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In magnetic particle imaging (MPI), system sensitivity can be enhanced by scanning the sample along a field free line (FFL) instead of a field free point (FFP). FFL MPI data can then be processed via system-matrix or projection-based reconstructions. Here, we compare the relative performance of these two approaches. We assume an ideal FFL (straight and homogeneous), which is translated and rotated in a two-dimensional field-of-view. We simulate the acquired data from a numerical vessel phantom for a broad range of noise levels. For the system-matrix reconstruction, we propose Alternating Direction Method of Multipliers (ADMM) to solve a constrained convex optimization problem. We also analyze the results of the nonnegative fused lasso (NFL) model to compare the performance of ADMM with one of the state-of-the-art system-matrix-based methods. For the projection-based reconstruction, we use the inverse Radon transform formulation with x-space reconstruction. System-matrix-based methods resulted in a higher structural similarity index and contrast compared to the x-space reconstruction method at the expense of longer reconstruction time. Artifacts occurred due to gridding errors for the x-space reconstruction. As expected, ADMM and NFL reconstructions yielded similar image quality.