Multi-Harmonic Gridded 3D Deconvolution (MH3D) for Robust and Accurate Image Reconstruction in MPI for Single Axis Drive Field Scanners

TL;DR

The paper addresses robust 3D MPI image reconstruction by introducing MH3D, a principled forward model and multi-harmonic deconvolution in the harmonic portrait domain. By gridding time-domain data into harmonic portraits and modeling each portrait as a convolution with a distinct PSF derived from Langevin derivatives, MH3D enables efficient, physics-informed 3D reconstructions with calibration and artifact-control built in. The approach demonstrates comparable or improved image quality relative to generalized model-based methods and single-harmonic reconstructions, with seconds-scale runtimes and practical benefits for data analysis and hardware debugging. The work also provides theoretical insights into the information content of MPI harmonics and practical strategies for downsampling, padding, phase calibration, and regularization, advancing robust 3D MPI imaging toward clinical translation.

Abstract

Objective: This work introduces a new magnetic particle imaging (MPI) reconstruction framework based on multi-harmonic 3D deconvolution (MH3D) of gridded portraits, offering a principled, model-driven approach to MPI imaging. Approach: MH3D defines a convolutional forward model using higher harmonic portraits, which are gridded images formed from filtered frequency-domain signal components. Each harmonic portrait is modeled as a convolution with a distinct PSF, closely approximated by derivatives of the Langevin function, and incorporates receive sensitivity and mesh downsampling for accurate modeling. We also introduce practical strategies for calibration, phase correction, and artifact reduction. Main Results: We validate the MH3D approach using analytic approximations, numerical simulations, and experimental phantom data. MH3D yields high-resolution 3D reconstructions on seconds-scale runtimes, improves image quality relative to common 3rd-harmonic-only reconstructions, and achieves image quality and resolution comparable to a generalized model-based method in simulations and phantom experiments. Significance: This work offers new theoretical insight into MPI signal structure, unveiling the methodological and theoretical underpinnings absent in earlier single-harmonic or heuristic methods, thereby supporting accurate and robust 3D imaging with excellent computational efficiency.

Figures (12)

• Figure 1: (a) and (b) show the 2nd and 3rd harmonic portraits at 4 consecutive $z$-slabs for a spiral phantom (shown in the bottom right of (b)) located in a plane at $z=0$. (c) demonstrates the FFP scanning trajectory for a single z-slab.
• Figure 2: MH3D reconstructions of an experimental data set with a phantom containing an array of small vials of equal SPION volume (80 $\mu$g per vial) across the entire FOV, which is intended to test the linearity and shift invariance of the MPI scanner and reconstruction. (a) CAD model of the test phantom. (b) Cross-sectional images of the reconstruction. (c)-(d) 3D volume renderings of the reconstruction.
• Figure 3: Comparison between model-based reconstruction sanders2025physics and MH3D for a simulated phantom. Minor differences arise from solver details and parameter tuning. The red arrow indicates some mild anisotropic blurring in the reconstruction inherently due to a $z$-axis only transmit. The blue arrow indicates where the circle passes through the $z=-2$ cm plane and appears as two dots. The bottom right images show the 3rd and 4th harmonic portraits at $z=-1$ cm. The top right image shows a 3D visualization of the simulated phantom.
• Figure 4: Illustration of harmonic PSF shapes. The right panels show harmonic portrait images from a 2D FFL Momentum (Magnetic Insight, Alameda, CA) scan with a z-axis drive and receive chain at 20 mT drive amplitude. The top row contains experimental measurements of a point source, while the middle and bottom rows compare these data with simulated and theoretical PSFs, showing good agreement across harmonics 2–4. The top left panel shows a 3D rendering of the second-harmonic PSF from our 3D FFP scanner, where red and blue denote positive and negative values, respectively. The bottom left panels plot 1D PSF cross-sections at $x=y=0$ for different drive amplitudes, alongside approximations based on derivatives of the Langevin function. All curves are scaled by their respective maxima.
• Figure 5: Imaging reconstruction mesh (left) showing the sampled locations shaded in gray. The forward model uses the full high resolution mesh, while only downsampling to the portrait slabs as the final operation. The 3rd harmonic PSF at $x=y=0$ is overlaid onto the mesh to demonstrate how discretizing the full forward model at only the scanned locations would undersample the PSFs, resulting in an inaccurate model. This is also demonstrated in the right images showing the PSFs at 1 mm resolution and 5 mm resolution in $z$. However, with multiple harmonics, the model can expand the information outward to positions between the measured locations.

Theorems & Definitions (9)

• Theorem 1
• Definition 1
• Corollary 1
• proof
• Proposition 1
• proof
• Theorem 2
• Proposition 2
• proof : Proof of Theorem \ref{['prop main']}