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SIAM J. on Imaging Sciences

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Issue 4 | 2011 | pp. 981-1233

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Volume 4, Issue 4, pp. 981-1233

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Select this Article		Image Inpainting Based on Coherence Transport with Adapted Distance Functions Thomas März SIAM J. Imaging Sci. 4, pp. 981-1000 (20 pages) Online Publication Date: October 20, 2011 Full Text: \| Download PDF
	+ -	Show Abstract We discuss an extension of our method image inpainting based on coherence transport. For the latter method the pixels of the inpainting domain have to be serialized into an ordered list. Until now, to induce the serialization we have used the distance to boundary map. But there are inpainting problems where the distance to boundary serialization causes unsatisfactory inpainting results. In the present work we demonstrate cases where we can resolve the difficulties by employing other distance functions which better suit the problem at hand.

Select this Article		How to Transform and Filter Images Using Iterated Function Systems Michael F. Barnsley, Brendan Harding, and Konstantin Igudesman SIAM J. Imaging Sci. 4, pp. 1001-1028 (28 pages) Online Publication Date: October 20, 2011 Full Text: \| Download PDF
	+ -	Show Abstract We generalize the mathematics of fractal transformations and illustrate how it leads to a new approach to the representation and processing of digital images, and consequent novel methods for filtering, watermarking, and encryption. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous representations. The underlying mathematics is established and some applications to digital imaging are described and exemplified.

Select this Article		A Mumford–Shah-Like Method for Limited Data Tomography with an Application to Electron Tomography Esther Klann SIAM J. Imaging Sci. 4, pp. 1029-1048 (20 pages) Online Publication Date: November 17, 2011 Full Text: \| Download PDF
	+ -	Show Abstract In this article the Mumford–Shah-like method of [R. Ramlau and W. Ring, J. Comput. Phys., 221 (2007), pp. 539–557] for complete tomographic data is generalized and applied to limited angle and region of interest tomography data. With the Mumford–Shah-like method, one reconstructs a piecewise constant function and simultaneously a segmentation from its (complete) Radon transform data. For limited data, the ability of the Mumford–Shah-like method to find a segmentation, and by that the singularity set of a function, is exploited. The method is applied to generated data from a torso phantom. The results demonstrate the performance of the method in reconstructing the singularity set, the density distribution itself for limited angle data, and also some quantitative information about the density distribution for region of interest data. As a second example limited angle region of interest tomography is considered as a simplified model for electron tomography (ET). For this problem we combine Lambda tomography and the Mumford–Shah-like method. The combined method is applied to simulated ET data.

Select this Article		Continuous Multiclass Labeling Approaches and Algorithms J. Lellmann and C. Schnörr SIAM J. Imaging Sci. 4, pp. 1049-1096 (48 pages) Online Publication Date: November 22, 2011 Full Text: \| Download PDF
	+ -	Show Abstract We study convex relaxations of the image labeling problem on a continuous domain with regularizers based on metric interaction potentials. The generic framework ensures existence of minimizers and covers a wide range of relaxations of the original combinatorial problem. We focus on two specific relaxations that differ in flexibility and simplicity—one can be used to tightly relax any metric interaction potential, while the other covers only Euclidean metrics but requires less computational effort. For solving the nonsmooth discretized problem, we propose a globally convergent Douglas–Rachford scheme and show that a sequence of dual iterates can be recovered in order to provide a posteriori optimality bounds. In a quantitative comparison to two other first-order methods, the approach shows competitive performance on synthetic and real-world images. By combining the method with an improved rounding technique for nonstandard potentials, we were able to routinely recover integral solutions within $1\%$–$5\%$ of the global optimum for the combinatorial image labeling problem.

Select this Article		Transient Wave Imaging with Limited-View Data Habib Ammari, Mark Asch, Lili Guadarrama Bustos, Vincent Jugnon, and Hyeonbae Kang SIAM J. Imaging Sci. 4, pp. 1097-1121 (25 pages) Online Publication Date: November 29, 2011 Full Text: \| Download PDF
	+ -	Show Abstract We consider for the wave equation the inverse problem of identifying locations of point sources and dipoles from limited-view data. Using as weights particular background solutions constructed by the geometrical control method, we recover classical imaging algorithms by appropriately averaging limited-view data. We show both analytically and numerically that if one can accurately construct the geometric control, then one can perform imaging with the same resolution when using limited-view data as when using full-view data.

Select this Article		Robust Video Restoration by Joint Sparse and Low Rank Matrix Approximation Hui Ji, Sibin Huang, Zuowei Shen, and Yuhong Xu SIAM J. Imaging Sci. 4, pp. 1122-1142 (21 pages) Online Publication Date: November 29, 2011 Full Text: \| Download PDF
	+ -	Show Abstract This paper presents a new patch-based video restoration scheme. By grouping similar patches in the spatiotemporal domain, we formulate the video restoration problem as a joint sparse and low-rank matrix approximation problem. The resulting nuclear norm and $\ell_1$ norm related minimization problem can also be efficiently solved by many recently developed numerical methods. The effectiveness of the proposed video restoration scheme is illustrated on two applications: video denoising in the presence of random-valued noise, and video in-painting for archived films. The numerical experiments indicate that the proposed video restoration method compares favorably against many existing algorithms.

Select this Article		Geometrically Guided Exemplar-Based Inpainting Frédéric Cao, Yann Gousseau, Simon Masnou, and Patrick Pérez SIAM J. Imaging Sci. 4, pp. 1143-1179 (37 pages) Online Publication Date: December 01, 2011 Full Text: \| Download PDF
	+ -	Show Abstract Exemplar-based methods have proven their efficiency for the reconstruction of missing parts in a digital image. Texture as well as local geometry are often very well restored by such methods. Some applications, however, require the ability to reconstruct nonlocal geometric features, e.g., long edges. In order to do so, we propose to first compute a geometric sketch, which is then interpolated and used as a guide for the global reconstruction. In comparison with other related approaches, the originality of our work relies on the following points: (1) The geometric sketch computation is parameter-free and based on level lines, which provides a complete, reliable, and stable representation of the image. (2) The completion of the geometric sketch is fully automatic. It is done using a new—and interesting on its own—geometric inpainting approach that interpolates level lines with Euler spirals. Euler spirals are natural curves for shape completion and have been used already for edge completion and inpainting. It is the first time, however, that these curves are used for completing the whole level lines structure. (3) The general reconstruction is performed using a guided version of a classical exemplar-based method. However, we do not constrain the exemplar-based reconstruction to strictly follow the geometric guide. We actually use a new metric between blocks that consists of the sum of the classical ${{\mathrm L}^2}$ metric between any two blocks of the general image plus an ${{\mathrm L}^2}$ metric between the corresponding blocks in the completed geometric image. This is equivalent to a Lagrangian relaxation of a strictly guided reconstruction. We discuss in the paper the details of the method and some related mathematical issues, and we illustrate its efficiency on several examples.

Select this Article		Theory of Waveform-Diverse Moving-Target Spotlight Synthetic-Aperture Radar Margaret Cheney and Brett Borden SIAM J. Imaging Sci. 4, pp. 1180-1199 (20 pages) Online Publication Date: December 13, 2011 Full Text: \| Download PDF
	+ -	Show Abstract We develop a theory for waveform-diverse moving-target synthetic-aperture radar in the case in which a single moving antenna is used both on transmission and on reception. We assume that the targets (scattering objects) are moving linearly, but we allow an arbitrary, known flight path for the antenna and allow it to transmit a sequence of arbitrary, known waveforms. A formula for phase-space (position and velocity) imaging is developed, and we provide a formula for the point-spread function of the corresponding imaging system. This point-spread function is expressed in terms of the ordinary radar ambiguity function. As an example, we show how the theory can be applied to the problem of estimating the errors that arise when target and antenna motion is neglected during the transit time of each pulse.

Select this Article		Mumford–Shah–Euler Flow with Sphere Constraint and Applications to Color Image Inpainting Jonas Haehnle and Andreas Prohl SIAM J. Imaging Sci. 4, pp. 1200-1233 (34 pages) Online Publication Date: December 15, 2011 Full Text: \| Download PDF
	+ -	Show Abstract Two fully discrete finite element–based algorithms to approximate the $L^2$ gradient flow of the Mumford–Shah–Euler functional for unit vector fields are proposed, analyzed, and compared. The first scheme uses a penalization strategy, and the second a Lagrange multiplier, to approximate and enforce the sphere constraint, respectively. Both schemes are applied to color image inpainting in the chromaticity and brightness color model and are also compared to inpainting with the standard Mumford–Shah functional, as well as channelwise red-green-blue inpainting.