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SIAM J. on Imaging Sciences

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2009

Volume 2, Issue 4, pp. 1003-1291


Imaging by Modification: Numerical Reconstruction of Local Conductivities from Corresponding Power Density Measurements

Y. Capdeboscq, J. Fehrenbach, F. de Gournay, and O. Kavian

SIAM J. Imaging Sci. 2, pp. 1003-1030 (28 pages)

Online Publication Date: October 14, 2009

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We discuss the reconstruction of the impedance from the local power density. This study is motivated by a new imaging principle which allows us to recover interior measurements of the energy density by a noninvasive method. We discuss the theoretical feasibility in two dimensions, and propose numerical algorithms to recover the conductivity in two and three dimensions. The efficiency of this approach is documented by several numerical simulations.

Incompressible, Quasi-Isometric Deformations of 2-Dimensional Domains

Gershon Wolansky

SIAM J. Imaging Sci. 2, pp. 1031-1048 (18 pages)

Online Publication Date: October 22, 2009

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This paper proposes a sensible definition of an optimal deformation between flat domains, as well as 2-dimensional surfaces. We assume that the surface is deformed by an area preserving deformation, and we look for an area preserving map which is as close to an isometry as possible. We also suggest a gradient descent flow for obtaining the optimal map.

Statistical Hypothesis Testing for Postreconstructed and Postregistered Medical Images

Eugene Demidenko

SIAM J. Imaging Sci. 2, pp. 1049-1067 (19 pages)

Online Publication Date: October 22, 2009

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Postreconstructed and postregistered medical images are typically treated as the raw data, implicitly assuming that those operations are error free. We question this assumption and explore how the precision of reconstruction and affine registration can be assessed by the image covariance matrix and confidence interval, called the confidence eigenimage, using a statistical model-based approach. Various hypotheses may be tested after image reconstruction and registration using classical statistical hypothesis testing vehicles: Is there a statistically significant difference between images? Does the intensity at a specific location or area of interest belong to the "normal" range? Is there a tumor? Does the image require rigid registration? We illustrate statistical hypothesis testing with three examples: breast computed tomography, breast near infrared linear reconstruction, and brain magnetic resonance imaging.

On the Application of the Monge–Kantorovich Problem to Image Registration

O. Museyko, M. Stiglmayr, K. Klamroth, and G. Leugering

SIAM J. Imaging Sci. 2, pp. 1068-1097 (30 pages)

Online Publication Date: November 04, 2009

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A problem of image registration is considered in the context of optimal mass transportation. The properties and limitations of an optimal image transportation are analyzed. A modified formulation of this approach is proposed in order to overcome the morphing effect. Finally, a fast and simple scale-space approach for the new formulation is introduced, and numerical examples are presented.

Compressive Sensing by Random Convolution

Justin Romberg

SIAM J. Imaging Sci. 2, pp. 1098-1128 (31 pages) | Cited 3 times

Online Publication Date: November 04, 2009

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This paper demonstrates that convolution with random waveform followed by random time-domain subsampling is a universally efficient compressive sensing strategy. We show that an $n$-dimensional signal which is $S$-sparse in any fixed orthonormal representation can be recovered from $m\gtrsim S\log n$ samples from its convolution with a pulse whose Fourier transform has unit magnitude and random phase at all frequencies. The time-domain subsampling can be done in one of two ways: in the first, we simply observe $m$ samples of the random convolution; in the second, we break the random convolution into $m$ blocks and summarize each with a single randomized sum. We also discuss several imaging applications where convolution with a random pulse allows us to superresolve fine-scale features, allowing us to recover high-resolution signals from low-resolution measurements.

Cahn–Hilliard Inpainting and a Generalization for Grayvalue Images

Martin Burger, Lin He, and Carola-Bibiane Schönlieb

SIAM J. Imaging Sci. 2, pp. 1129-1167 (39 pages)

Online Publication Date: November 04, 2009

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The Cahn–Hilliard equation is a nonlinear fourth order diffusion equation originating in material science for modeling phase separation and phase coarsening in binary alloys. The inpainting of binary images using the Cahn–Hilliard equation is a new approach in image processing. In this paper we discuss the stationary state of the proposed model and introduce a generalization for grayvalue images of bounded variation. This is realized by using subgradients of the total variation functional within the flow, which leads to structure inpainting with smooth curvature of level sets.

An Efficient Primal-Dual Method for $L^1$TV Image Restoration

Yiqiu Dong, Michael Hintermüller, and Marrick Neri

SIAM J. Imaging Sci. 2, pp. 1168-1189 (22 pages)

Online Publication Date: November 04, 2009

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Image restoration based on an $\ell^1$-data-fitting term and edge preserving total variation regularization is considered. The associated nonsmooth energy minimization problem is handled by utilizing Fenchel duality and dual regularization techniques. The latter guarantee uniqueness of the dual solution and an efficient way for reconstructing a primal solution, i.e., the restored image, from a dual solution. For solving the resulting primal-dual system, a semismooth Newton solver is proposed and its convergence is studied. The paper ends with a report on restoration results obtained by the new algorithm for salt-and-pepper or random-valued impulse noise including blurring. A comparison with other methods is provided as well.

Detection of Intensity and Motion Edges within Optical Flow via Multidimensional Control

Christoph Brune, Helmut Maurer, and Marcus Wagner

SIAM J. Imaging Sci. 2, pp. 1190-1210 (21 pages)

Online Publication Date: November 11, 2009

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In this paper, we propose a new optimization approach for the simultaneous computation of optical flow and edge detection therein. Instead of using an Ambrosio–Tortorelli type energy functional, we reformulate the optical flow problem as a multidimensional control problem. The optimal control problem is solved by discretization methods and large-scale optimization techniques. The edge detector can be immediately built from the control variables. We provide three series of numerical examples. The first shows that the mere presence of a gradient restriction has a regularizing effect, while the second demonstrates how to balance the regularizing effects of a term within the objective and the control restriction. The third series of numerical results is concerned with the direct evaluation of a TV-regularization term by introduction of control variables with sign restrictions.

Anisotropic Cheeger Sets and Applications

Vicent Caselles, Gabriele Facciolo, and Enric Meinhardt

SIAM J. Imaging Sci. 2, pp. 1211-1254 (44 pages)

Online Publication Date: November 11, 2009

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The main purpose of this paper is to develop the mathematical analysis of anisotropic total variation problems with a degenerate metric and the computation of the associated Cheeger sets. We illustrate our analysis with the computation of Cheeger sets with respect to different anisotropic norms of relevance in applications to image processing. In particular, we describe the computation of global minima of geodesic active contour models, and we illustrate the use of Cheeger sets for the problem of edge linking.

Compressive Imaging: An Application

Robert Muise

SIAM J. Imaging Sci. 2, pp. 1255-1276 (22 pages)

Online Publication Date: November 25, 2009

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We consider the application of compressive imaging theory to the problem of wide-area persistent surveillance. While the compressive sensing theory enjoys significant research attention, mainly because of the possibilities for orders-of-magnitude increases in signal/image processing applications, the application areas for compressive imaging have not kept pace due to the lack of an optical architecture which could directly improve current sensing capabilities. There are now cases in the literature and under study where optical architectures have been developed which require the incorporation of compressive imaging in order to perform the indicated exploitation application. This paper utilizes one such architecture to show a dramatic (two orders of magnitude) increase in performance for the application of wide-area persistent surveillance. This application together with its architecture is described as a field-of-view (FOV) multiplexing imager, and its relation to compressive imaging is discussed and exploited for increased field-of-regard (FOR) imaging. A simulated example is given in the last section with qualitatively impressive results. The optical architecture of FOV multiplexing, while showing a significant performance increase over current capabilities, also opens some interesting research questions.

Compressive Imaging of Subwavelength Structures

Albert C. Fannjiang

SIAM J. Imaging Sci. 2, pp. 1277-1291 (15 pages)

Online Publication Date: December 16, 2009

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The problem of imaging extended targets (sources or scatterers) is formulated in the framework of compressed sensing with emphasis on subwavelength resolution. The proposed formulation of the problems of inverse source/scattering is essentially exact and leads to the random partial Fourier measurement matrix in the case of periodic targets. In the case of square-integrable targets, the proposed sampling scheme in the Littlewood–Paley wavelet basis block-diagonalizes the scattering matrix with each block in the form of a random partial Fourier matrix corresponding to each dyadic scale of the target. The resolution issue is analyzed from two perspectives: stability and the signal-to-noise ratio (SNR). The subwavelength modes are shown to be typically unstable unless the measurement is carried out in near field. The number of the stable modes typically increases as the negative $d$th (the dimension of the target) power of the distance between the target and the sensors/source (in the unit of wavelength). he resolution limit is shown to be inversely proportional to the SNR in the high SNR limit. Numerical simulations are provided to validate the theoretical predictions.
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