Fig 1.
Experimental data illustrating proposed image reconstruction.
(Left) The measured signal is filtered and velocity compensated before gridding to partial fov images. The partial fov) images become the input to the optimization problem. (Right) The optimization problem formulation of dc recovery is illustrated. The forward model A consists of the S and D operators, where S is the segmentation operator and D is the dc removal operator. The initial estimated image is the zero vector, ρ0 = 0. The estimated image, ρ, is calculated and updated with each step of the iterative proximal gradient solver [29]. The optimization problem is formulated in Eq 6.
Fig 2.
Partial field of view gridding detail.
The received signal is interpolated to partial fov images using the ffr trajectory. Each x-axis traversal is broken into a separate partial fov image. Varying colors delimit each partial fov image. The sinusoidal pattern in the trajectory is formed due to the simultaneous x-axis shift field and the z-axis drive field.
Fig 3.
Experimental MPI data from a double helix phantom.
The 3d dataset was reconstructed using the previous dc recovery method and the proposed method. Both datasets are shown as maximum intensity projection images with no deconvolution. Images reconstructed with the proposed method contain less background haze and fewer artifacts. The imaging phantom was constructed by wrapping two 0.6mm id tubes injected with Micromod Nanomag mip mnps around an acrylic cylinder of od 2.7 cm. The total imaging time was 10 min with a fov of 4.5 cm by 3.5 cm by 7.5 cm (x,y,z).
Fig 4.
Experimental MPI data from a coronary artery phantom.
Images were reconstructed with the proposed reconstruction formulation and contrasted to the previous 1d dc recovery as well as no dc recovery. The imaging phantom was created by 3d printing an abs plastic coronary artery model. The reconstructed 3d dataset is shown as maximum intensity projection images. With no dc recovery, many image intensity dropouts are evident. These dropouts are corrected with dc recovery algorithms. The optimized 3d recovery contains fewer artifacts (solid arrow) and less background haze than the prior algorithm. Light deconvolution can be used to remove remaining background haze present in the reconstructed signal; however, deconvolution can lead to image dropouts (dashed arrow). The total imaging time was 10 min with a fov of 4.5 cm by 3.5 cm by 9.5 cm (x,y,z).
Fig 5.
Field free point MPI system photo.
This 7Tm-1 ffp mpi system was used to experimentally demonstrate the effectiveness of the 3d optimized reconstruction.
Fig 6.
Experimental data of a coronary artery phantom from Fig 4 at different angles.
The 3d volume-rendered datasets were reconstructed using the proposed method with deconvolution. The total imaging time was 10 min with a fov of 4.5 cm by 3.5 cm by 9.5 cm (x,y,z).
Fig 7.
Singular values and right singular vectors, V, were calculated on A for a 1d problem where 15 pixels overlapped between adjacent partial fovs and the partial fov width was 20.
The singular vectors represent the spatial z-axis and are shown in absolute value. The singular values demonstrate well-posed nature of the proposed reconstruction problem.
Table 1.
Sparse matrix versus matrix-free operator computation time and ram requirements.
Fig 8.
Condition number variation with overlap.
The condition number is calculated on the matrix A with the dc singular vector removed (reduced A) for a 1d problem with a partial fov width of 20. The trend curve is a least-squares fit to the calculated condition numbers and illustrates the general trend of improved condition number with increased partial fov overlap.