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2010

Volume 3, Issue 4, pp. 703-1145

† Special Section on Optimization in Imaging Sciences

Stabilization of Flicker-Like Effects in Image Sequences through Local Contrast Correction

Julie Delon and Agnès Desolneux

SIAM J. Imaging Sci. 3, pp. 703-734 (32 pages)

Online Publication Date: October 07, 2010

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In this paper, we address the problem of the restoration of image sequences which have been affected by local intensity modifications (local contrast changes). Such artifacts can be encountered particularly in biological or archive film sequences, and are usually due to inconsistent exposures or sparse time sampling. In order to reduce such local artifacts, we introduce a local stabilization operator, called LStab, which acts as a time filter on image patches and relies on a similarity measure which is robust to contrast changes. Thereby, this operator is able to take motion into account without relying on a sophisticated motion estimation procedure. The efficiency of the stabilization is shown on various sequences. The experimental results compare favorably with state-of-the-art approaches.

Deblurring of One Dimensional Bar Codes via Total Variation Energy Minimization

Rustum Choksi and Yves van Gennip

SIAM J. Imaging Sci. 3, pp. 735-764 (30 pages)

Online Publication Date: October 14, 2010

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Using total variation-based energy minimization we address the recovery of a blurred (convoluted) one dimensional (1D) bar code. We consider functionals defined over all possible bar codes with fidelity to a convoluted signal of a bar code and regularized by total variation. Our fidelity terms consist of the $L^2$ distance either directly to the measured signal or preceded by deconvolution. Key length scales and parameters are the $X$-dimension of the underlying bar code, the size of the supports of the convolution and deconvolution kernels, and the fidelity parameter. For all functionals, we establish parameter regimes (sufficient conditions) wherein the underlying bar code is the unique minimizer. We also present some numerical experiments suggesting that these sufficient conditions are not optimal and the energy methods are quite robust for significant blurring.

Smoothing Nonlinear Conjugate Gradient Method for Image Restoration Using Nonsmooth Nonconvex Minimization

Xiaojun Chen and Weijun Zhou

SIAM J. Imaging Sci. 3, pp. 765-790 (26 pages)

Online Publication Date: October 14, 2010

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Image restoration problems are often converted into large-scale, nonsmooth, and nonconvex optimization problems. Most existing minimization methods are not efficient for solving such problems. It is well known that nonlinear conjugate gradient methods are preferred to solve large-scale smooth optimization problems due to their simplicity, low storage, practical computation efficiency, and nice convergence properties. In this paper, we propose a smoothing nonlinear conjugate gradient method where an intelligent scheme is used to update the smoothing parameter at each iteration and guarantees that any accumulation point of a sequence generated by this method is a Clarke stationary point of the nonsmooth and nonconvex optimization problem. Moreover, we present a class of smoothing functions and show their approximation properties. This method is easy to implement without adding any new variables. Three image restoration problems with different pixels and different regularization terms are used in numerical tests. Experimental results and comparison with the continuation method in [M. Nikolova et al., SIAM J. Imaging Sci., 1 (2008), pp. 2–25] show the efficiency of the proposed method.

Fast Algorithms for Source Identification Problems with Elliptic PDE Constraints

Santi S. Adavani and George Biros

SIAM J. Imaging Sci. 3, pp. 791-808 (18 pages)

Online Publication Date: October 21, 2010

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We present algorithms for the solution of a class of source identification problems for systems governed by elliptic partial differential equations (PDEs) on two-dimensional regular geometries. The state is the solution of the PDE, which is driven by an unknown source field. Given observations of the state, we seek to reconstruct the source field. We consider the cases of full domain observations and boundary observations. The problem is formulated as a least-squares PDE-constrained optimization problem. We use a reduced space approach in which we “invert” the associated Hessian using a preconditioned conjugate gradient (PCG) algorithm. Using standard Fourier analysis, we derive analytical solutions for the case in which the governing PDE has constant coefficients. Based on these solutions, we construct preconditioners that accelerate the convergence of PCG in the case of variable-coefficient elliptic PDE constraints. We performed numerical experiments to show the effectiveness of the preconditioners for variable coefficients with different contrasts and smoothness properties. We observed mesh-independent and $\beta$-independent convergence for different cases of the variable coefficients. The computational complexity of solving the source identification problem is $\mathcal{O}(N\log N)$. The construction of the preconditioner costs $\mathcal{O}(N^{3/2})$, where $N$ is the discretization size for the source.

Cross Correlation and Deconvolution of Noise Signals in Randomly Layered Media

Josselin Garnier and Knut Sølna

SIAM J. Imaging Sci. 3, pp. 809-834 (26 pages)

Online Publication Date: October 28, 2010

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It is known that cross correlation of waves generated by noise sources, propagating in an unknown medium and recorded by a sensor array, can provide information about the medium. In this paper the medium is a three-dimensional small-scale randomly layered medium with slow macroscopic variations. The main objective here is to set forth a framework for analysis of cross correlations of waves generated by noise sources and propagating in such a medium and, moreover, to use this framework to design estimators for macroscale medium features. The noise sources are located at the bottom of a random medium slab and generate a random wave field that is scattered by the rapid random fluctuations of the medium and then recorded at the surface. Taking into account the pressure release boundary conditions at the surface, this situation corresponds to the so-called daylight configuration. The analysis is carried out in the asymptotic framework where the typical wavelength is small compared to the scale of the macroscopic variations of the background medium and large compared to the decoherence length of the microscopic random fluctuations of the medium. It is shown that the cross correlation of the waves recorded at the surface contains statistically stable information about the macroscopic background medium.

A Probabilistic Contour Observer for Online Visual Tracking

Ibrahima J. Ndiour, Jochen Teizer, and Patricio A. Vela

SIAM J. Imaging Sci. 3, pp. 835-855 (21 pages)

Online Publication Date: October 28, 2010

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This paper presents an online, recursive filtering strategy for contour-based tracking. Approaching the tracking problem from an estimation perspective leads to an observer design for the visual track signal associated with an individual target in an image sequence. The track state of the observer is decomposed into group and shape components that describe the gross location and the nonrigid shape, respectively, of the object. A probabilistic representation describes the shape nonparametrically. The constitutive components of the observer are detailed, which include a dynamical prediction model and a correction mechanism. Incorporating the probabilistic observer into the tracking process leads to improved performance and segmentations. The improvements are validated through application of the observer to recorded imagery with evaluation via objective measures of quality.

Analysis and Generalizations of the Linearized Bregman Method

Wotao Yin

SIAM J. Imaging Sci. 3, pp. 856-877 (22 pages)

Online Publication Date: October 28, 2010

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This paper analyzes and improves the linearized Bregman method for solving the basis pursuit and related sparse optimization problems. The analysis shows that the linearized Bregman method has the exact regularization property; namely, it converges to an exact solution of the basis pursuit problem whenever its smooth parameter $\alpha$ is greater than a certain value. The analysis is based on showing that the linearized Bregman algorithm is equivalent to gradient descent applied to a certain dual formulation. This result motivates generalizations of the algorithm enabling the use of gradient-based optimization techniques such as line search, Barzilai–Borwein, limited memory BFGS (L-BFGS), nonlinear conjugate gradient, and Nesterov's methods. In the numerical simulations, the two proposed implementations, one using Barzilai–Borwein steps with nonmonotone line search and the other using L-BFGS, gave more accurate solutions in much shorter times than the basic implementation of the linearized Bregman method with a so-called kicking technique.

KDE Paring and a Faster Mean Shift Algorithm

Daniel Freedman and Pavel Kisilev

SIAM J. Imaging Sci. 3, pp. 878-903 (26 pages)

Online Publication Date: November 02, 2010

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The kernel density estimate (KDE) is a nonparametric density estimate which has broad application in computer vision and pattern recognition. In particular, the mean shift procedure uses the KDE structure to cluster or segment data, including images and video. The usefulness of these twin techniques—KDE and mean shift—on large data sets is hampered by the large space or description complexity of the KDE, which in turn leads to a large time complexity of the mean shift procedure that is superlinear in the number of points. In this paper, we propose a sampling technique for KDE paring, i.e., the construction of a compactly represented KDE with much smaller description complexity. We prove that this technique has good properties in that the pared-down KDE so constructed is close to the original KDE in a precise mathematical sense. We then show how to use this pared-down KDE to devise a considerably faster mean shift algorithm, whose time complexity we analyze formally. Experiments show that image and video segmentation results of the proposed fast mean shift method are similar to those based on the standard mean shift procedure, with the typical speed-up several orders of magnitude for large data sets. Finally, we present an application of the fast mean shift method to the efficient construction of multiscale graph structures for images, which can be used as a preprocessing step for more sophisticated segmentation algorithms.

Anisotropic $\alpha$-Kernels and Associated Flows

Micha Feigin, Nir Sochen, and Baba C. Vemuri

SIAM J. Imaging Sci. 3, pp. 904-925 (22 pages)

Online Publication Date: November 16, 2010

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The Laplacian raised to fractional powers can be used to generate scale spaces as was shown in recent literature by Duits, Felsberg, Florack, and Platel [$\alpha$ scale spaces on a bounded domain, in Scale Space Methods in Computer Vision, L. D. Griffin and M. Lillholm, eds., Lecture Notes in Comput. Sci. 2695, Springer, Berlin, Heidelberg, 2003, pp. 494–510] and Duits, Florack, de Graaf, and ter Haar Romeny [J. Math. Imaging Vision, 20 (2004), pp. 267–298]. In this paper, we study the anisotropic diffusion processes by defining new generators that are fractional powers of an anisotropic scale space generator. This is done in a general framework that allows us to explain the relation between a differential operator that generates the flow and the generators that are constructed from its fractional powers. We then generalize this to any other function of the operator. We discuss important issues involved in the numerical implementation of this framework and present several examples of fractional versions of the Perona–Malik and Beltrami flows along with their properties.

Visual Multiple-Secret Sharing by Circle Random Grids

Shyong Jian Shyu and Kun Chen

SIAM J. Imaging Sci. 3, pp. 926-953 (28 pages)

Online Publication Date: November 30, 2010

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We design innovative algorithms for visual multiple-secret sharing using circle or cylinder random grids in this paper. Formal validations, security analyses, and computer implementations are discussed to demonstrate the correctness and feasibility of our algorithms. As compared to the schemes developed in conventional visual cryptography, our design delivers three significant advantages: (1) it is capable of sharing multiple (instead of only one or two) secret images in two shares; (2) it does not result in any extra pixel expansion so that the sizes of the secret image and the encrypted shares are exactly the same; and (3) it is simple and easy to implement. These advantages broaden the potential applicability and flexibility of visual secret sharing schemes.

Bochner Subordination, Logarithmic Diffusion Equations, and Blind Deconvolution of Hubble Space Telescope Imagery and Other Scientific Data

Alfred S. Carasso

SIAM J. Imaging Sci. 3, pp. 954-980 (27 pages)

Online Publication Date: November 30, 2010

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Generalized Linnik processes and associated logarithmic diffusion equations can be constructed by appropriate Bochner randomization of the time variable in Brownian motion and the related heat conduction equation. Remarkably, over a large but finite frequency range, generalized Linnik characteristic functions can exhibit almost Gaussian behavior near the origin, while behaving like low exponent isotropic Lévy stable laws away from the origin. Such behavior matches Fourier domain behavior in a large class of real blurred images of considerable scientific interest, including Hubble space telescope imagery and scanning electron micrographs. This paper develops a powerful blind deconvolution procedure based on postulating system optical transfer functions (otfs) in the form of generalized Linnik characteristic functions. The system otf and “true” sharp image are then reconstructed by solving a related logarithmic diffusion equation backward in time, using the blurred image as data at time $t=1$. The present methodology significantly improves upon previous work based on system otfs in the form of Lévy stable characteristic functions. Such improvement is validated by the substantially smaller image Lipschitz exponents that ensue, confirming increased fine structure recovery. These results resolve the unexplained appearance of exceptionally low Lévy stable exponents in previous work on the same class of images. The paper is illustrated with striking enhancements of gray-scale and colored images.

Image Sharpening via Sobolev Gradient Flows

J. Calder, A. Mansouri, and A. Yezzi

SIAM J. Imaging Sci. 3, pp. 981-1014 (34 pages)

Online Publication Date: December 02, 2010

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Motivated by some recent work in active contour applications, we study the use of Sobolev gradients for PDE-based image diffusion and sharpening. We begin by studying, for the case of isotropic diffusion, the gradient descent/ascent equation obtained by modifying the usual metric on the space of images, which is the $L^2$ metric, to a Sobolev metric. We present existence and uniqueness results for the Sobolev isotropic diffusion, derive a number of maximum principles, and show that the differential equations are stable and well-posed both in the forward and backward directions. This allows us to apply the Sobolev flow in the backward direction for sharpening. Favorable comparisons to the well-known shock filter for sharpening are demonstrated. Finally, we continue to exploit this same well-posed behavior both forward and backward in order to formulate new constrained gradient flows on higher order energy functionals which preserve the first order energy of the original image for interesting combined smoothing and sharpening effects.

A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science

Ernie Esser, Xiaoqun Zhang, and Tony F. Chan

SIAM J. Imaging Sci. 3, pp. 1015-1046 (32 pages)

Online Publication Date: December 14, 2010

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We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed by Zhu and Chan in [An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, CAM Report 08-34, UCLA, Los Angeles, CA, 2008] to a broader class of convex optimization problems. In addition, we survey several closely related methods and explain the connections to PDHG. We point out convergence results for a modified version of PDHG that has a similarly good empirical convergence rate for total variation (TV) minimization problems. We also prove a convergence result for PDHG applied to TV denoising with some restrictions on the PDHG step size parameters. We show how to interpret this special case as a projected averaged gradient method applied to the dual functional. We discuss the range of parameters for which these methods can be shown to converge. We also present some numerical comparisons of these algorithms applied to TV denoising, TV deblurring, and constrained $l_1$ minimization problems.
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Special Section on Optimization in Imaging Sciences

Guillermo Sapiro, Editor-in-Chief

SIAM J. Imaging Sci. 3, pp. 1047-1047 (1 page)

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Many image analysis problems can be formulated as estimation problems that require minimization of a discrete or discretized energy function. Commonly used energy functions are based on models in physics, geometry, or statistics. With some exceptions, however, many formulations correspond to NP-hard optimization problems. Recently there has been a surge in research in image sciences where either known or newly developed combinatorial optimization algorithms were successfully applied to a wide spectrum of problems in imaging. There are several aspects of this development. Some of the new combinatorial algorithms could guarantee globally optimal solutions for certain special cases, and several such special classes were identified in image. Furthermore, new optimization methods with proven performance guarantees have been developed and applied to a much wider spectrum of more difficult imaging problems. Imaging science has been enjoying both from the use of advanced optimization techniques and the development of new techniques with contributions beyond imaging problems.
This special section includes a number of representative papers in this topic. The diversity of these papers showcases the large spectrum of optimization topics and challenges currently addressed by the imaging community: These papers also show a unique characteristic of this area in imaging sciences: the combination of deep theoretical contributions with very relevant practical applications. The tools being used for proving the important theoretical results and the actual applications are very broad and diverse as well.
I would like to thank the guest editors—Professor Endre Boros, Professor Yuri Boykov, Professor Jerome Darbon, and Professor Philip Torr—as well as the special section advisor, Professor Andrew Blake, for their work with this section. Special thanks go also to the SIAM staff for the support for this section, as well as to the SIIMS editorial board for guidance.

Global Interactions in Random Field Models: A Potential Function Ensuring Connectedness

Sebastian Nowozin and Christoph H. Lampert

SIAM J. Imaging Sci. 3, pp. 1048-1074 (27 pages)

Online Publication Date: December 21, 2010

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Markov random field (MRF) models, including conditional random field models, are popular in computer vision. However, in order to be computationally tractable, they are limited to incorporating only local interactions and cannot model global properties such as connectedness, which is a potentially useful high-level prior for object segmentation. In this work, we overcome this limitation by deriving a potential function that forces the output labeling to be connected and that can naturally be used in the framework of recent maximum a posteriori (MAP)-MRF linear program (LP) relaxations. Using techniques from polyhedral combinatorics, we show that a provably strong approximation to the MAP solution of the resulting MRF can still be found efficiently by solving a sequence of max-flow problems. The efficiency of the inference procedure also allows us to learn the parameters of an MRF with global connectivity potentials by means of a cutting plane algorithm. We experimentally evaluate our algorithm on both synthetic data and on the challenging image segmentation task of the PASCAL Visual Object Classes 2008 data set. We show that in both cases the addition of a connectedness prior significantly reduces the segmentation error.

Global Optimization for One-Dimensional Structure and Motion Problems

Olof Enqvist, Fredrik Kahl, Carl Olsson, and Kalle Åström

SIAM J. Imaging Sci. 3, pp. 1075-1095 (21 pages)

Online Publication Date: December 21, 2010

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We study geometric reconstruction problems in one-dimensional retina vision. In such problems, the scene is modeled as a two-dimensional plane, and the camera sensor produces one-dimensional images of the scene. Our main contribution is an efficient method for computing the global optimum to the structure and motion problem with respect to the $L_{\infty}$ norm of the reprojection errors. One-dimensional cameras have proven useful in several applications, most prominently for autonomous vehicles, where they are used to provide inexpensive and reliable navigational systems. Previous results on one-dimensional vision are limited to the classification and solving of minimal cases, bundle adjustment for finding local optima, and linear algorithms for algebraic cost functions. In contrast, we present an approach for finding globally optimal solutions with respect to the $L_{\infty}$ norm of the angular reprojection errors. We show how to solve intersection and resection problems as well as the problem of simultaneous localization and mapping (SLAM). The algorithm is robust to use when there are missing data, which means that all points are not necessarily seen in all images. Our approach has been tested on a variety of different scenarios, both real and synthetic. The algorithm shows good performance for intersection and resection and for SLAM with up to five views. For more views the high dimension of the search space tends to give long running times. The experimental section also gives interesting examples showing that for one-dimensional cameras with limited field of view the SLAM problem is often inherently ill-conditioned.

Optimization by Stochastic Continuation

Marc C. Robini and Isabelle E. Magnin

SIAM J. Imaging Sci. 3, pp. 1096-1121 (26 pages)

Online Publication Date: December 21, 2010

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Simulated annealing (SA) and deterministic continuation are well-known generic approaches to global optimization. Deterministic continuation is computationally attractive but produces suboptimal solutions, whereas SA is asymptotically optimal but converges very slowly. In this paper, we introduce a new class of hybrid algorithms which combines the theoretical advantages of SA with the practical advantages of deterministic continuation. We call this class of algorithms stochastic continuation (SC). In a nutshell, SC is a variation of SA in which both the energy function and the communication mechanism are allowed to be time-dependent. We first prove that SC inherits the convergence properties of generalized SA under weak assumptions. Then, we show that SC can be successfully applied to optimization issues raised by the Bayesian approach to signal reconstruction. The considered class of energy functions arises in maximum a posteriori estimation with a Markov random field prior. The associated minimization task is NP-hard and beyond the scope of popular methods such as loopy belief propagation, tree-reweighted message passing, and graph cuts and its extensions. We perform numerical experiments in the context of three-dimensional reconstruction from a very limited number of projections; our results show that SC can substantially outperform both deterministic continuation and SA.

Global Solutions of Variational Models with Convex Regularization

Thomas Pock, Daniel Cremers, Horst Bischof, and Antonin Chambolle

SIAM J. Imaging Sci. 3, pp. 1122-1145 (24 pages)

Online Publication Date: December 21, 2010

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We propose an algorithmic framework for computing global solutions of variational models with convex regularity terms that permit quite arbitrary data terms. While the minimization of variational problems with convex data and regularity terms is straightforward (using, for example, gradient descent), this is no longer trivial for functionals with nonconvex data terms. Using the theoretical framework of calibrations, the original variational problem can be written as the maximum flux of a particular vector field going through the boundary of the subgraph of the unknown function. Upon relaxation this formulation turns the problem into a convex problem, although in a higher dimension. In order to solve this problem, we propose a fast primal-dual algorithm which significantly outperforms existing algorithms. In experimental results we show the application of our method to outlier filtering of range images and disparity estimation in stereo images using a variety of convex regularity terms.
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