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2012

Volume 5, Issue 1, pp. 1-482


Image Denoising Using Mean Curvature of Image Surface

Wei Zhu and Tony Chan

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].

Dictionary Learning for Noisy and Incomplete Hyperspectral Images

Zhengming Xing, Mingyuan Zhou, Alexey Castrodad, Guillermo Sapiro, and Lawrence Carin

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.

Adaptive Compressed Image Sensing Using Dictionaries

Amir Averbuch, Shai Dekel, and Shay Deutsch

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.

Fast Algorithms for Image Reconstruction with Application to Partially Parallel MR Imaging

Yunmei Chen, William Hager, Feng Huang, Dzung Phan, Xiaojing Ye, and Wotao Yin

SIAM J. Imaging Sci. 5, pp. 90-118 (29 pages)

Online Publication Date: January 24, 2012

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This paper presents two fast algorithms for total variation–based image reconstruction in a magnetic resonance imaging technique known as partially parallel imaging (PPI), where the inversion matrix is large and ill-conditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more easily solved subproblems. The first method reduces the image reconstruction problem to an unconstrained minimization problem, which is solved by an alternating proximal minimization algorithm. One phase of the algorithm solves a total variation (TV) denoising problem, and the second phase solves an ill-conditioned linear system. Linear and sublinear convergence results are given, and an implementation based on a primal-dual hybrid gradient (PDHG) scheme for the TV problem and on a Barzilai–Borwein scheme for the linear inversion is proposed. The second algorithm exploits the special structure of the PPI reconstruction problem by decomposing it into one subproblem involving Fourier transforms and another subproblem that can be treated by the PDHG scheme. Numerical results and comparisons with recently developed methods indicate the efficiency of the proposed algorithms.

Convergence Analysis of Primal-Dual Algorithms for a Saddle-Point Problem: From Contraction Perspective

Bingsheng He and Xiaoming Yuan

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.

A Variational Approach for Sharpening High Dimensional Images

Michael Möller, Todd Wittman, Andrea L. Bertozzi, and Martin Burger

SIAM J. Imaging Sci. 5, pp. 150-178 (29 pages)

Online Publication Date: January 24, 2012

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Earth-observing satellites usually not only take ordinary red-green-blue images but also provide several images including the near-infrared and infrared spectrum. These images are called multispectral, for about four to seven different bands, or hyperspectral, for higher dimensional images of up to 210 bands. The drawback of the additional spectral information is that each spectral band has rather low spatial resolution. In this paper we propose a new variational method for sharpening high dimensional spectral images with the help of a high resolution gray-scale image while preserving the spectral characteristics used for classification and identification tasks. We describe the application of split Bregman minimization to our energy, prove convergence speed, and compare the split Bregman method to a descent method based on the ideas of alternating directions minimization. Finally, we show results on Quickbird multispectral as well as on AVIRIS hyperspectral data.

Coherence Pattern–Guided Compressive Sensing with Unresolved Grids

Albert Fannjiang and Wenjing Liao

SIAM J. Imaging Sci. 5, pp. 179-202 (24 pages)

Online Publication Date: February 23, 2012

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Highly coherent sensing matrices arise in discretization of continuum imaging problems such as radar and medical imaging when the grid spacing is below the Rayleigh threshold. Algorithms based on techniques of band exclusion (BE) and local optimization (LO) are proposed to deal with such coherent sensing matrices. These techniques are embedded in the existing compressed sensing algorithms, such as Orthogonal Matching Pursuit (OMP), Subspace Pursuit (SP), Iterative Hard Thresholding (IHT), Basis Pursuit (BP), and Lasso, and result in the modified algorithms BLOOMP, BLOSP, BLOIHT, BP-BLOT, and Lasso-BLOT, respectively. Under appropriate conditions, it is proved that BLOOMP can reconstruct sparse, widely separated objects up to one Rayleigh length in the Bottleneck distance independent of the grid spacing. One of the most distinguishing attributes of BLOOMP is its capability of dealing with large dynamic ranges. The BLO-based algorithms are systematically tested with respect to four performance metrics: dynamic range, noise stability, sparsity, and resolution. With respect to dynamic range and noise stability, BLOOMP is the best performer. With respect to sparsity, BLOOMP is the best performer for high dynamic range, while for dynamic range near unity BP-BLOT and Lasso-BLOT with the optimized regularization parameter have the best performance. In the noiseless case, BP-BLOT has the highest resolving power up to certain dynamic range. The algorithms BLOSP and BLOIHT are good alternatives to BLOOMP and BP/Lasso-BLOT: they are faster than both BLOOMP and BP/Lasso-BLOT and share, to a lesser degree, BLOOMP's amazing attribute with respect to dynamic range. Detailed comparisons with the algorithms Spectral Iterative Hard Thresholding (SIHT) and the frame-adapted BP demonstrate the superiority of the BLO-based algorithms for the problem of sparse approximation in terms of highly coherent, redundant dictionaries.

A Butterfly Algorithm for Synthetic Aperture Radar Imaging

Laurent Demanet, Matthew Ferrara, Nicholas Maxwell, Jack Poulson, and Lexing Ying

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.

Almost Local Metrics on Shape Space of Hypersurfaces in $n$-Space

Martin Bauer, Philipp Harms, and Peter W. Michor

SIAM J. Imaging Sci. 5, pp. 244-310 (67 pages)

Online Publication Date: March 06, 2012

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This paper extends parts of the results from [P. W. Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74–113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type $M$ or the orbifold of immersions from $M$ to $\mathbb R^n$ modulo the group of diffeomorphisms of $M$. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions: $G_f(h,k) = \int_{M} \Phi(Vol(f),Tr(L))\bar{g}(h, k) vol(f^*\bar{g})$, where $\bar{g}$ is the Euclidean metric on $\mathbb R^n$, $f^*\bar{g}$ is the induced metric on $M$, $h,k\in C^\infty(M,\mathbb R^n)$ are tangent vectors at $f$ to the space of embeddings or immersions, where $\Phi:\mathbb R^2\to \mathbb R_{>0}$ is a suitable smooth function, $Vol(f) = \int_M vol(f^*\bar{g})$ is the total hypersurface volume of $f(M)$, and the trace $Tr(L)$ of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of $\Phi$ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.

Perspective Shape from Shading: Ambiguity Analysis and Numerical Approximations

Michael Breuß, Emiliano Cristiani, Jean-Denis Durou, Maurizio Falcone, and Oliver Vogel

SIAM J. Imaging Sci. 5, pp. 311-342 (32 pages)

Online Publication Date: March 08, 2012

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In this paper we study a perspective model for shape from shading and its numerical approximation. We show that an ambiguity still persists, although the model with light attenuation factor has previously been shown to be well-posed under appropriate assumptions. Analytical results revealing the ambiguity are complemented by various numerical tests. Moreover, we present convergence results for two iterative approximation schemes. The first is based on a finite difference discretization, whereas the second is based on a semi-Lagrangian discretization. The convergence results are obtained in the general framework of viscosity solutions of the underlying partial differential equation. In addition to these theoretical and numerical results, we propose an algorithm for reconstructing discontinuous surfaces, making it possible to obtain results of reasonable quality even for complex scenes. To this end, we solve the constituting equation on a previously segmented input image, using state constraint boundary conditions at the segment borders.

Edge Detection Filter based on Mumford–Shah Green Function

Sasan Mahmoodi

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.

A Total Variation–Based JPEG Decompression Model

K. Bredies and M. Holler

SIAM J. Imaging Sci. 5, pp. 366-393 (28 pages)

Online Publication Date: March 08, 2012

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We propose a variational model for artifact-free JPEG decompression. It is based on the minimization of the total variation over the convex set $U$ of all possible source images associated with given JPEG data. The general case where $U$ represents a pointwise restriction with respect to an $L^2$-orthonormal basis is considered. Analysis of the infinite dimensional model is presented, including the derivation of optimality conditions. A discretized version is solved using a primal-dual algorithm supplemented by a primal-dual gap-based stopping criterion. Experiments illustrate the effect of the model. Good reconstruction quality is obtained even for highly compressed images, while a graphics processing unit (GPU) implementation is shown to significantly reduce computation time, making the model suitable for real-time applications.

Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks

Mario Micheli, Peter W. Michor, and David Mumford

SIAM J. Imaging Sci. 5, pp. 394-433 (40 pages)

Online Publication Date: March 15, 2012

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This paper deals with the computation of sectional curvature for the manifolds of $N$ landmarks (or feature points) in $D$ dimensions, endowed with the Riemannian metric induced by the group action of diffeomorphisms. The inverse of the metric tensor for these manifolds (i.e., the cometric), when written in coordinates, is such that each of its elements depends on at most $2D$ of the $ND$ coordinates. This makes the matrices of partial derivatives of the cometric very sparse in nature, thus suggesting solving the highly nontrivial problem of developing a formula that expresses sectional curvature in terms of the cometric and its first and second partial derivatives (we call this Mario's formula). We apply such a formula to the manifolds of landmarks, and in particular we fully explore the case of geodesics on which only two points have nonzero momenta and compute the sectional curvatures of 2-planes spanned by the tangents to such geodesics. The latter example gives insight into the geometry of the full manifolds of landmarks.

Approximating Symmetric Positive Semidefinite Tensors of Even Order

Angelos Barmpoutis, Jeffrey Ho, and Baba C. Vemuri

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.

One Shot Inverse Scattering via Rational Approximation

Martin Hanke

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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