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SIAM J. on Imaging Sciences

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2012

Volume 5

partial Issue 2 | 2012 | pp. 483-651

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Volume 5, Issue 2 (partial)

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Select this Article		Multivalued Geodesic Ray-Tracing for Computing Brain Connections Using Diffusion Tensor Imaging N. Sepasian, J. H. M. ten Thije Boonkkamp, B. M. Ter Haar Romeny, and A. Vilanova SIAM J. Imaging Sci. 5, pp. 483-504 (22 pages) Online Publication Date: April 04, 2012 Full Text: \| Download PDF
	+ -	Show Abstract 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.

Select this Article		A Semismooth Newton Method for Nonlinear Parameter Identification Problems with Impulsive Noise Christian Clason and Bangti Jin SIAM J. Imaging Sci. 5, pp. 505-536 (32 pages) Online Publication Date: April 12, 2012 Full Text: \| Download PDF
	+ -	Show Abstract 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.

Select this Article		The Natural Vectorial Total Variation Which Arises from Geometric Measure Theory Bastian Goldluecke, Evgeny Strekalovskiy, and Daniel Cremers SIAM J. Imaging Sci. 5, pp. 537-563 (27 pages) Online Publication Date: April 12, 2012 Full Text: \| Download PDF
	+ -	Show Abstract 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.

Select this Article		Multistatic Imaging of Extended Targets Habib Ammari, Josselin Garnier, Hyeonbae Kang, Mikyoung Lim, and Knut Sølna SIAM J. Imaging Sci. 5, pp. 564-600 (37 pages) Online Publication Date: April 12, 2012 Full Text: \| Download PDF
	+ -	Show Abstract 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.

Select this Article		Sobolev Gradients and Image Interpolation Parimah Kazemi and Ionut Danaila SIAM J. Imaging Sci. 5, pp. 601-624 (24 pages) Online Publication Date: May 01, 2012 Full Text: \| Download PDF
	+ -	Show Abstract We present here a new image inpainting algorithm based on the Sobolev gradient method in conjunction with the Navier–Stokes model. The original model of Bertalmío, Bertozzi, and Sapiro [Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2001, pp. 355–362] is reformulated as a variational principle based on the minimization of a well-chosen functional by a steepest descent method using Sobolev gradients. This new theoretical framework offers an alternative to the direct solving of a high-order PDE, with the practical advantage of an easier and more flexible computer implementation. In particular, the proposed algorithm does not require any constant tuning or advanced knowledge of numerical methods for Navier–Stokes equations (slope limiters, dynamic relaxation for Poisson equation, anisotropic diffusion steps, etc.). Using a straightforward finite difference implementation, we demonstrate, through various examples for image inpainting and image interpolation, that the novel algorithm is faster than the original implementation of the Navier–Stokes model, while providing results of similar quality. This paper also provides the mathematical theory for the analysis of the algorithm. Using an evolution equation in an infinite dimensional setting, we obtain global existence and uniqueness results as well as the existence of an $\omega$-limit. This formalism is of more general interest and could be applied to other image processing models based on variational formulations.

Select this Article		A Robust Computational Algorithm for Inverse Photomask Synthesis in Optical Projection Lithography Siu Kai Choy, Ningning Jia, Chong Sze Tong, Man Lai Tang, and Edmund Y. Lam SIAM J. Imaging Sci. 5, pp. 625-651 (27 pages) Online Publication Date: May 17, 2012 Full Text: \| Download PDF
	+ -	Show Abstract Inverse lithography technology formulates the photomask synthesis as an inverse mathematical problem. To solve this, we propose a variational functional and develop a robust computational algorithm, where the proposed functional takes into account the process variations and incorporates several regularization terms that can control the mask complexity. We establish the existence of the minimizer of the functional, and in order to optimize it effectively, we adopt an alternating minimization procedure with Chambolle's fast duality projection algorithm. Experimental results show that our proposed algorithm is effective in synthesizing high quality photomasks as compared with existing methods.