Adaptive finite element methods for the solution of inverse problems in optical tomography

Bangerth, Wolfgang; Joshi, Amit

doi:10.1088/0266-5611/24/3/034011

Cited by 82 publications

(44 citation statements)

References 75 publications

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“…In that work, these artifacts are minimized by modeling and treating the approximation error within the Bayesian framework. In the area of FDOT, it has been shown that using adaptively refined meshes results in better accuracy and higher resolution in reconstructed images than that of uniform meshes when computational resources are constrained [18], [19]. In [18], the image reconstruction problem is formulated in a PDE-constrained optimization framework, and the mesh for solving this optimization problem is updated and refined based on the criterion suggested in dual weighted residual framework [20].…”

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confidence: 99%

“…In the area of FDOT, it has been shown that using adaptively refined meshes results in better accuracy and higher resolution in reconstructed images than that of uniform meshes when computational resources are constrained [18], [19]. In [18], the image reconstruction problem is formulated in a PDE-constrained optimization framework, and the mesh for solving this optimization problem is updated and refined based on the criterion suggested in dual weighted residual framework [20]. In [21], an algorithm to identify and resolve intersection of tetrahedral elements was developed to achieve fast and robust parameter mapping between the adaptively refined/derefined meshes of PDE-based forward and inverse problems.…”

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confidence: 99%

“…In [19] this algorithm was utilized for FDOT image reconstruction within a dual adaptive mesh scheme that are independently refined based on a posteriori error estimates. Note that in all of these studies, with the exception of [10], [11], [16], [18], [20], [22], the mesh refinement criterion is consid-ered separately for each problem disregarding its impact on the solution of the other problem.…”

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confidence: 99%

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Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise

Zhou

Yazıcı

2011

IEEE Trans. on Image Process.

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Abstract-In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration and the discretization of the forward and inverse problems. In this paper, based on these error estimates, we develop two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT, with the objective of error reduction in the reconstructed optical images due to discretization while keeping the size of the discretized forward and inverse problems within the allowable limits. We present three-dimensional numerical simulations to demonstrate the improvements in accuracy, resolution and detectability of small heterogeneities in reconstructed images provided by the use of the new adaptive mesh generation algorithms. Finally, we compare our algorithms both analytically and numerically with the existing conventional adaptive mesh generation algorithms.Index Terms-Adaptive meshing algorithms, error analysis, fluorescence diffuse optical tomography.

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Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise

Zhou

Yazıcı

2011

IEEE Trans. on Image Process.

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“…Among different regularization methods, the Tikhonov regularization is one of the most popular regularization methods and has been widely applied in resolving FMT problems [8,9]. It adds an L2-norm constraint of the solution to the original problem, and aims to reduce highfrequency noise in the reconstructed results.…”

Section: Introductionmentioning

confidence: 99%

Fast and robust reconstruction for fluorescence molecular tomography via a sparsity adaptive subspace pursuit method

Ye¹,

Chi²,

Xue³

et al. 2014

Biomed. Opt. Express

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Fluorescence molecular tomography (FMT), as a promising imaging modality, can three-dimensionally locate the specific tumor position in small animals. However, it remains challenging for effective and robust reconstruction of fluorescent probe distribution in animals. In this paper, we present a novel method based on sparsity adaptive subspace pursuit (SASP) for FMT reconstruction. Some innovative strategies including subspace projection, the bottom-up sparsity adaptive approach, and backtracking technique are associated with the SASP method, which guarantees the accuracy, efficiency, and robustness for FMT reconstruction. Three numerical experiments based on a mouse-mimicking heterogeneous phantom have been performed to validate the feasibility of the SASP method. The results show that the proposed SASP method can achieve satisfactory source localization with a bias less than 1mm; the efficiency of the method is much faster than mainstream reconstruction methods; and this approach is robust even under quite ill-posed condition. Furthermore, we have applied this method to an in vivo mouse model, and the results demonstrate the feasibility of the practical FMT application with the SASP method.

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“…In contrast, most attempts at solving inverse problems have favored non-adaptive methods. There has been recently a growing trend of research into the use of adaptive inverse problem solvers (see, e.g., [9,10,11,12,13,14,15]), but there remains a considerable gap between the state of the art forward model solvers and the strategies used in inverse problems. This discrepancy is mainly caused by the difficulties with obtaining and using derivative information in adaptive simulations.…”

Section: Introductionmentioning

confidence: 99%

Space–time adaptive solution of inverse problems with the discrete adjoint method

Alexe

Sandu

2014

Journal of Computational Physics

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Abstract. Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, h/p-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem. Space-time adaptive solution of inverse problems with the discrete adjoint method2

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Adaptive finite element methods for the solution of inverse problems in optical tomography

Cited by 82 publications

References 75 publications

Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise

Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise

Fast and robust reconstruction for fluorescence molecular tomography via a sparsity adaptive subspace pursuit method

Space–time adaptive solution of inverse problems with the discrete adjoint method

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