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Inverse Problems and Imaging

August 2017 , Volume 11 , Issue 4

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Landmark-guided elastic shape analysis of human character motions
Martin Bauer, Markus Eslitzbichler and Markus Grasmair
2017, 11(4): 601-621 doi: 10.3934/ipi.2017028 +[Abstract](3276) +[HTML](70) +[PDF](1044.8KB)

Motions of virtual characters in movies or video games are typically generated by recording actors using motion capturing methods. Animations generated this way often need postprocessing, such as improving the periodicity of cyclic animations or generating entirely new motions by interpolation of existing ones. Furthermore, search and classification of recorded motions becomes more and more important as the amount of recorded motion data grows.

In this paper, we will apply methods from shape analysis to the processing of animations. More precisely, we will use the by now classical elastic metric model used in shape matching, and extend it by incorporating additional inexact feature point information, which leads to an improved temporal alignment of different animations.

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials
Liliana Borcea, Fernando Guevara Vasquez and Alexander V. Mamonov
2017, 11(4): 623-641 doi: 10.3934/ipi.2017029 +[Abstract](3636) +[HTML](60) +[PDF](2535.7KB)

We propose a discrete approach for solving an inverse problem for the two-dimensional Schrödinger equation, where the unknown potential is to be determined from the Dirichlet to Neumann map. In the continuum, the problem for absorptive potentials can be transformed with the Liouville identity into a conductivity inverse problem. Its discrete analogue is to find a resistor network matching the measurements, and is well understood. Here we use a discrete Liouville identity to transform its solution to that of Schrödinger's problem. The discrete Schrödinger potential given by the discrete Liouville identity can be used to reconstruct the potential in the continuum in two ways. First, we can obtain a direct but coarse reconstruction by interpreting the values of the discrete Schrödinger potential as averages of the continuum Schrödinger potential on a special sensitivity grid. Second, the discrete Schrödinger potential may be used to reformulate the conventional nonlinear output least squares formulation of the inverse Schrödinger problem. Instead of minimizing the boundary measurement misfit, we minimize the misfit between discrete Schrödinger potentials. This results in a better behaved optimization problem converging in a single Gauss-Newton iteration, and gives good quality reconstructions of the potential, as illustrated by the numerical results.

Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery
Yun Cai and Song Li
2017, 11(4): 643-661 doi: 10.3934/ipi.2017030 +[Abstract](4169) +[HTML](82) +[PDF](506.2KB)

In this paper, we study the theoretical properties of iteratively reweighted least squares algorithm for recovering a matrix (IRLS-M for short) from noisy linear measurements. The IRLS-M was proposed by Fornasier et al. (2011) [17] for solving nuclear norm minimization and by Mohan et al. (2012) [31] for solving Schatten-\begin{document}$p$\end{document} (quasi) norm minimization (\begin{document}$0 < p≤q1$\end{document}) in noiseless case, based on the iteratively reweighted least squares algorithm for sparse signal recovery (IRLS for short) (Daubechies et al., 2010) [15], and numerical experiments have been given to show its efficiency (Fornasier et al. and Mohan et al.) [17], [31]. In this paper, we focus on providing convergence and stability analysis of iteratively reweighted least squares algorithm for low-rank matrix recovery in the presence of noise. The convergence of IRLS-M is proved strictly for all \begin{document}$0 < p≤q1$\end{document}. Furthermore, when the measurement map \begin{document}$\mathcal{A}$\end{document} satisfies the matrix restricted isometry property (M-RIP for short), we show that the IRLS-M is stable for \begin{document}$0 < p≤q1$\end{document}. Specially, when \begin{document}$p=1$\end{document}, we prove that the M-RIP constant \begin{document}$δ_{2r} < \sqrt{2}-1$\end{document} is sufficient for IRLS-M to recover an unknown (approximately) low rank matrix with an error that is proportional to the noise level. The simplicity of IRLS-M, along with the theoretical guarantees provided in this paper, make a compelling case for its adoption as a standard tool for low rank matrix recovery.

On the lifting of deterministic convergence rates for inverse problems with stochastic noise
Daniel Gerth, Andreas Hofinger and Ronny Ramlau
2017, 11(4): 663-687 doi: 10.3934/ipi.2017031 +[Abstract](3193) +[HTML](61) +[PDF](489.7KB)

Both for the theoretical and practical treatment of Inverse Problems, the modeling of the noise is crucial. One either models the measurement via a deterministic worst-case error assumption or assumes a certain stochastic behavior of the noise. Although some connections between both models are known, the communities develop rather independently. In this paper we seek to bridge the gap between the deterministic and the stochastic approach and show convergence and convergence rates for Inverse Problems with stochastic noise by lifting the theory established in the deterministic setting into the stochastic one. This opens the wide field of deterministic regularization methods for stochastic problems without having to do an individual stochastic analysis for each problem.

Non-convex TV denoising corrupted by impulse noise
Yoon Mo Jung, Taeuk Jeong and Sangwoon Yun
2017, 11(4): 689-702 doi: 10.3934/ipi.2017032 +[Abstract](4250) +[HTML](90) +[PDF](1157.5KB)

We propose a non-convex type total variation model for impulse noise removal by incorporating TV and the quasi-norm \begin{document}$\ell_q $\end{document}, \begin{document}$0 < q < 1 $\end{document}. Since the proposed model is non-convex and non-smooth, an iteratively reweighted algorithm is adapted and combined with a linearized ADMM. The convergence of the proposed algorithm is established and numerical results are given to illustrate the validity and efficiency of the proposed model.

Convergence of the gradient method for ill-posed problems
Stefan Kindermann
2017, 11(4): 703-720 doi: 10.3934/ipi.2017033 +[Abstract](3482) +[HTML](76) +[PDF](419.6KB)

We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for nonlinear operator equations is a special instance of this framework, and the gradient method then reduces to Landweber iteration. The main result of this article is a proof of weak and strong convergence under new nonlinearity conditions that generalize the classical tangential cone conditions.

Numerical optimization algorithms for wavefront phase retrieval from multiple measurements
Ji Li and Tie Zhou
2017, 11(4): 721-743 doi: 10.3934/ipi.2017034 +[Abstract](3305) +[HTML](65) +[PDF](2467.5KB)

Wavefront phase retrieval from a set of intensity measurements can be formulated as an optimization problem. Two nonconvex models (MLP and its variant LS) based on maximum likelihood estimation are investigated in this paper. We derive numerical optimization algorithms for real-valued function of complex variables and apply them to solve the wavefront phase retrieval problem efficiently. Numerical simulation is given with application to three test examples. The LS model shows better numerical performance than that of the MLP model. An explanation for this is that the distribution of the eigenvalues of Hessian matrix of the LS model is more clustered than that of the MLP model. We find that the LBFGS method shows more robust performance and takes fewer calculations than other line search methods for this problem.

Increasing stability for the inverse source scattering problem with multi-frequencies
Peijun Li and Ganghua Yuan
2017, 11(4): 745-759 doi: 10.3934/ipi.2017035 +[Abstract](4090) +[HTML](72) +[PDF](364.0KB)

Consider the scattering of the two-or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source scattering problem which is to reconstruct the source function. Our results show that increasing stability can be obtained for the inverse problem by using only the Dirichlet boundary data with multi-frequencies.

On a spatial-temporal decomposition of optical flow
Aniello Raffaele Patrone and Otmar Scherzer
2017, 11(4): 761-781 doi: 10.3934/ipi.2017036 +[Abstract](4547) +[HTML](69) +[PDF](1655.8KB)

In this paper we present a decomposition algorithm for computation of the spatial-temporal optical flow of a dynamic image sequence. We consider several applications, such as the extraction of temporal motion features and motion detection in dynamic sequences under varying illumination conditions, such as they appear for instance in psychological flickering experiments. For the numerical implementation we are solving an integro-differential equation by a fixed point iteration. For comparison purposes we use a standard time dependent optical flow algorithm, which in contrast to our method, constitutes in solving a spatial-temporal differential equation.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6




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