Understanding the combined effect of $k$-space undersampling and transient states excitation in MR Fingerprinting reconstructions

Magnetic resonance fingerprinting (MRF) is able to estimate multiple quantitative tissue parameters from a relatively short acquisition. The main characteristic of an MRF sequence is the simultaneous application of (a) transient states excitation and (b) highly undersampled $k$-space. Despite the promising empirical results obtained with MRF, no work has appeared that formally describes the combined impact of these two aspects on the reconstruction accuracy. In this paper, a mathematical model is derived that directly relates the time varying RF excitation and the $k$-space sampling to the spatially dependent reconstruction errors. A subsequent in-depth analysis identifies the mechanisms by which MRF sequence properties affect accuracy, providing a formal explanation of several empirically observed or intuitively understood facts. New insights are obtained which show how this analytical framework could be used to improve the MRF protocol.