 Review
 Open Access
 Published:
Recent developments of the reconstruction in magnetic particle imaging
Visual Computing for Industry, Biomedicine, and Art volume 5, Article number: 24 (2022)
Abstract
Magnetic particle imaging (MPI) is an emerging molecular imaging technique with high sensitivity and temporalspatial resolution. Image reconstruction is an important research topic in MPI, which converts an induced voltage signal into the image of superparamagnetic iron oxide particles concentration distribution. MPI reconstruction primarily involves system matrix and xspacebased methods. In this review, we provide a detailed overview of the research status and future research trends of these two methods. In addition, we review the application of deep learning methods in MPI reconstruction and the current open sources of MPI. Finally, research opinions on MPI reconstruction are presented. We hope this review promotes the use of MPI in clinical applications.
Introduction
Magnetic particle imaging (MPI) is a novel molecular imaging modality that can image superparamagnetic iron oxide particles (SPIOs) with high temporalspatial resolution and sensitivity in a noninvasive manner [1,2,3,4]. In 2005,Gleich and Weizenecker [1] first proposed the physical principles and wide application prospects of MPI. The imaging principle of MPI is related to the nonlinear magnetization response of SPIOs in a magnetic field. First, a gradient field generated by a permanent magnet or energized coil is used to create a fieldfree region (FFR). Subsequently, by superimposing an oscillating drive field, the FFR moves through the field of view (FOV) [5, 6]. The excitation field influences SPIOs near the FFR to undergo a nonlinear magnetization response to generate a voltage signal in the receiving coils (Fig. 1 A). When SPIOs enter the magneticfield saturation region, SPIOs are not magnetized to produce voltage signals (Fig. 1 B). The concentration distribution of the SPIOs can be further obtained by analyzing the reconstruction algorithm from the signal to the image [7, 8].
Compared with current powerful imaging techniques, MPI has significant advantages. Computed tomography (CT) and magnetic resonance imaging (MRI) can achieve a submillimeter spatial resolution, but their sensitivity is limited, resulting in poor specificity at the molecular level [4]. Positron emission tomography (PET) and singlephoton emission computed tomography (SPECT) have high sensitivity but a spatial resolution of approximately 3 mm. In addition, these applications are limited by the short halflife of radioactive tracers in cell tracking or other several research fields that require long observations [9]. Optical imaging technology can achieve a high sensitivity and spatialresolution imaging, but its imaging depth is limited to 23 cm [10, 11]. MPI is expected to overcome the limitations of molecular imaging technology in terms of imaging depth, sensitivity, resolution, and radiation, thereby becoming a new trend in the development of highend medical imaging that represents the international academic frontier of the development of modern medical imaging [4, 12,13,14]. Table 1 presents quantitative comparisons between different imaging modalities.
As a new molecular imaging technology, MPI plays an increasingly important role in many preclinical biomedical studies. MPI has been proven to achieve a highly sensitive detection of approximately 250 cells in vivo in cell tracking studies [15]. MPI has provided the highest imaging sensitivity in multimodality dynamic observations of gliomas [16, 17]. MPI can also detect and visualize the homing of breast tumor cells with high sensitivity [18]. MPI has also been successfully applied in cardiovascular and cerebrovascular imaging [19, 20] and neuroimaging [21]. These advances fully demonstrate the imaging advantages of the high sensitivity and specificity of MPI, demonstrating the great potential and value of this new molecular imaging technology.
In recent years, the imaging theory and instruments for MPI have been continuously improved and developed. In 2008, instead of moving the object mechanically in a horizontal direction, Gleich et al. [22] achieved fast twodimensional (2D) imaging by superposing a horizontal drive field. In the same year, they proposed a fieldfree line (FFL), which is an effective encoding scheme for MPI. The results of the simulations demonstrated an obvious improvement in image quality compared with the existing fieldfree point (FFP) [23,24,25,26]. Subsequently, novel singlesided MPI equipment was proposed by Sattel et al. [27], which overcame the limitations of specimen size. In 2009, they achieved the first threedimensional (3D) realtime in vivo MPI [2]. In 2013, the world’s first commercial MPI scanner (based on FFP) was released by Bruker Biospin. In 2014, the Magnetic Insight company from the United States unveiled the world’s second commercial MPI scanner (based on FFL) [4]. In 2015, a superspeed traveling wave MPI scanner was proposed to realize superspeed imaging of magnetic particle flows [28]. In 2016, Franke et al. [29] designed a hybrid system that combined MRI and MPI. In 2019, Gräser et al. [30] designed an MPI scanner with the size of the human brain. In 2020, scientists from Turkey proposed an opensided MPI scanner system based on the FFL scheme [31]. In 2021, a novel handheld MPI scanner was designed for the intraoperative imaging of breast nodules [32]. Figure 2 shows the development timeline of MPI.
With the continuous development of MPI scanners, abundant research on MPI reconstruction has been conducted to enhance the imaging performance of the instruments. MPI reconstruction algorithms are primarily of two types of methods: system matrix (SM) and xspace. SMbased MPI reconstruction is an important field [33,34,35]. The acquisition of SMs in SMbased MPI reconstructions is an important research topic. Currently, three methods have been reported for obtaining SMs: measurementbased SM, sparse recovery SM, and modelbased SM. Owing to the complex magnetic environment and magnetization behavior of SPIOs, the development of an accurate physical model remains a challenging problem. Therefore, the first reported, and the most accurate method, is measurementbased SM acquisition, which involves moving a delta sample with a robot at all positions of the entire FOV [1, 36]. Despite its high accuracy, the measurement process is time consuming. In addition, the measured SM requires a lot of memory, which leads to inefficient reconstruction. Therefore, sparse recovery methods and modelbased SM acquisition have been proposed [37, 38]. In addition to the SM acquisition method, a reconstruction strategy based on SM is an important research topic. Further details regarding this are covered later in this review.
Xspace methods are another important component of MPI reconstruction [39,40,41,42]. This study reviews the basic theory and improved algorithms developed in recent years. In addition, with the current development of artificial intelligence (AI), strategies based on deep learning networks have also shined in the field of MPI reconstruction. We review the relevant research in detail. Furthermore, there is an increasing number of MPI research groups around the world, as well as many opensource data, programs, and software platforms, which are also discussed in this review. Finally, the conclusions and outlook of MPI are presented.
SMbased MPI reconstruction
The objective of MPI reconstruction is to transform the induced voltage signal into a spatial concentration distribution of the SPIOs. A SM is a mapping of the two items, considering the complex magnetic field and SPIOs properties [7, 36, 43]. The linear mapping between the induced voltage signal \(u_k\) ( Fourier coefficients of the timedomain signal u(t)) and SPIO concentration \(c(\mathbf {r})\) can be described as follows:
where \(\mathbf {r}\) represents the spatial position, and \(s_k(\mathbf {r})\) denotes the system function. K denotes the total number of frequency components. By sampling all N positions of the FOV, we obtain the following linear relationship:
where \(\mathbf {S}\in \mathbb {C}^{K\times N}\) denotes the SM. \(\mathbf {u} \in \mathbb {C}^{K\times 1}\) and \(\mathbf {c} \in \mathbb {R}^{N\times 1}\) denote the voltage and SPIOs concentration vector, respectively.
SMbased reconstruction plays an important role in MPI reconstruction. This section describes the commonly used SM acquisition methods and reconstruction strategies based on SM.
Acquiring the SM
Measurementbased methods
A tedious calibration procedure must be performed for the measurementbased SM acquisition method. Figure 3 shows the calibration procedure for the measurementbased method. It measures the induced voltage signal at all positions in the FOV with a delta sample to obtain the SM. Even a mediumsized image of 34\(\times\)28\(\times\)20 can require up to six hours to measure the SM [2, 37]. Therefore, a highquality image requires a timeconsuming and tedious calibration procedure. In addition, when the experimental conditions change, such as changes in particle species and magnetic field intensity, the SM must be recalibrated. In addition, cooling is required during calibration because the coils overheat after prolonged system operation [44].
Sparse recovery methods
Owing to the complexity and time requirements of the measurementbased SM acquisition method, many studies have proposed sparse recovery methods to obtain a complete SM using only a subset of all calibration scans. In 2013, Knopp and Weber [37] proposed using the sparsity of SMs based on certain basis transformations (such as the discrete Fourier transform, discrete cosine transform or discrete Chebyshev transform) for compressed sensing (CS) reconstruction to significantly reduce the number of calibration scans. Each row of the SM is handled separately. Taking the kth row as an example, an inverse problem must be solved:
where \(\mathbf {\Phi }\in \mathbb {C}^{N\times N}\) denotes a basis transformation matrix, \(\mathbf {x}\in \mathbb {C}^{N\times 1}\) denotes the optimization results, and \(\mathbf {P}\) represents the undersampled matrix, which denotes the position index of the points sampled from the kth row \(\mathbf {s}_k\). \(\mathbf {y}_k\) denotes the undersampled measurement: \(\mathbf {y}_k=\mathbf {P}\mathbf {s}_k\). Several algorithms have been proposed to solve this inverse problem [37]. The sparse recovery method based on the standard CS can reduce the sampling rate by 10% [37].
Based on CS theory, many improved sparse recovery algorithms have also been extended. Weber and Knopp [45] proposed the use of SM symmetry to further compress SMs (Fig. 4). Grosser et al. [44] proposed the use of lowrank tensors to represent SMs, which allows SM recovery even when the sampling rate is reduced to 2%. Ilbey et al. [46] presented a framework for a coded calibration scene. Compared with the standard CS method, the image quality is significantly improved under the same signalnoise ratio (SNR) and sampling rate [46]. Sparse recovery algorithms can significantly reduce the calibration time and guarantee the quality of image reconstruction, which has attracted increasing attention in the field of MPI reconstruction.
Modelbased methods
In addition to the measurementbased and sparse recovery methods, Knopp et al. [38] proposed a modelbased method in 2010. They used the model of a signal chain to generate SMs. This approach consisted of three steps.
1) Signal encoding: the principle of the signal chain is used to model signal encoding. In MPI, the relationship between the timedomain induced voltage u(t) in the receiving coil and SPIO concentration \(c(\mathbf {r})\) is as follows:
where \(\mu _0\) represents the permeability of a vacuum, and \(\mathbf {M}(\mathbf {r},t)\) denotes the unit magnetization of the SPIOs. \(\mathbf {p}(\mathbf {r})\) is the sensitivity profile of the recording coil. Because filtering is easier in the frequency domain, processing is generally performed in the frequency domain to increase the accuracy of the modeledinduced voltage:
where \(a_k\) is the transfer function. Comparing Eqs. (1), (4), and (5), the modelbased system function is obtained as follows:
2) Particle model: Langevin is the most commonly used model for describing the magnetization of SPIOs:
with
where \(K_B\) and T are the Boltzmann constant and temperature, respectively. m denotes the magnetic moment, and \(\mathbf {e_H}=\mathbf {H}(\mathbf {r},t)/\Vert \mathbf {H}(\mathbf {r},t)\Vert _2\) denotes the direction of the magnetic field strength. D represents the diameters of the SPIOs.
3) Transfer function: \(a_k\) can be obtained by minimizing the following equation:
where k denotes the frequency index, \(s_k^{meas}\) denotes the measured system function, and \(\hat{s}_k^{model}=s_k^{model}/a_k\). \(a_k\) can be computed as follows.
In principle, a measured system function with a single position is sufficient for obtaining \(a_k\). This approach demonstrates the feasibility of modeling instead of measuring SMs [38]. Subsequently, Knopp et al. [47] verified the accuracy of a modelbased SM acquisition method on 2D data. Modelbased SM acquisition methods can accurately model the system function based on a few calibration scans, and the accuracy can be increased by improving the underlying physical model.
SMbased MPI reconstruction based on the regularization strategy
After obtaining the SM, the concentration distribution of the SPIOs can be obtained by solving Eq. (2) using a series of optimization algorithms. Owing to the illposedness of the inverse problem, a regularization strategy was introduced to improve the accuracy of image reconstruction. Tikhonov regularization is the most commonly used method and is widely used in MPI reconstruction owing to its fast and simple implementation [3]:
where \(\lambda\) denotes the regularization parameter that penalizes the solution with a large Euclidean norm. \(\mathbf {W}\) denotes the weighting matrix used to normalize the elements of the SM.
Both direct and iterative methods are used to solve the Tikhonov regularization. Examples of direct methods are Cholesky decomposition or singular value decomposition (SVD) [48, 49]. Direct methods have several advantages. For example, SVD can flexibly tune the regularization parameters with only a small amount of computational effort. However, the complete SM must typically be stored in memory while solving Eq. (11), which is unfeasible for a huge SM of 3D MPI.
Therefore, iterative methods have emerged with a high demand in MPI, leading to less memory and computational effort in several cases. Popular iterative methods include the conjugate gradient method [48] and Kaczmarz method [50, 51]. The Kaczmarz method is a rowaction method that operates on rows independently. Convergence has been reported to occur within ten iterations of the Kaczmarz method [52]. Therefore, the stateoftheart method for SMbased reconstruction is the Tikhonov regularization solved by the Kaczmarz method [53]. In recent years, many Kaczmarzrelated Tikhonov regularization solutions have emerged in the field of MPI reconstruction [3, 50, 54].
Although Tikhonov regularization is simple and easy to implement, it does not use the inherent spatial neighborhood structure priori to improve the image quality. Therefore, to introduce the neighborhood structure,Storath et al. [55] proposed nonnegative fused LASSO regularization and achieved edgepreserving and noisereducing reconstruction for MPI. The form of the fused LASSO regularization is as follows:
where \(TV(\mathbf {c})\) denotes the total variation that promotes a sparse edge. \(\Vert \mathbf {c}\Vert _1\) represents the \(L_1\) norm. \(\alpha\) and \(\beta\) are regularization parameters. Storath et al. [55] also introduced nonnegativity to model a (12) and defined it as nonnegative fused LASSO regularization. Compared with Tikhonov regularization, nonnegative fused LASSO regularization is more robust to Gaussian noise. In addition, the reconstructed images showed sharp boundaries and uniformity. Regularization technology plays an important role in the field of SMbased MPI reconstruction and is an important approach to further improve image quality.
Xspace methods for MPI reconstruction
Xspace methods are another important research topic for MPI reconstruction [39]. Xspace and SMbased reconstruction differ significantly. Specifically, xspace can achieve a realtime reconstruction by dividing the velocity of the FFR based on the linear shiftinvariant (LSI) system.
Mathematically, the theory of onedimensional (1D) xspace in MPI can be described as follows:
where \(\dot{\mathbf {r}}_s(t)\) denotes the velocity of the FFR (In this review, we consider FFP as an example.). \(\mathbf {p(r)}\) is the sensitivity of the receiver coil, and G denotes the gradient strength. \(h(r)=\frac{d\mathbf {M}}{d\mathbf {H}}\) represents the point spread function (PSF), which is the derivative of the Langevin function:
The raw MPI image was obtained using a simple twostep velocitycompensation process. MPI reconstruction is performed by gridding the induced voltage signal to the velocity of the FFP:
where \(\hat{c}(\mathbf {r})\) denotes the raw MPI image, which can be described as the convolution of the real MPI image \(c(\mathbf {r})\) with a PSF.
The MPI voltage signal can be regarded as a sampling operation in the raw MPI image \(\hat{c}(\mathbf {r})\) at the instantaneous position of the FFP. Therefore, \(\hat{c}(\mathbf {r})\) can be reconstructed by the velocity compensation and meshing of the MPI voltage signal based on the position of the FFP.
In 2011, Goodwill and Conolly [56] extended 1D xspace theory to 2D and 3D xspace reconstruction. They proved that 3D MPI is also an LSI imaging process and derived the 3D PSF of MPI. Furthermore, they verified the feasibility of the 3D xspace method on a theoretical basis and conducted 2D MPI verification based on an xspace MPI scanner. However, the quality of reconstructed images must be improved. A year later, Goodwill et al. [57] proposed the projection xspace MPI theory based on FFL scanning. They used permanent magnets to generate the FFL for the first time and provided the first FFLbased MPI images. Fast 3D MPI scans were performed using a FFLbased projection xspace to further improve image quality and sharpness. However, classical xspace methods ignore the relaxation effects of SPIOs and assume SPIOs are instantaneously magnetized with the magnetic field, possibly diminishing the MPI signal and causing artifacts in the reconstructed images [58, 59].
Xspace methods based on relaxation models
Classical xspace methods use the Langevin model under the adiabatic hypothesis to describe the magnetization process of SPIOs:
However, in practice, SPIOs cannot operate in an ideal adiabatic environment. In a nonadiabatic environment, SPIOs change with the magnetization delay, which is often referred to as relaxation effects or relaxation time. Therefore, errors occur when using the Langevin model to describe the magnetization process of SPIOs under adiabatic assumptions, leading to blurred and inaccurate xspace reconstructed images (Fig. 5 A).
Several research groups have proposed improved xspace methods based on relaxation models to account for the relaxation effects and accurately describe the magnetization process. Croft et al. [60] theoretically and experimentally demonstrated the blurring of xspace images and deterioration of resolution and SNR caused by relaxation effects. Furthermore, they proposed a simple relaxation model called the Debye model to achieve a desirable signal strength and resolution (Fig. 5 B). A differential equation in the Debye model is as follows:
where \(\mathbf {M}_D(\mathbf {r},t)\) denotes the Debye model that describes nonadiabatic magnetization. \(\tau\) denotes the relaxationtime constant. Furthermore, by solving this differential equation, the following temporal convolution formulation can be obtained:
where p(t) denotes the Heaviside function, with \(\tau >0\). Nonadiabatic magnetization can be described as the temporal convolution of the adiabatic magnetization and convolution kernel q(t) (q(t) denotes the relaxation process). Correspondingly, the nonadiabatic raw image is described as the temporal convolution of the adiabatic raw image and convolution kernel:
\(\hat{c}_{L}\) denotes a raw image based on the Langevin model (ideal adiabatic conditions). The nonadiabatic Xspace method based on the Debye relaxation model showed excellent consistency with the measured signals from the Berkeley MPI scanner [60].
The relaxation effect is caused by Néel and Brownian rotations of the SPIOs. In MPI, two rotations are coupled, and both rotations have their own relaxation times. However, the Debye model uses a relaxation time constant \(\tau\) to represent all rotations, which cannot describe the dynamic magnetization process or rotation mechanism in a complete and accurate manner [61]. Therefore, in recent years, many theoretical models have been developed to accurately describe the relaxation effects of SPIOs. Löwa et al. [62] proposed that the magnetization behavior of SPIOs should be quantified as a function of the excitation magnetic field, properties of SPIOs, and surrounding environment. At present, the LandauLifshitzGilbert model is the most comprehensive model for exploring the relaxation effect [63]. However, an obvious error is observed between the simulation and measurement results, and significant computational power is required to generate the solution [61]. The FokkerPlank model explores the dependence of the relaxation time on the excitation field [64]. However, the solutions did not couple the Néel and Brownian rotations [65, 66]. The complex relaxation effect has an important influence on the reconstruction results of the xspace; therefore, conducting relevant research is crucial.
AI methods for MPI reconstruction
The development of deep learning technology based on AI has inspired new approaches to MPI reconstruction. Currently, AI is primarily applied in MPI reconstruction [67, 68], SM recovery [69], and image postprocessing [70].
In 2017, Chae [67] proposed the reconstruction of MPI images using a singlelayer fully connected network (Fig. 6 A). The input of the network is the voltage \(\mathbf {u}\) in the frequency domain, and the output is the concentration distribution of the SPIOs \(\mathbf {c}\): \(\mathbf {c}=\delta (\mathbf {W}\mathbf {u})\). \(\delta\) is the activation function, and \(\mathbf {W}\) represents the weighting matrix. The author constructed 30000 training datasets and 1000 test sets using 1D simulation. The test images were all reconstructed well by the trained network. The results showed that a neural network structure is expected to be a better tool for MPI image reconstruction because of its capacity to overcome the low incoherence of the inverse kernel through the classification property. Subsequently, Dittmer et al. [68] proposed a novel reconstruction framework based on a deep image prior (DIP). The basic idea of DIP is to use an untrained neural network that implicitly encodes a priori to perform a reliable reconstruction [71]. Compared with traditional variational and iterative regularization, it has significant advantages in terms of image quality [72].
The SM recovery problem was transformed into a superresolution reconstruction problem for deep learning approaches. Baltruschat et al. [69] proposed the use of a 3DSM recovery network (3DSMRnet) to recover a 3D SM with an undersampling rate of less than 1.6% (Fig. 6 B). Their results showed that 3DSMRnet is superior to the CS method in terms of SM recovery quality, reconstructed image quality, and performing time. In addition, Güngör et al. [73] proposed an SM superresolution method based on transformer network architecture (TranSMS) (Fig. 6 C). Compared with stateoftheart CS and deep learning baselines, TranSMS enables lowerror SM recovery and highquality MPI image reconstruction.
The advantage of deep learning technology in the field of image processing also inspired MPI image postprocessing. Shang et al. [70] designed an endtoend dualsampling convolutional neural network to improve the spatial resolution of MPI (FDSMPI) (Fig. 6 D). The results of the simulation, phantom, and in vivo experiments demonstrated the advantages of FDSMPI in improving MPI image resolution. Deep learning technology plays an important role in signal and image processing. It has been applied to many medical modalities such as CT, MRI, and ultrasound. This will definitely shine in benefit MPI in the future.
Current open sources for MPI
Many countries, such as Germany, United States, Turkey, China, Japan, and South Korea, have conducted research on MPI. More importantly, many groups have created open sources that allow novice groups to conduct MPIrelated research as quickly as possible. In this section, we summarize widely used open sources.
Knopp et al. [72] published an article about Open MPI data in 2020, which provides experimental MPI and related data processing code for free. This benefits research groups without MPI systems to understand and analyze MPI experimental data. This dataset is stored in the MDF format, which is an opendocument standard format. A preclinical MPI scanner (Bruker, Ettlingen) was used to acquire the MPI data, including three different phantoms and three different imaging sequences, as shown in Fig. 7. The CAD data of the phantoms are also provided. Furthermore, the Open MPI data also contain the SM based on the calibrations for different mesh densities (Fig. 7 B). Several groups have conducted research on image reconstruction, SM recovery, and other aspects based on Open MPI data [44, 68, 69].
In addition, Tian’s team from the Chinese Academy of Sciences has conducted several MPI studies. They set up an MPI website (MPILabs, http://mpilab.net/en/simulation) that exhibits relevant academic research, opensource experimental data, an algorithm framework, and a software platform. Notably, the software platform MPIRF developed in Python 3.8 integrates SMbased and xspace algorithms to realize MPI reconstruction from the voltage signal of SPIOs to the image [74].
Another public MPI site, openSource MPI (https://osmpi.github.io), is offered by Harvard Medical School and Massachusetts Institute of Technology. It publishes the MPI system design and basic components, as well as the associated auxiliary simulation algorithms. These MPI open sources have greatly promoted the development of MPI worldwide.
Conclusions and outlook
As a novel imaging technology, MPI has wide research and application prospects. This study primarily reviews MPI reconstruction techniques, the SMbased and xspace methods. For these, we provide a detailed review of the SM acquisition and reconstruction methods based on a regularization strategy. In addition, the xspace reconstruction theory and corresponding improved algorithms based on the relaxation model were described. We also reviewed recent deep learning applications in MPI reconstruction and current open sources of MPI.
This review demonstrates the multifaceted nature of MPI. The main challenge for SMbased reconstruction in the future may involve obtaining the SM more quickly and accurately through modeling, measurement, or simulation; this requires a detailed understanding of the SM. In addition, the introduction of prior knowledge and deep learning technology to further improve the reconstruction performance of SMbased reconstruction remains a research focus. In terms of xspace methods, establishing a more accurate magnetization model to describe the relaxation or hysteresis effect is an important challenge to overcome. Furthermore, multicontrast or multicolor reconstruction based on the specific physical properties of SPIOs, as well as sequence development and corresponding image reconstruction based on novel hardware design, are important challenges and interesting hot spots.
In the past 20 years, because of the tireless efforts of scientists worldwide, MPI has progressed considerably in equipment development, magnetic particle preparation, and reconstruction algorithm research, which have further promoted the application of MPI in clinical settings. We expect this review to provide references for MPI researchers and promote the development and future clinical applications of MPI.
Availability of data and materials
All data generated or analyzed in this study are included in the published article [and its supplementary information files].
Abbreviations
 MPI:

Magnetic particle imaging
 SPIO:

Superparamagnetic iron oxide particle
 FFR:

Fieldfree region
 FOV:

Field of view
 CT:

Computed tomography
 MRI:

Magnetic resonance imaging
 PET:

Positron emission tomography
 SPECT:

Singlephoton emission computed tomography
 FFL:

Fieldfree line
 FFP:

Fieldfree point
 3D:

Threedimensional
 SM:

System matrix
 CS:

Compressed sensing
 SNR:

Signalnoise ratio
 SVD:

Singular value decomposition
 LSI:

Linear shiftinvariant
 PVT:

Paraventricular thalamic nucleus
 1D:

Onedimensional
 2D:

Twodimensional
 AI:

Artificial intelligence
 DIP:

Deep image prior
 3DSMRnet:

3DSM recovery network
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This work was supported in part by the National Key Research and Development Program of China, Nos. 2017YFA0700401 and 2017YFA0205200; the National Natural Science Foundation of China, Nos. 62027901, 81827808, 81527805, and 81671851; the CAS Youth Innovation Promotion Association, No. 2018167; the CAS Key Technology Talent Program; and the Project of HighLevel Talents Team Introduction in Zhuhai City, No. Zhuhai HLHPTP201703.
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Yin, L., Li, W., Du, Y. et al. Recent developments of the reconstruction in magnetic particle imaging. Vis. Comput. Ind. Biomed. Art 5, 24 (2022). https://doi.org/10.1186/s42492022001205
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DOI: https://doi.org/10.1186/s42492022001205
Keywords
 Magnetic particle imaging
 Image reconstruction
 System matrix
 Xspace