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Inverse Problems & Imaging

October 2018 , Volume 12 , Issue 5

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Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation
Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib and Said Raghay
2018, 12(5): 1055-1081 doi: 10.3934/ipi.2018044 +[Abstract](202) +[HTML](133) +[PDF](1905.73KB)
Abstract:

This paper proposes a stable numerical implementation of the Navier-Stokes equations for fluid image registration, based on a finite volume scheme. Although fluid registration methods have succeeded in handling large deformations in various applications, they still suffer from perturbed solutions due to the choice of the numerical implementation. Thus, a robust numerical scheme in the optimization step is required to enhance the quality of the registration. A key challenge is the use of a finite volume-based scheme, since we have to deal with a hyperbolic equation type. We propose the classical Patankar scheme based on pressure correction, which is called Semi-Implicit Method for Pressure-Linked Equation (SIMPLE). The performance of the proposed algorithm was tested on magnetic resonance images of the human brain and hands, and compared with the classical implementation of the fluid image registration [13], in which the authors used a successive overrelaxation in the spatial domain with Euler integration in time to handle the nonlinear viscous. The obtained results demonstrate the efficiency of the proposed approach, visually and quantitatively, using the differences between images criteria, PSNR and SSIM measures.

Mitigating the influence of the boundary on PDE-based covariance operators
Yair Daon and Georg Stadler
2018, 12(5): 1083-1102 doi: 10.3934/ipi.2018045 +[Abstract](173) +[HTML](86) +[PDF](1239.96KB)
Abstract:

Gaussian random fields over infinite-dimensional Hilbert spaces require the definition of appropriate covariance operators. The use of elliptic PDE operators to construct covariance operators allows to build on fast PDE solvers for manipulations with the resulting covariance and precision operators. However, PDE operators require a choice of boundary conditions, and this choice can have a strong and usually undesired influence on the Gaussian random field. We propose two techniques that allow to ameliorate these boundary effects for large-scale problems. The first approach combines the elliptic PDE operator with a Robin boundary condition, where a varying Robin coefficient is computed from an optimization problem. The second approach normalizes the pointwise variance by rescaling the covariance operator. These approaches can be used individually or can be combined. We study properties of these approaches, and discuss their computational complexity. The performance of our approaches is studied for random fields defined over simple and complex two- and three-dimensional domains.

Using generalized cross validation to select regularization parameter for total variation regularization problems
You-Wei Wen and Raymond Honfu Chan
2018, 12(5): 1103-1120 doi: 10.3934/ipi.2018046 +[Abstract](263) +[HTML](75) +[PDF](1676.42KB)
Abstract:

The regularization approach is used widely in image restoration problems. The visual quality of the restored image depends highly on the regularization parameter. In this paper, we develop an automatic way to choose a good regularization parameter for total variation (TV) image restoration problems. It is based on the generalized cross validation (GCV) approach and hence no knowledge of noise variance is required. Due to the lack of the closed-form solution of the TV regularization problem, difficulty arises in finding the minimizer of the GCV function directly. We reformulate the TV regularization problem as a minimax problem and then apply a first-order primal-dual method to solve it. The primal subproblem is rearranged so that it becomes a special Tikhonov regularization problem for which the minimizer of the GCV function is readily computable. Hence we can determine the best regularization parameter in each iteration of the primal-dual method. The regularization parameter for the original TV regularization problem is then obtained by an averaging scheme. In essence, our method needs only to solve the TV regulation problem twice: one to determine the regularization parameter and one to restore the image with that parameter. Numerical results show that our method gives near optimal parameter, and excellent performance when compared with other state-of-the-art adaptive image restoration algorithms.

Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations
Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette and Frank Wübbeling
2018, 12(5): 1121-1155 doi: 10.3934/ipi.2018047 +[Abstract](249) +[HTML](199) +[PDF](2945.43KB)
Abstract:

This paper discusses the properties of certain risk estimators that recently regained popularity for choosing regularization parameters in ill-posed problems, in particular for sparsity regularization. They apply Stein's unbiased risk estimator (SURE) to estimate the risk in either the space of the unknown variables or in the data space. We will call the latter PSURE in order to distinguish the two different risk functions. It seems intuitive that SURE is more appropriate for ill-posed problems, since the properties in the data space do not tell much about the quality of the reconstruction. We provide theoretical studies of both approaches for linear Tikhonov regularization in a finite dimensional setting and estimate the quality of the risk estimators, which also leads to asymptotic convergence results as the dimension of the problem tends to infinity. Unlike previous works which studied single realizations of image processing problems with a very low degree of ill-posedness, we are interested in the statistical behaviour of the risk estimators for increasing ill-posedness. Interestingly, our theoretical results indicate that the quality of the SURE risk can deteriorate asymptotically for ill-posed problems, which is confirmed by an extensive numerical study. The latter shows that in many cases the SURE estimator leads to extremely small regularization parameters, which obviously cannot stabilize the reconstruction. Similar but less severe issues with respect to robustness also appear for the PSURE estimator, which in comparison to the rather conservative discrepancy principle leads to the conclusion that regularization parameter choice based on unbiased risk estimation is not a reliable procedure for ill-posed problems. A similar numerical study for sparsity regularization demonstrates that the same issue appears in non-linear variational regularization approaches.

Recovery of seismic wavefields by an lq-norm constrained regularization method
Fengmin Xu and Yanfei Wang
2018, 12(5): 1157-1172 doi: 10.3934/ipi.2018048 +[Abstract](172) +[HTML](117) +[PDF](998.25KB)
Abstract:

Reconstruction of the seismic wavefield from sub-sampled data is an important problem in seismic image processing, this is partly due to limitations of the observations which usually yield incomplete data. In essence, this is an ill-posed inverse problem. To solve the ill-posed problem, different kinds of regularization technique can be applied. In this paper, we consider a novel regularization model, called the \begin{document}$l_2$\end{document}-\begin{document}$l_{q}$\end{document} minimization model, to recover the original geophysical data from the sub-sampled data. Based on the lower bound of the local minimizers of the \begin{document}$l_2$\end{document}-\begin{document}$l_{q}$\end{document} minimization model, a fast convergent iterative algorithm is developed to solve the minimization problem. Numerical results on random signals, synthetic and field seismic data demonstrate that the proposed approach is very robust in solving the ill-posed restoration problem and can greatly improve the quality of wavefield recovery.

On finding a buried obstacle in a layered medium via the time domain enclosure method
Masaru Ikehata and Mishio Kawashita
2018, 12(5): 1173-1198 doi: 10.3934/ipi.2018049 +[Abstract](125) +[HTML](76) +[PDF](469.0KB)
Abstract:

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.

Retinex based on exponent-type total variation scheme
Lu Liu, Zhi-Feng Pang and Yuping Duan
2018, 12(5): 1199-1217 doi: 10.3934/ipi.2018050 +[Abstract](156) +[HTML](69) +[PDF](1598.94KB)
Abstract:

Retinex theory deals with compensation for illumination effects in images, which has a number of applications including Retinex illusion, medical image intensity inhomogeneity and color image shadow effect etc.. Such ill-posed problem has been studied by researchers for decades. However, most exiting methods paid little attention to the noises contained in the images and lost effectiveness when the noises increase. The main aim of this paper is to present a general Retinex model to effectively and robustly restore images degenerated by both illusion and noises. We propose a novel variational model by incorporating appropriate regularization technique for the reflectance component and illumination component accordingly. Although the proposed model is non-convex, we prove the existence of the minimizers theoretically. Furthermore, we design a fast and efficient alternating minimization algorithm for the proposed model, where all subproblems have the closed-form solutions. Applications of the algorithm to various gray images and color images with noises of different distributions yield promising results.

Capped $\ell_p$ approximations for the composite $\ell_0$ regularization problem
Qia Li and Na Zhang
2018, 12(5): 1219-1243 doi: 10.3934/ipi.2018051 +[Abstract](112) +[HTML](74) +[PDF](548.34KB)
Abstract:

The composite \begin{document}$\ell_0$\end{document} function serves as a sparse regularizer in many applications. The algorithmic difficulty caused by the composite \begin{document}$\ell_0$\end{document} regularization (the \begin{document}$\ell_0$\end{document} norm composed with a linear mapping) is usually bypassed through approximating the \begin{document}$\ell_0$\end{document} norm. We consider in this paper capped \begin{document}$\ell_p$\end{document} approximations with \begin{document}$p>0$\end{document} for the composite \begin{document}$\ell_0$\end{document} regularization problem. The capped \begin{document}$\ell_p$\end{document} function converges to the \begin{document}$\ell_0$\end{document} norm pointwisely as the approximation parameter tends to infinity. We first establish the existence of optimal solutions to the composite \begin{document}$\ell_0$\end{document} regularization problem and its capped \begin{document}$\ell_p$\end{document} approximation problem under conditions that the data fitting function is asymptotically level stable and bounded below. Asymptotically level stable functions cover a rich class of data fitting functions encountered in practice. We then prove that the capped \begin{document}$\ell_p$\end{document} problem asymptotically approximates the composite \begin{document}$\ell_0$\end{document} problem if the data fitting function is a level bounded function composed with a linear mapping. We further show that if the data fitting function is the indicator function on an asymptotically linear set or the \begin{document}$\ell_0$\end{document} norm composed with an affine mapping, then the composite \begin{document}$\ell_0$\end{document} problem and its capped \begin{document}$\ell_p$\end{document} approximation problem share the same optimal solution set provided that the approximation parameter is large enough.

Stability estimates in tensor tomography
Jan Boman and Vladimir Sharafutdinov
2018, 12(5): 1245-1262 doi: 10.3934/ipi.2018052 +[Abstract](154) +[HTML](79) +[PDF](452.99KB)
Abstract:

We study the X-ray transform \begin{document}$I$\end{document} of symmetric tensor fields on a smooth convex bounded domain \begin{document}$Ω\subset{\mathbb R}^n$\end{document}. The main result is the stability estimate \begin{document}$\|^{s}f\|_{L^2}≤ C\|If\|_{H^{1/2}}$\end{document}, where \begin{document}$^{s}f$\end{document} is the solenoidal part of the tensor field \begin{document}$f$\end{document}. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.

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