Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging

Erb, Wolfgang; Weinmann, Andreas; Ahlborg, Mandy; Brandt, Christina; Bringout, Gaël; Buzug, Thorsten M.; Frikel, Jürgen; Kaethner, Christian; Knopp, Tobias; März, Thomas; Möddel, Martin; Storath, Martin; Weber, Alexander

doi:10.1088/1361-6420/aab8d1

Cited by 15 publications

(15 citation statements)

References 52 publications

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“…A first step towards a theoretical understanding of this problem, while excluding the temporal dependence, can be found in [39]. A one-dimensional problem setup considering the time-dependency of the measurements was analyzed in [11] for one particular trigonometric excitation. Furthermore, the multi-dimensional imaging problem for different dynamic magnetic field patterns was analyzed in the context of inverse problems regarding the best possible degree of ill-posedness [23].…”

Section: Appendix Iii: Magnetic Particle Imaging (Mpi)mentioning

confidence: 99%

Regularization by Architecture: A Deep Prior Approach for Inverse Problems

et al. 2019

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The present paper studies the so called deep image prior (DIP) technique in the context of inverse problems. DIP networks have been introduced recently for applications in image processing, [50], also first experimental results for applying DIP to inverse problems have been reported [51]. This paper aims at discussing different interpretations of DIP and to obtain analytic results for specific network designs and linear operators. The main contribution is to introduce the idea of viewing these approaches as the optimization of Tiknonov functionals rather than optimizing networks. Besides theoretical results, we present numerical verifications for an academic example (integration operator) as well as for the inverse problem of magnetic particle imaging (MPI). The reconstructions obtained by deep prior networks are compared with state of the art methods.

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Section: Appendix Iii: Magnetic Particle Imaging (Mpi)mentioning

confidence: 99%

Regularization by Architecture: A Deep Prior Approach for Inverse Problems

et al. 2019

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“…MPI is a new biomedical imaging modality that images the distribution of SPIONs. Some researchers adopt mathematical models to describe the physical process, like the equilibrium model [ 25 ] or its variations, the x-space approach [ 3 , 26 , 27 , 28 ], Chebyshev polynomials [ 29 ], and analytic inversion formulas [ 30 , 31 ]. The state-of-the-art reconstruction techniques adopt the experimentally calibrated forward operators, called the SM-based image reconstruction [ 32 , 33 ].…”

Section: The Theory Of Sm-based Mpimentioning

confidence: 99%

The Reconstruction of Magnetic Particle Imaging: Current Approaches Based on the System Matrix

et al. 2021

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Magnetic particle imaging (MPI) is a novel non-invasive molecular imaging technology that images the distribution of superparamagnetic iron oxide nanoparticles (SPIONs). It is not affected by imaging depth, with high sensitivity, high resolution, and no radiation. The MPI reconstruction with high precision and high quality is of enormous practical importance, and many studies have been conducted to improve the reconstruction accuracy and quality. MPI reconstruction based on the system matrix (SM) is an important part of MPI reconstruction. In this review, the principle of MPI, current construction methods of SM and the theory of SM-based MPI are discussed. For SM-based approaches, MPI reconstruction mainly has the following problems: the reconstruction problem is an inverse and ill-posed problem, the complex background signals seriously affect the reconstruction results, the field of view cannot cover the entire object, and the available 3D datasets are of relatively large volume. In this review, we compared and grouped different studies on the above issues, including SM-based MPI reconstruction based on the state-of-the-art Tikhonov regularization, SM-based MPI reconstruction based on the improved methods, SM-based MPI reconstruction methods to subtract the background signal, SM-based MPI reconstruction approaches to expand the spatial coverage, and matrix transformations to accelerate SM-based MPI reconstruction. In addition, the current phantoms and performance indicators used for SM-based reconstruction are listed. Finally, certain research suggestions for MPI reconstruction are proposed, expecting that this review will provide a certain reference for researchers in MPI reconstruction and will promote the future applications of MPI in clinical medicine.

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“…They examine and compare the ill-posedness of the reconstruction problem for different dimensions. A mathematical analysis of the 1D model is provided by Erb, Weinmann et al [7]. They investigate properties like the illposedness and discover an exponential singular value decay of the reconstruction problem.…”

Section: Introductionmentioning

confidence: 99%

“…Fig 7. The three versions of the one-peak phantom differ only in the width of the concentration peak of voxel r 5 , while the remaining voxels have a constant tracer concentration of zero…”

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confidence: 99%

Modeling Magnetic Particle Imaging for Dynamic Tracer Distributions

Brandt

Schmidt

2021

Sens Imaging

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Magnetic Particle Imaging (MPI) is a promising tracer-based, functional medical imaging technique which measures the non-linear magnetization response of magnetic nanoparticles to a dynamic magnetic field. For image reconstruction, system matrices from time-consuming calibration scans are used predominantly. Finding modeled forward operators for magnetic particle imaging, which are able to compete with measured matrices in practice, is an ongoing topic of research. The existing models for magnetic particle imaging are by design not suitable for arbitrary dynamic tracer concentrations. Neither modeled nor measured system matrices account for changes in the concentration during a single scanning cycle. In this paper we present a new MPI forward model for dynamic concentrations. A static model will be introduced briefly, followed by the changes due to the dynamic behavior of the tracer concentration. Furthermore, the relevance of this new extended model is examined by investigating the influence of the extension and example reconstructions with the new and the standard model.

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Mathematical analysis of the 1D model and reconstruction schemes for magnetic particle imaging

Cited by 15 publications

References 52 publications

Regularization by Architecture: A Deep Prior Approach for Inverse Problems

Regularization by Architecture: A Deep Prior Approach for Inverse Problems

The Reconstruction of Magnetic Particle Imaging: Current Approaches Based on the System Matrix

Modeling Magnetic Particle Imaging for Dynamic Tracer Distributions

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