)

Online Publication Date: September 09, 2010

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In this paper we propose an automatic grid resolution to compute the vector field distance transform of triangle meshes. This implicit representation is useful in performing many mesh processing operations such as mesh fusion and mesh filtering in object surface reconstruction. Setting an appropriate grid resolution is mandatory for obtaining good quality results from these operations. Grid resolution is usually set experimentally by operators until satisfying results are obtained. An automated process is demonstrated that improves this parameter setting step. We also introduce a new marching cube triangulation adaptation to the vector field distance transform. Unlike the previous adaptation, our algorithm allows vertex interpolation on cube faces instead of only on cube edges. We compare our new design to the previous one using a reliable error metric evaluation. Results show our triangulation outperforms the previous one in terms of mesh quality and runtime performances. The new algorithm takes advantage of the more accurate vector field definition to better approximate the implicit isosurface.

A Formal $\Gamma$-Convergence Approach for the Detection of Points in 2-D Images

Daniele Graziani, Laure Blanc-Féraud, and Gilles Aubert

SIAM J. Imaging Sci. 3, pp. 578-594 (17 pages)

Online Publication Date: September 09, 2010

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We propose a new variational model to locate points in 2-dimensional biological images. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. In order to provide numerical experiments we replace this energy with a sequence of more treatable functionals by means of the notion of $\Gamma$-convergence.

Compressed Remote Sensing of Sparse Objects

Albert C. Fannjiang, Thomas Strohmer, and Pengchong Yan

SIAM J. Imaging Sci. 3, pp. 595-618 (24 pages)

Online Publication Date: September 14, 2010

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The linear inverse source and scattering problems are studied from the perspective of compressed sensing. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a log-square factor with overwhelming probability. Important contributions include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the incoherence effect induced by random antenna locations. The predictions of theorems are confirmed by numerical simulations.

Multiscale Photon-Limited Spectral Image Reconstruction

Kalyani Krishnamurthy, Maxim Raginsky, and Rebecca Willett

SIAM J. Imaging Sci. 3, pp. 619-645 (27 pages)

Online Publication Date: September 29, 2010

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This paper studies photon-limited spectral intensity estimation and proposes a spatially and spectrally adaptive, nonparametric method for estimating spectral intensities from Poisson observations. Specifically, our method searches through estimates defined over a family of recursive dyadic partitions in both the spatial and spectral domains, and finds the one that maximizes a penalized log likelihood criterion. The key feature of this approach is that the partition cells are anisotropic across the spatial and spectral dimensions, so that the method adapts to varying degrees of spatial and spectral smoothness, even when the respective degrees of smoothness are not known a priori. The proposed approach is based on the key insight that spatial boundaries and singularities exist in the same locations in every spectral band, even though the contrast or perceptibility of these features may be very low in some bands. The incorporation of this model into the reconstruction results in significant performance gains. Furthermore, for spectral intensities that belong to the anisotropic –Besov function class, the proposed approach is shown to be near-minimax optimal. The upper bounds on the risk function, which is the expected squared Hellinger distance between the true intensity and the estimate obtained using the proposed approach, matches the best possible lower bound up to a log factor for certain degrees of spatial and spectral smoothness. Experiments conducted on realistic data sets show that the proposed method can reconstruct the spatial and the spectral inhomogeneities very well even when the observations are extremely photon-limited (i.e., less than 0.1 photon per voxel).

Learning the Morphological Diversity

Gabriel Peyré, Jalal Fadili, and Jean-Luc Starck

SIAM J. Imaging Sci. 3, pp. 646-669 (24 pages) | Cited 1 time

Online Publication Date: September 29, 2010

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This article proposes a new method for image separation into a linear combination of morphological components. Sparsity in fixed dictionaries is used to extract the cartoon and oscillating content of the image. Complicated texture patterns are extracted by learning adapted local dictionaries that sparsify patches in the image. These fixed and learned sparsity priors define a nonconvex energy, and the separation is obtained as a stationary point of this energy. This variational optimization is extended to solve more general inverse problems such as inpainting. A new adaptive morphological component analysis algorithm is derived to find a stationary point of the energy. Using adapted dictionaries learned from data allows one to circumvent some difficulties faced by fixed dictionaries. Numerical results demonstrate that this adaptivity is indeed crucial in capturing complex texture patterns.

Clifford Bundles: A Common Framework for Image, Vector Field, and Orthonormal Frame Field Regularization

Thomas Batard

SIAM J. Imaging Sci. 3, pp. 670-701 (32 pages)

Online Publication Date: September 29, 2010

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The aim of this paper is to present a new framework for regularization by diffusion. The methods we develop in what follows can be used to smooth multichannel images, multichannel image sequences (videos), vector fields, and orthonormal frame fields in any dimension. From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and so-called generalized Laplacians. Sections are regularized from heat equations associated with generalized Laplacians, the solutions being approximated by convolutions with kernels. Then, the behavior of the diffusion is determined by the geometry of the vector bundle, i.e., by the metric of the base manifold and by a connection on the vector bundle. For instance, the heat equation associated with the Laplace–Beltrami operator can be considered from this point of view for applications to images and video regularization. The main topic of this paper is to show that this approach can be extended in several ways to vector fields and orthonormal frame fields by considering the context of Clifford algebras. We introduce Clifford–Beltrami and Clifford–Hodge operators as generalized Laplacians on Clifford bundles over Riemannian manifolds. Laplace–Beltrami diffusion appears as a particular case of diffusion for degree 0 sections (functions). Dealing with base manifolds of dimension 2, applications to multichannel image, two-dimensional vector field, and orientation field regularization are presented.
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