COMPRESSED SENSINg AND SPARSE RECONSTRuCTION IN MPI

The reconstruction of a particle concentration in Magnetic Particle Imaging (MPI) usually involves the solution of a linear system of equations [1]. The acquisition and storage of the system matrix is challenging in practice: The acquisition time is proportional to the discretisation of the field of view (FOV). Assuming optimistic conditions, the system matrix acquisition lasts about two days for a spatial grid containing 64x64x64 voxels. Such a system matrix has a memory size of about 300 GB and therefore, it cannot be stored in the main memory of the computer, which is necessary for the application of fast reconstruction algorithms. As it has been recently shown, the acquisition time of a system matrix can be reduced significantly using compressed sensing methods [2]. It suffices to undersample the system matrix by acquiring only 20 % of the voxels in the FOV [3]. These points are randomly selected, acquired, transformed into a sparse domain and finally the sparse system matrix is reconstructed. Storing the system matrix in its sparse domain reduces the memory requirements. The reconstruction of the particle concentration may happen in the sparse domain as well [4]. The system functions that are the spatial distributions of one frequency component of the system matrix, can be sorted by their signal to noise ratio (SNR). Those system functions, whose SNR is above the noise level, are selected, transformed and used for further reconstruction. Applying a low SNR threshold of 10, the amount of selected system functions reduces to 3 %. For undersampled system functions the SNR can be estimated by the mixing order of the corresponding frequency [5]. The frequencies obtained in MPI can be displayed as a linear combination of the exciting frequencies. The sum of the coefficients of this linear combination determines the mixing order. A small mixing order corresponds to a high SNR. After selecting and transforming the system functions into a sparse domain, they are reconstructed and compressed. One of the main properties of the sparse domain is that many values are zero or near zero. Those values do not need to be stored as their impact on the reconstruction of the particle concentration is negligible. Discarding 90 % of the smallest values, there is no visible difference in the reconstruction. Applying both the compression and the SNR threshold, the system matrix shrinks to 0.03 % of its original size and can be stored in the main memory. Finally, undersampling cannot only be used to reduce the acquisition time of a system matrix, but also to improve the spatial resolution of an MPI measurement. The resolution depends on the discretisation of the FOV. Discretising a FOV with NxN pixels, the reconstruction results in an image of NxN pixels as well. When undersampled by a factor of 0.25, a FOV of 128x128 pixels can be fully acquired in the same time as a FOV of 64x64 pixels. As the size of the FOV is the same in both cases, undersampling the system matrix doubles the spatial resolution in two spatial directions.